Mechanical Engineering

International Review of

Mechanical Engineering (IREME) Contents Effect of the Resin Type on the Acoustic Activity and the Mechanical Behavior of E Glass/Polymer Resin ±55° Filament Wound Pipes Under Axial Loading by Ated Ben Khalifa, Mondher Zidi

792

Mechanical Stress Analysis in a Dynamic Graphite-Graphite Contact: Consequences on Wear by Y. Mouadji, A. Bouchoucha, M. A. Bradai, H. Zaidi

798

Fracture Toughness Transferability in Pipe with External Oriented Defect by B. El Hadim, H. El Minor, M. El Hilali

804

Experimental Studies of Film Boiling Phenomena on Carbon Heated Surface by S. Illias, M. A. Idris, M. Z. M. Zain

812

Activated Carbon for Drying Compressed Air for Low Pressure Applications by N. S. Senanayake, D. M. K. K. Dissanayake

818

Analytical Solution for Chemically Reacting Free Convective Couple Stress Fluid in an Annulus with Soret and Dufour Effects by D. Srinivasacharya, K. Kaladhar

823

Computational Analysis of Mixed Convection Over Heated, Vertical Rectangular Fin Array, at Richardson Number of Unity by J. P. Shete, N. K. Sane, S. Pavithran

834

Use of Flat Mini Heat Pipes for the Thermal Management of High Dissipative Electronic Packages for Avionic Equipments by M. C. Zaghdoudi, C. Tantolin, C. Sarno

843

Application of Lattice Boltzmann Method for Lid Driven Cavity Flow by Fudhail Abdul Munir, Mohd Irwan Mohd Azmi , Mohd Rody Mohd Zin, Mohd Azli Salim, Nor Azwadi C. Sidik

856

Mechanical Investigation of the Nozzles Attached to Pressure Vessels by M. M. S. Fakhrabadi, V. Norouzifard, B. Dadashzadeh, M. Dadashzadeh

862

Deposition of Solid Particles in Convective Flow Over Backward-Facing Step Under the Effect of Radiative Heat Transfer by V. Golkarfard, S. A. Gandjalikhan Nassab, A. B. Ansari

867

Numerical Study of the Mixing of Co-Axial Jets by Fernando M. S. P. Neves, Jorge M. M. Barata, André R. R. Silva

876

Experimental Investigation of Cavitation in a Centrifugal Pump with Double-Arc Synthetic Blade Design Method by Spyridon D. Kyparissis, Dionissios P. Margaris

884

Modeling of Surface Finish and MRR in Low Cost Internal Grinding by K. Kishore, V. V. Satyanarayana, P. V. Gopal Krishna, G. Kiran Kumar

893

Numerical Study of Entropy Generation in Laminar Forced Convection Flow Over Inclined Backward and Forward Facing Steps in a Duct by M. Atashafrooz , S. A. Gandjalikhan Nassab, A. B. Ansari

898

(continued)

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

Inverse Problem of Rocket Nozzle Throat for Estimating Inner Wall Heat Flux by Broydon– Fletcher–Goldfarb–Shanno & Conjugate Gradient Method by H. Khoshkam, M. Alizadeh

908

Study and Simulation of Thermal Buckling in a Thin Rectangular FGM Plate by Mahdi Hamzehei, Mostapha Raki

918

Experimental Investigation of Different Heat Recovery Systems in Leisure Center and its Effect on CO2 Emission by M. M. Abo Elazm, A. F. Elsafty

927

An Exact Solution for the Vibration FGM Hollow Cylindrical Shell Based on High Order Theory Under Free-Simply Support Boundary Conditions by M. Setareh, M. R. Isvandzibaei

933

Using Homotopy Analysis Method to Determine Profile for Disk Cam by Means of Optimization of Dissipated Energy by Hamid M. Sedighi, Kourosh H. Shirazi

941

Path-Whispering in a Virtual Environment by Fawaz Y. Annaz

947

Modeling, Control and Analysis of a Serial and Parallelogram Lower Member Mechanism by Alvaro Uribe, João Rosário, Luciano Frezzatto

952

Parametric Study of Electro-Hydraulic Servo Valve Using a Piezo-Electric Actuator by S. F. Rezeka, A. Khalil, A. Abdellatif

961

The Effect of the Fibre Orientation on the Failure Load of Face Sheets Composite Sandwich Beams by F. Bourouis, F. Mili

968

An Overview on Thermal Barrier Coating (TBC) Materials and its Effect on Engine Performance and Emission by Pankaj N. Shrirao, Anand N. Pawar, Atul B. Borade

973

Process Robustness in a Dimensional Testing Laboratory by Caterina Poustourli, Vrassidas I. Leopoulos

979

Analytical Solution to Transient Temperature Field in Semi-Infinite Body Caused by Moving Ellipsoidal Heat Source by Aniruddha Ghosh, N. K. Singh, Somnath Chattopadhyaya

987

Research Methodology of an Integrated Approach for Thermal Mapping of Hot Section Components of Gas Turbine Engines by Sachin V. Bhalerao, A. N. Pawar, Atul B. Borade

993

The Pseudo Radiation Energy Amplifier (PREA) by A. Boucenna

1000

EXTRACTED BY ICOME 2011 VIRTUAL FORUM – 2ND INTERNATIONAL CONFERENCE ON MECHANICAL ENGINEERING A Simplified Method for Thermally-Induced Volumetric Error Compensation by Yuxia Lu, M. N. Islam

1006

Fatigue and Brinelling Evaluation of ASME Extraction Pressure Vessel Closure with Locking Ring by A. M. Senthil Anbazhagan, M. Dev Anand

1013

The Modeling of the 2D Continuum with Non-Linearities by Jiří Podešva

1020

Flying Qualities Estimation Methods for Small Unconventional Aircraft by P. Hospodář, P. Vrchota, A. Drábek

1026


International Review of Mechanical Engineering (I.RE.M.E.), Vol. 5, N. 5 July 2011

Effect of the Resin Type on the Acoustic Activity and the Mechanical Behavior of E Glass/Polymer Resin ±55° Filament Wound Pipes Under Axial Loading Ated Ben Khalifa, Mondher Zidi Abstract – Acoustic emission is one of the most innovative of non-destructive testing techniques because makes it possible to follow in real time the damage of composite materials. The aim of this paper is to study the effect of variation in the resin type on the mechanical behavior and the acoustic activity of the filament wound pipe under monotonous axial loading. The results obtained show that the mechanical behavior as well as the acoustic activity of the filament wound pipe change for each type of studied resin. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Filament Wound Pipe, Resin Type, Tensile Test, Mechanical Behavior, Acoustic Activity.

I.

Godin et al [10] further explored the clustering of acoustic emission signals collected during tensile tests on unidirectional glass fiber/polyester composite using supervised and unsupervised classifiers. Notwithstanding their scientific acumen these studies do not seem to have sufficiently focused on the use of the acoustic emission activity during tensile test on FWP. This paper is an attempt to apply some implications of the very promising of these studies in FWP. The present research aims at the assessment, through experimental investigation, of the effect of resin type on the mechanical behavior and acoustic emission activity of FWP E-glass/polymer-resin with a ± 55° ° winding angle under axial loading by AE. Final failure levels will be investigated, and the results obtained will be presented by means of stress/strain curves and AE curves.

Introduction

Acoustic emission (AE) is a non-destructive test which can be used in different fields with an aim to studies the mechanical behavior of materials [1]-[9]-[10], to control industrial equipment [2], to monitoring cutting tools [3] or to analyze the damage mode of filament wound pipes (FWP). Previous studies have predominantly tended to either use mechanical tests to determine the mechanical behavior of FWP [4]-[5] or investigate the damage mode using AE of a unidirectional glass/polyester composite [9]-[10]. Jinbo Bai et al [4] have investigated the mechanical behavior of E glass-fiber/epoxy-resin ±55° filament-wound pipes under pure tensile loading and concluded that the mechanical behavior can be divided into four zones: an elastic behavior, transverse cracking, failure of the matrix, and a final failure of the tubes. Rousseau J. et al [5] stated that the effect of the degree of weaving on the growth damage in wound pipes is negligible as far as the off-axis of loading. After studying the damage behavior of multidirectional reinforced glass/epoxy pipes under biaxial monotonic loading, Martens and Ellyin [6] show that the important mechanical properties of a composite pipe are strength, stiffness and durability. Lokman et al [7] stated that, at high loads, fiber failure is important and that it controls the final damage, while at low loads, the failure is controlled by matrix damage. Mertiny et al [8] studied multi-angle winding and found that it has an important effect on the strength of tubular composite structure. Huguet et al [9] were among the first researchers to study the use of acoustic emission to identify damage modes in unidirectional glass fiber reinforced polyester and to state that the principal damage mode is fibermatrix decohesion.

II.

Production of Specimens

The filament wound pipe was produced by the CTRA Tunisia Co. E glass direct roving and vinylester or polyester resins were used for this production. The winding angle of ±55° was manufactured using a CNC winding machine. Filament wound composite pipes with 12 layers were produced with the following dimensions: 9m length, 80mm inside diameter and 5mm average thickness. These structures were cured for 2 hours at 135°C on a mandrel in a slow motion rotary oven. After pulling out the mandrel, the pipes were post-cured for 2 hours at 150 °C. The pipes were then cut into test lengths of 300mm using a diamond wheel cutting saw. Mechanical properties of the matrix and reinforcement materials are given in Table I. The dimensions of the specimen are reproduced in Figure 1.

Manuscript received and revised June 2011, accepted July 2011


792

Ated Ben Khalifa, Mondher Zidi

Component

TABLE I MECHANICAL PROPERTIES OF THE MATRIX AND REINFORCEMENT MATERIALS Tensile Modulus Tensile Density (g/ml) Tensile Strength (MPa)

(GPa)

Elongation (%)

Vinylester resin

86

3.2

5.0-6.0

1.046(at 25°C)

Polyester resin

70

3.6

2.2

1.17(at 20°C)

E glass fiber

1970

78.794

-

-

Linear density (Tex)

-

2400

two parametric channels for other transducers, such as strain gauge, pressure and load [13]. The AEwin for PCI2 software enables us to plot curves that represent the acoustic activity response of FWP according to a stress/strain curve. The clamping of the two sensors (C1 and C2) and the PCI-2 system is shown in Figure 3(a) and Figure 3(b), respectively. A constant crosshead speed of 10 mm/min was maintained during the tensile test coupled to the A.E. acquisition.

Fig. 1. Dimensions of specimens

III. Experimental Protocol Figure 2 shows the clamping of the specimens to a special device manufactured according to the ASTM-D 2105-01 standard test [11]. This device is clamped to a tensile machine SHIMADZU UH-F30A as shown in Figure 3(a). The tensile machine mode used is an imposed displacement with a 10 mm-min-1 displacement rate. A burn-off test according to the standard test T57-518 [12] is performed to determine the fiber rate (%) for each tube. The average value is 60±2%.

. Figs. 3. (a) Assembly of the device system on the testing machine and clamping of A.E. sensors in filament tube; (b) PCI-2 system

IV.

Results

IV.1. Tensile Test A series of tests of four tubes for each type (VT1, VT2, VT3 and VT4 for vinylester-resin type, and PT1, PT2, PT3 and PT4 for polyester-resin type) is performed to determine the average values of mechanical properties. These values are given in Table II. The stress/strain curve shown in Figure 4 summarizes the mechanical behavior of the glass/vinylester and polyester resin ± 55° FWP. These curves are divided into three zones. In the elastic zone, the behavior is the same. The second zone is similar at the beginning but is stopped at 48 MPa as an average value for the FWP with polyester resin, and we observe that the curve remains linear until 4% of axial strain and then we observe a drop of axial stress. This linearity is associated with the elongation of the molecular sequence. This behavior is different from the usual behavior of the FWP vinylester resin because in this zone, which begins at 55MPa of axial stress, we observe a sharp fall of axial stress. This is related to matrix cracking. For the last zone, we observe that for FWP with polyester resin, this zone starts at 8.5% as an average value of axial strain, but it starts at only 4% for the FWP with vinylester resin.

Fig. 2. Clamping of the pipe specimen

When the acoustic emission was studied, an acoustic emission acquisition system produced by the Physical Acoustic Corporation was used. The acquisition system contains two channels of Acoustic Emission (connected with two sensors and two preamplifiers) for simultaneous waveforms and feature processing and AEwin software. In addition to these two AE channels, the system also has


International Review of Mechanical Engineering, Vol. 5, N. 5

793

Ated Benn Khalifa, Mon ndher Zidi

FWP under axiaal monotonic loading is studied. s First,, statiistic analysis is performed to identify th he sensor (C1 or C2) C that presennts the best accquisition of signals s and att the same time to identify tthe amplitudee distributionn rang ge. Figure F 6(a) (foor vinylester rresin) and Fig gure 6(b) (forr poly yester resin) show s that the second senso or C2 is moree reprresentative theen the first sennsor C1 in a number n of AE E Hitss. Table T III recaapitulates the numbers of AE Hits forr each h sensor andd each resinn type. We contend thatt poly yester resin prresents more eevents that vin nylester resin.

Figures 5 present tow w damaged FWP: first with vinylester resin (Fig. 5(a))) and the secoond with polyyester resin (Fig. 5((b)). These reesults are in liine with the states of-the-art finndings of thee researches mentioned inn the introduction. However, thhere are indicees obtained inn the experimentall process poiinting the poossibility of there t being a fourrth zone. The existence off the latter caan be shown only after a conductinng the necessaary AE analyssis. IV.2. Acooustic Emissioon In this parrt of the reseaarch, the effecct of the resin type on the acousstic activity of o E glass/pollymer resin ± 55°

TABLE II THEE AVERAGE VALU UES OF THE MECH HANICAL PROPER RTIES Young moddulus (GPa)

Fracture strength (MPa)

Fraccture strain (%)

TUBES VT

PT

VT

PT

VT

PT

1

11.65

12.25

54.38

47.71

3.56

8.2

2

11.73

12.34

54.66

47.84

3.48

8.35 5

3

11.57

11.95

54.28

47.25

3.77

8.1

4

11.37

12.45

53.85

48.1

3.73

6 8.46

Avverage values

11.58

12.25

54.29

47.73

3.64

8.28 8

±0.14

±0.16 6

Gap type

±0.22

±0.21

±0.34

±0.36

Figg. 4. Stress/strainn curves glass/pollymer resin ± 55° FWP under axiaal monotonic loadding

(a)

(b) r (b) with pollyester resin Figs. 5. Daamaged FWP: (a) with vinylester resin;

Copyright © 20011 Praise Worthyy Prize S.r.l. - Alll rights reserved

Internationnal Review of Mecchanical Engineeering, Vol. 5, N. 5

794


TA ABLE III NUMBERS OF AE HITS FOR EACH A SENSOR AND D EACH RESIN TYPE Y Resin Type

Vinylester

Polyesterr

C1

35599

68503

C2

86378

113088

craccking (in this zone, z amplituude value is ab bout the rangee of 45 4 and 55 dB B) but when stress level increases, i wee obseerve a transveerse cracking as a second mechanism m off dam mage. The lim mit of axial strain is about 3.2 3 %. At thiss limiit, the third zone starts and we ob bserve that a sign nificant numbeer of counts iss recorded. In this zone, thee distrribution of am mplitude rangee is between 50 5 and 70 dB.. Thiss amplitude range r is assocciated with matrix m failure.. Thee last zone starts s at the value of 6..7% as axiall defo ormation in association with a variaable acousticc activ vity. This acctivity presennts a variablee number off coun nts which chaaracterizes the fiber failure. Figure F 7(b) prresents a cum mulative analy ysis strain vs.. Cou unts/stress currve of glass/vvinylester resiin ±55° FWP P poly yester resin. For F these currves, we obseerve a similarr acou ustic activity in the first aand second zone z stated inn Figu ure 7(a). Afteer the value of 2.3% of axial a strain, a third d zone starts. In this zone, we observe an a increase off acou ustic activityy. This acousstic activity is related ass show wn in the mecchanical behavvior to an elon ngation of thee mollecular sequeence. This aacoustic activ vity remainss flucctuating until the beginningg of the fourth h zone whichh startts with the value v of 10.22% of axial strain. s In thiss fourrth zone, thee acoustic acctivity presents a variablee coun nts which chaaracterize fiberr failure. Finally, F Tablee V presents a correlation between thee acou ustic emissionn parameters of the two cases of ourr stud dy based on mechanical m parrameters. We can thereforee concclude that thhe resin typee has a serio ous effect onn mecchanical and acoustic a param meters.

AE Hits

For the distribution d off amplitude raange, we obsserve that most AE E Hits are in the range betw ween 45 db annd 75 db for the toow type of reesin as shownn in the Tablee IV. This distribuution of ampllitude range is i associated with matrix crackiing and interfa facial debondinng [9]-[14]. Acoustic emission e cum mulative analyysis according to a strain vs. couunts/stress currve of glass/poolymer resin ±55° ± FWP for eacch channel waas plotted. In this t analysis, only the acoustic activity recorrded by the seecond sensor C2 C is used becausse it is moree representatiive than the first sensor C1. Figure 7(aa) presents thee cumulative analyses a strainn vs. Counts/stresss curve of glass/vinylesterr resin ±55° FWP F vinylester reesin. We nottice that acoustic responsse is divided into four zones. In the first zonne, we noticee that acoustic actiivity is missinng. This zonee is relative too the elastic zone where no dam mage is obserrved. This zonne is limited to thhe value of 1.2% of axial strain. We nootice that acoustic activity startss at this valuee and we recorrd an increasing nuumber of couunts accordingg to the rise of the stress level. At first thiis zone is relative to matrix m

Resin type Vinylester Polyester

Amplitudee(dB) Hits

(45--50] 272213 105587

TABLE IV DISTRIBUTTION OF AMPLITU UDE RANGE (50-555] (55-60 0] (60-65] 263880 21317 138499 96222 8115 5 6147

(65-70] 8513 4993

(70-75] 4641 3466

(75-80] 2605 1690

TABLE V CORRELATTION BETWEEN THE ACOUSTIC EMISSION M PARAMETERS BASED ON MECHANICAL PA ARAMETERS Tubes witth resin type Viinylester Polyester M Mechanical Fracture stress (MPa) ( 54.2 29 (±0.34) 47.73 (±0.36)) p parameters Fracture strainn (%) 3.64 4 (±0.14) 8.28 (±0.16)) 439 Counts 1119 AE E parameters Amplitude (ddB) 79 83

(a)

(b)

a monotonic loading: l (a) vinyllester resin; (b) po olyester resin Figs. 6. Ampplitude distributioon range of glass//polymer resin ± 55° FWP under axial



795


(a)

(b) Figs. 7. Cum mulative analysess strain vs. Countts/stress curve of glass/vinylester resin r ±55° FWP polyester p resin unnder axial monoto onic loading: yester resin (a) vinylesster resin; (b) poly

V.

the filament wouund pipe into tthree zones: propagation p off craccks in the matrix m or m microscopic crracks to thee interfaces fiber/m matrix, matrix cracking, and d fiber failure.. Thee effect of ressin type on thhe mechanicaal behavior iss thuss stated in the difference bbetween the values of thee fraccture strength and fracture sstrain. We cam me to realize,, how wever, that tennsile test cannnot accuratelly delimit thee

C Conclusion

In this papper, we studieed the effect of the resin typpe on the acoustic activity andd the mechannical behavioor of filament wouund glass ± 555° pipe under pure p tensile. As has already beenn shown byy the researcchers mentioned abbove, the tensile test dividdes the damagge of



796


damage behaavior becausee there is an elastic e zone which w remains invisible in stresss/strain curvess. For this reaason, AE is used too resolve this problem. p Scrutiny of A.E stressses vs. counnts /strain cuurves shows that thhere is an elaastic zone whiich presents a non damaged FW WP. So we conclude thaat the tensile test divides the mechanical m b behavior of thhe FWP into four zones and not into three zones as statted by the stuudies mentioned inn literature on the topic [4]. We have been able to observe that,, for each typpe of resin, there is a special response off acoustic acttivity which corressponds to a speecial damage. The acouustic emissionn can therefore be an opttimal method to coontrol the behhavior of the FWP under axial a loading.

P renforcéé au verre textilee, Préimprégnés,, [12] NF T57-518. Plastique Teneur en veerre et en chaarge-Méthode par p calcinations,, Normalisation française,Octobre fr e 1987. [13] PCI-2 based AE E system user's m manual rev 3 partt #: 6301 – 1000;; Physical Acousstics Corporationn, Princeton Jun nction, Nj; Aprill 2007. [14] Nechad H. Evaaluation de l’enddommagement et de la rupture dee matériaux hétérrogènes par ultrrasons et émissiion acoustique : Estimation de laa durée de vie reestante, Ph.D. disssertation Institutt National des Sciiences Appliquéees de Lyon 2004.

Acknow wledgementts

n 1977 in Soussee Ated Ben Khaalifa was born in Tunisia. He obbtained his Bacheelor of Science inn mechanical engineering from Higher H School off Techniques of Tun nis in 2001. Sciences and T He received hiis Master’s degreee in Mechanics,, Materials and P Processes from th he Higher Schooll of Sciences andd Techniques of Tunis T in 2005. He has been tteaching Mechan nical Engineeringg at the Preparatory Insstitute for Engineeers of Monastir in Tunisia sincee 2006 6. His current c research interests includee: Composite Maaterials and non-distru uctive testing by Acoustic Emissioon.

A Authors’ infformation Univ versity of Monastiir, LGM M, ENIM, Av. Ibnn Eljazzar, 5019 9 Monastir, Tunissia. Tel: (+216) 73 500 277 Fax: (+216) 73 500 512, (+216) 73 5055 866 E-maail: [email protected]

e his thhanks to Mr. Kais The authoor wishes to express Amara, technnical managerr at the compaany CTRA Tuunisia and his technnical staff forr their valuabale assistancee and input provideed for this work.

Refferences [1]

G. Kalogiaannakis, J. Quinttelier, P. De Baetts, J.Degrieck, D. D Van Hemelrijckk , Identification of wear mechaniisms of glass/polyyester composites by means of accoustic emission. Wear xxx (2007)) xxx– xxx. wound [2] HILL, E. V. K., Predinctiing burst pressurres in filament-w mission data, Matterials composite pressure vesselss by acoustic em Evaluationn, 1992, vol 5, 122, pp 1439-1445 [3] P.Kulandaaivelu, S.Sundaraam, P.Senthil Kuumar, Neural Neetwork Based Weear Monitoring of Single Pointt Cutting Tool using Acoustic Emission E Techniqques, IREME, Jaanuary 2011, Voll. 5 N. 1, pp. 52-558 [4] Bai et al, Mechanical behhavior of ±55° filament-wound f glassfiber/epoxxy-resin tubes: I-microstructural analyses, mechanical behavior and a damage mechhanisms of compposite tubes underr pure tensile loaading, pure interrnal pressure, annd combined loaading, Compositees science and tecchnology 51(19977) 141-153. [5] J. Rousseaau, D. Perreux, N. N VerdieÁre, Thee influence of wiinding patterns on o the damage behavior of filament-wound fi pipes, Compositees Science and Teechnology 59 (19999) 1439±1449. [6] Martens M, Ellyin F. Biaxial monotoonic behavior of a mposites: Part A 2000; multidirectional glass fiberr epoxy pipe. Com 31:1001–114. [7] Gemi L ett al, Progressive fatigue failure beehavior of glass/eepoxy (±75°) fillament wound pipes under puure internal preessure; Materials and Design 30 (22009) 4293–42988. mental investigatiion on [8] Mertiny P, Ellyin F, Hothaan A. An experim the effect of multi angle filament windinng on the strenggth of tubular composite structuures. Compos Sci Technol (22004); 64(1):1. U of [9] Huguet S,, Godin N, Gaerttner R, Salmon L, Villard D. Use acoustic emission e to ideentify damage modes m in glass fibre reinforcedd polyester. Comppost Sci Technol (2002) ( ;62:1433––44. [10] Godin N, Huguet S., Gaaertner R., Salm mon L. ; Clusterinng of e signals collected durring tensile testts on acoustic emission unidirectioonal glass/polyesster composite using u supervisedd and unsupervissed classifiers. NDT&E N International 37 (2004) 253– 264. [11] D-2105-011, Standard Tesst Method for Longitudinal Tensile T Properties of “Fiberrglass” (Glasss-FiberReinfforced A Society ty for Thermosettting-Resin) Pipe and Tube, American Testing Materials Ma (ASTM) designation: d D21105-01.

Dr. Mondher Zidi is a Professor of Mechanicall Engineering att the National Engineering Schooll of Monastir inn Tunisia. He ob btained his MScc and PhD in Mechanical En ngineering from m France) .he thenn “Ecole Centraale de Lyon” (F integrated thhe Mechanicaal Engineeringg Department at the National Eng gineering Schooll ontributes to thee of Monastir inn Tunisia. He co teaching of Triibology and Com mposite Materials. Dr. Zidi’s Z research arreas include: Natturals fibers, com mposite materialss and microindentation m tests.



797


Mechanical Stress Analysis in a Dynamic Graphite-Graphite Contact: Consequences on Wear Y. Mouadji1, A. Bouchoucha², M. A. Bradai3, H. Zaidi4

Abstract – In order to evaluate the wear of tribological pairs, it is important to know the type and the magnitude of the mechanical stress imposed at the dynamic contact. To do this, a numerical modell based on quasi-analytical solutions has been developed using a Matlab program. This modell can take into account the evolution of the mechanical stress generated in a dynamic contact as a function of the loading conditions and the coefficient of friction. This tool allows, by exploiting the calculation results, to obtain stress fields in every point under contact. To analyze these stress fields, we have considered the solution of the semi-infinite plane problem given by McEwen [1] and Johnson [2], where the contact pressure of Hertz [3] is limited to a circular contact area of radius a. The discussion of the results is based on the correlation between the limit values of the contact stress and the experimental results of the coefficient of friction and wear. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Friction, Wear, Stress Field, Contact Modell, Graphite, Speed

We will determine the stress states in the plane and more specifically the stress state the most severe, so we will be able to evaluate the characteristics magnitude of the mechanic (σxx, σe, τmax) in order to analyze the degradation of materials. As for the modelling part of this study, it was done through programming using Matlab, which requires low computation time. For this, we considered the solution of the semi-infinite plane problem given by McEwen [1], with a contact pressure limited to a circular contact area of radius a. In addition, to control the maximum number of parameters related to contact, we conducted some tests to determine experimentally the coefficient of friction, as a function of load, from which we calculate the stress with the slightest error.

Nomenclature P E Rccomp ν ρ Ft E* σxx σyy σzz τxz µ a

Normal load (N) Young modulus (N/m²) Stress to compression (MPa) Poisson's ratio The density (kg/m3) Tangential load (N) The equivalent Young's modulus (N/m²) The normal stresses in the x direction (MPa) The normal stresses in the y direction (MPa) The normal stresses in the z direction (MPa) The tangential stress (MPa) The coefficient of frection The contact radius (mm)

I.

Introduction

II.

The investigation of the stress distribution in tribological pairs is considered as one of the most significant problems of tribology [1]-[11]. Indeed, in the vicinity of the contact surface, significant damages are observed on the surface by wear or on the subsurface by cracks and failure. To calculate the stress field in a sliding contact, Diao and others [4] have analyzed a 2D slidingcontact under an elliptical pressure distribution. On the other hand, Stephens and others [5] have analyzed a 2D sliding contact of a coating under a cylindrical indenter. In our case, we have considered a sphere-plane dry contact in the two states, static and dynamic. It is therefore to address the influence of the normal load P and the tangential load Ft.

Friction and Wear Tests

In this experimental part, the goal is to highlight the influence of the load and the sliding velocity parameters on friction and wear. The obtained results are used to determine the stresses issued from the dynamic contact. II.1.

Experimental Device

In order to study the influence of the mechanical parameters on friction and wear, we have made tests using, in open air, a pin-disk tribometer (Fig. 1). The disk is 100 mm outside diameter and 15 mm in thickness. The pin, leaning against the disk by a normal load P, has a cylindrical shape of 20 mm length and 8 mm in diameter. It is mounted on the articulated arm in order to achieve a permanent contact pin-disk. The rotation of the arm



798

Y. Mouadji, A. Bouchoucha, M. A. Bradai, H. Zaidi

around the horizontal axis allows the application of the normal load on the pin. The disk is initially polished using a sandpaper grade 2500. The coefficient of friction is directly estimated by an acquisition system. The wear is evaluated by the weighing method before and after each test.

Fig. 2. Stress distribution for elastic contact between a sphere and a plane under the action of a normal load

Fig. 1. Pin-disk tribometer

II.2.

The Used Material

The used material is graphite (MY3D). This type of graphite can be applied in mechanical seals. The following table gives its mechanical characteristics. TABLE I MATERIAL CHARACTERISTICS Graphite MY3D Properties E N/m² 31.109 Rccomp MPa 120 ν 0,3 Hardness Shore 75 ρ kg/m3 2800

Fig. 3. General diagram of the problem

The contact radius a is given by: (1)

III. Modelling

with E* defined by the expression:

The components of the stress field on a contact area are usually induced by the normal load P and tangential load Ft, applied to the contact. To better understand the phenomena at the interface, we must evaluate the stress field affecting the carrying capacity of the contact. For this, the two bodies in contact are modelled by elastic semi-spaces. The Hertzian contact [3], quasi-static and frictionless (Fig. 2) fits in the following assumptions: - The bodies in contact are assumed limited by surfaces having a common normal and a common tangent plane; - The surfaces are perfectly smooth; - There is no friction between the two surfaces in contact; - The material of both bodies are homogeneous and isotropic; - The dimensions of the contact area are small compared to the radius of the principal curvature of the surfaces in contact. Consider now, a normal load P applied in the z direction and a tangential force Ft applied in the direction of displacement x on an elastic sphere of radius R. This sphere is located on a semi-infinite plane (Fig. 3).

(2) The objective of this part is not to give an exhaustive description of the mechanic of contact. For more details, see the works that refer to this area [2], [6]. By accepting the above simplifications for the expressions of the stresses tensors at any point in the plane, we used the formulation given by McEwen [1]. Thus, we obtain the equations for normal stresses at each point of the material:

(3)

(4)

(5)



799


(6)

with:

(7)

(8) and µ is the coefficient of friction determined experimentally as a function of the load P (Fig. 4).

Fig. 5. Plot of principal stresses values σxx, σyy, σzz and τxz at x = 0 as a function of depth z

IV.1. Distribution of Shear Stresses Note that in the static case (Fig. 6), the point of Hertz is not the only critical zone of maximum shear. The parallel Plans at the contact surface are subjected to shear τxz induced by the contact pressure and having extrema at depth z = 0,8a. In the static case, the maximum shear stress is closer to the surface of contact than it is in the dynamic case (Table II). The calculation program realized in Matlab allows the study of the variation of the shear stress in depth and at the vertical of the point of support of the load.

Fig. 4. Experimental curve of the coefficient of friction in a graphitegraphite dynamic contact as a function of the normal load

IV.

Results and Discussion

The preceding equations allow us to trace and examine the evolution of each component of the stresses tensor. On the other hand, we know that at the surface, the normal stresses are equal to the contact pressure. But, often this type of calculation is insufficient, since failure of the contact surfaces is originated from flaking caused by shear stresses within the material. In addition, in elastic deformation, the calculation of principal stresses, in the material along the perpendicular to the plane of the ellipse of support, shows that the curves σxx(z), σyy(z) and σzz(z) do not progress parallel to each other (Fig. 5). Their differences taken in pairs give the value of the tangential stresses which is different depending on the considered level. Thus, the tricercle of Mohr associated with these stresses allows the defining of the shear stresses at different heights: (9) In addition, to apply the criterion of Tresca, τemax must be ≤ 0.5·Re. We can therefore choose the material corresponding to this criterion. Fig. 6. Contours of shear stress of static contact (µ = 0, P = 30 N)



800


Figure 7 shows the changes of τxz as a function of the load P for different values of μ determined experimentally.

are crucial in the case of ductile materials and propagate perpendicularly to it [7]. TABLE II VALUES OF MAXIMUM CONSTRAINTS SHEAR FOR DIFFERENT LOADS τxzmax Shear areas PN a mm µ τxzmax MPa (x=0) MPa mm 5 0,0959 0,3750 11,43 27,9 -0,18 10 0,1209 0,2864 14,9 37,0 -0,25 15 0,1384 0,2534 16,9 43,4 -0,28 20 0,1523 0,2500 20,8 53,0 -0,33 30 0,1743 0,2257 24,1 63,3 -0,35 30 0, 1743 0,7 74,7 166,7 -0,36 30 0, 1743 0 0 32,9 -0,14

IV.2. Principal Stresses Figure 9 shows that for all values of Ft , the maximum of the normal stress σxx is always on surface at the back of the pin. Initially, for the static case, this maximum is located on the rim of the contact zone (at the center when µ = 0) (Fig. 8). Furthermore, by comparing these curves that illustrate the stress distribution σxx for several values of the tangential force Ft, we can see a tension zone that forms at the back and a compression zone formed simultaneously on the right (Fig. 9). Indeed, the greater the tangential force (Table III), the greater the value of this stress increases on the back of the pin and drops in front of it. For all these values of Ft , the maximum of σxx is always found on the surface on the left of the pin.

Fig. 7. Contours of τxz for different values of coefficient of friction

We note, moreover, that the shear stress increases with the increase of the product µ·P. Indeed, the maximum original point having a value of 0.8·a, in the static case, moves slightly towards the depth and much more forward when µ·P and the shear stress at the surface increase simultaneously. The result is a relatively low wear (Fig. 10). As soon as the coefficient of friction exceeds the value of 0.7 (a value which is not reached experimentally), the shear stress at the surface is 74.7 MPa and is higher than the maximum shear τmax = 0.5 Re. This point characterizes the irreversible damage. It occurs then deformations induced by the sliding, which

Fig. 8. Contours of Principal constraints of static contact (µ = 0, P = 30 N)



801


IV.3. Contribution of Stresses to Wear The change of wear mechanism is related to the plastic deformation of the underlying metal and the contact temperature, strongly affected by the normal load P and the sliding velocity v. The curves of Figure 10 show that the wear, according to the load, increases up to P = 10 N, then it becomes relatively stable. This is due to the activation of the metal surfaces and the degree of oxidation. The thicknesses of oxide layers are low but sufficient to handle the shear at the interface [7]. The wear may be caused by the fact that the stresses allowable by the material skin are actually below the limits of the substrate. This type of phenomenon may be related to physicochemical changes of the surface with temperature [8]. The wear can also be due to the fatigue of the surface under repeated impact in elastic regime causing progressive wear. Some works [5] show that the width of the worn contact area tends to the width of the contact calculated, in elastic regime, between a cylinder and a plane. Figure 11 gives a typical example of the worn area. Some calculations (Table III) show that the radius of the contact area calculated by Hertz theory is 0.1523 mm and the radius of the worn area is about eight times that value, so 1.2 mm. The work of the asperities deformation, in sliding, causes an increase in the frictional force and the deformations result in the phenomenon of wear accompanied with a mutual metallic transfer between the facing surfaces [5]. It is also extremely important to consider the evolution of roughness in time of contact. Indeed, during the test when the surfaces wear occur, the formation and agglomeration of debris cause a profound change of the contact (Fig. 11). The consequence of this latter phenomenon is to reduce the surface roughness and reduce the abrasive effect of hard particles interposed at the interface. Under these conditions, the oxidation of the contact surface, its partial disruption by mechanical action, its deposition on the path and its mixing with the debris of carbon are the dominant mechanisms [9]. The mixture (graphite-oxide) acts as a lubricant and reduces friction and wear [10]. Fig. 9. Contours of principal stress σxx for different coefficients of friction

PN

TABLE III VALUES OF PRINCIPAL STRESSES FOR DIFFERENT LOADS σxx max en traction σxx max en a mm µ MPa compression MPa

5

0,0959

0,3750

41,5

-45,6

10

0,1209

0,2864

60,7

-58,9

15

0,1384

0,2534

77,3

-66,6

20

0,1523

0,2500

93,4

-81,8

30

0,1743

0,2257

120,2

-93,7

30

0, 1743

0,7

188,5

-295,7

30

0, 1743

0

109,5

0

Fig. 10. Variation of wear rate, tangential force and maximum value of shear stresses in a graphite-graphite dynamic contact as a function of the normal load



802


[2]

Johnson K. L., «Contact mechanics», Cambridge University Press, (1985). [3] Hertz H., «On the elastic contact of elastic solids», J. Reine Angew. (1881), Math.92, 156-171. [4] Diao D. F., Kato K., Hayahi K., «The maximum tensile stress on a hard coating under sliding friction», Tribol. Int. Vol 27, p267272, (1994). [5] Stephens L. S., Liu Y., Melrtis E. I., «Finite element analysis of the initial yielding behavior of a hard coating/substrate system wiith functionally graded interface under indentation and friction», ASME J. Tribol. Vol 122, p 381-387, (2000). [6] Hills D. A., Nowell D., «Sackfield A., Butterworth-Heineman», Books, Mechanics of elastic contact, (1993). [7] Djamai A., Zaidi H., Villechaise B., «Modélisation numérique 3D pour le calcul des contraintes de contact sous une sphère rigide en contact sur un revêtement élastique mince», Tribologie et Conception Mécanique, (2004), pp. 269-283. [8] Rocchi J., «Couplage entre modelisations et experimentations pour étudier le role de l’oxydation et des solicitations mécaniques sur la rhéologie et les débits de troisième corps solide : cas d’un contact de géométrie conforme», Thèse de doctorat. INSA de Lyon, (2005), pp.96. [9] Bouchoucha A., Chakroud S., Paulmier D., «Influence of oxygen on the tribological behavior on friction and wear in the couple cooper-steel crossed by an electrical current », Tribotest Journal 11-1 (2004) 11-27. [10] Zhongliang H., Zhenhua C., Jintong X., Guoyun D., «Effect of PV factor on the wear of carbon brushes for micromotors», Wear 265 (2008) 336-340. [11] Jullien A., Meurisse M. H., Berthier Y., «Determination of tribological history and wear through visualization in lubricated contact using a carbon-based composite», Wear, 194(1-2) (1996), 116-125.

In addition, superficial changes involve modification to the stresses scope and allowable deformations of the material, thus leading to a detachment of particles by brittling surface of the material (particles of size ranges from several nanometers to several micrometers) [11].

Fig. 11. Optical image of a worn area of graphite, P=20N and V=1m/s

V.

Conclusion

The numerical calculation of the stresses field from the sphere-plane contact modelling has been established thanks to the experimental values of the coefficient of friction as a function of the normal load. It appears from our study the following points: – The critical shear necessary to the deformation of the material in the sub-layer descends in depth from the surface, as the static load applied increases. This results in an increase in the thickness of the deformed area and the wear of the pin; – The general calculation of solicitations shows that the first laminating, initially observed in the sub-layer for a contact without friction, gradually descends in depth, as the tangential force increases; – The coefficient of friction is a parameter that has a considerable influence on the stresses distribution in the dynamic contact; – When the coefficient of friction is greater than 0.7, the irreversible damage (of type abrasive on surface) produce plastic deformation induced by sliding. However, the maximum shear is located at the center of the contact; – The effect of the tangential force is to increase the compressive stress at the front of the contact area and to increase the tensile to the rear of this area. The position of the maximum principal stress of traction is always at the back of the perimeter of contact; – Whatever the value of the coefficient of friction (different from zero) is, the principal stress of the tensile increases with the increasing tangential force and may even exceed the maximum contact pressure at the origin point.

Authors’ information 1

Centre Universitaire de Khenchela Institut Sciences et Technique, Algerie. E-mail: [email protected] 2

Laboratoire de Mécanique, Campus Chaabet-Ersas, Faculté des Sciences de l’Ingénieur, Université Mentouri Constantine, Algerie. E-mail: [email protected] 3

Laboratoire de technologie des matériaux et génie des procédés, Faculté de technologie, Université de Béjaia. Algérie. [email protected] 4

Laboratoire LMS (UMR-6610-CNRS), SP2MI, Téléport 2, Boulevard Marie et Pierre Curie, Université de Poitiers, BP 30179, 86962 Futuroscope Chasseneuil Cedex, France. Y. Mouadji Juin 1986: Baccalauréat technique Mathématiques, lycée technique Tewfik Khaznadar, Constantine. - Octobre 1992: Ingéniorat d'état en Génie Mécanique, Université de Constantine, Institut de Génie Mécanique. - Novembre 2000: Magister en Génie Mécanique, Université de Constantine, Département de Génie Mécanique.

References [1]

Mc Even E., «Stresses in elastic cylinders in contact along a generatrix (including the effect of tangential friction)», Journal Phil. Mag. 40, (1949) 454.



803


Fracture Toughness Transferability in Pipe with External Oriented Defect B. El Hadim, H. El Minor, M. El Hilali

Abstract – The problem of pipelines subjected to external defects caused by foreign scratch objects or gouges is treated. The external oriented defect represented by a blunt notch in the pipe under internal pressure has been considered and an elastic-plastic finite element method is applied to this survey. The critical notch stress intensity factor (or toughness of material) is evaluated for various outside diameters of pipe and various notch defect orientations. The fracture toughness transferability is treated by using the stress triaxiality and introducing of a new transferability parameter called pt. application of this method has been made on bending specimens extracted from the pipe. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: External Oriented Defect, Pipe, Fracture Toughness, Transferability Parameter

The fracture of gas pipelines produced by the external defects, e.g.; gouges, foreign scratch objects were the cause of more than a third of failures in service. This impact on pipe under internal pressure is represented by a blunt notch defect. The fracture mechanics concept applied to the analysis of structures which contain cracks or flaws is well known in literature [1]-[2]. The problem of fracture in pipelines is treated in several papers [3]-[4] and the stress intensity factor is the most used criteria to characterize the fracture phenomena [5]-[6]. In this paper, we are interested in the external oriented defect in the pipe by using the fracture mechanics analysis. The problem is treated by considering various outside diameters of pipe and various defect orientations. The survey is based on the volumetric method as a local fracture criterion for evaluating the critical notch stress intensity factor or so called the fracture toughness of material. Transferability of fracture toughness is an important problem of structural design because the properties measured in some reference conditions of geometry or constraint have to be modified to be applied in other similar conditions in structure design. Fracture toughness transferability (FTT) is made by the way of a new transferability parameter called pt. The local stress distribution was calculated by the Castem software 2000. It is well known that ductile fracture is sensitive to stress triaxiality. In this work, β is used as a measure of stress triaxiality. This parameter is defined as the ratio of the hydrostatic stress over the equivalent Von Mises stress:

Nomenclature a A% D E K n pt t c X eff

Defect depth Relative elongation Outside diameter Modulus of elasticity Hardening coefficient Hardening exponent Transferability parameter Wall thickness Critical effective distance

Xn

Distance at the end of zone III Stress triaxiality Critical maximum of triaxiality

β

c β max c σ eff

Critical Effective stress

σ max σU σY σ θθ max

Maximum stress Ultimate tensile stress Yield stress Maximum hoop stress

λ ν ρ χ (r )

Defect orientation Poisson’s ratio Notch radius Relative stress gradient

I.

Introduction

Pipelines have been used as one of low cost methods for oil and gas transmission. But they are exposed to a certain number of environmental aggressions.

β=


σh

σ eq,VM

(1)


804

B. El Hadim, H. El Minor, E. Hilali

TABLE III GEOMETRICAL CHARACTERISTICS OF SPECIMEN

where:

σh =

σ xx + σ yy + σ zz

ρ (mm)

D=2Re (mm)

t (mm)

λ (°)

3

0.75

88.9

5.49

0; 15; 30; 45

0.75

114.3

6.02

0; 15; 30; 45

0.75

168.3

7.11

0; 15; 30; 45

0.75

219.1

8.18

0; 15; 30; 45

and: 1

σ eq ,VM =

2

(σ 1 − σ 2 ) 2 + (σ 1 − σ 2 ) 2 + (σ 1 − σ 2 ) 2 II.

Material IV.

The material is an ASTM A106 Gr. B steel. His mechanical and chemical proprieties are listed in Table I and Table II.

Stress distributions around the notch defect have been converted into so called notch stress intensity factor using the notch fracture mechanics and particularly the volumetric method. The volumetric method [7] is a local fracture criterion, which supposes that the fracture process requires a certain fracture volume. This volume is assumed as a cylinder with effective distance at its diameter. The elastic-plastic stress distribution along the ligament is plotted in the bi-logarithmic diagram as can be seen in Fig. 2. Three distinct zones in the diagram can be distinguished: - Zone I: the elastic-plastic stress opening stress increases and attains a peak value. - Zone II: the elastic-plastic stress drops gradually in the elastic regime. - Zone III: starts at a certain distance which is named the effective distance. It represents linear behaviour in the bi-logarithmic diagram.

TABLE I CHEMICAL PROPRIETIES OF ASTM A106 Gr.B C

Mn

Si

Cr

Ni

Mo

S

P

0.3

0.32

0.12

0.2

0.14

0.18

0.03

0.03

TABLE II MECHANICAL PROPRIETIES OF ASTM A106 Gr.B E (GPa)

ν

σY (MPa)

σU (MPa)

A%

n

K (MPa)

210

0.3

351

562

30

0.146

848

Notch Stress Intensity Factor and Volumetric Method

III. Geometrical Characteristics To evaluate the critical notch stress intensity factor in pipe under internal pressure means that the critical loading pressure must be determined. But, it seems to be complicated and expensive. Then, we considered specimens extracted from the pipe. Specimen geometry is given in Fig. 1. Four outside diameters and for each diameter four notch orientations are considered as shown in Table III. The defect depth is equal to one half of pipe wall thickness for that we have an helical oriented notch.

Fig. 2. Schematic elastic-plastic stress distribution along notch ligament and stress intensity concept

The notch stress intensity factor [N.S.I.F] is defined as function of effective distance and effective stress: Fig. 1. Specimen geometry with oriented blunt notch

K ρ = σ eff


2π X eff

(2)


805


The numerical analysis in specimen with external defect needs to determinate the stress distribution in the vicinity of notch. The part modelled in 3D analysis is meshed by quadrangular elements with eight nodes. Computing was carried out on Castem software 2000 [9]. By using the maximum hoop stress as failure criterion [10], the stress distribution (σrr, σθθ, σrθ, σzz, σrz and σθz) was simulated along the notch (LET in Fig. 4) in several sections (Fig. 5) in order to determinate the solicitation mode of fracture. Those sections are normal to the notch.

The effective distance corresponds to the minimum of the relative stress gradient which given as:

χ (r ) =

1

σ yy ( r )

∂σ yy ( r ) ∂r

(3)

The effective stress is considered as the average volume of the stress distribution over the effective distance. However stresses are multiplied by a weight function in order to take into account the influence of stress gradient due to geometry and loading mode. The effective stress is defined as:

σ eff =

V.

1 X eff

X eff

∫0

σ yy ( r ) × (1 − r × χ ( r ) ) dr

(4)

Finite Element Analysis

For the specimen, we looked for a mode of loading which gives the same stress distribution on the vicinity of notch tip that in pipe. The stress distribution in pipe is treated in a previous article [8]. An example is given in Figure 3.

Fig. 4. The long blunt notch model and mesh density

Fig. 3. visualization of the maximal hoop stress in different sections along the half notch of pipe ( λ=30°) [8] Fig. 5. Perpendicular sections to the notch



806


After several simulations, the results concerning specimen bending mode with a loading distributed to the center seems to be adequate (Fig. 6).

(a) Notch parallel to the specimen axis (λ=0°)

Fig. 6. Loading mode of the specimen

The results of numerical simulations exhibit that for all notch orientations, we have the following conclusions: - The hoop stress is maximal along the notch tip for all the considered sections (Fig. 7). This means that the specimen is solicited to the opening mode of fracture (Mode I). - The shear stresses are negligible and the hoop stress σθθ which is maximal at the middle of the notch is predominant (Figs. 8). Then, one can conclude that the fracture is governed by the hoop stress and the micro cracks appear at the first hot point (P061) of ligament where σθθ is maximal. According to the above conclusions, which are the same that for the pipe [8], we will be interested in the radial direction (L051 in Fig. 4) where the crack is expected to occur first and grow up radially.

(b) Notch oriented to the specimen axis (λ=30°) Figs. 8. Stress distribution along the half notch of specimen for two different orientations

VI.

Experimental Tests

The specimen is submitted to a loading distributed to the center in bending mode as indicated in Fig. 9. To achieve this loading mode, a support has been conceived to be fixed on the machine (Fig. 10 and Fig. 11). Four various diameters are considered, and for each diameter, we have four notch orientations. Two tests are performed for each case, and then the total number of tested specimens is 32 (Table IV). To evaluate the critical notch stress intensity factor (C.N.S.I.F) K ρc , we need to find the critical load which produces the fracture of specimen. All values of critical loads after tests are listed in Table V.

Fig. 7. Visualization of the maximal hoop stress in different sections along the half notch in specimen (λ=00° and λ=30°)

Fig. 9. Loading mode of the specimen



807


TABLE V CRITICAL LOADS AFTER TESTS Outside diameter/ thickness D/t (mm/mm) (Standard)

88.9/5.49

114.3/6.02

168.3/7.11 Fig. 10. Schema of installation

219.1/8.18

Notch orientation λ (°)

Test N°1 F1 (N)

Test N°2 F2 (N)

Average load FM (N)

00 15 30 45 00 15 30 45 00 15 30 45 00 15 30 45

2560 3100 3700 4380 2850 3300 4200 4560 3020 3700 4100 4800 3300 3660 4300 4900

2640 2900 3600 4620 2950 3420 4140 4800 3100 3500 4000 5040 3100 3740 4580 5240

2600 3000 3650 4500 2900 3360 4170 4680 3060 3600 4050 4920 3200 3700 4440 5070

Fig. 11. Support installed in the machine TABLE IV NOMBER OF SPECIMENS TESTED

Fig. 12. Stress distribution in the bi-logarithmic diagram

Notch orientation λ (°)

Outside diameter/ thickness D/t (mm/mm)

VII.

00

15

30

45

219,1/ 8,18

2

2

2

2

168.3/ 7,11

2

2

2

2

114,3/ 6,02

2

2

2

2

88.9/ 5,49

2

2

2

2

The calculation of the critical notch stress intensity factors (C.N.S.I.F) K ρc is obtained by using the formulae 2 in section IV. c For that, the critical effective stress σ eff and the c critical effective distance X eff are also computed by

using the numerical analysis. The values of K ρc in radial direction for various

Notch Stress Intensity Factor Calculation

outside diameters of specimen and for various notch orientations are summarized in Table VI. One can note that the Critical Notch Stress Intensity Factor K ρc is sensitive to the diameter of specimen and

The critical notch stress intensity factors are computed using the maximum hoop stress versus ligament in the bi-logarithmic diagram (Fig. 12). The stress distribution along the radial direction (P061-P071) is taken into account due to the considerations mentioned above. The effective distance corresponds to the minimum of relative gradient stress and is used to specify the effective stress.

notch orientations as can be seen respectively on Figures 13 and 14. K ρc increases with diameter and decreases with notch orientation.



808


c β max and corresponds to the distance X βc max . After, it decreases along the remainder of the ligament.

TABLE VI CRITICAL NOTCH STRESS INTENSITY FACTORS FOR VARIOUS OUTSIDE DIAMETERS AND NOTCH ORIENTATIONS Critical Outside Critical Critical notch stress diameter / effective effective intensity Angle thickness distance stress factor D/t λ (°) c c σ eff X eff c (mm/mm) Kρ (MPa) (mm) (Standard) (MPa.m0.5) 00 323.353 0.175 10.722 15 308.290 0.150 9.464 88.9/5.49 30 228.261 0.100 5.722 45 189.236 0.050 3.354 00 435.229 0.200 15.428 15 397.641 0.175 13.186 114.3/6.02 30 300.378 0.150 9.222 45 205.671 0.100 5.155 00 499.177 0.425 25.795 15 479.528 0.400 24.040 168.3/7.11 30 395.848 0.250 15.689 45 275.866 0.125 7.731 00 502.407 0.700 33.319 15 496.445 0.625 31.110 219.1/8.18 30 417.971 0.325 18.888 45 325.555 0.175 10.795

Fig. 16. Example of evolution of Critical Stress Triaxiality along the ligament at notch tip c The maximum of triaxiality β max is sensitive to the specimen diameter and the notch orientation. It increases with diameter and decreases with notch orientation (Table VII).

TABLE VII CRITICAL MAXIMUM OF TRIAXIALITY FOR VARIOUS OUTSIDE DIAMETERS AND NOTCH ORIENTATIONS Critical Critical Outside diameter distance on maximum of Angle / thickness maximum of triaxiality D/t (mm/mm) λ (°) triaxiality c (Standard) c β max (mm) X β (mm) max

88.9/5.49 Fig. 13. Evolution of Critical Notch Stress Intensity Factor with non dimensional outside diameter 114.3/6.02

168.3/7.11

219.1/8.18

00 15 30 45 00 15 30 45 00 15 30 45 00 15 30 45

2.293 2.193 1.845 1.393 2.342 2.248 1.899 1.445 2.373 2.294 1.976 1.495 2.624 2.473 2.014 1.590

0.475 0.450 0.425 0.350 0.475 0.475 0.475 0.400 0.550 0.550 0.500 0.400 0.475 0.475 0.525 0.475

Evolution of critical notch stress intensity factor K ρc

Fig. 14. Evolution of Critical Notch Stress Intensity Factor with notch orientation

c is plotted versus critical maximum of triaxiality β max and given in Figure 16. It can be noted that fracture toughness increases with this parameter but this evolution remains sensitive to the specimen diameter. c We conclude that the maximum of triaxiality β max is not adequate as transferability parameter. An improvement has been made using a new transferability parameter pt based on a geometrical correction. This parameter is given as follows:

VIII. Fracture Toughness Transferability Stress triaxiality has been chosen as a transferability parameter because ductile fracture is there sensitive [11][12]. For all considered diameters, the critical stress triaxiality distribution at notch tip (Fig. 15) increases until a maximum which for the critical event is called



809


c pt = β max

(

D c c X β max ⋅ X eff t

)

The evolution of transferability parameter pt with the inverse of the non dimensional diameter (D/t)-1 is given in Figure 18.

(5)

Evolution of critical notch stress intensity factor K ρc versus pt transferability parameter is shown in Figure 17.

Fig. 18. Evolution of Transferability Parameter versus the inverse of non dimensional diameter

IX.

Fig. 16. Evolution of Critical Notch Stress Intensity Factor versus critical maximum of triaxiality

Discussion

The most recent transferability parameters are based on stress distribution at defect tip. They are mainly T stress [13], Q parameter [14] and Q* parameter [15]. The Q parameter of Ruggieri and Dodds [4]O is defined as the difference of the relative opening stress value at non dimensional distance r ⋅ σ y / J = 2 of two distribution; a reference distribution generally for small scale yielding (SSY) and the current one: Q=

Fig. 17. Evolution of Critical Notch Stress Intensity Factor versus pt

All data for every specimen diameter and notch orientation merge into a unique linear curve: (6)

−2

⎛D⎞ − B (λ ) ⋅ ⎜ ⎟ ⎝ t ⎠

−1

SSY

(8)

It has been seen by finite element that in this case the distances of maximum opening stress for the two constraint situations coincide (Figure 19). This common distance is considered as the characteristic distance in a local fracture criterion. If one multiply the relationship (16) by ρ ⋅ X c and

The parameters introduced into the transferability parameter pt depend on non dimensional diameter D/t and notch orientation λ. After results analysis, pt can be written in the following form: ⎛ D⎞ ⎛D⎞ pt ⎜ λ , ⎟ = A ( λ ) ⋅ ⎜ ⎟ ⎝ t ⎠ ⎝ t ⎠

σY

The validity of Q is limited to homothetic distribution given by the following rule: the stress gradient between the two non dimensional distances (5) and (1) is less than 10%: Q(1) − Q( 5) ≤ 0 ,1 gradQ = (9) 4

transferability parameter

K ρc = 4.4513 pt − 0.043

σ yy − (σ yy )

+ C ( λ ) (7)

considers pure brittle fracture: Q=

where:

σ yy ⋅ ρ ⋅ X c − (σ yy )

σY ⋅ ρ ⋅ X c

A ( λ ) = 0.2623 λ 3 − 17.228 λ 2 + 41.14 λ + 10713 B ( λ ) = 0.0308 λ 3 − 2.0747 λ 2 + 7.4917λ + 1261.5

Q=

C ( λ ) = 0.0009λ 3 − 0.0620 λ 2 + 0.2535 λ + 40.173


SSY

KC − K IC

σY ⋅ ρ ⋅ X c

⋅ ρ ⋅ Xc

(10)

(11)


810


[6]

[7] [8]

[9]

[10] [11] Fig. 19. Definition of Q parameter and characteristics used in a local fracture criterion [16]

[12]

Q is then a simple relative difference between critical stress intensity factors. This means that Q parameter gives another description of the relative fracture toughness which gives a strong limitation to its interest and justifies the use of the parameter pt .

[13]

[14]

[15]

X.

Conclusion [16]

In this study, external defect in pipeline is treated. To facilitate this survey, several specimens are extracted from pipes with various diameters. The oriented blunt notch defect is examined in order to evaluate the Critical Notch Stress Intensity Factor or so called fracture toughness. The fracture toughness transferability by using the actual parameters such as T stress or Q parameter gives no satisfactory results. Q parameter cannot be used in only restricted conditions in quasi elastic case. T stress can be used only in pure elastic conditions. Q* parameter measured on notched specimens can be used for elastic-plastic case, but similarly to Q parameter is a relative difference of critical notch stress intensity factors. For these reasons, we propose to introduce a new transferability parameter which is based on stress triaxiality in vicinity of notch tip.

Authors’ information Equipe de recherche au Laboratoire d’Ingénierie des Procédés de l’Energie et de l’Environnement (LIPEE), ENSA, Agadir, Morocco. Brahim El Hadim, Teacher in the Department of Mechanical Manufacturing at the Technical School of Settat, Morocco. He holds a degree in mechanical engineering in 1993, a 3rd round (DESA) in Mechanics of Structures in 2005 and actually prepares a doctorate in science. His interest lies in fracture mechanics engineering.

Hassan El Minor, Professor Ability in Mechanics and Calculation of Structures at the “ Ecole Nationale des Sciences Appliquées” (ENSA) of Agadir, Morocco. His interest lies in fracture mechanics and manufacturing engineering.

References [1]

[2]

[3]

[4]

[5]

Hole Using Finite Element Method, International Review of Mechanical Engineering, (2008). D. Ouinas, A. Hebbar, J. Viña, Evaluation of the Stress Intensity Factor in a Structure Repaired with an Elliptical Composite Patch, International Review of Mechanical Engineering, (2007). G. Pluvinage, Fracture and Fatigue Emanating from stress concentrators (Dordrecht: Kluwer Academic Publishers; 2003).). B. El Hadim and col, Evaluation of Notch Stress Intensity Factor in pipe with external oriented defect, International Review of Mechanical Engineering, Vol.5 N1, pp7-11, (2011). Castem 2000, Logiciel d’éléments finis développé par le département des études mécaniques et thermiques du commissariat français de l’énergie atomique C.E.A. H. El minor, Rupture fragile en mode mixte I+II, amorcée à partir d’entaille, Université Mohamed V- Agdal, Rabat, Maroc, (2002). J. R. Rice and D. M. Tracy, Journal of Mechanics and Physics of Solids 26, pp. 163-186, (1969). M. Biel-Golaska, Analysis of cast steel fracture mechanisms for different states of stress, Fatigue & Fracture of Engineering Materials & Structures 21, pp. 965-975, (1998). C. Betegon, J. W. Hancock, Two parameter characteristics of elastic-plastic crack tip fields, J. Appl. Mech. 58, pp. 104-101, (1991). C. Ruggieri, X. Gao, R.H. Dodds; Transferability of elastic-plastic fracture toughness using the Weibull stress approach: Significance of parameter calibration, Engineering Fracture Mechanics 67, pp. 101-117, (2000). I. Elayachi, O. Akourri and G. Pluvinage, Fracture Toughness Transferability for Notched Specimens, International Conference on Advance in Mechanical Engineering and Mechanics Hammamet, Tunisia, (2006). O. Akourri, I. Elayachi, Stress Triaxiality as Fracture Toughness Transferability Parameter for Notched Specimens, International Review of Mechanical Engineering, (2007).

S. G. Jallouf, Approche Probabiliste du Dimensionnement Contre le Risque de Rupture, Université de Paul Verlaine de Metz et Université d’Alep, France, (2006). V. Le Corre, Etude de la compétition Déchirure ductile/Rupture fragile : application à la tenue mécanique des tubes en acier CMn et de leurs joints soudés, L’Ecole Centrale de Lille et l’Université des Sciences et Technologies de Lille, France, (2006). D. Zelmati, A. Amirat, Finite Element Method Analyses of Fracture Toughness of API X70 Pipeline Steels, International Review of Mechanical Engineering, (2008). F. Rahimi, I. Shafieenejad, Computation of Tension Intensity Coefficients in Cracked Thin-Walled Pipes, International Review of Mechanical Engineering, (2009). M. Souiyah, A. Alshoaibi, A. Muchtar, A. K. Ariffin, Stress Intensity Factor Evaluation for Crack Emanating from Circular-

El Mokhtar Hilali, Professor Ability in Mechanics and Calculation of Structures at the “ Ecole Nationale des Sciences Appliquées” (ENSA) of Agadir, Morocco. His interest lies in fracture mechanics and manufacturing engineering.



811


Experimental Studies of Film Boiling Phenomena on Carbon Heated Surface S. Illias1, M. A. Idris2, M. Z. M. Zain1

Abstract – The purpose of this research is to study and examine the nucleate-boiling and filmboiling phenomena on carbon heated surface, which occur when a water droplet collided with the heated surface at a very high temperature (100ºC-420ºC). When the surface temperature reaches a maximum value, the critical superheated surface is suddenly covered with a vapor layer. Because of the vapor layer’s lower thermal conductivity, this vapor layer insulates the surface. This condition of vapor film insulating the surface from the liquid characterizes film-boiling. The carbon boiling curve that obtained from the experiment is examined in order to study the relation between carbon boiling curve and Leidenfrost effect. According to the Leidenfrost effect, liquids cannot touch a surface with a temperature above their boiling point because evaporation forms a cushion of vapor preventing contact. The higher the surface’s temperature above the boiling point of the liquid, the more rapid evaporation occurs. The carbon material was heated in order to study this droplet dispersion and bounding phenomenon in the droplet collision boiling system. The phenomenon was photographed by using a high-speed camera (10,000 fps) from the horizontal direction. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Nucleate-Boiling, Film-Boiling, Droplet, Thermal Conductivity, Heated Surface

I.

In contrast, Takano and Kobayashi [9] found that the temperature range expanded toward a higher value in testing a ceramic surface of low thermal conductivity. For water drops on a heating surface with a temperature range from the nucleate-boiling state to the transitionboiling state, Makino and Michiyoshi [10][11], obtained an empirical formula that correlated the liquid-solid contact time with the heating surface temperature, thermal properties, and drop diameter. This research is based on the idea that if a surface heating the liquid is significantly hotter than the liquid, then film boiling will occur, where a thin layer of vapor which has low thermal conductivity, insulates the surface. This condition of a vapor film insulating the surface from the liquid characterizes film-boiling. The surface material was heated in order to catch this film boiling phenomenon, and the behavior on the solid-liquid interface at the moment of the droplet collision was photographed from the horizontal direction by using a high speed camera. In this study, focusing on the range of the heating surface temperature (100ºC - 420ºC), namely the range from the vicinity of the maximum boiling rate point of the droplet up to the film-boiling, the liquid-solid contact behavior on the heating surface was caught. From the data received, a unique droplet bounding phenomena during film-boiling period was examined.

Introduction

Unlike bubble generation from the cavity of the solid heating surface, large numbers of minute bubbles are generated in a liquid layer by the spontaneous nucleation, when the liquid layer, which contacted the heated surface, is rapidly heated. The spontaneous nucleation boiling is also called fluctuation nucleation. Existence of this fluctuation nucleation boiling phenomenon and the generation behavior are found by Skripov [1], and afterwards it has been clarified by many researches, Baumeister, K . J ., and Simon, F.F [2], Inada and Yang [3] and Nishio and Hirata [4]. This research focused on the fact of the fluctuation nucleation boiling is generated at the moment when a water droplet collides with the solid surface heated at the fixed temperature. In this time, the boiling phenomenon, in which large number of minute liquid particles intensely scatters to the atmosphere, namely the miniaturization-boiling phenomenon named by Inada et al. [5][6], was observed. Temple-Pediani [7] investigated the effects of heating surface materials in the temperature range surrounding the maximum evaporation rate point, which diminished with an increase in the thermal conductivity of a heating material. The effects of thermal conductivity of heating surface also have been investigated by Raupeeh [8] using biofuel for gas turbine application.



812

S. Illias, M. A. Idris, M. Z. M. Zain

II.

Figure 2 shows a carbon material which is used during the experiment. This carbon material has a dimension of 30mm diameter and 8mm height. The contact angle of water droplet on heated surface is about 75º. The thermal property value for carbon material is 0.6029 x 10-4 [Ws1/2/m2K].

Experimental Apparatus

Figure 1 illustrates a schematic of the experimental apparatus. The heating surface is a disc of 30mm diameter and 8mm height, and it is retained levelly and is heated from its periphery by an electric cartridge type heater. Degassed and distilled water was used as a droplet. The water droplet diameter was retained 4.0mm, and falling height was kept at 65mm. This height is in the range that the droplet itself does not disintegrate by the collision energy of the droplet. The temperature of the droplet was kept at 16ºC by the circulation of the tab water in the circumference of the nozzle. The temperature of the heating surface was measured by attaching the thermocouples with ceramics adhesive at two points on the surface. The water droplet behavior was photographed in real time and was recorded by a frame rate of 10,000 fps at 1/10,000 shutter speeds.

III. Results and Discussion Figure 3 and Figure 4 shows the droplet boiling behavior on carbon heated surface at the temperature of 140ºC and 160 ºC. From the experiment, it is observed that a weak nucleation boiling occurred at the temperature of 140ºC and 160ºC on carbon heated surface. The captured image shows the difference between both nucleation-boiling on carbon heated surface. From Figure 3, it is observed that the water droplet begins to form a hat shape during 2.0ms and it becomes flat on the surface during 6.0ms up to 14.0ms. This phenomenon continues for a few milliseconds (ms) and after that the water droplet started to form a mountain shape at about 19.0ms. This phenomenon continues until it becomes flat again at 46.0ms. Figure 4 shows a droplet boiling behavior at 160 ºC. From Figure 4, it is observed that the nucleate-boiling becomes more aggressive compared with 140 ºC. At the temperature of 160 ºC, the water droplet have shown a difference in shape when it touches a hot surface. The droplet still continues to form a hat shape during this period but the elapsed time between both is difference. It is observed that the droplet takes a longer time from a hat shape until it becomes flat again. During the experiment, it is also observed that only nucleate-boiling and filmboiling occurred on carbon heated surface. Figure 5 shows the droplet boiling behavior on carbon heated surface at the temperature of 260ºC. At this period, it is observed that the film- boiling begins to take place on carbon heated surface. From the image captured, it is observed that droplet bounding phenomena begins at 15ms and continue until 43ms when the water droplet touches the surface again. This phenomenon continues about 28ms above the carbon heated surface. Although the film-boiling has already started, a weak dispersion from the mother droplet still occurs which means that the film-boiling is still not in perfect condition. Figure 6 shows the droplet bounding phenomena on carbon heated surface at the temperature of 340˚C. The numerical value shown under each photograph is the elapsed time after the droplet collision. When the surface temperature reaches a maximum value, the critical superheated, vapor begins to form faster than liquid can reach the surface. Thus, the heated surface suddenly becomes covered with a vapor layer. Because of the vapor layer’s lower thermal conductivity, this vapor layer insulates the surface. This condition of vapor film insulating the surface from the liquid characterizes filmboiling.

1: High speed video camera 2: Main memory 3: TV monitor 4: TV monitor controller 5: Computer 6: Pen recorder 7: Degassed and distilled water 8: Circulating water at constant temperature 9: Halogen lamp 10: Droplet 11: Shutter 12: Cartridge heater 13: Thermocouple 14: Material disc Fig. 1. Experimental Apparatus

4 mm

75˚

Fig. 2. Carbon Heated Surface



813


Fig. 3. Nucleate boiling phenomena on carbon heated surface at the temperature of 140˚C

Fig. 4. Nucleate boiling phenomena on carbon heated surface at the temperature of 160˚C

Fig. 5. Film boiling phenomena on carbon heated surface at the temperature of 260˚C




814




From Figure 6, it is observed that the droplet begins to bounds at 26ms and continues to float up in the air until 43ms when the droplet once again touches the surface. This means that the droplet floating phenomena on the carbon heated surface continues at about 17ms as shown in Figure 6. The highest level of droplet bounding height occurred at 35.0ms on carbon heated surface at the temperature of 420ºC. The bounding height is approximately 3mm as shown in Figure 8. This bounding phenomena on heated surface is highly caused by the vapor pressure that released from the hot surface. A high vapor pressure is produced when the droplet approaches to the hot surface. This bounding phenomena of the droplet is largely due to the large repulsive force produced by the vapor pressure from the hot surface. From Figure 6, Figure 7 and Figure 8, it is assumed that a very hot vapor layer has been generated on the heating surface. During this period, the droplet floats on a vapor cushion and the droplet has no direct contact with carbon heated surface. Figure 9 shows the relation between elapsed time and bounding height of water droplets that occurred on carbon heated surface. It is observed that a weak filmboiling begins to form at the temperature of 260ºC on carbon heated surface. From Figure 9, it is also observed that the bounding phenomena starts at the temperature of 260ºC and continues until the surface reaches a high temperature of 420ºC.

Droplet bounding on hot surface is very unique phenomena and only occurred during film-boiling period. From Figure 9, it is observed that the highest level of droplet bounding take places on carbon heated surface at the temperature of 420ºC. The droplet shape changes spontaneously due to free vibration period of the droplet. Droplet shape movements and changes can clearly be observed through the captured image shown in Figure 6, Figure 7 and Figure 8.

Fig. 9. Bounding height on carbon heated surface



815


Figure 11 shows the relationship between initial wall temperature and droplet floating period on carbon heated surface. From Figure 11, it is observed that the droplet floating time increase from 260 ºC up to 320 ºC on carbon heated surface. Then the time decrease when the surface temperature reaches 340 ºC and 360 ºC. At the temperature of 340 ºC, it is observed that the droplet begins to bounds at approximately 26ms and it keeps on floating above the carbon surface until 43ms when the droplet touches the surface again. The floating time during this period is about 17ms as shown in Figure 6. The longest period of droplet bounding phenomena on carbon heated surface occur at the temperature of 420 ºC which is about 46ms. The droplet begins to bounds at about 19ms and keeps on floating until it touches the surface again at 65ms as shown in Figure 8. It is also observed that the maximum bounding height occur at the same temperature of 420 ºC. Although, the graph in Figure 11 shows a pattern of increase and decrease in time(ms), but it is difficult to find correlation between both.

Fig. 10. Relationship between temperature change and droplet evaporation lifetime curve of carbon heated surface

Figure 10 shows the relation between temperature change and droplet evaporation lifetime curve of carbon heated surface. From Figure 10, it is observed that a week nucleation-boiling begins at the temperature of 100ºC and continue until the surface temperature reach 220ºC on carbon heated surface. The curve on graph starts to fall down when the temperature reach 260ºC which means that the film-boiling phenomena already begins to take place. From the graph, it can be conclude that this unique film- boiling phenomena on carbon heated surface starts at the temperature of 260 ºC and continue until the surface reaches a high temperature of 420 ºC. The carbon S-shaped graph also agrees closely with the Leadenfrost effect. The Leidenfrost point signifies the onset of stable film- boiling. It represents the point on the boiling curve where the heat flux is at the minimum and the surface is completely covered by a vapor blanket. Heat transfer from the surface to the liquid occurs by conduction and radiation through the vapor.

IV.

Conclusion

1. Only nucleate-boiling and film-boiling occurred on carbon heated surface. 2. The maximum droplet bounding height occurred on carbon heated surface at the temperature of 420ºC which is approximately 3mm. 3. Droplet evaporation lifetime curve on carbon heated surface agrees closely with the Leadenfrost effect graph. 4. The longest period of droplet floating phenomena occurred on carbon heated surface at the temperature of 420 ºC which is approximately 46ms.

Acknowledgements The authors would like to thank Universiti Malaysia Perlis for awarding a research grant to continue this research project (grant number: 9001-00183). The authors also would like to thank School of Manufacturing Engineering, Universiti Malaysia Perlis for supporting this research.

References [1] [2]

[3]

[4] Fig. 11. Relationship between initial wall temperature and droplet loating period on carbon heated surface


Skripov,V . P ., Metastable Liquid, John Wiley, New York (1974). Baumeister, K. J., and Simon, F. F., “Leidenfrost Temperature – Its Correlations for Liquids Melts, Cryogens, Hydrocarbons, and Water,” ASME Journal of Heat Transfer, Vol. 95, pp.166-173, 1973. Inada, S. and Yang, W. –J. Heat Transfer in Two-Phase Flow, Heat Transfer 2002, Vol 3, Experimental Technique, pp. 389, 2002. Nishio, S., and Hirata, M., “Study on the Leidenfrost Temperature (2nd Report, Behavior of Liquid-Solid Contact Surface and Leidenfrost Temperature), “ Trans. JSME, Vol.44, pp.1335-1346, 1978.


816


[5]

Inada, S., and Yang, W, -J., Visualization of Liquid-Solid Contact Process of Drop Impinging on a Hot Quartz Plate by Lazer Holographic Interferometry, Proceeding, ASME-JSME Thermal Engineering Joint Conference, Vol.3, pp. 361-364, 1991. [6] Inada, S., Miyasaka, Y., Sakumoto, M., and Mogi, I., “Evaporation Behaviors of a Water Droplet and Heat Transfer on a Heated Surface Contaminated by Scale Adherence, “Trans. JSME, Vol. 54, pp. 2121-2127, 1988. [7] Temple-Pediani, R.W., “Fuel Drop Vaporization Under Pressure on a Hot Surface,” Proc. IMechE, Vol. 184, pp. 677-690, 1969. [8] N. N. Raupeeh, M. N. M. Jaafar, M. A. A. Arizal, M. S. A. Ishak, Development of Gas Turbine Spray Test Rig for Biofuel, International Review of Mechanical Engineering, Vol. 5 N. 1, pp. 106-112, 2011. [9] Takano, T., and Kobayashi, K. “Vaporization Behavior of a Single Droplet Impinging on a Heated Surface With a FlameSprayed Ceramic Coating,” Heat Transfer-Japanese Research, Vol. 20, pp 1-17, 1991. [10] Makino, K and Michiyoshi, I., .The behavior of a water droplet on heated surface. J. Heat Mass Transfer, 27-5, pp. 781-791, 1984. [11] Makino, K. and Michiyoshi, I., Discussion of Transient Heat Transfer to a Water Droplet on Heated Surfaces under Atmospheric Pressure, International Journal of Heat and Mass Transfer, 30 (9), pp. 1895-1905, 1987.


Lecturer, School of Manufacturing Engineering, Universiti Malaysia Perlis, PO Box 77, Pejabat Pos Besar, 01000 Kangar, Perlis, Malaysia. E-mail: [email protected] 2 Lecturer, School of Material Engineering, Universiti Malaysia Perlis, PO Box 77, Pejabat Pos Besar, 01000 Kangar, Perlis, Malaysia.



817


Activated Carbon for Drying Compressed Air for Low Pressure Applications N. S. Senanayake1, D. M. K. K. Dissanayake2 Abstract – This paper presents the results of a study on the use of activated carbon for dehydration of compressed air. Compressed air consists of moisture and other impurities that are detrimental to smooth operation of machine tools and equipment. Usually dehydration is done centrally by different methods such as refrigeration drying, deliquescent drying, membrane drying, and adsorption drying. In adsorption drying, silica gel and activated alumina are widely used and these materials are relatively expensive compared to activated carbon which is locally produced using different sources of biomass. The study proved that activated carbon adsorbent beds can successfully be used to dry compressed air at 4 – 8 bar to meet the industry requirements. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Activated Carbon, Drying of Compressed Air, Adsorption Drying

I.

Liquid water and water vapor removal increases the efficiency of air operated equipment, prevents corrosion and clogging, extends the service life of pneumatic components, prevents air line freeze-ups and reduces product rejects. Water could exist in compressed air in three phases, i.e. liquid, aerosol (mist) and vapor (gas). The most noticeable and removable are water and aerosol which can be removed by high efficiency filtration together with refrigeration dryers. Water vapor is more difficult to remove and requires the use of a desiccant dryer with high efficiency filters.

Introduction

Compressed air is an essential power source in many industries today. It is used in various types of pneumatic machines and tools for instance; textile garment knitting machines, molding machines, pneumatic conveyers, hand drills, lifting equipments etc. Compressed air is produced by compressing atmospheric air and contains all the impurities those are available in the atmosphere. These impurities include moisture, dust and dirt particle smaller than the air intake filter rating. In addition, oil contamination occurs at the intake location or in an oil lubricated compressor. At high temperatures, oils break down to form acids and mixing with particulates, forms sludge. Oils can also act like water droplets. These impurities and resulting condensate rapidly wears tools and pneumatic machinery, blocks valves and orifices causing costly maintenance. It also erodes pipes and valves resulting in product contamination. Treatment of compressed air is an important process before it is introduced to sensitive and expensive pneumatic machines and tools for smooth operation in their life span. Compressed air treatment is done at various stages depending upon the type of pneumatic machines and tools in the network and the purpose for which compressed air is used. Air filters, pre-filters, after-filters, micro-filters, dryers, lubricators, oil separators, activated carbon filters, cyclone separators are used for compressed air treatment process. Most compressors, nowadays have an after cooler as a standard fitting to remove the water. The after-cooler reduces the water content of the air by around 68%. The compressed air dryers reduce the water vapor concentration and can prevent further liquid water formation in air lines.

II.

Methods of Drying Compressed Air

Four methods are commonly used in the industry to remove the water in the compressed air, namely refrigeration drying, membrane drying, deliquescent drying and adsorption drying. Refrigeration drying is based on the principle that when the temperature of the compressed air is lowered the moisture holding capacity is reduced. Hence by lowering the temperature the water vapor in the compressed air is condensed and removed. Membrane materials permeable to water vapor are an excellent medium for producing dry air from standard compressed air. The membrane module consists of bundles of hollow membrane fibers, each permeable to water vapor. As the compressed air passes through the center of these fibers, water vapor permeates through the walls of the fiber. A small portion of the dry air (purge flow) is redirected along the outside of each hollow fiber, carrying away the moisture-laden air which is then exhausted to room atmosphere. The remainder of the dry air is piped to the application. Membrane dryers can be



818

N. S. Senanayake, D. M. K. K. Dissanayake

purification of fats, oils, beverages, decolorisation of cane and beet sugar solutions and in many more applications [7]. Charcoal was initailly the only raw material for producing activated carbon, but it has been partly replaced due to price considerations and the limited availability of charcoal, by other carbon materials (FAO, 1985). In 2003, Martinze et al. [8] reported a study on the production of activated carbon from agricultural industry resudues which gave activated carbon that were comparable with the commercial activated carbon. The production of activated carbons from agricultural wastes converts unwanted, surplus agricultural waste, of which billions of kilograms are produced annually, to useful valuable adsorbants [9]. The adsorption capacities of gas filters containing activated carbon are specified by the break-through time, and the break-through time is affected by temperature, flow rate, and moisture content. Among these factors, moisture content has been the focus of many quantitative researches works [10]. In the present study possibility of using an activated carbon bed for the dehydration of compressed air of relatively low pressure was investigated as an alternative to the refrigeration drying and other adsorbents used in the industry.

conveniently located near the point-of use and can supply clean dry compressed air with low pressure dew points (from +4 to - 40oC). These dryers are costly, especially under light load conditions and membranes are also highly susceptible to oil and dirt, which causes the membranes to break down easily. As the structure is microscopic, membranes cannot be cleaned, and has to be replaced. Deliquescent drying involves chemicals (salts) usually in tablet form. As the compressed air passes through the salt, the salt attracts water and dissolves into brine that can be drained off. Desiccant dryers use substances or desiccants that attract and hold water. Two types of desiccants (adsorbents) are commonly used, namely activated alumina and silica gel. These substances do not dissolve when they absorb water. They fill up as much as water as they can hold and maintain their shape. Once they are full they need to be replaced or be regenerated. Reactivation of adsorbent materials is very difficult, but adsorbents an easily be reactivated by heating the bed of material to a certain temperature [1]. These dryers are used primarily where very low dew points are required. Silica gel and activated alumina are the common substances used in desiccant or adsorbent dryers. Silica gel and zeolites have been utilized for dehumidification processes in industrial and residential applications for their great pore surface area and good moisture adsorption capacity [2]. Silica gel is an amorphous form of silicon dioxide. It will adsorb up to 40% of its own weight in water vapor with an adsorption efficiency of approximately 35%. Gently heating silica gel will drive off the adsorbed moisture and leave it ready for reuse. These are available in a wide range of sizes suitable for use with a wide range of applications. Activated alumina is an aluminum oxide which is highly porous and exhibits tremendous surface area. Because activated alumina has a higher capacity for water than that for silica gel at elevated temperatures it is mainly used as a desiccant for warm gasses including air. Activated alumina has a moisture adsorption capacity of 42% at 100% RH and 7.5 % at 10% RH [3]. Activated carbon is a form of carbon that has been processed to make it extremely porous and thus to have a very large surface area avaiable for adsorption of mositure. Due to its high degree of microporosity, just 1 gram of activated carbon has a surface area in excess of 500 m2 . Activated carbon is used in a variety of applications in commercial level and dometic level. Compared to other adsorbants activated carbon or char coal is abondant in many coutries and it can be prodcued easily by various methods. Activated carbon has been used in many industrial applications for instance, removal of SO2 from humidified flue gas [4], and removal of NH3-N, NO2-N and UV254 organics from water [5]. Bingnan (2010) [6] repoted a study on the H2S adsorption capacity of the activated carbon from coke oven gas. Activated carbon is also used for the purification of drinking water,

III. Materials and Method A bed of activated carbon was constructed as shown in Fig. 1.

Fig. 1. Experimental setup for testing

Fig. 2 and Fig. 3 show the cylindrical pipe before filling up with activated carbon and after activated carbon was packed up inside it respectively. The cylindrical column fabricated from galvanized pipe of diameter 4 inch was loosely filled with 1kg of activated carbon supplied by HayCarb (Plc), a local company that produces activated carbon.



819


saturated adsorbent bed was then supplied with dehydrated air coming through a second adsorbent bed. The relative humidity of outlet air was recorded until the relative humidity became stable at high level.

IV.


Following are the observations made with regard to activated carbon adsorbent bed at 4 bar pressure. The relative humidity was 29%. TABLE II EXPERIMENTAL RESULTS WITH REGARD TO ACTIVATED CARBON ADSORBENT BED AT 4 BAR PRESSURE Inlet pressure (bar)

4

Inlet flow rate (cfm)

8.2 o

Ambient temperature ( C) Fig. 2. Cylindrical pipe before packing with activated carbon

The activated carbon used had the following properties (Table I). TABLE I PROPERTIES OF THE ADOPTED ACTIVATED CARBON Black, powder

Bulk Density

25-30 lbs/Ft3

Crush strength

20 lbs

Adsorption capacity: 100% RH

39% /wt

90% RH

35.5 % /wt

60% RH

30% /wt

Abrasion loss (%/wt)

< 0.2

Inlet compressed air temperature( C)

28

Desiccant bed height (mm)

305

Desiccant bed diameter (mm)

102

Initial mass of activated carbon (kg)

1

Final mass of activated carbon(kg)

1.36

Breaking point (min)

30

Length of unused bed (mm)

76

Maximum pressure drop (bar)

0.3

Pressure drop at breaking point (bar)

0.2

Based on the mass of activated carbon before and after adsorption the adsorption capacity of the activated carbon was calculated as 36%. This confirmed the manufacturer’s specification of 35.5% at 90% relative humidity at atmospheric pressure. The relative humidity ratio was plotted against the time corresponding to the two pressures (4bar and 8bar) as shown in Fig. 4 and Fig. 5. The value of c is the relative humidity of the air at outlet and co is the relative humidity of inlet air. According the Fig. 5 the break through time is 30min at which the outlet relative humidity begins to rise steadily. In a time period of 40min the relative humidity of compressed air leaving the bed reached 20% (5.8% RH) of the inlet humidity at 4bar pressure. For intended applications 6% RH with corresponding pressure due point of -20oC is acceptable to avoid serious corrosion problems in the pipe net work. So, for many applications the adsorbent bed could be used for 40min, and thereafter it needs to be regenerated. The unused bed length was estimated as 76mm using the following equation when the bed needed to be regenerated: ⎛ t ⎞ LUB = ⎜ 1 − *b ⎟ L ⎝ t ⎠

Fig. 3. Cylindrical pipe after packed with activated carbon

Color and form

30 o

Compressed air at 4bar and 8bar was passed through the adsorbent bed and relative humidity at the inlet and outlet of the bed was measured using RH meters in until outlet relative humidity reached the inlet conditions. After the outlet air stream became stable close to the relative humidity of the inlet air, the supply of compressed air was stopped. The mass of activated carbon after ceasing the air flow was measured. The

where: LUB=Length of unused bed (m)



820


tb = Time for breakthrough (min) t* = Allowed adsorption time (min) L = Length of bed (m)

Fig. 7. Pressure drop variation with time at 8bar at inlet

V.

Conclusion

The tests carried out at 4bar and 8bar pressure showed the possibility of using activated carbon to dehydrate compressed air. The break through time was found to be 30min and in order to satisfy the minimum moisture level requirement in compressed air (i.e.6% relative humidity) the time of drying was 40min. At this stage the moisture adsorption by the activated carbon was 36% as against the values reported by Abiko et al. [11] in 2010 which was 50%. Further the same authors in a separate study reported time durations taken by activated carbon samples to reach saturation at different relative humidity values. They reported slightly less than 60min time duration at 60% relative humidity of air. According to a study on silica gel beds the time taken to reduce the relative humidity to 20% (c/co = 0.2) of the inlet air was reported as nearly 50min [2]. As far as these studies are concerned the moisture breakthrough time for activated carbon and the time duration for adsorbing moisture of air until relative humidity reaches to an acceptable limit (i.e. 20% of the ambient) was found to be slightly less which will not have a significant effect on the use of activated carbon as a moisture adsorbent in compressed air. The maximum pressure drop occurred was acceptable to meet the industry requirements. At the same time, the cost of drying of one kilogram was SLR 22.00, which could be further reduced by regeneration and reusing the adsorbent. Suitable equipment needs to be designed for the requirement of the industry and with an automated regeneration mechanism most suitable for small scale industries in developing countries.

Fig. 4. Breakthrough curves for adsorption

Fig. 5. Breakthrough curves for regeneration

The drop in pressure experienced was very small, due to the loose filling of the activated carbon bed as shwin in Fig. 6 and Fig. 7. The maximum drop with 4bar inlet pressure occurred during the drying period was 0.3bar (7.5%). At 8bar pressure this was 0.5bar (6.25%). Taking the price of 1kg of activated carbon as SLR 200.00 (Sri Lankan Rupee), simple computation gives that the material cost of drying amounts to SLR 22.00 (0.20US$) for one cubic meter of compressed air. This is not considering the advantage gained through regeneration of the adsorbent bed.

References [1]

[2]

[3] [4]

Fig. 6. Pressure drop variation with time at 4bar inlet


A.K.M. Iqbal Hussain, Reactivation of Silica Gel Moist Air Dehumidifier, Proc. 1st International Conference on the Developments in Renewable Energy Technology (ICDRET 2009), December 17 – 19, Dhaka, Bangladesh, 2009, pp.65-68. K.S. Chang, H.C. Wang, T.W. Chung, Effect of regeneration on the adsorption dehumidification process packed silica gel beds, Applied Thermal Engineering, Vol.24,pp.735-742, 2004. www.bhindsut.com, Adsorbent desiccants, M.A.Daley, C.L. Mangun, J.A.DeBarr, S.Riha, A.A.Lizzio, G.L. Donnals, and J.Economy, Adsorption of SO2 onto Oxidized and


821


N. S. Senanayake graduated in Mechanical Engineering in 1985. He joined the Open University of Sri Lanka (OUSL) in 1986 as an Assistant Lecturer in Mechanical Engineering. In 1996, He obtained his PhD from the University of Cranfield, United Kingdom in the area of Food Processing Machine Development. At present N. S. Senanayake works as a Senior Lecturer- Grade I in Mechanical Engineering at the OUSL. He is the Program Facilitator for the Sri Lankan students following the Worldwide M.Sc. program in Sustainable Energy Engineering conducted by the Royal Institute of Technology (KTH) in Sweden. His current research includes development of Open and Distance learning methods for engineering degree programmes and sustainable use of energy in the manufacturing industries.

Heat-treated Activated Carbon Fibers (ACFS), Carbon, Vol.35, No.3, pp.411-417, 1997. [5] Yin Yan-e, Hu Zhong-hua, Shen Xin-qiang, Removal of NH3-N, NO2-N and UV254 organics by biological activated carbon fiber, Proc. 2nd International Conference on Bioinformatics and Biomedical Engineering (iCBBE 2008), Shanghai, china, May 1618, 2008, pp.3591-3595. [6] R.E.N. Bingnan, Thermodynamic Analysis of Adsorption of H2S on Modified Activated Carbon, Proc. 4th International Conference on Bioinformatics and Biomedical Engineering (iCBBE), Chengdu, China, June 18 – 20, 2010, pp. 1-3. [7] FAO, Industrial charcoal making, Forestry Paper 63, Corporate Document Repository, Food and Agriculture Organization of the United Nations, 1985. [8] M.L.Martinez, L. Moiraghi, M. Agnese, C. Guzman, Making and Some Properties of Activated Carbon Produced From Agricultural Industrial Residues From Argentina, The Journal of Argentine Chemical Society, Vol.91, No. 4/6, 2003, pp. 103-108. [9] D. Mohan, K.P. Singh, Single and Multi-component adsorption of cadmium and zinc using activated carbon derived from bagasse – an agricultural waste, Water Research, 36, 2002, pp. 2304-2318. [10] H.Abiko, M. Furuse, and T.Takano, Reduction of Adsorption Capacity of Coconut Shell Activated Carbon for Organic Vapors Due to Moisture Contents, Industrial Health, Vol.48, 2010, pp.427-437. [11] H.Abiko, M. Furuse, and T.Takano, Quantitative Evaluation of the Effect of Moisture Contents of Coconut Shell Activated Carbon Used for Respirators on Adsorption Capacity for Organic Vapors, Industrial Health, Vol.48, 2010, pp.52 – 60.

D. M. K. K. Dissanayake graduated in Mechanical Engineering 2010, from the Open University of Sri Lanka. He at present works as a Senior Production Executive at the Renuka Group (Kandy Plantations Limited), a leading company producing desiccated coconut. Dissanayake has also worked as an Assistant Engineer at Matrix (pvt) Ltd, Sri Lanka for four years. His current research interests are industry automation and matters related to optimized utilization of compressed air in the manufacturing industry.


Corresponding Author, Senior Lecturer, Department of Mechanical Engineering, Faculty of Engineering Technology, Open University of Sri Lanka, Nugegoda, Sri Lanka. E-mail: [email protected] Telephone +94 11 2881314 Fax:+94 11 2822737 2 Senior Production Executive, Renuka Group (Kandy Plantation Limited), Giriulla, Sri Lanka. E-mail: [email protected] Telephone: +94 772908014



822


Analytical Solution for Chemically Reacting Free Convective Couple Stress Fluid in an Annulus with Soret and Dufour Effects D. Srinivasacharya1, K. Kaladhar2

Abstract – The purpose of this research is to study and examine the effects of the cross diffusion (namely the Soret and Dufour effects) in the presence of chemical reaction on fully developed natural convection heat and mass transfer of a couple stress fluid in an annulus formed by two circular cylinders. The governing non-linear partial differential equations are transformed into a system of ordinary differential equations using similarity transformations. The resulting equations are then solved for approximate analytical series solutions using Homotopy Analysis Method (HAM). Profiles of dimensionless velocity, temperature and concentration are shown graphically for various values of Dufour number, Soret number, Couple stress parameter and chemical reaction parameter. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Free Convection, Couple Stress Fluid, Soret and Dufour Effect, Heat and Mass Transfer, HAM, Chemical Reaction

Superscript

Nomenclature Br Cp Df Dm f g GrT GrC k1 K KT p Pr Re S Sr T u

'

Brinkman number Specific heat at constant pressure Dufour number Mass diffusivity Reduced stream function Dimensionless microrotation Temperature Grashof number Mass Grashof number Rate of Chemical reaction Chemical reaction parameter Thermal conductivity of the fluid pressure Prandtl number Reynolds number Couple stress fluid parameter Soret number Temperature Velocity components in the φ direction

I.

µ ν ρ Ω

Introduction

There has been widespread interest in the study of natural convection of a fluid in a cylindrical annulus between two vertical concentric cylinders subject to axial rotation. This motivation is primarily due to the extensive range of practical applications such as electrical machineries where heat transfer occurs in the annular gap between the rotor and stator, growth of single silicon crystals and other rotating systems. The flow in a rotating annulus can be considered as a possible analogue of the motion of a planetary atmosphere. The first numerical study of free convection in a rotating annulus was conducted by Williams [1]. He made an excellent analysis of axisymmetric flows in the annular geometry for different combinations of physical parameters. The accuracy of the numerical study was further verified with experimental observations [2] for the same geometry and are found to be in good agreement. de Vahl Davis et al. [3] made a numerical study of natural convection in a vertical annular cavity by considering two different cases. In the first case the inner and lower surfaces are heated and rotating while the other two surfaces are stationary and cooled. In the second case the inner and upper surfaces are heated, rotating and other surfaces are cooled and stationary. The present trend in the field of chemical reaction analysis is to give a mathematical model for the system to predict the reactor performance. A large amount of research work has been reported in this field. Chemical

Greek symbols α βT, βC λ η1 θ

Differentiation with respect to λ

Thermal diffusivity Coefficients of thermal and Solutal expansion Similarity variable The coupling material constant Dimensionless temperature Dimensionless concentration Dynamic viscosity Kinematic viscosity Density of the fluid Angular velocity



823

D. Srinivasacharya, K. Kaladhar

reaction can be described as either heterogeneous or homogeneous processes, which depends on whether it occurs at an interface or as a single-phase volume reaction. Research on combined heat and mass transfer with chemical reaction and thermophoresis effect can help to design for chemical processing equipment, formation and dispersion of fog, distribution of temperature and moisture over agricultural fields as well as groves of fruit trees, damage of crops due to freezing, food processing and cooling towers. Cooling towers are the cheapest way to cool large quantities of water. In particular, the study of heat and mass transfer with chemical reaction is of considerable importance in chemical and hydrometallurgical industries. For example, formation of smog is a first-order homogeneous chemical reaction. Considering the emission of NO2 from automobiles and other smoke-stacks, NO2 reacts chemically in the atmosphere with unburned hydrocarbons (aided by sunlight) and produces peroxyacetyl nitrate, which forms an envelope can be termed as photochemical smog. Mass transfer in concentric rotating cylinders with surface chemical reaction in the presence of Taylor vortexes considered by Ashar et al. [4]. Recently, Pop et al. [5] investigated the steady fully developed mixed convection flow in a vertical channel with first-order chemical reaction. Shateyi et al. [6] considered the two-dimensional flow of an incompressible viscous fluid through a non-porous channel with heat generation and a chemical reaction. When heat and mass transfer occur simultaneously in a moving fluid, the relations between the fluxes and the driving potentials are of a more integrate nature. It has been observed that an energy flux can be generated not only by temperature gradients but also by concentration gradients. The energy flux caused by a concentration gradient is termed the diffusion-thermo (Dufour) effect. On the other hand, mass fluxes can also be created by temperature gradients and this embodies the thermaldiffusion (Soret) effect. In most of the studies related to heat and mass transfer process, Soret and Dufour effects are neglected on the basis that they are of a smaller order of magnitude than the effects described by Fourier’s and Fick’s laws. But these effects are considered as second order phenomena and may become significant in areas such as hydrology, petrology, geosciences, etc. The Soret effect, for instance, has been utilized for isotope separation and in mixture between gases with very light molecular weight (H2, He) and of medium molecular weight (N2, air). The Dufour effect was recently found to be of order of considerable magnitude such that it cannot be neglected [7]. Kafoussias [8] presented the local similarity solution for combined free-forced convective and Mass Transfer Flow Past a Semi-infinite Vertical Plate. Dursunkaya and Worek [9] studied diffusionthermo and thermal-diffusion effects in transient and steady natural convection from a vertical surface, whereas Kafoussias and Williams [10] presented the same effects on mixed convective and mass transfer

transfer steady laminar boundary layer flow over a vertical flat plate with temperature dependent viscosity. Hsiao [11] studied heat and mass mixed convection for viscoelastic fluid past a stretching sheet with ohmic dissipation through a porous space later he presented numerical calculation heat and mass transfer of a micropolar fluids flow with magnetic and radiation effects to past a stretching sheet [12]. During recent years the study of convection heat and mass transfer in non-Newtonian fluids has received much attention and this is because the traditional Newtonian fluids cannot precisely describe the characteristics of the real fluids. Ziabakhsh and Domairry [13] have obtained the solution for natural convection of the Rivlin-Ericksen fluid of grade three between two infinite parallel vertical flat plates. Sajid et al. [14] studied fully developed mixed convection flow of a viscoelastic fluid between permeable parallel vertical walls using HAM. Steady heat transfer boundary layer solutions of polar fluid presented by Ferdows et al. [15]. In addition, progress has been considerably made in the study heat and mass transfer in magneto hydrodynamic flow of nonNewtonian fluids due to its application in many devices, like the MHD power generator, aerodynamics heating, electrostatic precipitation and Hall accelerator etc. Different models have been proposed to explain the behavior of non-Newtonian fluids. Among these, couple stress fluids introduced by Stokes [16] have distinct features, such as the presence of couple stresses, body couples and non-symmetric stress tensor. The couple stress fluid theory presents models for fluids whose microstructure is mechanically significant. The effect of very small microstructure in a fluid can be felt if the characteristic geometric dimension of the problem considered is of the same order of magnitude as the size of the microstructure. The main feature of couple stresses is to introduce a size dependent effect. Classical continuum mechanics neglects the size effect of material particles within the continua. This is consistent with ignoring the rotational interaction among particles, which results in symmetry of the force-stress tensor. However, in some important cases such as fluid flow with suspended particles, this cannot be true and a size dependent couple-stress theory is needed. The spin field due to microrotation of freely suspended particles set up an antisymetric stress, known as couple-stress, and thus forming couple-stress fluid. These fluids are capable of describing various types of lubricants, blood, suspension fluids etc. The study of couple-stress fluids has applications in a number of processes that occur in industry such as the extrusion of polymer fluids, solidification of liquid crystals, cooling of metallic plate in a bath, and colloidal solutions etc. Stokes [16] discussed the hydromagnetic steady flow of a fluid with couple stress effects. A review of couple stress (polar) fluid dynamics was reported by Stokes [17]. Recently Srinivasacharya and RamReddy [18] considered the effect of double stratification on mixed convection in a



824


micropolar fluid saturated non-darcy porous medium. The homotopy analysis method [19] was first proposed by Liao in 1992, is one of the most efficient methods in solving different types of nonlinear equations such as coupled, decoupled, homogeneous and nonhomogeneous. Also, HAM provides us a great freedom to choose different base functions to express solutions of a nonlinear problem [20]. The application of the Homotopy Analysis Method (HAM) in engineering problems is highly considered by scientists, because HAM provides us with a convenient way to control the convergence of approximation series, which is a fundamental qualitative difference in analysis between HAM and other methods. Later Liao [21] presented an optimal homotopy analysis approach for strongly nonlinear differential equations. Recent developments of HAM, like convergence of HAM solution, Optimality of convergence control parameter discussed by Turkyilmazoglu [23],[24]. In this paper, we have investigated the Soret and Dufour effects on steady free convective heat and mass transfer flow between circular annulus in couple stress fluid with chemical reaction. The Homotopy Analysis method is employed to solve the governing nonlinear equations. Convergence of the derived series solution is analyzed. The behavior of emerging flow parameters on the velocity, temperature and concentration is discussed.

II.

⎡ ∂ 2T

α⎢

⎣ ∂r

+

(

)

⎡ ∂ 2T 1 ∂T ⎤ ⎢ 2 + ⎥+ r ∂r ⎦ ⎣ ∂r

(5)

where u is the velocity component of the fluid in the direction of φ, p is the pressure, ρ is the density, g is the acceleration due to gravity, µ is the coefficient of viscosity, βT is the coefficient of thermal expansion, βC is the coefficient of solutal expansion, α is the thermal diffusivity, Dm is the mass diffusivity, CP is the specific heat capacity, CS is the concentration susceptibility, Tm is the mean fluid temperature, KT is the thermal diffusion ratio, η1 is the additional viscosity coefficient which specifies the character of couple-stresses in the fluid and k1 is the rate of chemical reaction: ∇12u =

∂ ⎡1 ∂ ( ru )⎤⎥ ∂r ⎢⎣ r ∂r ⎦

The boundary conditions are given by

Mathematical Formulation

u − µ ∇12

η1 ρ CP

⎡ ∂ 2C 1 ∂C ⎤ Dm KT Dm ⎢ 2 + ⎥+ r ∂r ⎦ Tm ⎣ ∂r −k1 ( C − Ca ) = 0

Consider an incompressible, laminar free convective couple stress fluid between two coaxial concentric circular cylinders of radii a and b (a < b). Choose the cylindrical polar coordinate system (r, φ, z) with z-axis as the common axis for both cylinders. The inner is at rest and the outer cylinder is rotating with constant angular velocity (Ω). The flow being generated due to the rotation of the outer cylinder. The fluid properties are assumed to be constant except for density variations in the buoyancy force term. In addition, the Soret and Dufour effects with chemical reaction are considered. The flow is a natural convection caused by buoyancy forces. With the above assumptions and Boussinesq approximations with energy and concentration, the equations governing the steady flow of an incompressible couple stress fluid are:

η1 ∇14

2

2 2 1 ∂T ⎤ µ ⎡⎛ ∂u ⎞ u ∂u ⎛ u ⎞ ⎤ 2 − + ⎢⎜ ⎥+ ⎟ ⎜ ⎟ ⎥+ r ∂r ⎦ ρ CP ⎣⎢⎝ ∂r ⎠ r ∂r ⎝ r ⎠ ⎥⎦ (4) 2 2 ⎡ ⎤ D K C C 1 ∂ ∂ ∇12 u + m T ⎢ 2 + ⎥=0 CS CP ⎣ ∂r r ∂r ⎦

+

u = 0 at r = a and u = bΩ at r = b

(6a)

∇12u = 0 at r = a and r = b

(6b)

T = Ta ,C = Ca at r = a and T = Tb , C = Cb at r = b

(6c)

The boundary condition (6a) corresponds to the classical no-slip condition from viscous fluid dynamics. The boundary conditions (6b) imply that the couple stresses are zero at the surfaces. Introducing the following similarity transformations: r =b λ,u =

Ω

λ

f (λ ) ,

T − Ta = (Tb − Ta )θ , C − Ca = ( Cb − Ca ) φ

(7)

∂u =0 ∂ϕ

(1)

in equations (3)-(4), we get the following nonlinear system of differential equations:

∂p u = ∂r r 2

(2)

1 iv S 2 ⎡⎢λ 2 f ( ) + 2λ f ''' ⎤⎥ − λ f '' + ⎣ ⎦ 4 1 GrT 1 GrC λθ − λφ = 0 − 16 Re 16 Re

u − g βT (T − Ta ) − g βC ( C − Ca ) = 0 (3)


(8)


825


⎛ ⎝

3 2⎞ f ⎟+ 4 ⎠ (9) 3 2 λ θ '' + λ θ ' = 0 3 2

such that:

λ 3 θ '' + λ 2 θ ' + Br ⎜ λ 2 ( f ' ) − λ f f ' + 2

+4BrS 2 λ 3 ( f '' ) + Pr D f 2

(

(

K ( λφ '' + φ ' ) + Sc Sr ( λθ '' + θ ' ) − Sc φ = 0 4

where ci(i = 1, 2, ..., 6) are constants. Introducing nonzero auxiliary parameters h1, h2 and h3 we develop the zeroth-order deformation problems as follow:

(10)

where primes denote differentiation with respect to λ D K (T − T ) µ CP Ωb alone, Re = , Pr = , Sr = m T b a , ν Tm ( Cb − Ca ) ν KT K=

k1 b

2

ν

GrC =

, GrT =

g βT (Tb − Ta ) b3

ν

g βC ( Cb − Ca ) b

ν

2

2

3

, Df =

, Br =

(1 − p ) L1 ⎣⎡ f ( λ ; p ) − f0 ( λ )⎦⎤ = ph1 N1 ⎣⎡ f ( λ ; p )⎦⎤ (15) (1 − p ) L2 ⎡⎣θ ( λ ; p ) − θ0 ( λ )⎤⎦ = ph2 N 2 ⎡⎣θ ( λ ; p )⎤⎦ (16)

µΩ , K f (Tb − Ta ) 2

Dm KT ( Cb − Ca )

ν CS CP (Tb − Ta )

(1 − p ) L2 ⎡⎣φ ( λ ; p ) − φ0 ( λ )⎤⎦ = ph3 N3 ⎡⎣φ ( λ ; p )⎤⎦

and

1 η1 is the couple stress parameter, The effects of b µ couple-stress are significant for large values of S(= l/b),

f ( λ0 ; p ) = 0, f '' ( λ0 ; p ) = 0, f (1; p ) = b, f '' (1; p ) = 0

θ (1; p ) = 1, φ (1; p ) = 1

of the molecular dimensions of the liquid, it will vary greatly for different liquids. For example, the length of a polymer chain may be a million times the diameter of water molecule [16]. Therefore, there are all the reasons to expect that couple-stresses appear in noticeable magnitudes in liquids with large molecules. where primes denote differentiation with respect to λ. Boundary conditions (4) in terms of f, θ and become: f '' = 0, θ = 0, φ = 0

at

f = b,

f '' = 0, θ = 1, φ = 1

at

(18)

θ ( λ0 ; p ) = 0 ,φ ( λ0 ; p ) = 0 ,

η1 is the material constant. If l is a function µ

f = 0,

(17)

subject to the boundary conditions:

S=

where l =

)

L1 c1 + c2 λ + c3λ 2 + c4 λ 3 = 0 , L2 ( c5 + c6 λ ) = 0 (14)

)

where p [0, 1] is the embedding parameter and the nonlinear operators N1 , N2 and N3 are defined as: N1 ⎡⎣ f ( λ , p ) ,θ ( λ , p ) ,φ ( λ , p ) ⎤⎦ =

(

)

= S 2 λ 2 f ( iv ) + 2 λ f ''' + 1 1 GrT − λ f '' − 4 16 Re

λ = λ0 (11) λ =1

λθ −

(19) 1 GrC 16 Re

λφ

N 2 ⎡⎣ f ( λ , p ) ,θ ( λ , p ) ,φ ( λ , p ) ⎤⎦ = = λ 3θ '' + λ 2θ ' + 4 Br S 2 λ 3 ( f '' ) + 2

⎛a⎞ where λ0 = ⎜ ⎟ ⎝b⎠

2

3 2 3 ⎛ ⎞ + Br ⎜ λ 2 ( f ' ) − λ f ' + f 2 ⎟ + 2 4 ⎝ ⎠

(

+ Pr D f λ 3φ '' + λ 2φ '

III. The HAM Solution of the Problem For HAM solutions, we choose the initial approximations of f(η ), θ(η) and ϕ (η) as follows: f0 ( λ ) =

b ( λ − λ0 ) , 1 − λ0

λ − λ0 λ − λ0 θ0 ( λ ) = , φ0 ( λ ) = 1 − λ0 1 − λ0

∂4 ∂λ

4

, L2 =

∂2 ∂λ 2

)

N3 ⎡⎣ f ( λ , p ) ,θ ( λ , p ) ,φ ( λ , p ) ⎤⎦ = 1 = λφ '' + φ ' − K Sc φ + Sr Sc ( λθ '' + θ ' ) 4

(21)

For p = 0 we have the initial guess approximations:

(12)

f ( λ ; 0 ) = f 0 ( λ ) , θ ( λ ; 0 ) = θ 0 ( λ ) , φ ( λ ; 0 ) = φ0 ( λ ) (22)

and choose the auxiliary linear operators: L1 =

(20)

When p = 1, equations (15) – (17) are same as (8) – (10) respectively, therefore at p = 1 we get the final solutions:

(13)

f ( λ;1) = f ( λ ) , θ ( λ;1) = θ ( λ ) , φ ( λ ;1) = φ ( λ )


(23)


826


∞

Hence the process of giving an increment to p from 0 to 1 is the process of f(λ;p) varying continuously from the initial guess f0(λ) to the final solution f(λ) (similar for θ(λ;p) and ϕ (λ; p)). This kind of continuous variation is called deformation in topology so that we call system Eqs. (15)-(18), the zeroth-order deformation equation. Next, the mth-order deformation equations follow as: L1 ⎣⎡ f m ( λ ) − χ m f m −1 ( λ ) ⎦⎤ =

(λ )

(24)

L2 ⎡⎣θ m ( λ ) − χ m θ m −1 ( λ ) ⎤⎦ = h2 Rmθ ( λ )

(25)

h1 Rmf

L2 ⎡⎣φm ( λ ) − χ m φm −1 ( λ ) ⎤⎦ = h3 Rm ( λ ) φ

θ ( λ; p ) = θ0 ( λ ) + ∑ θ m ( λ ) p m

(33)

m =1 ∞

φ ( λ; p ) = φ0 ( λ ) + ∑ φm ( λ ) p m

(34)

m =1

in which h1, h2 and h3 are choosen in such a way that the series (32) – (34) are convergent at p = 1. Therefore we have from (23) that: ∞

f n ( λ ) = f ( λ; p ) = f0 ( λ ) + ∑ f m ( λ )

(35)

m =1

(26)

∞

θ n ( λ ) = θ ( λ; p ) = θ0 ( λ ) + ∑ θ m ( λ )

with the boundary conditions:

(36)

m =1

f m ( λ0 ) = 0 , f m (1) = 0, f m'' ( λ0 ) = 0 , f m'' (1) = 0 ,

θ m ( λ0 ) = 0 ,θ m (1) = 0 ,φm ( λ0 ) = 0,φm (1) = 0

∞

(27)

φn ( λ ) = φ ( λ; p ) = φ0 ( λ ) + ∑ φm ( λ )

(37)

m =1

where: Rmf

=S

2

(λ

2

f

( iv )

)

+ 2 λ f ''' +

1 1 GrT − λ f '' − 4 16 Re

(

for which we presume that the initial guesses to f, θ and the auxiliary linear operators L and the non-zero auxiliary parameters h1, h2 and h3 are so properly selected that the deformation f(λ;p), θ(λ;p) and (λ;p) are smooth enough and their mth-order derivatives with respect to p in equations (35)-(37) exist and are given respectively by:

λθ −

)

1 GrC 16 Re

λφ

(

(28)

)

Rmθ = λ 3θ '' + λ 2 θ ' + Pr D f ( λ 3φ '' + λ 2 φ ' + ⎡ m −1 ' ⎤ 3 m −1 + Br ⎢ λ 2 f m −1− n f n' − f m −1− n f n' ⎥ + 2 n =0 ⎣ n =0 ⎦

∑

3 + Br 4

∑

m −1

fm ( λ ) =

(29)

m 1 ∂ f (λ; p) m! ∂p m

p =0

m −1

∑ fm−1−n fn + 4 Br S 2 λ 3 ∑ f ''

n =0

n =0

m−1−n

f n''

1 Rm = λφ '' + φ ' + Sr Sc ( λθ '' + θ ' ) − K Sc φ 4 φ

θm ( λ ) = (30)

φm ( λ ) =

and, for m being integer:

χm = 0

for

m ≤1

=1

for

m >1

1 ∂ mφ ( λ ; p ) m! ∂p m

p =0

p =0

It is clear that the convergence of Taylor series at p = 1 is a prior assumption, whose justification is provided via a theorem [23], so that the system in (35)-(37) holds true. The formulae in (35)-(37) provide us with a direct relationship between the initial guesses and the exact solutions. All the effects of interaction of the chemical reaction as well as of the heat and mass transfer, Soret and Dufour effects and couple stress flow field can be studied from the exact formulas (35)-(37). Moreover, a special emphasize should be placed here that the mthorder deformation system (24)-(27) is a linear differential equation system with the auxiliary linear operators L whose fundamental solution is known.

(31)

The initial guess approximations f0(λ), θ0(λ) and 0(λ), the linear operators L1, L2 and the auxiliary parameters h1, h2 and h3 are assumed to be selected such that equations (15) – (18) have solution at each point p [0, 1] and also with the help of Taylors series and due to eq. (22), f(λ;p), θ(λ; p) and (λ; p) can be expressed as: ∞

f ( λ; p ) = f0 ( λ ) + ∑ f m ( λ ) p m

m 1 ∂ θ (λ; p) m! ∂p m

(32)

m =1



827


IV.

(

The expressions for f, θ and contain the auxiliary parameters h1, h2 and h3. As pointed out by Liao [17], the convergence and the rate of approximation for the HAM solution strongly depend on the values of auxiliary parameter h. For this purpose, h-curves are plotted by choosing h1, h2 and h3 in such a manner that the solutions (32)-(34) ensure convergence [17]. Here to see the admissible values of h1, h2 and h3, the h-curves are plotted for 15th-order of approximation in Figs. 1-3 by taking the values of the parameters Pr = 0.71, Sc = 0.22, Br = 0.5, Re = 2, GrT = 10, GrC = 10, S = 1.0, K = 0.1, Df = 0.03 and Sr = 2.0. It is clearly noted from Fig. 1 that the range for the admissible values of h1 is −1.4 < h1< −0.45. From Fig. 2, it can be seen that the h-curve has a parallel line segment that corresponds to a region −1.5 0), convective heat flux at x=0 with constant Heat transfer coefficients at the constant temperature. The following hypotheses have been taken into account: 1- Thermo-physical properties are assumed constant. 2- Heat transfer is one-dimensional, because of the throats thickness. 3- Heat transfer coefficients are constant. 4- Radiation is not important, the convection heat transfer is more important than radiation heat transfer. Under these conditions, the heat transfer process in the specimen can be described by the following system of equations: 0 ,

0

1

0

1

,

0 0

,

,

1

(1a) (1b)

0 0

(1c) (1d)

Here k, ρ, and c are the thermal conductivity, density, heat, and capacity, respectively. The governing equation is parabolic and the solution for the above heat conduction problem is solved by using finite volume method [23].

Fig. 1. The problem geometry

III. Simulated Inexact Measurement The measured temperature data must contain measurement errors. In order to compare the results for situations involving random measurement errors, a normally distributed uncorrelated error with zero mean and constant standard deviation are considered. The simulated inexact measurement data Y could be expressed as: (2)



909

H. Khoshkam, M. Alizadeh

To perform the iterations according to equation (4a), a (l,t) step size βn and the gradient of the functional need to be computed. In order to develop expressions for the determi-nation of these two quantities, a ‘sensitivity problem’ and an ‘adjoint problem’ are constructed as described below.

where and Y are the solution of the direct problem with an exact boundary heat flux q(l,t) and the measured temperature, respectively. Furthermore, ω is the random variable with normal distribution, zero mean and unitary standard deviation and for the 99% confidence bound we have -2.576 < ω < 2.576 [10].

IV.1.1. Sensitivity Problem

IV.

The Inverse Problem

The sensitivity problem is obtained from the original direct problem defined by equations (1a)–(1d) in the following manner: It is assumed that when q(l, t) undergoes a variation ∆ ,T is perturbed by T + ∆T . Then replacing in the direct problem q by q + ∆q and T by T + ∆T, subtracting the resulting expressions from the direct problem and neglecting the second-order terms, the following sensitivity problem for the sensitivity function ∆T are obtained:

For the inverse problem, the boundary heat flux at x = l is regarded as being unknown, but everything else in equations (1a)–(1d) is known. In addition, temperature readings at x=0 are considered available. Let the temperature reading taken by sensors at x = 0 be denoted by Y(0,t), it is noted that the measured temperature Y(0,t) contain measurement errors. Then the inverse problem can be stated as follows: by utilizing the above mentioned measured temperature data Y(0,t) , estimate the unknown boundary heat flux q(l,t). The solution of the present inverse problem is to be obtained in such a way that the following functional is minimized:

,

0,

∆

∆

Here T(0,t) are the estimated or computed temperatures at x=0 and time t , is the final time. These quantities are determined from the solution of the direct problem give previously by using an estimated heat flux for the exact q(l, t).

,

where βn is the search step size in going from iteration n to iteration 1, and (l,t) is the direction of descent (i.e. search direction) given by: ,

p

,

∆

(5a)

0

1

0

(5b)

,

0

,

0 (5c)

1

0

0,

0, ;

(5d)

(6)

∆

0, ;

(7)

0,

l, t

(4b) where T(0,t;qn) is the solution of the direct problem using estimated heat flux for exact q(l, t) at x = 0 and time t. The sensitivity functions ∆T(0,t;qn) are taken as the solutions of problem (5a) at the measured position x = 0 and time t by letting ∆q= . The search step size βn is determined by minimizing the functional given by equation (7) with respect to βn. The following expression results:

, which is a conjugation of the gradient direction at iteration n and the direction of descent pn-1(l,t) at iteration n-1. The conjugate coefficient is determined from: ,

,

∆

0

where qn+1 is replaced by the expression given in equation (4a). If temperature 0, ; linearize by a Taylor expansion, equation (6) takes the form:

(4a)

0,1, …

,

,

1

0, ;

The following iterative process based on the CGM is now used for the estimation of unknown heat flux q(l, t) by minimizing the functional S[q(l, t)]: ,

0

0

The solution for the above sensitivity problem is solved by using finite volume method [20]. The functional S(qn+1) for iteration n + 1 is obtained by rewriting equation (3) as:

IV.1. CGM

,

, ∆

(3)

0,

∆

∆

0

(4c)



910


0,

0, ∆

∆

0,

variables as η=t-tf. Then finite volume method [20] can be used to solve the above adjoint problem. Finally, the following integral term is left:

(8)

0,

IV.1.2. Adjoint problem

∆

To obtain the adjoint problem, equation (1a) is multiplied by the Lagrange multiplier (or adjoint function) λ(x,t) and the resulting expression is integrated over the corresponding space and time domains. Then the result is added to the right hand side of equation (3) to yield the following expression for the functional S[q(l,t)]:

, ∆

0,

0,

,

, The variation ∆S is obtained by perturbing q by ∆q and T by ∆T in equation (9), subtracting the resulting expression from the original equation (9) and neglecting the second-order terms. We thus find:

2

0,

0,

∆

0,

In equation (10), the double integral terms integrated by parts; the boundary conditions of sensitivity problem are utilized. The vanishing of integrands leads to the following adjoint problem for determination of λ(x,t): λ

λ

λ ,

0 λ ,

2 λ ,

,

1

0

(11a)

0

1

0

(11b)

0,

Step 4

Knowing λ(l,t), compute S' [q(l,t)] from equation (13).

Knowing S' [q(l,t)], compute γn from equation (4c) and the direction of descent pn from equation (4b). Step 6 Set ∆q = pn and solve the sensitivity problem (5a)-(5d) to obtain ∆T(x,t). Step 7 Knowing ∆T(x,t), compute the search step size from equation (8). Step 8 Knowing the search step size βn and the direction of descent , compute the new estimate qn+1(l,t) from equation (4a), and return to step 1. Step 5

0, (11c) 0&

(13)

Step 1 Solve the direct problem (1a)-(1d) and compute T(x,t), based on qn(l,t). Step 2 Check the stopping criterion (14a). Continue if not satisfied. Step 3 Knowing T(x,t) and measured temperature Y(x,t), solve the adjoint problem (11a)-(11d) and compute λ(l,t).

are the the the

0

,

The iterative procedure for the CGM can be summarized as follows: Suppose an initial guess (l,t) is available for the function q(l,t). Set n=0 and then:

∆

∆

(12b)

IV.1.3. Computational Algorithm

(10) λ x, t

,

A comparison of equations (12a) and (12b) leads to the following expression for the gradient of functional [q(l,t)] of the functional S[q(l,t)]:

λ x, t

,

∆

+ (9)

∆

(12a)

From definition [21], the functional increment can be presented as:

∆ ,

,

0 Stopping criterion if the problem contains no measurement errors is specified as:

0

1

0

(11d)

,

(14a)

where ε is a small-specified number. However, the measured temperature data must contain measurement errors, the following expression is obtained for stopping criteria ε [22]:

The adjoint problem is different from the standard initial value problems in the final time conditions at time t = tf is specified instead of the customary initial condition. However, this problem can be transformed to an initial value problem by the transformation of the time Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


911


components. All the components are gathered inside a vector as:

(14b) IV.2. BFGS

,

The steepest descent method, the conjugate gradient method, the Newton method, and the variable metric method (VMM), all belong to the gradient based class of unconstrained optimization techniques. However, VMM has superior characteristics in relation to the others[22]. The variable metric method is very stable and continues to progress towards the minimum even when dealing with highly distorted and eccentric functions. Zhang et al. [23] demonstrated mathematically that for a strictly convex quadratic objective function, the generated iterative sequence of VMM converges to the unique solution of the problem globally and super linearly. It is show that the method will yield the optimal solution with a desired degree of accuracy. The iterative procedure for the Broydon–Fletcher–Goldfarb–Shanno (BFGS) can be summarized as follows: Step 1

Step 2

Step 3

Step 4 Step 5

Step 6

Step 7

(15)

Pulse sensitivity coefficient of measured temperature of the kth Sensor with respect to each component of vector is defined as: X(

,

,

)=

,

for

=1,2,...,M

(16)

Governing equations for sensitivity coefficients are : obtained with respect to each 1

(17a)

(17b)

Find the pulse sensitivity coefficients for each components of by solving Eqs. (17a)–(17d) in the entire time domain. Start with an initial guess for (as initial base point) and with a M M positive definite symmetric matrix H1 (M is total number of unknowns). Usually H1 is taken as the identity matrix I. Set iteration number as i=1. Compute the gradient of the objective function, f ; at the base point ; and define

1 0 X( ,

(17c)

)=0

for

=1, 2 , ... , M

(17d)

The above equation should be solved M times by using finite volume method for every =1, 2 , ... , M in order to compute X( , , ). in order to estimate the components of the unknown vector, a ''sum of squares of errors function'' must be minimized:

search direction as: f Normalize by its magnitude: / Find the optimal step length λ in the direction and move to the next ''base point'' using: λ for optimality. If Test the new base point is optimal, terminate the iteration process. Otherwise, go to step (7). Update the ((H)) matrix:

,

,

,

(18)

In the above definition, K is total number of sensors, Y is the measured temperature at sensor location of , and T is the calculated temperature utilizing the direct heat conduction model (Eq. (1)) based on a given or assumed vector for . is in fact the objective function which must be minimized. Gradient of the objective function used in BFGS has the form of:

λ

,

and:

Step 8

,…,

,…,

(19)

Considering the definition of (from Eq. (18)), and using the sensitivity coefficients obtained from solving as: Eqs. (17a)-(17d) one can write

Set the new iteration number i=i+1, and go to step 3.

If ''M'' to be total number of time steps that cover the entire time domain in which heat flux is to be estimated. The unknown heat flux q (t) is discretized into M time Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


912


2

,

2

,

,

,

X

,

The following computational parameters are chosen for the numerical experiments:

,

,

,

X

,

,

,

,

X

,

,

T0=25 °C T∞ =25°C l =0.01 m k =138 W/( m K) α=5.369×10-5 m2 /s h=5000 W/( m2 K)

. . . 2

,

Here α is the thermal diffusivity of the material. Besides, the space and time increments used in numerical calculations are taken as ∆x =0.00001m (i.e.1000 grid points in space) and ∆t =1s (i.e.30 grid points for = 30s). We now present below tree numerical test cases in determining q(l,t)by the inverse analysis using the CGM & BFGS.

In step (5) of BFGS, the optimal step size (λ ) in the λ direction of is a value of λ that minimizes with respect to λ i.e., λ is the root of the following equation [3]: λ λ

0

Numerical Test-Case 1: The unknown transient boundary heat flux q(l,t) is assumed, applied at x = l in the following form:

(20)

For the stopping criteria (step 8); in this work f is used as the stopping criteria, In the case of non-noisy data, is an arbitrary small number. However, in the case of noisy data, should be chosen based on the iterative regularization method in order to reduce sensitivity of the solution to the random noise errors. The main idea in the iterative regularization is to stop the iterative procedure close but not exactly at the optimum point. Then, it will tend to regularize the solution and to damp out the destructive effects of random noises in data [22]:

,

252.5 10

1.547 3.26

10 10

3.475

(22)

The relative root mean square error (eRMS) for the estimated q(l, t) is defined as: 1

∑

, 1

∑

,

100%

(23)

,

(21) where I and N represent the index of discrete time and total number of measurements, respectively, while , denote the estimated values of heat flux.

where K and A are total number of sensors and noise amplitude respectively, while M is the total number of time steps.

V.

TABLE I ROOT MEAN SQUARE ERROR AND CONVERGENCE CRITERIA FOR ESTIMATING HEAT FLUX TO THE THIRD DEGREE POLYNOMIAL Number of Run time || || eRMS Iterations (s) at final


Our simulations define from Eqs. (1a)–(1d) that estimate the strength of the boundary heat flux. To illustrate the accuracy of the CGM and BFGS in predicting boundary heat flux q(l,t) with the present inverse analysis three different boundary heat flux functions over temporal domain; namely, a third degree polynomial function, triangular function and a step function are adopted to illustrate the numerical modeling. The exact temperature and the heat flux used in the following examples are selected so that these functions can satisfy Eqs. (1a)– (1d). One of the advantages of using the IHCP to solve the inverse problems is that, the initial guesses of the unknown quantities can be chosen arbitrarily. In all the test cases considered here the initial guesses of q(l,t) is taken as: q (l,t)Initial = 0

o

BFGS σ ≈ 0. C

24

9.69

10.66

3.21*10

o σ ≈ 0. C

14

9.66

9.57

0.0051

o BFGS σ = 3. C

12

13.16

8.11

3.12*10

6

11.75

4.15

0.0049

14

34.02

8.55

4.35*10

5

14.91

3.44

0.0055

CGM

CGM

o σ = 3. C

BFGS

o σ = 10. C o

CGM σ = 10. C

Figs. 2 and Table II show that, in the third degree polynomial heat flux, if the measurement error for the temperatures, measured by sensor, are σ = 0 and σ = 3 , BFGS and CGM methods converge very rapidly and exact enough to the real heat flux and is not so sensitive to the measurements error. Nevertheless, in σ = 10 ,



913


BFGS have a large root mean square error when the method estimate the third degree polynomial heat flux. Very good match between the actual and estimated values testifies the accuracy of the CGM method in large measurement error.

errors, it appears that the CGM stalls after approximately five or six iterations, whereas, the BFGS is able to reduce the objective function continuously, although with a slow rate. TABLE II ROOT MEAN SQUARE ERROR AND CONVERGENCE CRITERIA FOR ESTIMATING TRIANGULAR HEAT FLUX Number || ||at eRMS Run time of final (s) Iterations o

BFGS σ ≈ 0. C

22

12.57

11.06

o σ ≈ 0. C

14

12.08

10.72

o BFGS σ = 3. C

10

12.75

8.26

o σ = 3. C

5

12.25

3.86

14

38.63

8.19

4

12.43

2.85

CGM

CGM

4.83*10 0.010 3.67*10 0.011

o

BFGS σ = 10. C o σ = 10. C

CGM Fig. 2(a). The estimation of the third degree polynomial heat flux with 0

6.27*10 0.011

Numerical Test-Case 2: The unknown boundary heat flux q(l,t) is assumed, applied at x = l in the following form: 0 10 , 10

10 5

29 5

10

0

1

1

16

16

(24)

30

The results show that the inverse solutions obtained by CGM and BFGS remain stable and regular as the measurement errors are increased until 3, after than the BFGS method has a large error and long running time, but CGM method remain stable and has a good running time and only 4 iteration. The previous results appear in this numerical test case again, it can be confirm the last result. In present study, a comparison between the running time of the BFGS and CGM method is in continuance.

Fig. 2(b). The estimation of the third degree polynomial heat flux with 3

Fig. 2(c). The estimation of the third degree polynomial heat flux with 10 Fig. 3(a). The estimation of the triangular heat flux with 0

The BFGS has the ability to reduce to values smaller as compared to the CGM. In large measurement



914


TABLE III ROOT MEAN SQUARE ERROR AND CONVERGENCE CRITERIA FOR ESTIMATING STEP HEAT FLUX Number of Run || || at eRMS Iterations time (s) final o

BFGS σ ≈ 0. C

31

0.037

10.92

o σ ≈ 0. C

17

0.35

11.94

o BFGS σ = 3. C

16

8.35

8.42

3.39*10

9

7.39

6.30

0.0040

17

29.27

8.91

4.41*10

6

15.58

4.14

0.0042

CGM

o

CGM σ = 3. C

2.10*10 0.0039

o

BFGS σ = 10. C CGM Fig. 3(b). The estimation of the triangular heat flux with 3

Figs. 4-6 (a & b) show the estimate temperature history at x = l by BFGS and CGM have a good accuracy in 3 but when the error is increased to 10, the CGM method have stable and good results but the BFGS met hod have some error in estimate temperature. In BFGS the most important error appears in low temperature that is not important to design and we can use both of the method for nozzle designing but CGM have better accuracy and stability than BFGS in large temperature measurement error. Total time for estimating the third degree polynomial heat Flux with 0, 3, 10 by CGM is 17.34s and by BFGS is 27.32s. For the triangular heat flux by CGM is 17.43s by BFGS is 27.51s. For the step heat flux by CGM is 22.38s by BFGS is 28.25s. Total times show that the CGMs rate of converges is better than BFGS. for estimating the third degree polynomial Total heat Flux with 0 , 3, 10 by CGM is 36.32 and by BFGS are 56.87. For the triangular heat flux by CGM is 36.76 with the BFGS is 63.77. For the step heat flux with the CGM is 23.32 with the BFGS is 37.65. for the CGM and BFGS shows that the Total CGM is more accurate than the BFGS, when measurement temperatures have error.

Fig. 3(c). The estimation of the triangular heat flux with 10

The estimated heat flux components using BFGS and non-noisy data need 22 iterations were performed to satisfy stopping criterion but CGM need only 14 iterations were performed to have a better estimation. Numerical Test-Case 3: The unknown boundary heat flux q(l,t) is assumed, applied at x = l in the following form: ,

0 3.6

0 10

16 16

30

o σ = 10. C

(25)

In this numerical test case, results are same to previous test when the errors are large, but in low measurement error, the BFGS converge more rapidly than the CGM, and with a better accuracy. This results show that in the step function without any measurement errors the BFGS is better method for estimating than CGM. In order to utilize the aforementioned inverse methods for designing and analyzing of rocket nozzle, it will be shown in the following section, the estimated flux by BFGS and CGM leads to the same temperature distribution as the real flux. However, the temperature was estimated by the CGM has more accuracy than the BFGS.

Fig. 4(a). The estimation of the step heat flux with 0



915


Fig. 6(a) Fig. 4(b). The estimation of the step heat flux with 3

Fig. 6(b) Fig. 4(c). The estimation of the step heat flux with 10

Figs. 6. (a)-(b) exact and estimate temperature history at x = l with 3 and 10 respectively (the triangular heat flux)

Fig. 5(a)

Fig. 7(a)

Fig. 7(b)

Fig. 5(b)

Figs. 7. (a)-(b) exact and estimate temperature history at x = l with 3 and 10 respectively (the step flux)

Figs. 5. (a)-(b) exact and estimate temperature history at x= l with 3 and 10 respectively (the third degree polynomial heat flux)



916


VI.

Conclusion

Inverse heat conduction problem algorithms based on BFGS and CGM were formulated in this paper. The CGM and BFGS were successfully applied for the solution of the inverse heat conduction problem for determining the unknown transient boundary heat flux by utilizing simulated temperature obtained from the boundary with low temperature measurement error. From the numerical test cases in this study it is concluded that the inverse solution obtained by using the technique of CGM is not sensitive to the measurement error but BFGS is sensitive to large measurement errors. It is clear that both of the methods could be used for estimating the nozzle throat inner wall heat flux, but CGM is more accurate and stable than BFGS and converges very rapidly.

[16]

[17]

[18]

[19]

[20]

[21]

[22]

References [1]

[2]

[3]

[4]

[5]

[6]

[7] [8]

[9] [10]

[11]

[12]

[13]

[14]

[15]

Tsung-Chien Chen, Chiun-Chien Liu, Inverse estimation of heat flux and temperature on nozzle throat-insert inner contour, International Journal of Heat and Mass Transfer, 51 (2008) 3571–3581. H. N. Wang, J. H. Wang, A numerical investigation of ablation and transpiration cooling using the local thermal non-equilibrium model [R], in: Proceeding of the 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Sacramento, California, July 9–12, AIAA-(2006)-5264. J. H. Wang, H. N. Wang, J. G. Sun, J. Wang, Numerical simulation of control ablation by transpiration cooling, Heat Mass Transfer 43 (2007) 471–478. Junxiang Shi, Jianhua Wang, Inverse problem of transpiration cooling for estimating wall heat flux by LTNE model and CGM method, International Journal of Heat and Mass Transfer 52 (2009) 2714–2720 G. Stolz, Jr., Numerical Solutions to an Inverse Problem of Heat Condu-ction for Simpl Shapes, ASME J. Heat Transfer, vol.82, (1960) pp. 20-26. E. M. Sparrow, A. Haji-Sheikh, T. S. Lundgren, The Inverse Problem in Transient Heat Conduction, ASME J. Appl. Mech, vol. 86,(1964) pp. 369-375. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston, Washington, (1977). Ch. H. H., H. H. Wu, An Inverse Hyperbolic .Heat Conduction Problem in Estimating Surface Heat Flux by the Conjugate Gradient Method, J. Phys. D: Appl. Phys. vol. 39, (2006) pp. 4087-4096. J. V. Beck, B. Blackwell, C. R. St. Clair, Inverse Heat Conduction-Ill Posed Problem, Wiley, New York, 1985. Ch. H. H., H. H. Wu, An Inverse Hyperbolic .Heat Conduction Problem in Estimating Surface Heat Flux by the Conjugate Gradient Method, J. Phys. D:. Appl. Phys., vol. 39, (2006) pp.4087-4096. Junxiang Shi, Jianhua Wang, Inverse problem of transpiration cooling for estimating wall heat flux by LTNE model and CGM method, International Journal of Heat and Mass Transfer 52 (2009) 2714–2720. Yun Ky Hong, Seung Wook Baek, Inverse analysis for estimating the unsteadyinlet temperature distribution for two-phase laminar flow in a channel, Int.J.Heat Mass Transfer 49 (2006) 1137–1147. J. H. Bae, J. M. Hyun, H. S. Kwak, Mixed convection from a multiblock heater in a channel with imposed thermal modulation, Number, Heat Transfer Part A 45 (2004) 329–345. J. G. Bauzin, N. Laraqi, Simultaneous estimation of frictional heat flux and two thermal contact parameters for sliding contacts, Numer. Heat Transfer 45 (4) (2004) 313–328. S. Abboudi, A. Artioukhine, Two dimensional computational estimation of transient boundary conditions for a flat specimen.

[23]

[24]

[25] [26]

[27]

[28]

[29] [30]

In: Proceedings of the fourth International Conference on Inverse Problems Engineering: Theory and Practice, June 13–18, 2002, Rio, Brasil, ASME 2003. Linhua L, Heping T, Qizheng Y. Inverse radiation problem of temperature filed in three dimensional rectangular furnaces. Int Commun Heat Mass Transfer (1999); 26: 239–48. Daun KJ, Morton DP, Howell JR. Geometric optimization of radiant enclosures containing specular surfaces. ASME J Heat Transfer (2003); 125:845–51. Franca FR, Howell J, Ezekoye OA, Morales JC. Inverse design of thermal systems. In: Hartnett JP, Irvine TF, editors. Advances in heat transfer, vol. 36. New York: Elsevier; (2002). p. 1–110. S. S Rao Optimization; theory and applications, 2nd edn (9th reprint). New Age International (P) Limited Publishers, New Delhi (1995). Luk ̌ an L, Spedicato E., Variable metric methods for unconstrained optimization and nonlinear least squares. J Compute Appl Math (2000) 124: 61–95 M. Prudhomme, S. Jasmin, Determination of a heat source in porous medium with convective mass diffusion by an inverse method, International Journal of Heat and Mass Transfer 46 (2003) 2065–2075. S. Ebrahimkhah, M. Alizadeh, M. Sh. Mazidi, The function estimation of the unknown bondary heat flux in the invers heat conduction problems, International Journal of Energy & Technology 2 (14) (2010) 1–9. A.Pourshaghagh et al.Comparison of four different versions of the variable metric method for solving inverse heat conduction problems,Heat Mass Tran.(‘07)43:285–294. H. U. Ugwu, S. N. Ojobor, Design and Fabrication of Thermal Conductivity Measuring Equipment, International Review of Mechanical Engineering, Vol. 5 N. 1 (2011) 134-142. S. V. Patankar, Numerical Heat Transfer and Fluid Flow McGraw-Hill New York, (1980). Huang C H., Chen, A three-dimensional inverse forced convection problem in estimating surface heat flux by conjugate gradient method, Int. J. Heat Mass Transfer 43 (2000) 3171–81. F. Kowsary, A. Behbahaninia, A. Pourshaghaghy, Transient heat flux function estimation utilizing the variable metric method, International Communications in Heat and Mass Transfer 33 (2006) 800–810. A. M. Alshoaibi, A. K. Ariffin, Finite Element Modeling of Fatigue Crack Propagation Using a Self Adaptive Mesh Strategy, International Review of Mechanical Engineering, Vol. 2 n. 4 (2008) 537 – 544. Z. Z. Zhang, D. H. Cao, J. P. Zeng, Property of a class of variable metric methods, Applied Mathematics Letters 17 (2004) 437–442. Md. Moslemuddin Fakir, S. Basri, R. Varatharajoo, A. A. Jaafar, A. S. Mohd. Rafie, D. L. A.Majid, Comparison of Optimum Finite Element Method vs. Differential Quadrature Method in Two-dimensional Heat Transfer Problem, International Review of Mechanical Engineering, Vol. 2 n. 3 (2008) 483 – 488.

Authors’ information School of Mechanical Engineering, Iran University of Science and Technology, E-mails: [email protected] [email protected] H. Khoshkam (Corresponding author) was born in Tehran in 1986. He got his bachelor’s degree in fluid mechanical engineering at Guilan university of Rasht in 2008. Right now, he is MS student in mechanical engineering at Iran University of Science and technology. His major interest is in heat transfer. M. Alizadeh got his bachelor’s degree in mechanical engineering at Amirkabir University of Technology, of Iran, He got his bachelor’s degree in computer at Frankfurt, Germany, He got his MS degree in mechanical engineering at Amirkabir University of Technology, of Iran and He got his PHD degree at Berlin, Germany. Right now, he is Assistant Professor at Iran University of Science and technology. His major interest is in Non-Newtonian fluids and Hydrodynamic.



917


Study and Simulation of Thermal Buckling in a Thin Rectangular FGM Plate Mahdi Hamzehei, Mostapha Raki

Abstract – Equilibrium and stability equations of a rectangular plate made of functionally graded material (FGM) under thermal loads are derived, based on the higher order shear deformation plate theory. Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. The derived equilibrium and stability equations for functionally graded plates (FGPs) are identical to the equations for laminated composite plates with 51 layers. A buckling analysis of a functionally graded plate under one type of thermal loads is carried out and results in closed form solutions, uniform temperature rise and gradient through the thickness are considered, and the buckling temperatures are derived. The critical buckling temperature relations are reduced to the respective relations for functionally graded plates with a linear composition of constituent materials and homogeneous plates The results are compared with the critical buckling temperatures obtained for functionally graded plates ANSYS software (FEM) given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of functionally graded plates. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Stability of Plate, Thermal Buckling, Rectangular Plate, Functionally Graded Material, Higher Order Shear Deformation Theory

[1] in 1984, have attracted much interest as heat shielding materials for aircraft, space vehicles and other engineering applications. Functionally graded materials are composite materials, which are microscopically inhomogeneous, and the mechanical proper-ties vary smoothly or continuously from one surface to the other. It is this continuous change that results in gradient properties in functionally graded materials. Typically, these materials are made from a mixture of metal and ceramic, or a combination of different metals. Unlike fiber matrix composites which have a strong mismatch of m a,b echanical properties across the interface of two discrete materials bonded together and may result in debonding at high temperatures, functionally graded materials have the advantage of being able to survive environment with high temperature gradient, while maintaining their structural integrity. The ceramic material provides high temperature resistance due to its low thermal conductivity, while the ductile metal component prevents fracture due to thermal stresses. Furthermore, a mixture of ceramic and metal with a continuously varying volume fraction can be easily manufactured. In view of the advantages of functionally graded materials, a number of investigations dealing with thermal stresses had been published in the scientific literature. In recent years, Tanigawa et al. [2] derived a one dimensional temperature solution for a

Nomenclature a,b

Plate length and width E ( z ) ,Ec ,E m Elasticity modulus of FGM, ceramic and metal Plate thickness h m,n Number of half waves in x- and ydirections Stress resultants Ni ,M i ,Qi K ( z ) ,K c ,K m Thermal conductivity of FGM, ceramic and metal x, y,z Rectangular cartesian coordinates u,v,w Displacement components Temperature T Critical buckling temperature change ∆Tcr α ( z ) ,α c ,α m Coefficient of thermal expansion FGM, ceramic Shear strains metal ε xy ,ε xz ,ε yz

ε x ,ε y

Normal strains

σ x ,σ y

Normal stresses

I.

Introduction

In recent years, functionally graded materials (FGMs) which named by a group of material scientists in Japan



918

Mahdi Hamzehei, Mostapha Raki

non-homogeneous plate in transient state and also optimized the material composition by introducing a laminated composite model. Analytical formulation and numerical solution of the thermal stresses and deformations for axisymmetrical shells of FGM subjected to thermal loading due to fluid was obtained by Takezono et al. [3]. Aboudi et al. developed a new kind of higher order shear deformation theory for functionally graded materials that explicitly couples the micro-structural and macrostructural effects [4]. The response of a functionally graded ceramic-metal plate was investigated by Praveen and Reddy using a finite element model that accounts for the transverse shear strains, rotary inertia, and moderately large rotations in the Von Karman sense [5]. In Ref. [6], Reddy et al. developed the relationship between the bending solutions of the classical plate theory and the first order plate theory for functionally graded circular plates. Sumi studied the propagation and reflection of thermal and mechanical waves in FGMs under impulsive heat addition [7]. javaheri and Eslami reported mechanical and thermal buckling of rectangular functionally graded plates (FGPs) based on the classical plate theory [8,9]. hey used energy method and reached to the closed form solutions. They derived equilibrium and stability equations for functionally graded plates are identical to the equations for laminated composite plates. They have also investigated thermal buckling of FGPs based on the higher order displacement field [10]. They obtained buckling loads by solving the system of five stability equations. Motivated by Javaheri, Lanhe studied thermal buckling of moderately thick rectangular FGPs based on the first order shear deformation theory [11]. Considerable research has also been performed on the analysis of the stresses and deformations of functionally graded structures. However, Buckling analyses of FGM structures are scarce in the open literature. A formulation of the stability problem for FGM plates was presented by Birman [12] where a micro-mechanical model was employed to solve the buckling problem for a rectangular plate subjected to uniaxial compression. The stability of a functionally graded cylindrical shell under axial harmonic loading was investigate by Ng et al. [13]. Recently, Wu et al. presented the thermal buckling analysis of a simply supported thin rectangular FGM plate based on the classical plate theories [16]. In that paper, initially consider an FGM rectangular thin flat plate of length a , width b , and thickness h , subjected to the thermal loads. The material properties are assumed to vary as a power form of thickness coordinate variable, the linear stability equations are derived using the critical equilibrium method, and then the closed form of solutions for the linear stability equations is presented. We also investigated the influence of neutral plane

deformation, the aspect ratio, the relative thickness, and the graded index of the plate on the critical buckling temperature difference. In view of the fact that one solutions to buckling of linear variational thickness plates under thermal loads exist, an attempt is made to solve the thermal buckling problem of a functionally graded plate with moderately thickness and simply supported boundary conditions. In this paper, the stability equations are established based on the higher order shear deformation theory then five equations are combined into one governing equation with respect to w by eliminating the other variables. At last, the analytical solution for this equation is presented and the influence of transverse shear deformation on buckling is discussed. In this study, one kinds of thermal loading, uniform temperature rise and gradient through the thickness are considered.

II.

Functionally Graded Plate

FGMs are typically made from a mixture of ceramics and metal or a combination of different metals. The ceramic constituent of the material provides the hightemperature resistance due to its low thermal conductivity. The ductile metal constituent, on the other hand, prevents fracture caused by stresses due to high temperature gradient in a very short period of time. Further, a mixture of a ceramic and a metal with a continuously varying volume fraction can be easily manufactured.

III. Material Properties Consider a rectangular plate made of a mixture of metal and ceramic. The material in top surface and in bottom surface is metal and ceramic respectively. The modulus of elasticity E, the coefficient of thermal expansion α and the Poisson’s ratio ν are assumed as:

E ( z ) = EcVc + Em (1 − Vc ) ,

. E ( z ) = α cVc + α m (1 − Vc ) ,ν ( z ) = ν 0

(1)

where Em and α m denote the elastic moduli and the . coefficient of thermal expansion of metal respectively; Ec and α c denote the elastic moduli and the coefficient of thermal expansion of ceramic respectively; VC denotes the volume fraction of the ceramic and is assumed as a power function as follows:


k

⎛ 2z + h ⎞ Vc = ⎜ ⎟ ,Vm = 1 − Vc ⎝ 2h ⎠

(2)


919


These equations can be reduced by satisfying the stress free conditions on the top and bottom face of the laminates, which are equivalent to h ε xz = ε yz = 0 at z = ± 2 : .

where z is the thickness coordinate variable; and − h ≤ z ≤ h , where h is the thickness of the plate 2 2 and k is the power law index that takes values greater than or equals to zero. Substituting Eq. (2) into Eq.(1), material properties of the FGM plate are determined, which are the same as the equations proposed by Praveen and Reddy [5]:

(

)

u = u0 + zu1 − v = v0 + zv1 −

k

⎛ 2z + h ⎞ E ( z ) = Em + Ecm ⎜ ⎟ , ⎝ 2h ⎠ k

⎛ 2z + h ⎞ α ( z ) = α cVc + α cm ⎜ ⎟ ,ν ( z ) = ν 0 ⎝ 2h ⎠

(3)

Ecm = Ec − Em ,α cm = α c − α m

IV.

(4)

We initially consider an FGM rectangular thin plate of length a , width b , and thickness h , subjected to the thermal loads. Rectangular Cartesian coordinates ( x, y,z ) are

0 1 ⎛ ε xz ⎞ ⎛ ε xz ⎞ 2 ⎛ k xz ⎞ ⎜⎜ ⎟⎟ = ⎜ 0 ⎟ + z ⎜ 1 ⎟ ⎜ k yz ⎟ ⎝ ε yz ⎠ ⎜⎝ ε yz ⎟⎠ ⎝ ⎠

0 ⎞ ⎛ ε xz ⎛ u + w0 ,x ⎞ ⎜ ⎟=⎜ 1 ⎟ 0 ⎜ ε yz ⎟ ⎝ u1 + w0 ,y ⎠ ⎝ ⎠

(5)

⎛ k0 ⎞ ⎛ u ⎞ 1,x ⎜ x ⎟ ⎜ ⎟ ⎜ k y0 ⎟ = ⎜ v1,y ⎟ ⎜ ⎟ ⎜ ⎟⎟ 0 ⎟ ⎜u ⎜ k xy 0 ,y + v0 ,x ⎠ ⎝ ⎝ ⎠ −4 ⎛ ( u1,x + w0 ,xx ) ⎜ 2 ⎛k ⎞ ⎜ 3h 2 x ⎜ ⎟ −4 ⎜ k y2 ⎟ = ⎜ v1,y + w0 ,yy ⎜ ⎜ ⎟ 3h 2 ⎜ 2 ⎜ k xy ⎟ ⎝ ⎠ ⎜ −4 u + v + 2 w 0.xy ⎜ 2 1,y 1,x ⎝ 3h ⎛ −4 ⎞ ⎛ k1xz ⎞ ⎜ 3h 2 ( u1 + w0 ,x ) ⎟ ⎟ ⎜ ⎟=⎜ ⎜ k1yz ⎟ ⎜ −4 ⎟ ⎝ ⎠ ⎜ v1 + w0 ,y ⎟ ⎝ 3h 2 ⎠

ε yz = v,z + w,y where ε x and ε y are the normal strains and ε xy ,ε xz , and

ε yz are the shear strains. Here u,v and w denote the displacement components in the x, y and z directions, respectively, and a comma indicates the partial derivative. According to the higher order shear deformation theory, used in the present study is based on the following displacement:

(

)

(

u ( x, y,z ) = u0 ( x, y ) + zu1 ( x, y ) + + z 2 u2 ( x, y ) + z 3u3 ( x, y )

(8)

1 ⎛ ⎞ u0 ,x + w02,x ⎜ ⎟ ⎛ ⎞ 2 ⎟ ⎜ ⎟ ⎜ 1 2 ⎟ ⎜ ⎟=⎜ v0 ,y + w0 ,y ⎟ 2 ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ε xy ⎝ ⎠ ⎜ u0 ,y + v0 ,x + w0 ,x w0 ,y ⎟ ⎜ ⎟ ⎝ ⎠

ε xz = u,z + w,x

(

(6)

+ z v2 ( x, y ) + z v3 ( x, y ) 2

( v1 + w0,y ) ,w = w0

ε x0 ε y0

ε x = u,x + w,x2

v ( x, y,z ) = v0 ( x, y ) + zv1 ( x, y ) +

3h 2

(7)

where:

assumed for derivations. According to the first order shear deformation theory, the strains of the plate can be expressed as:

4 z3

( u1 + w0 ,x )

⎛ k x2 ⎞ ⎛ ε x ⎞ ⎛ ε x0 ⎞ ⎛ k x0 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 0 3 2 ⎜ε y ⎟ = ⎜ ε y ⎟ + z ⎜ ky ⎟ + z ⎜ ky ⎟ ⎜ ⎟ ⎜⎜ ⎟⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ 2 ⎟ ⎜ k xy ⎝ ε xy ⎠ ⎜⎝ ε xy ⎟⎠ ⎜⎝ k xy ⎟⎠ ⎝ ⎠

Equilibrium and Stability Equations

1 2 1 ε y = v,y + w,y2 2 ε xy = u,y + v,x + w,x w,y

3h 2

Substituting Eqs. (7) into nonlinear strain displacement relations (5) gives the kinematic relations as:

where:

4 z3

3

)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(9)

)

Hooke’s law for a plate is defined as:

w ( x, y ) = w0 ( x, y )



920


σx = σy =

E ( z) 1 −ν 02

E (z)

⎡ε y + ν 0ε x − (1 + ν 0 ) α ( z ) T ⎤⎦ 1 −ν 02 ⎣

σ xy =

The stability equations of the plate may be derived by the adjacent equilibrium criterion [17]. Assume that the equilibrium state of a FGP under mechanical or thermal loads is defined in terms of the displacement components u0 ,v0 and w0 . The displacement components of a neighboring stable state differ by u1 ,v1 and w1 with respect to the equilibrium position. Thus, the total displacements of a neighboring state are:

⎡⎣ε x + ν 0ε y − (1 + ν 0 ) α ( z ) T ⎤⎦

σ yz =

E ( z)

2 (1 + ν 0 ) E ( z)

2 (1 + ν 0 )

ε xy ,σ xz =

E ( z)

2 (1 + ν 0 )

(10)

ε xz ,

ε yz

u =u00 + u10 ,v =v00 + v10 ,w=w00 + w10

The forces and moments per unit length of the plate expressed in terms of the stress components through the thickness are:

u1 =u10 + u11 ,v1 =u10 + u11

(15)

Similarly, the force resultants of a neighboring state may be related to the state of equilibrium as:

h

( Ni ,M i ) = ∫− h2 σ i (1,z ) dz

(11)

2

i = x, y,xy

Substituting Eqs. (3), (5), and (10) into Eqs. (11), gives the constitutive relations as (12):

(

)

(

( N y ,M y )

)

) )

( N xy ,M xy )

) )

0 0 ⎡( E1 ,E2 ) ε xy + ( E2 ,E3 ) k xy 1 ⎢ = 2 2 (1 + ν 0 ) ⎢ + ( E4 ,E5 ) k xy ⎣

Qi → Qi 0 + Qi1

i = x, y,xy

(16)

force increments corresponding to u1 ,v1 and w1 .The stability equations may be obtained by substituting Eqs. (15) and (16) in Eq. (14) Upon substitution, the terms in the resulting equations with superscript 0 satisfy the equilibrium condition and therefore drop out of the equations. Also, the nonlinear terms with superscript 1 are ignored because they are small compared to the linear terms. The remaining terms form the stability equations as:

⎡( E1 ,E2 ) ε y0 + ν 0ε x0 + ⎤ ⎢ ⎥ ⎢ + E ,E k 0 + ν k 0 + ⎥ ( ) 1 ⎢ y 2 3 0 x ⎥ = 2 ⎢ 1 −ν 0 + ( E ,E ) k 2 + ν k 2 + ⎥ ⎢ ⎥ y 4 5 0 x ⎢ ⎥ ⎣⎢ − (1 + ν 0 ) (φ1 ,φ2 ) ⎦⎥

( (

i = x, y,xy i = x, y,xy

where N x1 ,N y1 and N xy1 represent the linear parts of the

⎡( E1 ,E2 ) ε x0 + ν 0ε 0y + ⎤ ⎢ ⎥ ⎢ + E ,E k 0 + ν k 0 + ⎥ 1 ⎢ ( 2 3) x 0 y ⎥ ( N x ,M x ) = 2 ⎢ 1 −ν 0 + ( E ,E ) k 2 + ν k 2 + ⎥ ⎢ ⎥ x 4 5 0 y ⎢ ⎥ ⎢⎣ − (1 + ν 0 ) (φ1 ,φ2 ) ⎥⎦

( (

Ni → N i 0 + Ni1 M i → M i 0 + M i1

N x1,x + N xy1,y = 0 ,N xy1,x + N y1,y = 0 M x1,x + M xy1,y = 0,M xy1,x + M y1,y = 0 N x 0 w10 ,xx

+⎤ ⎥ ⎥ ⎦

+ 2 N xy 0 w10 ,xy

+

N y 0 w10 ,yy

(17)

=0

The superscript 1 refers to the state of stability and the superscript 0 refers to the state of equilibrium conditions.

where:

( E1 ,E2 ,E3 ,E4 ,E5 ) = ∫

h

2 −h 2

(1,z,z

2

3

,z ,z

4

V.

) E ( z ) dz

The initial uniform temperature of the plate is assumed to be T1 .The plate is simply supported along the edges in bending and rigidly fixed in extension. Under these boundary conditions, the temperature can be uniformly raised to a final value T2 such that the plate buckles [9].To find the critical buckling temperature difference, ∆T = T2 − T1 the pre-buckling thermal stresses should be found. Solving the membrane form of equilibrium equations, using the method developed by Meyers [18] in conjunction with Galerkin’s formulation, gives the prebuckling force resultants:

h

(φ1 ,φ2 ) = ∫− h2 (1,z ) α ( z ) T ( x, y,z ) E ( z ) dz

(13)

2

h

µ = ∫− h2 α 2T 2 ( x, y,z ) E ( z ) dz 2

The nonlinear equations of equilibrium according to Von Karman’s theory are given by: N x,x + N xy ,y = 0,N xy ,x + N y ,y = 0 M x,x + M xy ,y = 0,M xy ,x + M y ,y = 0

Buckling of FGP Under Uniform Temperature Rise

(14)

N x w0 ,xx + 2 N xy w0 ,xy + N y w0 ,yy = 0



921


N x0 =

φ1 φ ,N x 0 = 1 ,N xy 0 = 0 1 −ν 0 1 −ν 0

⎡⎛ mπ ⎞ 2 1 −ν 0 ⎛ nπ ⎞ 2 ⎤ k11 = E1 ⎢⎜ ⎟ + ⎜ ⎟ ⎥ 2 ⎝ b ⎠ ⎥⎦ ⎣⎢⎝ a ⎠ E (1 + ν 0 ) ⎛ mπ ⎞ ⎛ nπ ⎞ k12 = 1 ⎜ ⎟⎜ ⎟ 2 ⎝ a ⎠⎝ b ⎠ 3 2 4 E ⎡⎛ mπ ⎞ ⎛ mπ ⎞ ⎛ nπ ⎞ ⎤ k13 = − 25 ⎢⎜ + ⎟ ⎜ ⎟⎜ ⎟ ⎥ 3h ⎢⎣⎝ a ⎠ ⎝ a ⎠ ⎝ b ⎠ ⎥⎦ 2 2 4 E ⎞ ⎡⎛ mπ ⎞ 1 −ν 0 ⎛ nπ ⎞ ⎤ ⎛ k14 = ⎜ E2 − 24 ⎟ ⎢⎜ + ⎟ ⎜ ⎟ ⎥ 2 ⎝ b ⎠ ⎥⎦ 3h ⎠ ⎢⎣⎝ a ⎠ ⎝ 2E ⎞ ⎛E ⎛ mπ ⎞⎛ nπ ⎞ k15 = ⎜ 2 − 24 ⎟ (1 + ν 0 ) ⎜ ⎟⎜ ⎟ 2 3 h ⎝ a ⎠⎝ b ⎠ ⎝ ⎠ k21 = −k12

(18)

The simply supported boundary condition is defined as (19): w10 ( x, 0 ) = w10 ( x,b ) = w10 ( 0, y ) = w10 ( a, y ) = 0 u10 ( x, 0 ) = u10 ( x,b ) = v10 ( 0 , y ) = v10 ( a, y ) = 0 u10 ( x, 0 ) = u10 ( x,b ) = v11 ( 0, y ) = v11 ( a, y ) = 0 M y ( x, 0 ) = M y ( x,b ) = M x ( 0 , y ) = M x ( a, y ) = 0

The following approximate solution is seen to satisfy both the differential equation and the boundary conditions: u10 ( x, y ) = u11 ( x, y ) = v10 ( x, y ) = v11 ( x, y ) = w10

m

⎡1 −ν 0 ⎛ mπ ⎞ 2 ⎛ nπ ⎞ 2 ⎤ k22 = E1 ⎢ ⎜ ⎟ +⎜ ⎟ ⎥ ⎣⎢ 2 ⎝ a ⎠ ⎝ b ⎠ ⎦⎥

n

∑ ∑ u0mn cos α x sin β y

k23 = −

∑ ∑ u1mn cos α x sin β y

k24

m =1 n =1 m n

m =1 n =1 m n

∑ ∑ v0mn sin α x cos β y

4E ⎛ k25 = ⎜ E2 − 24 3h ⎝

(20)

m =1 n =1 m n

∑∑ v1mn sin α x cos β y

2

2 2 16 E7 ⎡⎛ nπ ⎞ ⎛ mπ ⎞ ⎤ ⎛ E3 E1 8E5 ⎞ k 33 = + − − 4 ⎟ (1 −ν 0 ) ⎢ ⎥ −⎜ ⎜ ⎟ ⎜ ⎟ h ⎠ 9h 2 ⎢⎣⎝ b ⎠ ⎝ a ⎠ ⎥⎦ ⎝ h 2 2 2 2 ⎡⎛ nπ ⎞ 2 ⎛ mπ ⎞ 2 ⎤ ⎡ ⎛ mπ ⎞ ⎛ nπ ⎞ ⎤ 2 × ⎢⎜ ⎟ +⎜ ⎟ ⎥ + 1 −ν 0 ⎢ N x 0 ⎜ ⎟ + N y0 ⎜ ⎟ ⎥ ⎝ a ⎠ ⎝ b ⎠ ⎥⎦ ⎢⎣⎝ b ⎠ ⎝ a ⎠ ⎥⎦ ⎢⎣ E 8E ⎞ ⎛ 4E ⎛ mπ ⎞ ⎛ 16 E7 4 E5 ⎞ k34 = − ⎜ 23 − 1 − 45 ⎟ (1 −ν 0 ) ⎜ − ⎟+⎜ ⎟ 2 h ⎠ ⎝ a ⎠ ⎝ 9h 4 3h 4 ⎠ ⎝ h

m =1 n =1

(

mπ nπ , β= m,n = 1, 2,3,... ,where m and a b n are number of half waves in x and y directions,

where α =

respectively,

and

( u0mn ,u1mn ,v0mn ,v1mn ,w0mn )

are

)

⎡⎛ mπ ⎞3 ⎛ mπ ⎞ ⎛ nπ ⎞ 2 ⎤ × ⎢⎜ ⎟ +⎜ ⎟⎜ ⎟ ⎥ ⎣⎢⎝ a ⎠ ⎝ a ⎠ ⎝ b ⎠ ⎦⎥

constant coefficients. Substituting Eqs. (20) into the stability equations (17) and using the kinematic and constitutive relations yield a system of five homogeneous equations for u0 mn ,u1mn ,v0 mn ,v1mn , and w0 mn , that is: ⎛ u0 ,mn ⎞ ⎜ ⎟ ⎜ v0 ,mn ⎟ ⎡⎣ kij ⎤⎦ ⎜ w0 ,mn ⎟ = 0 ⎜ ⎟ ⎜ u1,mn ⎟ ⎜v ⎟ ⎝ 1,mn ⎠

2 2 ⎞ ⎡⎛ nπ ⎞ 1 −ν 0 ⎛ mπ ⎞ ⎤ + ⎢ ⎜ ⎟ ⎥ ⎟ ⎜ b ⎟ 2 ⎝ a ⎠ ⎥⎦ ⎠ ⎠ ⎣⎢⎝

k31 = k13 ,k32 = k23

m =1 n =1 m n

( x, y ) = ∑ ∑ w0mn sin α x sin β y

2 3 4 E4 ⎡⎛ mπ ⎞ ⎛ nπ ⎞ ⎛ nπ ⎞ ⎤ + ⎢⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎥ 3h 2 ⎢⎣⎝ a ⎠ ⎝ b ⎠ ⎝ b ⎠ ⎥⎦ 2E ⎞ ⎛E ⎛ mπ ⎞ ⎛ nπ ⎞ = ⎜ 2 − 24 ⎟ (1 + ν 0 ) ⎜ ⎟⎜ ⎟ ⎝ a ⎠⎝ b ⎠ ⎝ 2 3h ⎠

E 8E ⎞ ⎛ 4E ⎛ nπ ⎞ ⎛ 16 E7 4 E5 ⎞ k35 = − ⎜ 23 − 1 − 45 ⎟ (1 −ν 0 ) ⎜ − ⎟+⎜ ⎟ 2 h ⎠ ⎝ b ⎠ ⎝ 9h 4 3h 4 ⎠ ⎝ h ⎡⎛ nπ ⎞3 ⎛ nπ ⎞ ⎛ mπ ⎞2 ⎤ × ⎢⎜ ⎟ +⎜ ⎟⎜ ⎟ ⎥ ⎣⎢⎝ b ⎠ ⎝ b ⎠ ⎝ a ⎠ ⎦⎥ k41 = −k14 ,k42 = − k24 ,k43 = − k34

(21)

2 2 ⎛ 8 E 16 E7 ⎞ ⎡⎛ mπ ⎞ 1 −ν 0 ⎛ nπ ⎞ ⎤ k44 = − ⎜ 25 − − + E ⎢ 3⎟ ⎜ ⎟ ⎜ ⎟ ⎥ 2 ⎝ b ⎠ ⎥⎦ 9h 4 ⎝ 3h ⎠ ⎢⎣⎝ a ⎠ E 8E ⎞ ⎛ 4E − ⎜ − 23 + 1 + 45 ⎟ (1 −ν 0 ) 2 h ⎠ ⎝ h

in which kij is a symmetric matrix with the components (22):



922


Substituting pre-buckling forces form Eqs.(17) into the relation of k33 and setting kij = 0 to obtain the nonzero solution, the value of ∆Tcr is found as: ∆Tcr =

b Kd ⎡ n 2b 2 ⎤ π 2 (1 + ν 0 ) hµ* ⎢ 2 + n 2 ⎥ K c ⎣ a ⎦

⎡

( Emα cm + Ecmα m )

⎣

k +1

+

(23)

∆Tcr =

Ecmα cm ⎤ ⎥ (24) 2k + 1 ⎦

K d = det kij ,i, j = 1, 2 ,3

s×

∫−h( x ) / 2

(25)

+k14 k25 k41k52 + k15 k21k44 k52 + k11k24 k45 k52 + +k15 k22 k41k54 + k11k25 k42 k54 + k12 k21k45 k54 + +k12 k24 k41k55 + k14 k21k42 k55 + k11k22 k44 k55 + −k14 k25 k42 k51 − k15 k22 k44 k51 − k12 k24 k45 k51 + −k15 k24 k41k52 − k11k25 k44 k52 − k14 k21k45 k52 + −k12 k25 k41k54 − k15 k21k42 k54 − k11k22 k45 k54 + −k14 k22 k41k54 − k11k24 k42 k55 − k12 k21k44 k55

The critical temperature difference ∆Tcr is obtained for the values of m and n that make the preceding expression a minimum. By setting the power law index equal to one ( k = 1) ,

(29)

Illustration

Em = 70Gpa

α m = 23 ×

10 c

EC = 380Gpa

−5

α C = 7.4 ×

0

k m = 204 W mk

difference of homogeneous plates [16]-[23].

10 c

−5

0

kC = 204 W mk

5

5

(B)

Numerical Analysis with ANSYS k

FGPs likes layers composite with 51 layers has become modeled in ANSYS with little difference mechanical properties in layers. Element Type has applied is Solid186 which has good result with little fault in compare with solve analytic. For accounting the critical temperature difference first you should first account the critical boundary force= s × N i ,i = x, y

, i = x, y

TABLES I (A) CONSTITUENT MATERIALS , (B) CRITICAL BUCKLING TEMPERATURE OF THE FGP UNDER UNIFORM TEMPERATURE RISE DUE TO THE ANSYS (A) AND FIRST ORDER (F) THEORIES WITH RESPECT TO k AND b / a (A) Aluminum Alumina

Eq. (23) is reduced to the critical temperature difference for an FGP with a linear composition of ceramics and metal. In addition, by setting the power law index equal to zero ( k = 0 ) Eq. (23) is reduced to the critical temperature

( Ni )cr

E ( z ) α ( z ) dz

To illustrate the proposed approach, a ceramic-metal FGP is considered. Variation of the critical temperature difference ∆Tcr versus the aspect ratio b / a , and power law index k are listed for three loading cases in Tables I the values of the critical temperature difference ∆Tcr obtained by the method developed in the present article based on higher order theory are compared with respective values obtained FEM method with ANSYS software. The combination of materials consists of aluminum and alumina. Constituent materials are presented in Table I(a). In Table I(b) the results of buckling analysis for the plate under uniform temperature rise are presented. Tables I show that the buckling temperature increases by the increase of the aspect ratio b / a and decreases with increase of the power law index k from 1 to 10. Figure 1 shows 51 different material types.

K c = k15 k24 k42 k52 + k12 k25 k44 k51 + k14 k22 k45 k51 +

(1 −ν 0 )( Ni )cr

h( x ) / 2

VII.

and (26):

VI.

(28)

The value of ∆Tcr is found as:

where:

φ1 , i = x, y (1 −ν 0 )

2

µ* = ⎢ Emα m +

Ni =

1 5 10

A H A H A H

b

=1 a 7.955 7.939 7.273 7.26 7.476 7.462

b

=2 a 19.848 19.835 18.147 18.132 18.647 18.636

b

=3 a 39.633 39.624 36.214 36.203 37.212 37.2

b

=4 a 67.263 67.25 61.403 61.395 63.077 63.068

b

=5 a 102.641 102.634 93.614 93.605 96.132 96.12

Figure 2 shows the displacement of FGM plate under 4 natural frequencies which is for the FGPs with these Figure 3 specifications k = 1,b / h = 100 , ∆T = 200 . shows the Von Mises Stress of FGM plate under 4 natural frequency which is for the FGPs with these

(27)

where s: coefficient Frequency and boundary force Ni :



923


specifications k = 1,b / h = 100 , ∆T = 200 .

FREQ 1

Fig. 1. 51 Different Material Types

FREQ 1

FREQ 2

FREQ 2

FREQ 3

FREQ 3

FREQ 4 FREQ 4 Fig. 3. Von Mises Stress of FGM Plate Under 4 First Natural Frequencies

Fig. 2. Displacement FGM Plate Under 4 First Natural Frequency



924


• With addition number layers and also addition meshing the model presented exactly quantity ∆Tcr will take. • With increasing layears and meshing we will get the best result. • With increasing k FGPs Von Mises Stress decrease with increasing Natural Frequency. • With increasing k displacement FGPs decrease with increasing Natural Frequency.

The Young’s modulus, coefficient of thermal the Young’s modulus, coefficient of thermal expansion and thermal conductivity presented in the previous table. Poisson’s ratio is chosen to be 0.3. The plate is assumed to be simply supported on all four edges. It is interesting to note that the buckling temperatures for homogeneous plates ( k = 0 ) are considerably higher than those for the FGPs

( k〉 0 )

especially for the

comparatively longer and thicket plates. The critical buckling temperatures obtained based on classical plate theory are noticeably greater than values obtained based on higher order shear deformation theory. The differences are considerable for long and thin plates.

Acknowledgements This research extracted from Mr. Raki Msc thesis, so the authors would like to express their gratitude to department of mechanical engineering of Islamic Azad University, Ahvaz Branch for providing support for this study.

VIII. Conclusion In the present article, equilibrium and stability equations for rectangular simply supported FGPs are obtained. The derivation is based on the higher order shear deformation theory, with the assumption of power law composition for the constituent materials. The buckling analysis of FGPs under one types of thermal loading are presented. Closed form solutions for the critical buckling temperatures of plates are presented. It is concluded that the equilibrium and stability equations are identical to the corresponding equations for laminated composite plates. The critical buckling temperature differences ∆Tcr for the FGPs are generally lower than the corresponding values for homogeneous plates. Functionally graded plates have many of the same advantages as heat resistant material, but it is important to check their strength due to the thermal buckling. The critical buckling temperature difference ∆Tcr for FGPs is increased by increasing the aspect ratio b / a . In the present article, equilibrium and stability equations for rectangular simply supported FGPs are obtained. The derivation is based on the higher order shear deformation theory, with the assumption of power law composition for the constituent materials. The buckling analysis of FGPs under one types of thermal loadings are presented. Closed form solutions for the critical buckling temperatures of plates are presented. It is concluded that • The equilibrium and stability equations are identical to the corresponding equations for laminated composite plates. • The critical buckling temperature differences ∆Tcr for the FGPs are generally lower than the corresponding values for homogeneous plates. Functionally graded plates have many of the same advantages as heat resistant material, but it is important to check their strength due to the thermal buckling. • The critical buckling temperature difference ∆Tcr for FGPs is increased by increasing the aspect ratio b / a . • The higher order shear deformation theory underestimates the buckling load compared with ANSYS software (FEM) theory.

References [1] [2]

[3]

[4]

[5] [6]

[7]

[8]

[9] [10]

[11]

[12]

[13]

[14] [15] [16]

[17]


Koizumi M. FGM activities in Japan. Composites, Vol 28 , 1997. Tanigawa Y, Matsumoto M, Akai T. Optimization of material composition to minimize thermal stresses in non-homogeneous plate subjected to unsteady heat supply. Japanese Society of Mechanical Engineering Int J., Ser A, Vol 40(1) pp.84–93. 1997. Takezono S, Tao K, Inamura E. Thermal stress and deformation in functionally graded material shells of revolution under thermal loading due to fluid. Japanese Society of Mechanical Engineering Int J, Ser A, Vol 62, No 594, pp. 474–81, 1996. Aboudi J, Pindera M, Arnold SM. Coupled higher-order theory for functionally grade composites with partial homogenization. Compos Eng. Vol. 5, No. 7, pp. 771–92, 1995. Praveen GN, Reddy JN. Nonlinear transient thermal plates. Int J Solids Structure, Vol. 35, No. 33, pp.4457–76, 1998. Reddy JN, Wang CM, Kitipornchai S. Axisymmetric bending of functionally graded circular and annular plates. Eur J Mech A/ Solids, Vol. 18, No.1, pp.185–99, 1999. Sumi N. Numerical solution of thermal and mechanical waves in functionally graded materials[C]. Third International Congress on Thermal Stresses, Branti Zew, Krakow, Poland, pp. 569–72, 1999. Javaheri R, Eslami MR. Buckling of functionally graded plates under inplane compressive loading. ZAMM, Vol. 82, No. 4, pp. 277–83, 2002. Javaheri R, Eslami MR. Thermal buckling of functionally graded plates. AIAA J, Vol.40, No.1, pp.62–9, 2002. Javaheri R, Eslami MR. Thermal buckling of functionally graded plates based on higher order theory. J Thermal Stresses, Vol. 25, pp. 603–25, 2002. Lanhe W. Thermal buckling of a simply supported moderately thick rectangular FGM plate. Compos Struct., Vol. 64, pp.211–8 2004. Birman V. Buckling of functionally graded hybrid composite plates. Proceedings of the 10th conference on engineering mechanics, Boulder, CO, pp. 1199–1202, 1995. Ng TY, Lam KY, Liew KM. Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading. Int J Solids Struct, Vol.38, No.9, pp.1295–309, 2001. Tauchert TR. Thermal buckling of thick antisymmetic angle ply laminates. J Thermal Stresses, Vol. 10, No.1, pp.113–24, 1987. Thornton EA. Thermal buckling of plates and shells, Appl Mech Rev., Vol. 46, No.10, pp. 485–506, 1993. Wu Lanhe, Wang Libin, Liu Shuhong. On thermal buckling of a simply supported rectangular FGM plate. Chin J Eng Mech, in press [in Chinese]. Brush DO, Almroth BO. Buckling of bars, plates, and shells. New-York: McGraw-Hill; 1975.


925


[18] C. A. Meyers and M. W. Hyer, Thermal Buckling and Postbuckling of Symmetrically Laminated Composite Plates, J. Thermal Stresses, vol. 14, pp. 519-540, 1991. [19] M. R. Isvandzibaei, M. Setareh, A Functionally Graded Cylindrical Shell for Analysis Vibration with Effects Unsymmetrical Boundary Conditions, IREME, Vol. 5 . pp. 465473, 2011. [20] S. Kiran Aithal, S. Narendranath, Vijay Desai, P. G. Mukunda, Characterization of Al-Si Functionally Graded Material using Centrifuge Casting Method, IREME, Vol. 3. n. 5, pp. 632-639, 2009. [21] M.R.Isvandzibaei, Analysis Free Vibration of FGM Cylindrical Shells under Clamped-Simply Support Boundary Conditions, IREME, Vol. 5 N. 1, pp. 71-78, 2011. [22] G. M. Vörös, Buckling and Vibration of Stiffened Plates, IREME, Vol .1 n. 1, pp. 49 – 60, 2007. [23] A. Shahrjerdi, M. Bayat, F. Mustapha, S. M. Sapuan, R. Zahari, Stress Analysis a Functionally Graded Quadrangle Plate Using Second Order Shear Deformation Theory, IREME, Vol. 4. n. 1, pp. 60-64, 2010.

Mahdi Hamzehei is assistant professor (Ph.D) of mechanical engineering at Islamic Azad university, Ahvaz branch, Ahvaz, Iran. He received his Ph.D from department mechanical engineering at Amirkabir University of technology (Tehran Polytechnics). He worked on hydrodynamics and heat transfer in a gassolid fluidized bed reactor experimentally and numerically. He received his M.Sc from Shiraz University in 2004. He worked on Measuring and Prediction of Temperature Distribution in a Spark Ignition Engine Piston and Cylinder Head at Actual Process in his master’s thesis. He obtained his bachelor, degree from Iran University of Science Technology (IUST) in 2001. He is a researcher member of Fluid Mechanics Research Center in Amirkabir University and Internal Combustion Engine Research Center in Shiraz University. He worked and supervised some industrial and academic projects in filed of Fluid mechanics, Heat transfer, Fluidization and combustion. Mr. Hamzehei has authored more than 35 papers in conference proceedings and five journal papers. Also he is a member of NPC, ISME and ICI. Mostapha Raki (Corresponding author) is Msc student in department of mechanical engineering. Ahvaz Branch, Islamic Azad University, Ahvaz, Iran. This research extracted from his thesis. For Msc he works on study of thermal buckling in rectangular FGM plate with variation thickness based on higher order shear deformation.

Authors’ information Department of Mechanical Engineering, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran. E-mail: [email protected]



926


Experimental Investigation of Different Heat Recovery Systems in Leisure Center and its Effect on CO2 Emission M. M. Abo Elazm, A. F. Elsafty Abstract – Nowadays the crisis of energy is one of the most important problems of the world. Building units are the first consumer in world energy. Because of this, in design of Green Buildings the most important challenge is reduction of energy consumption in buildings. Leisure centers are also known as high energy consumers, especially if they have large ice rinks and swimming pools. The aim of this paper is to investigate different heat recovery concepts, by using the heat rejected from the condenser of the refrigeration unit to heat the water of swimming pool especially in winter. An experimental model has been constructed to contribute in solving the problem of energy shortage. The model installed includes an ice rink, a swimming pool and two different condensers. A comparison between different cases of operation to obtain the maximum energy saving was carried out. The results show that the energy saving was about 30% by using both air and water cooled condensers in series compared to the air cooled condenser only. The results also showed a significant decrease in the time required for the ice formation by using both air and water cooled condensers. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Energy Saving, Heat Recovery, Ice Rinks, Leisure Centers

Indoor ice skating rinks in particular use a lot of water when making ice and it consumes a lot of electricity to keep it frozen. [2] One method of reducing the operating cost is to recover the wasted energy, resulting in lower utility bills. Energy recovery is the beneficial use of heating or cooling energy that would typically be rejected to the environment. Technologies that recover heating and/or cooling energy reduce the cost and the consumption of energy in commercial and institutional buildings. The recaptured energy may be useful for space heating, water heating, or other industrial processes. [3] The main objective is to develop a simple leisure model which includes ice rink and a swimming pool and run both ice skating rink and swimming pool model effectively and efficiently. Secondly, study the energy savings by applying a different heat recovery concepts, by using the heat rejected from the condenser of the refrigeration unit to heat the water of swimming pool especially in winter and to compare between the different cases to obtain the maximum energy saving.

Nomenclature Symbol

Description

Units

Q

Heating/cooling loads

kW

SY

Safety Factor

--

C

Specific Heat

kJ/kg oC

H

Latent Heat

kJ/kg

t

Temperature

o

m

Mass Flow Rate

kg/s

τ

time

s

A

Surface area

m2

h

Heat transfer coefficient

W/m2 oC

η

Efficiency

--

I.

C

Introduction

Leisure centers are considered as nice places for public enjoyment as well as practice several kinds of sports. Two places are the most popular locations inside leisure centers which are leisure ice that includes ice skating rinks for public skating and leisure pools that include many kinds of swimming pools. Leisure centers spend more than 40% of the operating cost on energy especially those places which have long opening hours and high demands for heat and electricity. [1] However, Ice skating is one of the most beloved winter activities.

II.

Theoretical Analysis

The objective of this analysis is to calculate the total cooling load of the ice skating rink. The total cooling load includes different loads which are: water freezing load, stainless steel cooling load, refrigeration to cool secondary coolant load and heat losses load.



927

M. M. Abo Elazm, A. F. Elsafty

III. Ice Skating Rink Cooling Load

mbrine =

To maintain the ice in the ice skating rink in the same phase, heat must be absorbed from the ice. This amount of heat is called the total cooling load (QR) which could be calculated from equation (1)[4]:

(

)

QR = SY factor × ( QF + QSS + QSR + QSL )

IV.

(

)

τ

mss × ⎡⎣CSS × ( t1SS − t2 SS ) ⎤⎦

τ

(

mbrine × ⎡⎣Cbrine × ( t1brine − t2brine ) ⎤⎦

τ

QWH =

(

)

mW ⎡Cw × twpool − t1 ⎤ ⎣ ⎦

(9)

τ

The convection losses from the free surface of swimming pool (Qloss-conv) could be calculated from equation (10):

(3)

(

Qloss -conv = h × Apool × twpool − tamb

V.

)

(10)

Experimental Test Rig

For conventional rinks, the ice which forms the skating surface is produced by spraying water on to the top of the rink base and passing a cooling fluid through a network of pipes installed inside it to remove heat. The range of the thickness of ice ranges between 25 30 mm. The ice temperature varies between (-3 and -10 °C). A vapor compression refrigeration unit is used. Two methods were investigated in ice rinks: direct and indirect methods. [8] The experimental set-up unit is shown in a photograph for the general view Figure (1) and schematic diagram Figure (2). The experimental test model includes three cycles: Ice rink cooling cycle, refrigerant cycle and the swimming pool heating cycle. In this test model, the capacity of brine tank is 80 liters, consists of brine 25% ethylene glycol + 75% water. By using 37 liter brine in evaporator tank, the size is accommodate for the brine circulation system ,volume changes created by changes in temperature, and also to allow the addition of fresh solution as required. The system consists of ice rink which made from stainless steel. The rink was formed from two plates, including guided channels between these plates for good heat transfer between ice pad and brine as shown in Figure (2).

(4)

(5)

Based on the time of formation of ice, the total cooling load QR , and using R- 502 P-h chart, the refrigerant mass flow rate (mref) could be calculated from equation (6)[7]: QR = mref × ( h1 − h4 )

(8)

The amount of heat needed for heating the swimming pool water (QWH) to the required temperature could be calculated from equation (9):

The losses from the free surface of ice rink (Qconv) could be calculated from equation (5) [6]:

Qconv = h × A × ( tamb − t3 )

)

(2)

where t2SS is the required stainless steel temperature, t1SS is initial temperature of the stainless steel and mSS is mass of ice rink form (stainless steel). The brine used in the experiment was 25% concentration ethylene glycol + 75% water, which was resulted in specific heat of brine Cbrine = 2.88 kJ/ kg .°C The amount of heat to be removed from the brine Secondary Coolant (QSR) could be calculated from equation (4):

QSR =

Swimming Pool Heating Load

QTH = SY factor × ( QWH + Qlosses )

where t1 is the initial water temperature, t2 is the Freezing temperature and t3 is the required ice temperature. Stainless steel was used in binding ice rink instead of concrete because stainless steel is easier to be formed and fabricated and more elastic than concrete. The heat load for stainless steel (QSS) could be calculated from equation (3):

QSS =

(7)

For heating the swimming pool water, heat must be added to raise the temperature of the water and keep it at the required temperature. The heating load (QTH )could be calculated from equation (8) [7]:

(1)

The Water freezing Load (QF) could be calculated from equation (2) [5]: mWF ⎡CW × t1 − t2 + H sf + Ci × ( t2 − t3 ) ⎤ ⎣ ⎦ QF =

QR Cbrine × ∆tbrine ×η

(6)

Since the evaporator capacity (cooling load) should be enough to cool and freeze ice rink indirectly by a secondary refrigerant (brine). The brine mass flow rate could be calculated from the following equation(7) [7]:



928

M. M. Abbo Elazm, A. F. F Elsafty

evaccuated from N2 and reecharged witth refrigerantt (R502). The brinne is filled in tthe evaporato or tank and itss volu ume is adjuusted to 37 Liters. Thee evaporatorr tem mperature is adjusted a at -11oC by th he evaporatorr therrmostat. Waterr is filled in thhe swimming pool and thenn the water is alloowed to pass through the water cooledd cond denser. The sw wimming poool temperature is adjusted att 30oC by the swim mming pool tthermostat ex xcept for casee (2). The main electrical power supply is switched on. Thee amb bient temperatture is measurred before staarting the test.. Presssure gauges and refrigerattion system teemperature iss checcked before sttarting the testt. Refrigeration R c cycle is then tturned on. Th he brine pumpp is tu urned on. Thhe brine flow rate is settleed. The waterr pum mp is turned on, o when the water conden nser is neededd for heating swim mming pool. The water flow rate iss settlled. When thee temperature inside the icee skating rinkk reacches (0 oC), the water sppraying proceess is started.. Diffferent param meters like refrigerant temperature,, swim mming pool water w temperatture, brine tem mperature andd ice thickness t are measured eveery 15 minutess.

Fig. 1. Schematicc diagram of the system s

VII.

Uncertaiinty Analyssis

The T uncertaintty analysis is done for both h temperaturee and electric energgy. The value of the uncertainty analysiss for the temperatuure is ±0.0002277 while thee value of thee uncertainty analyysis for the eneergy is ±0.000 0086.

VIII. Results an nd Discussiions The T obtained experimental e data are repreesented in thee follo owing figuress which are ussed to analysiss the variationn of temperatures t and power consumption with w differentt operrating periodss.

Fig. 2. Expperimental test rigg

The swim mming pool was w made of stainless s steell and has two holes, one for the supply of thee hot water andd the other for thhe return of cold water. A reciprocating Hermeticallyy sealed type compressor is used. The heat absorbed byy the refrigeraant from the ice rink andd the compressor is i rejected byy the condenseer. There aree two types of coondensers: airr-cooled conddenser and water w cooled conddenser. The experimental e set-up has been designed andd constructedd to carry outt the comparaative study betweeen four differrent cases of operation forr the swimming pool. The first case, by usinng only air coooled condenser without w heat recovery. r Thee second casee, by using water and a air cooledd condensers operating o in series without any control on thhe swimming pool temperaature. The third casse, by using water w and air cooled c condennsers operating in alternative. The T fourth casse, by using water w and air cooleed condensers operating in series s with control on swimmingg pool temperrature.

VI.

Fig. 3. Ice I thickness verssus time for all caases

Figure F 3 shoowed that thee ice formattion starts too incrrease its thiickness afterr different periods. p Thee max ximum ice thickness was 229 mm for both b cases (2)) and (4) while thee minimum icee thickness was w 24 mm forr casee (1). Figure F 4 show wed a decreaase in the icee temperaturee with h the operatingg period. The required ice temperature t iss -8 °C.

Tesst Procedurres

The steps of the experrimental workk begin as thee test rig was settleed up. The cyycle was charrged with nitroogen gas to be surre that there is no leakage. The cycle then is

Copyright © 20011 Praise Worthhy Prize S.r.l. - Alll rights reserved

Internationaal Review of Mecchanical Engineerring, Vol. 5, N. 5

929


It was fouund that the required periodd for (cases 2, 3, 4) was betweenn 1.6 hour and 2 hours. The minim mum period was w 1.6 hour in i (case 4). Itt was found that thhe steady staate temperature was the same, often 2 hourr, for cases (2), (3), and (4)) while the stteady state temperaature for case (1) was reachhed after 5 houurs.

Fig. 6. Comprressor temperaturre versus time forr all cases

Figure F 7 shhows the relationship between b thee swim mming pool water tempeerature and the t operatingg periiod. Itt was found that, at startiing the waterr temperaturee incrreases in the first f 30 minuutes for the diifferent cases.. Thee highest increease was for ccases (2, 3, an nd 4) followedd by case c (1). After A that all a cases reeached the steady statee tem mperature.

Fig. 4. Ice temperatuure versus time foor all cases

Figure 5 showed decrease d in the evaporator temperature with the opeerating periodd for the diffe ferent cases. It was fouund that, at staarting the tempperature decreeases in the first three hours for the diffferent cases. The quickest decrrease was for case (2). After thatt, all cases have h reachedd the steady state temperature at -11 °C C which waas controlledd by evaporator thhermostat.

Fig. 7. Swimming pool temperatture versus time for f all cases

Fig. 5. Evaporator E tempeerature versus tim me for all cases

Figure 6 shows the relationship between the compressor temperature t a the operatting period for the and different casees. It was found that at starting the compreessor temperature increases in the t first 30 minutes m at diffe ferent values for thee four cases. The higheest increase was w for case (3); followedd by case (1), thenn case (4) whhile the lowesst was at casee (2). After that all cases reacheed the steady state temperaature with differennt values. The maxim mum value waas about 70 oC at case (3) while w the minimum m value was abbout 45 oC at case c (2).

a cases Fig. 8. Energgy Consumption versus time for all

Figure F 8 show wed that enerrgy consumpttion increasess with h time for all cases. c Itt was found that after ffive hours, th he maximum m enerrgy consumpttion by the ssystem was 23.25 2 kWh att



930


case (1) while the minimuum energy coonsumption byy the system was 16.4 kWh at case (4). Beccause of usinng an electric heateer and the deccrease of the operating o periood of compressor and a water puump, it is recoommended too use case (4) as it i works for loower period and a higher ennergy saving.

Fig. 11. Fuuel savings (oil) vversus time for alll cases

Figure F 12 show wed that the C CO2 emissions reduced withh the operating periiod at differennt values for three t cases (2,, 3 an nd 4). The laargest reduction of CO2 em missions wass occu urred at case (4) and the m minimum redu uction of CO2 emissions was at case (1). Figg. 9. Energy savinng versus time forr all cases

Figure 9 showed s that thhe energy savved increases with different opeerating period for three casees. The higheest increase was w found for case (4) folloowed by case (2), then t case (3). It is noticed that t case (1) has h a great differennce with the other o cases. The best operating o conddition is in casse (4).

Fig. F 12. CO2 Emiission (using oil) reduction versus for all cases

IX.

Conclusion

An A experimental set-up has been designed d andd constructed to appply a heat reecovery concept, by usingg the heat rejected from the conndenser of thee refrigerationn unitt to heat the water w of swimm ming pool thro ough differentt fourr cases. The results r revealeed that. By using u only thee air cooled condeenser, the maxximum powerr required forr the ice skating rink was 23.255 kWh while by b using bothh air and water coooled condennsers, the abssorbed powerr redu uced to 16.4 kWh k and the energy saving is about 30%.. When W the airr and water cooled condeensers are inn seriees, the tempeerature of icee decreases qu uickly withinn firstt 2 hours to the required ice temperatture while byy usin ng both air and a water coooled condenssers operatingg togeether, the reequired time for freezing g the ice iss considerably reduuced. Minimum M operating time foor water pump p was reachedd wheen the air coolled condenserr was operatin ng all the timee and the water cooled conddenser operattes until thee swim mming pool temperature reaches 30 °C. Also thee com mpressor worrking time iis minimum when bothh cond densers opeerate togetheer in series producingg

Fig. 10. Cost savingg versus time for all cases

Figure 100 shows the relationship between the cost saved and thee operating peeriod. The savedd cost increasses accordingg to the operaating period at diffferent values for f three casess (2, 3, and 4). The maxim mum saved coost is 2.4 L.E after five houurs at case (4) whille the minimuum saved cost was 0 L.E at case (1). It is recom mmended to usse case (4) as it works for loower period and maximum m saveed cost. Figure 11 showed thatt the saved fuuel increases with operating period at differeent values forr three cases (2, 3 and 4). The highest incrrease was forr case (4). It was noticed thatt case (1) offers a greeat differencee in comparison with w the otherr cases. The recomm mended operatiing condition is i case (4).



931


maximum ennergy saving.

A Authors’ infformation Mech hanical Engineeriing Dept. Colleege of Engineerinng and Technologgy Arab b Academy for Science, Technnology and Marritime Transport,, P.O.B Box: 1029, Alex xandria, Egyp pt. E-maails: m.aboelazm@ @aast.edu [email protected]

Refferences [1]

Brambley M. R., and Wellls S. E., (1983). Energy conservvation mming pools Energy (oxford) d) UK, measures for indoor swim Volume 8, Issue 6, pp. 403--418. e [2] Seghouanii et al., (20008). “Predictionn of yearly energy requiremennts for indoor icee rinks” Energy and a Buildings jouurnal, volume 41, issue 5, pages (5500-511). [3] Long welll J.P, (2000). “Ennergy for the futuure” Fuel and Ennergy, volume 377, pp.87-87 (1). [4] ASHRAE,, (1990). Handbook; Refrigeeration Systems and Applicatioons, Chapter 34--IIce Rink. [5] Daoud ett al. (2008). Calculation of reefrigeration loadds by convectionn, radiation, andd condensation in ice rinks ussing a transient 3D 3 zonal model. Applied Thermall Engineering, Volume Vo 28, Issues 14-15, pp.1782-11790. [6] Incorpera and Frank, (20007). Fundamentaals of Heat and Mass Transfer. 6th Edition. [7] Somrani et e al. (2008). “H Heat transfer benneath ice rinks floors “Building and Environmentt, Volume 43, Issuue 10, PP. 1687-11698. c E Energy [8] Lenko Breendan, (2001). “IIce rink energy conservation”, Ice Publicaation. [9] R. Saim, S. Abboudi, B. Benyoucef, A. Azzi, Sepp-2007 Computatiion of Turbulent Forced Convectioon in Heat Exchaangers Equipped with the Transvverse Baffles, International Reviiew of 5 Mechanicaal Engineering (II.Re.M.E), Vol. (1 n. 5), pp. 588 – 594. [10] Hasim Altan, Jitka Mohhelnikova, Noveember 2009, Energy E a Carbon Redduction due to Renovated Builddings, Savings and Internationnal Review of Meechanical Engineeering (I.Re.M.E)), Vol. 3. n. 6, pp.. 825-832. [11] Hesham M. M Mostafa Tarekk F. Oda, (May-20010), Convectionn Heat Transfer and a Pressure Droop for Dilute Poolymer Solutions Flow Inside Multi-Channels M F Flat Tube, International Revieew of Mechanicaal Engineering (II.Re.M.E), Vol. (44 n. 4), pp. 381-3992.

E born inn Dr. Mohameed M. Abo Elazm Alexandria, Eggypt on the 6th of April-1976 andd received the P Ph.D. in Mechan nical Eng. From m Ain Shams Unniversity, Cairo, Egypt, in 2008.. The main reseearch interests aree simulation andd modeling of siingle and multiphase flow, CFD,, Renewable Ennergy Resourcess and Heat andd Mass Transfer.. Is now an Assisstant Professor inn the Mechanical M Eng. Department at thhe Arab Academy y for Science andd Tech hnology. Prof. Ahmed F. Elsafty born n in Alexandria,, Egypt, 12th of September 1967 and received thee Ph.D from Cooventry Universitty, UK in 2002.. The main ressearch interests are Renewablee Energy, Heat aand Mass Transffer, Refrigerationn and Air conditiioning, Modeling g and Simulation,, and Environmeental Protection. Is now Head off Mechanical E Eng. Departmen nt at the Arabb Academy for S Science and Techn nology.



932


An Exact Solution for the Vibration FGM Hollow Cylindrical Shell Based on High Order Theory Under Free-Simply Support Boundary Conditions M. Setareh, M. R. Isvandzibaei

Abstract – Study of the vibration of thin cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel is very important. The objective is to study the natural frequencies and the effects clamped-free boundary conditions on the natural frequencies of the functionally graded cylindrical shell. The study is carried out using third order shear deformation shell theory. The analysis is carried out using Hamilton’s principle. The governing equations of motion of functionally graded cylindrical shells are derived based on third order shear deformation shell theory. Results are presented on the frequencies characteristics and the effects of free-simply support boundary conditions on edge end functionally graded cylindrical shell. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Vibration, Cylindrical Shell, Free-Simply Support

Recently, the present authors presented studies on the influence of boundary conditions on the frequencies of a multi–layered cylindrical shell [6]. In all the above works, different thin shell theories based on Love– hypothesis were used. Vibration of cylindrical shells with ring support is considered by Loy and Lam [7]. The concept of functionally graded materials (FGMs) was first introduced in 1984 by a group of materials scientists in Japan [8]-[9] as a means of preparing thermal barrier materials. Since then, FGMs have attracted much interest as heat-shielding materials. FGMs are made by combining different materials using power metallurgy methods [10]. They possess variations in constituent volume fractions that lead to continuous change in the composition, microstructure, porosity, etc., resulting in gradients in the mechanical and thermal properties [11]-[12]. Vibration study of FGM shell structures is important. In this paper a study on the vibration of functionally graded cylindrical shells is presented. The FGMs considered are composed of stainless steel and nickel where the volume fractions follow a powerlaw distribution. The analysis is carried out using Hamilton’s principle. Results are presented on the frequencies characteristics and the effects of free-simply support boundary conditions on functionally graded cylindrical shell.

Nomenclature E k ,k ′, γ 2 , γ 3

Young's modulus Bending strains

ν σ11 ,σ 22 ,σ 12 , σ 13 ,σ 23 ρ

Poisson ratio Stress components

Qij

Mass density Reduced stiffness

N H ij

Power law exponent Higher-order stiffness

R n ,m L T h P ,Pi a

U1 , U 2 , U 3

Radius of the shell Wave numbers Length of the shell Temperature in Kelvin Thickness of the shell Material properties Position of the ring support Natural frequency Displacements of the shell

H ij

Higher-order stiffness

ω

I.

Introduction

Cylindrical shells often used as load bearing structures for aircrafts, ships and buildings. Understanding of vibration behavior of cylindrical shells is an important aspect for the successful applications of cylindrical shells. Researches on free vibrations of cylindrical shells have been carried out extensively [1]-[5].

II.

Functionally Graded Materials

For the cylindrical shell made of FGM the material properties such as the modulus of elasticity E , Poisson ratio ν and the mass density ρ are assumed to be



933

M. Setareh, M. R. Isvandzibaei

functions of the volume fraction of the constituent materials when the coordinate axis across the shell thickness is denoted by z and measured from the shell’s middle plane. The functional relationships between E , ν and ρ with z for a stainless steel and nickel FGM shell are assumed as [13]: N

⎛ 2Z + h ⎞ E = ( E1 − E2 ) ⎜ ⎟ + E2 ⎝ 2h ⎠

(2)

N

⎛ 2Z + h ⎞ ⎟ + ρ2 ⎝ 2h ⎠

ρ = ( ρ1 − ρ 2 ) ⎜

(8)

⎛ ⎞ ⎜ ⎟ ⎛ α3 ⎞ ∂ ⎜ U2 ⎟+ ∈23 = A2 ⎜ 1 + ⎟ ⎜ ⎟ ⎝ R2 ⎠ ∂α 3 ⎜ A ⎛ 1 + α 3 ⎞ ⎟ ⎜ 2⎜ R ⎟⎟ 2 ⎠⎠ ⎝ ⎝ ∂U 3 1 + ⎛ α 3 ⎞ ∂α 2 A2 ⎜ 1 + ⎟ ⎝ R2 ⎠

(9)

(1)

N

⎛ 2Z + h ⎞ ⎟ +ν 2 ⎝ 2h ⎠

ν = (ν 1 −ν 2 ) ⎜

⎛ ⎞ ⎜ ⎟ ⎛ α ⎞ ∂ ⎜ U1 ⎟+ ∈13 = A1 ⎜ 1 + 3 ⎟ ⎜ ⎛ α ⎞⎟ R ∂ α ⎝ 1 ⎠ 3 ⎜ A1 ⎜ 1 + 3 ⎟ ⎟ ⎜ R1 ⎠ ⎟⎠ ⎝ ⎝ ∂U 3 1 + ⎛ α ⎞ ∂α A1 ⎜ 1 + 3 ⎟ 1 R1 ⎠ ⎝

(3)

The strain-displacement relationships for a thin shell [14]: ⎡ ∂U1 U 2 ∂A1 A⎤ 1 ∈11 = + + U3 1 ⎥ ⎢ A2 ∂α 2 R1 ⎦ ⎛ α ⎞ ∂α A1 ⎜1 + 3 ⎟ ⎣ 1 R1 ⎠ ⎝ ∈22

⎡ ∂U 2 U1 ∂A2 A ⎤ + + U3 2 ⎥ ⎢ A1 ∂α1 R2 ⎦ ⎛ α ⎞ ∂α A2 ⎜ 1 + 3 ⎟ ⎣ 2 ⎝ R2 ⎠ 1

∈33 =

∂U 3 ∂α 3

⎛ ⎞ ⎛ α ⎞ A1 ⎜ 1 + 3 ⎟ ⎜ ⎟ R1 ⎠ ∂ ⎜ U1 ⎟ ∈12 = ⎝ ⎜ ∂ α ⎛ α ⎞ ⎛ α ⎞⎟ A2 ⎜ 1 + 3 ⎟ 2 ⎜⎜ A1 ⎜ 1 + 3 ⎟ ⎟⎟ R1 ⎠ ⎠ ⎝ R2 ⎠ ⎝ ⎝ ⎛ ⎞ ⎛ α ⎞ A2 ⎜1 + 3 ⎟ ⎜ ⎟ R2 ⎠ ∂ ⎜ U2 ⎟ + ⎝ ⎜ ∂ α ⎛ α ⎞ ⎛ α ⎞⎟ A1 ⎜ 1 + 3 ⎟ 1 ⎜⎜ A2 ⎜ 1 + 3 ⎟ ⎟⎟ R1 ⎠ ⎝ ⎝ ⎝ R2 ⎠ ⎠

A1 =

∂r ∂r , A2 = ∂α1 ∂α 2

(10)

(4 ) where A1 and A2 are the fundamental form parameters or Lame parameters, U1 , U 2 and U 3 are the displacement at any point ( α1 , α 2 , α 3 ), R1 and R2 are the radius of curvature related to α1 , α 2 and α 3 respectively. The third- order theory of Reddy used in the present study is based on the following displacement field:

(5 )

⎧U1 = u1 (α1 ,α 2 ) + α 3 ⋅ φ1 (α1 ,α 2 ) + ⎪ +α 32 ⋅ψ 1 (α1 ,α 2 ) + α 33 ⋅ β1 (α1 ,α 2 ) ⎪ ⎪ ⎨U 2 = u2 (α1 ,α 2 ) + α 3 ⋅ φ2 (α1 ,α 2 ) + ⎪ +α 32 ⋅ψ 2 (α1 ,α 2 ) + α 33 ⋅ β 2 (α1 ,α 2 ) ⎪ ⎪U = u α ,α 3( 1 2) ⎩ 3

(6 )

(7)

(11)

These equations can be reduced by satisfying the stress-free conditions on the top and bottom faces of the laminates, which are equivalent to ∈13 =∈23 = 0 at h . 2 Thus for third order theory:

Z =±

⎧U1 = u1 (α1 ,α 2 ) + α 3 ⋅ φ1 (α1 ,α 2 ) + ⎪ ⎛ u ∂u3 ⎞ ⎪ −C1 ⋅ α 33 ⎜ − 1 + φ1 + ⎟ ⎪ A1∂α1 ⎠ ⎝ R1 ⎪ ⎪ ⎨U 2 = u2 (α1 ,α 2 ) + α 3 ⋅ φ2 (α1 ,α 2 ) + ⎪ ⎛ u ∂u3 ⎞ ⎪ −C1 ⋅ α 33 ⎜ − 2 + φ2 + ⎟ ⎪ R2 A2 ∂α 2 ⎠ ⎝ ⎪ ⎪⎩U 3 = u3 (α1 ,α 2 )

Fig. 1. Geometry of a generic shell


(12)


934


where C1 =

4 2

0 ⎫ ⎧∈11 ⎧⎛ 1 ∂u1 u ∂A1 u3 ⎞ ⎫ + 2 + ⎟ ⎪ ⎪ ⎪ ⎪⎜ ⎪ ⎪ ⎪⎝ A1 ∂α1 A1 A2 ∂α 2 R1 ⎠ ⎪ ⎪ 0 ⎪ ⎪ ⎪ u ∂A2 u3 ⎞ ⎪ ⎪∈22 ⎪ ⎪⎛ 1 ∂u2 + 1 + ⎨ ⎬ = ⎨⎜ ⎬ ⎟ ⎪ ⎪ ⎪⎝ A2 ∂α 2 A1 A2 ∂α1 R2 ⎠ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪∈12 ⎪ ⎪ A2 ∂ ⎛⎜ u2 ⎞⎟ + A1 ∂ ⎛⎜ u1 ⎞⎟ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ A1 ∂α1 ⎝ A2 ⎠ A2 ∂α 2 ⎝ A1 ⎠ ⎭

(15)

⎧⎛ 1 ∂φ ⎫ φ2 ∂A1 ⎞ 1 ⎪ ⎧ k11 ⎫ ⎪⎜ A ∂α + A A ∂α ⎟ ⎪ 1 2 2 ⎠ ⎪ ⎪ ⎪⎝ 1 1 ⎪ ⎪⎪ ⎪⎪ ⎪⎪⎛ 1 ∂φ φ1 ∂A2 ⎞ ⎪ 2 k = + ⎨ 22 ⎬ ⎨⎜ ⎬ ⎟ ⎪ ⎪ ⎪⎝ A2 ∂α 2 A1 A2 ∂α1 ⎠ ⎪ ⎪ ⎪ ⎪⎛ ⎪ A ∂ ⎛ φ2 ⎞ A1 ∂ ⎛ φ1 ⎞ ⎞ ⎪ ⎩⎪ k12 ⎭⎪ ⎪⎜ 2 + ⎟ ⎜ ⎟ ⎜ ⎟ ⎟⎪ ⎪⎜ ⎩⎝ A1 ∂α1 ⎝ A2 ⎠ A2 ∂α 2 ⎝ A1 ⎠ ⎠ ⎭

(16)

.

3h Substituting Eq. (12) into nonlinear straindisplacement relation (4) - (9), it can be obtained for the third-order theory of Reddy: 0 ′ ⎫ ⎧∈11 ⎫ ⎧∈11 ⎫ ⎧ k11 ⎫ ⎧ k11 ⎪ ⎪ ⎪ ⎪ 0 ⎪⎪ ⎪ ⎪ ⎪ 3⎪ ′ ⎬ ⎨∈22 ⎬ = ⎨∈22 ⎬ + α 3 ⎨k22 ⎬ + α 3 ⎨k22 ⎪∈ ⎪ ⎪ 0 ⎪ ⎪k ⎪ ⎪k ′ ⎪ ⎩ 12 ⎭ ⎪⎩∈12 ⎪⎭ ⎩ 12 ⎭ ⎩ 12 ⎭

(13)

0 ⎧ 2 ⎫ ⎧ 3 ⎫ ⎧∈13 ⎫ ⎧⎪γ 13 ⎫⎪ 2 ⎪γ 13 ⎪ 3 ⎪γ 13 ⎪ α α = + + ⎨ ⎬ ⎨ 0 ⎬ 3⎨ 2 ⎬ 3⎨ 3 ⎬ ⎩∈23 ⎭ ⎪⎩γ 23 ⎪⎭ ⎪⎩γ 23 ⎪⎭ ⎪⎩γ 23 ⎪⎭

(14)

where:

⎧⎛ 1 ⎛ ∂u ⎫ ∂ 2 u3 ∂φ ∂A ∂u3 ⎞ ⎞ 1 ⎪⎜ ⎜ − ⎪ + 1+ − 2 1 +⎟ ⎟ 2 ⎪⎜ A1 ⎜⎝ R1∂α1 ∂α1 A1∂α1 A1 ∂α1 ∂α1 ⎟⎠ ⎟ ⎪ ⎪⎜ ⎪ ⎟ ∂u3 ⎞ ⎪⎜ ∂A1 1 ⎛ u2 ⎪ ⎟ ⎪⎜ + ∂α A A ⎜ − R + φ2 + A ∂α ⎟ ⎪ ⎟ 2 1 2 ⎝ 2 2 2 ⎠ ⎠ ⎪⎝ ⎪ ′ ⎫ ⎧k11 ⎪⎛ ⎛ ⎪ 2 ⎞ ⎞ ⎪ ⎪ ⎪⎜ 1 ⎜ − ∂u2 + ∂φ2 + ∂ u3 − ∂A2 ∂u3 ⎟ + ⎟ ⎪ ⎪⎪⎜ A2 ⎝⎜ R2 ∂α 2 ∂α 2 A2 ∂α 22 A22 ∂α 2 ∂α 2 ⎠⎟ ⎟ ⎪⎪ ⎪⎪ ⎪⎪ ′ ⎬ = −C1 ⎨⎜ ⎨k22 ⎬ ⎟ ∂u3 ⎞ ⎪ ⎪ ⎪⎜ ∂A2 1 ⎛ u1 ⎪ ⎟ ⎪ ⎪ ⎪⎜ + ∂α A A ⎜ − R + φ1 + A ∂α ⎟ ⎪ ⎟ 1 1 2 ⎝ 1 1 1⎠ ⎠ ′ ⎪⎭ ⎪⎩k12 ⎪⎝ ⎪ ⎪⎛ ⎛ 2 2 ⎞ ⎞ ⎪ ⎪⎜ A2 ⎜ − ∂ ⎜⎛ u2 ⎟⎞ + ∂ ⎜⎛ φ2 ⎟⎞ + 1 ∂ u3 − 1 ∂A2 ∂u3 ⎟ + ⎟ ⎪ ⎪⎜ A1 ⎜⎝ R2 ∂α1 ⎝ A2 ⎠ ∂α1 ⎝ A2 ⎠ A22 ∂α1∂α 2 A24 ∂α1 ∂α 2 ⎟⎠ ⎟ ⎪ ⎪⎜ ⎟⎪ ⎪⎜ A1 ⎛ ∂ ⎛ u1 ⎞ ∂ ⎛ φ1 ⎞ 1 ∂ 2u3 1 ∂A12 ∂u3 ⎞ ⎟ ⎪ − 4 ⎟⎟ ⎟ ⎪ ⎪⎜ + ⎜⎜ − ⎜ ⎟+ ⎜ ⎟+ 2 ⎪⎩⎝ A2 ⎝ R1∂α 2 ⎝ A1 ⎠ ∂α 2 ⎝ A1 ⎠ A1 ∂α1∂α 2 A1 ∂α 2 ∂α1 ⎠ ⎠ ⎪⎭ 0 ⎫ ⎧ ⎧γ 13 ⎛ u1 1 ∂u3 ⎞ ⎫ ⎪ ⎪ ⎪⎜ φ1 − + ⎟ ⎪ R1 A1 ∂α1 ⎠ ⎪ ⎪ ⎪ ⎪⎝ ⎨ ⎬=⎨ ⎬ u2 1 ∂u3 ⎞ ⎪ ⎪ ⎪ ⎪⎛ ⎪γ 0 ⎪ ⎪⎜ φ2 − R + A ∂α ⎟ ⎪ 2 2 2 ⎠⎭ ⎩ 23 ⎭ ⎩⎝ 2 ⎫ ⎧ ⎛ u1 ⎧γ 13 ⎪ ⎜ − + φ1 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎝ R1 ⎨ ⎬ = 3C1 ⎨ ⎪ ⎪ ⎪⎛ u2 ⎪γ 2 ⎪ ⎪⎜ − R + φ2 + 2 ⎩ 23 ⎭ ⎩⎝

where

( k ,k ′ ,γ

(18)

(ε 2

0

,γ 3

,γ 0

)

)

(17)

are the membranes strains and

are the bending strains, known as the

curvatures.

III. Formulation

∂u3 ⎞ ⎫ ⎟⎪ A1∂α1 ⎠ ⎪ ⎬ ∂u3 ⎞ ⎪ ⎟ A2 ∂α 2 ⎠ ⎪⎭

(19)

⎧ ⎛ u1 ∂u3 ⎞ ⎫ 3 ⎫ ⎧γ 13 ⎪ ⎜ − + φ1 + ⎟ ⎪ A1∂α1 ⎠ ⎪ ⎪ ⎪ ⎪ ⎝ R1 ⎪ ⎪ R1 ⎪ ⎪ ⎪⎪ ⎪⎪ ⎨ ⎬ = C1 ⎨ ⎬ ⎪ ⎪ ⎪ ⎛ − u2 + φ + ∂u3 ⎞ ⎪ ⎟ 2 ⎪ ⎪ ⎪⎜ R A2 ∂α 2 ⎠ ⎪ ⎝ 2 3 ⎪ ⎪⎩γ 23 ⎪ ⎪ ⎭ R2 ⎪⎩ ⎪⎭

(20)

Consider a cylindrical shell as shown in Fig. 2, where R is the radius, L the length and h the thickness of the shell. The reference surface is chosen to be the middle surface of the cylindrical shell where an orthogonal coordinate system x,θ ,z is fixed. The displacements of the shell with reference this coordinate system are denoted by U1 , U 2 and U 3 in the x,θ and z directions, respectively.



935


⎧ N11 ⎫ ⎧σ 11 ⎫ ⎧ M11 ⎫ ⎧σ 11 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ h ⎪ h ⎪ ⎪⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎨ N 22 ⎬ = ∫ 2h ⎨σ 22 ⎬ dα 3 , ⎨ M 22 ⎬ = ∫ 2h ⎨σ 22 ⎬ α 3 dα 3 (25) − − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2⎪ 2⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ N12 ⎭⎪ ⎩⎪σ 12 ⎭⎪ ⎩⎪ M12 ⎭⎪ ⎩⎪σ 12 ⎭⎪

⎧ P13 ⎫ ⎧σ 13 ⎫ ⎧ P11 ⎫ ⎧σ 11 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ h ⎪ h ⎪ ⎪ ⎪ ⎪⎪ ⎪⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪⎪ 3 ⎨ P22 ⎬ = ∫ 2h ⎨σ 22 ⎬ α 3 dα 3 , ⎨ ⎬ = ∫ 2h ⎨ ⎬ α 3 dα 3 (26) ⎪ ⎪ −2 ⎪ ⎪ ⎪ ⎪ −2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ P23 ⎪⎭ ⎪⎩σ 23 ⎪⎭ ⎩⎪ P12 ⎭⎪ ⎩⎪σ 12 ⎭⎪

Fig. 2. Geometry of a FGM cylindrical shell

For a thin cylindrical shell, the stress -strain relationship are defined as: ⎧σ11 ⎫ ⎡Q11 ⎪ ⎪ ⎢ ⎪σ 22 ⎪ ⎢Q12 ⎪ ⎪ ⎨σ 23 ⎬ = ⎢ 0 ⎪σ ⎪ ⎢ 0 ⎪ 13 ⎪ ⎢ ⎪⎩σ12 ⎪⎭ ⎢⎣ 0

0

0

Q22

0

0

0

Q44

0

0 0

0 0

Q55 0

Q12

⎧Q13 ⎫ ⎧ R13 ⎫ h ⎧σ 13 ⎫ h ⎧σ 13 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎨ ⎬=∫ h⎨ ⎬ dα 3 , ⎨ ⎬ = ∫ 2h ⎨ ⎬ α 3 dα 3 ⎪Q ⎪ − 2 ⎪σ ⎪ ⎪ R ⎪ − 2 ⎪σ ⎪ ⎩ 23 ⎭ ⎩ 23 ⎭ ⎩ 23 ⎭ ⎩ 23 ⎭

⎤ ⎧∈11 ⎫ ⎪ ⎪ 0 ⎥⎥ ⎪∈22 ⎪ ⎪ ⎪ 0 ⎥ ⎨∈23 ⎬ (21) ⎥ 0 ⎥ ⎪∈13 ⎪ ⎪ ⎪ Q66 ⎥⎦ ⎪⎩∈12 ⎪⎭ 0

IV.

Q11 = Q22 =

ν E Q12 = 1 −ν 2

1 −ν 2

Q44 = Q55 = Q66 =

The Equations of Motion for Vibration of a Generic Shell

The equations of motion for vibration of a generic shell can be derived by using Hamilton's principle which is described by:

For a isotropic cylindrical shell the reduced stiffness Qij ( i , j=1, 2 and 6) are defined as: E

δ∫

,

=∫

−h / 2

{

Qij 1

, α 3 , α 3 , α 33 ,α 34 ,α 35 ,α 36

(23)

} dα

K=

U=

(24)

2

2

2

(29)

α3

⎛ σ 11 ∈11 +σ 22 ∈22 +

⎞

∫ ∫ ∫ ⎜⎝ +σ12 ∈12 +σ13 ∈13 +σ 23 ∈23 ⎟⎠ dV

(30)

⎛ q1δ U1 + q2δ U 2 + ⎞ ⎟ A1 A2 dα1dα 2 ⎠ α1 α 2

(31)

α1 α 2 α 3

3

V=

materials. Here Aij denote the extensional stiffness, Dij the

∫ ∫ ⎜⎝ +q3δ U 3

The infinitesimal volume is given by:

bending stiffness, B ij the bending-extensional coupling Eij ,Fij ,Gij ,H ij

∫ ∫ ∫ ρ (U1 + U 2 + U 3 ) dV

1 2α

1 α2

where Qij are functions of z for functionally gradient

stiffness and

(28)

where K, Π ,U and V are the total kinetic, potential, strain and loading energies, t1 and t2 are arbitrary time. The kinetic, strain and loading energies of a cylindrical shell can be written as:

{ Aij ,Bij , Dij , Eij , Fij , Gij , H ij } = 2

( Π − K ) dt = 0,

Π = U −V

(22)

where E is the Young's modulus and ν is Poisson's ratio. Defining (24):

h/ 2

t2

t1

E 2 (1 + ν )

(27)

are the extensional,

dV = A1 A2 dα1dα 2 dα 3

bending, coupling, and higher-order stiffness. For a thin cylindrical shell the force and moment results are defined as:

(32)

with the use of Eqs. (11)-(20) and substituting into Eq. (28), we get the equations of motions a generic shell:



936


−

∂ ( N11 A2 ) ∂α1

(

)

2 Q ∂A2 ∂ N12 A1 ∂ ⎛ P11C1 A2 − − 13 A1 A2 − ⎜ ∂α1 ∂α1 ⎝ R1 A1∂α 2 R1

+ N 22

⎞ P22 C1 ∂A2 + ⎟+ R1 ∂α1 ⎠

CP AA ∂ ⎛ P12 C1 A12 ⎞ 1 3C1 R13 A1 A2 − 1 13 2 1 2 = ⎜⎜ ⎟⎟ + ∂α 2 ⎝ R1 ⎠ A1 R1 R1 ⎛ ⎡ ⎛ u ∂u3 ⎞ ⎞ ∂u3 ⎞ C1u1 ⎤ C φ C 2 ⎛ u I 3 + 1 1 I 4 − 1 ⎜ − 1 + φ1 + = − ⎜ u1 I 0 + φ1 I1 + ⎢ −C1 ⎜ − 1 + φ + + ⎥ ⎟ I6 ⎟ ⎟ ⎜ A1∂α1 ⎠ R1 ⎦⎥ R1 R1 ⎝ R1 A1∂α1 ⎠ ⎟⎠ ⎝ R1 ⎣⎢ ⎝ −

∂ ( N 22 A1 ) ∂α 2

(

(33)

)

2 Q ∂A1 ∂ N12 A2 ∂ ⎛ P22 C1 A1 ⎞ P11C1 ∂A1 − N11 + + 23 A1 A2 + + ⎜ ⎟− A2 ∂α1 R2 R2 ∂α 2 ∂α 2 ∂α 2 ⎝ R2 ⎠

CP AA ∂ ⎛ P12C1 A22 ⎞ 1 3C1 R23 − A1 A2 + 1 23 2 1 2 = ⎜ ⎟ ∂α1 ⎜⎝ R2 ⎟⎠ A2 R2 R2 ⎛ ⎡ ⎛ u ∂u3 ⎞ C1u2 ⎤ ∂u3 ⎞ ⎞ C φ C 2 ⎛ u = ⎜ u2 I 0 + φ2 I1 + 1 2 I 42 + ⎢ −c1 ⎜ − 2 + φ2 + + I 3 − 1 ⎜ − 2 + φ2 + ⎥ ⎟ ⎟ I6 ⎟ ⎜ R2 A2 ∂α 2 ⎠ R2 ⎥⎦ R2 ⎝ R2 A2 ∂α 2 ⎠ ⎟⎠ ⎢⎣ ⎝ R2 ⎝ +

(34)

2 ⎛ ∂ 2 ( P11C1 A2 / A1 ) A1 A2 A1 A2 ∂ ( P22 A1C1 / A2 ) ∂ ⎛ C1 P11 ∂A1 ⎞ N N + + + − + ⎜− ⎜ ⎟ 11 22 ⎜ ∂α 2 ⎝ A2 ∂α 2 ⎠ R1 R2 ∂α12 ∂α 22 ⎝ 2 2 ∂ ⎛ P22C1 ∂A2 ⎞ ∂ ( P12C1 ) ∂ ⎛ P12 C1 ∂A22 ⎞ ∂ ( P12 C1 ) ∂ ⎛ P12C1 ∂A12 ⎞ + − − − − ⎜⎜ 2 ⎟⎟ ⎜ ⎟+ ⎜ ⎟ ∂α1 ⎝ A1 ∂α1 ⎠ ∂α1∂α 2 ∂α 2 ⎝ A2 ∂α1 ⎠ ∂α1∂α 2 ∂α1 ⎜⎝ A12 ∂α 2 ⎟⎠

∂ ⎛ P13C1 A2 ⎞ ∂ ( Q23 A1 ) ∂ ( 3C1 R23 A1 ) + + ⎜ ⎟− ∂α1 ∂α1 ∂α1 ⎝ R1 ⎠ ∂α 2 ∂α 2 A ∂A ⎞ ⎞ ∂ ⎛ C1 P23 A1 ⎞ ∂ ⎛ P11C1 A2 ∂A1 ⎞ ∂ ⎛ − ⎜⎜ ⎟⎟ − ⎜⎜ P22 C1 12 2 ⎟⎟ ⎟⎟ = ⎜ ⎟− 2 ∂α 2 ⎝ R2 ⎠ ∂α1 ⎝ A1 ∂α1 ⎠ ∂α 2 ⎝ A2 ∂α 2 ⎠ ⎠ ⎛⎛ ⎡ ∂ ⎛ φ1 ⎞ ⎡ ∂ ⎛ u1 ⎞ ∂ ⎛ u2 ⎞ ⎤ ∂ ⎛ φ2 ⎞ ⎤ ∂ ⎛ u2 ⎪⎧ 2 = − ⎨u3 I 0 + C1 ⎢ ⎜ ⎟+ ⎜ ⎟ ⎥ I 4 − C1 I 6 ⎜⎜ ⎜⎜ − ⎜ ⎟+ ⎜ ⎟ ⎥ I 3 + C1 ⎢ ⎜ A A A A R α α α α ∂ ∂ ∂ ∂ ∂ ⎢⎣ 1 ⎝ 1 ⎠ ⎢⎣ 1 ⎝ 1 ⎠ 2 ⎝ 2 ⎠⎥ 2 ⎝ 2 ⎠⎥ 2 α 2 ⎝ A2 ⎪⎩ ⎦ ⎦ ⎝⎝ −

+

∂ ( Q13 A2 )

+

∂ ( 3C1 R13 A2 )

−

∂A ∂u3 ∂ ⎛ φ2 ⎞ 1 ∂ 2 u3 − 2 2 ⎜ ⎟+ 2 ∂α 2 ⎝ A2 ⎠ A2 ∂α 2 A1 ∂α 2 ∂α 2

−

∂ ( M 11 A2 ) ∂α1

+

∂ ( C1 P11 A2 ) ∂α1

−3C1 R13 A1 A2 + A1 A2 Q13 + −

−

⎞⎞ ⎟ ⎟⎟ + ⎠⎠

⎞ ⎛ ∂A1 ∂u3 ⎞ ⎪⎫ ∂ ⎛ u1 ⎞ ∂ ⎛ φ1 ⎞ 1 ∂ 2 u3 − ⎟⎟ + ⎜⎜ − ⎟⎟ ⎬ ⎜ ⎟+ ⎜ ⎟+ 2 2 ⎠ ⎝ R1∂α1 ⎝ A1 ⎠ ∂α1 ⎝ A1 ⎠ A1 ∂α1 A1 ∂α1 ∂α1 ⎠ ⎪⎭

+ M 22

(

) (

)

2 ∂ P12C1 A12 ∂A2 ∂A ∂ M12 A1 − C1 P22 2 − + + A1∂α 2 A1∂α 2 ∂α1 ∂α1

⎡ ⎛ C1 P13 u A1 A2 = − ⎢u1 I1 + φ1 I 2 − C1u1 I 3 + ⎜ −2C1φ1 + C1 1 + R1 R1 ⎝ ⎣⎢

(36)

1 ∂u3 ⎞ ⎤ C1 ∂u3 ⎞ 2⎛ u ⎟ I 4 + C1 ⎜ − + φ1 + ⎟ I6 ⎥ A1 ∂α1 ⎠ A1∂α1 ⎠ ⎥⎦ ⎝ R1

∂ ( M 22 A1 ) ∂α 2

+

∂ ( C1 A1 P22 ) ∂α 2

−3C1 R23 A1 A2 + A1 A2 Q23 + −

(35)

+ M 11

(

) (

)

2 ∂ P12C1 A22 ∂A1 ∂A ∂ M12 A2 − C1 P11 1 − + + ∂α 2 ∂α 2 A2 ∂α1 A2 ∂α1

⎡ ⎛ C1 P23 u A1 A2 = − ⎢u2 I1 + φ2 I 2 − C1u2 I 3 + ⎜ −2C1φ2 + C1 2 + R2 R2 ⎝ ⎣⎢

(37)

2 ∂u3 ⎞ ⎤ C1 ∂u3 ⎞ 2⎛u + φ2 + ⎟ I 4 + C1 ⎜ ⎟ I6 ⎥ A2 ∂α 2 ⎠ A2 ∂α 2 ⎠ ⎥⎦ ⎝ R2



937


For Eqs. (33) – (37) are defining as: Ii =

V.

h 2 h − 2

∫

1 R2 = a, = 0,A2 = a,A1 = 0,α 3 = α 3 ,α 2 = θ ,α1 = x (39) R

ρα 3i dα 3

(38)

Substituting relationship (39) into Eqs. (33)-(37) the equations of motions for vibration of cylindrical shell with the third-order theory of Reddy are converted to:

Equations of Motion for Vibration of Cylindrical Shell

a

∂u ∂N11 ∂N12 + = I 0u1 + ( I1 − C1 I 3 ) φ1 − C1 I 3 3 (40) ∂x ∂θ ∂x

The curvilinear coordinates and fundamental from parameters for a cylindrical shell are: ∂N 22 ∂P + C1 12 + Q23 − 3C1 R23 + C1 P23 = ∂θ ∂x ⎛ ⎛ ⎛C C C2 ⎞ C C2 ⎞ C 2 ⎞ ∂u = ⎜ I 0 + 2 1 I 3 + 12 I 6 ⎟ u2 + ⎜ I1 − C1 I 3 + 1 I 4 − 1 I 6 ⎟ φ2 − ⎜ 1 I 3 − 12 I 6 ⎟ 3 ⎜ ⎟ ⎜ ⎟ ⎜ a ⎟ ∂θ a a a a a ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ∂ 2 P12 C1 ∂ 2 P22 − 2 C 1 ∂x∂θ a ∂θ 2 ∂x 2 ∂Q13 ∂R13 ∂Q23 ∂R23 C1 ∂P23 −a + 3C1a − + 3C1 − = ∂x ∂x ∂θ ∂θ a ∂θ ∂u C ∂u ∂φ1 ⎛ C1 C 2 ⎞ ∂φ = −C1 I 3 1 − 1 I 3 2 + −C1 I 4 + C12 I 6 + ⎜ − I4 + 1 I6 ⎟ 2 + ⎟ ∂θ a ∂x a ∂θ ∂x ⎜⎝ a ⎠ −C1a

∂ 2 P11

+ N 22 −

(

−

C12 a2

(41)

)

I6

∂ 2 u3 C12 ∂ 2 u3 ∂u2 + C12 I 6 + I6 − u3 I 0 ∂θ a ∂x 2 ∂θ 2

−a

∂M11 ∂P ∂M 12 ∂P + C1a 11 − + C1 12 − 3C1 R13 a + aQ13 = ∂x ∂x ∂θ ∂θ

∂u3 = − I1u1 + C1 I 3u1 + − I 2 + 2C1 I 4 − C12 I 6 φ1 + C1 I 4 − C12 I 6 ∂x

(

) (

)

∂M 22 ∂P ∂M12 ∂P − C1 22 − a + C1a 12 + ∂θ ∂θ ∂x ∂x −3C1 R23 a + aQ23 + C1 R23 =

(42)

(43)

−

(44)

∂u C C ⎛ ⎞ = ⎜ − I1C1 I 3 − 1 I 4 ⎟ u2 + ( − I 2 + 2C1 I 4 ) φ2 − 1 I 4 3 a ⎠ a ∂θ ⎝

the amplitudes of the vibrations in the x,θ and z directions, φ1 and φ2 are the displacement fields for higher order deformation theories for a cylindrical shell, φ ( x ) is the axial function that satisfies the geometric

The displacement fields for a FG cylindrical shell and the displacement fields which satisfy these boundary conditions can be written as: ∂φ ( x )

cos ( nθ ) cos (ω t ) ∂x u2 = B φ ( x ) sin ( nθ ) cos (ω t ) u1 = A

u3 = Cφ ( x ) cos ( nθ ) cos (ω t )

boundary conditions. The axial function φ ( x ) is chosen as the beam function as: (45)

⎛ λm x ⎞ ⎛λ x⎞ + γ 2 cos ⎜ m ⎟ + ⎟ ⎝ L ⎠ ⎝ L ⎠

φ ( x ) = γ 1 cosh ⎜

∂φ ( x )

cos ( nθ ) cos (ω t ) ∂x φ2 = Eφ ( x ) sin ( nθ ) cos (ω t )

φ1 = D

⎛ ⎛λ x⎞ ⎛ λ x ⎞⎞ −ζ m ⎜ γ 3 sinh ⎜ m ⎟ + γ 4 sin ⎜ m ⎟ ⎟ ⎝ L ⎠ ⎝ L ⎠⎠ ⎝

(46)

where, A , B , C , D and E are the constants denoting Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


938


The geometric boundary conditions for free-simply support boundary conditions can be expressed mathematically in terms of φ ( x ) as:

Expanding this determinant, a polynomial in even powers of ω is obtained:

β 0ω10 + β1ω 8 + β 2ω 6 + β3ω 4 + β 4ω 2 + β 5 = 0 (49)

free boundary condition and simply support boundary condition:

φ '' ( 0 ) = φ ''' ( L ) = 0

where β i ( i = 0 ,1, 2 ,3, 4 ,5 ) are some constants. Eq. (49) is solved five positive and five negative roots are obtained. The five positive roots obtained are the natural angular frequencies of the cylindrical shell based thirdorder theory. The smallest of the five roots is the natural angular frequency studied in the present study. The FGM cylindrical shell is composed of Nickel at its inner surface and Stainless steel at its outer surface. The material properties for stainless steel and nickel, calculated at T = 300K , are presented in Table I.

(47)

φ ( 0 ) = φ '' ( L ) = 0

Substituting Eq. (45) into Eqs. (40) - (44) for third order theory we can be expressed:

(

)

det Cij − M ij ω 2 = 0

(48)

TABLE I PROPERTIES OF MATERIALS Stainless Steel

Coefficients E 201.04×109 0 3.079×10-4 -6.534×10-7 0 2.07788×1011

P0 P-1 P1 P2 P3

where P0 ,P−1 ,P1 ,P2

and

ν

ρ

0.3262 0 -2.002×10-4 3.797×10-7 0 0.317756

8166 0 0 0 0 8166

to the constituent materials. The material properties P of FGMs are a function of the material properties and volume fractions of the constituent material.


TABLE II COMPARISON OF NATURAL FREQUENCY (Hz) FOR A ISOTROPIC CYLINDRICAL SHELL L = 20.3cm , R = 5.08 cm , h = 0.25cm , E = 2.07788 × 1011 N m-2 ,ν = 0.317756 , ρ = 8166 kg m -3

m 1 2 3 4 5 6

Loy[15] 2043.8 5635.4 8932.5 11407.5 13253.2 14790.0

M.R.Isvandzibaei [16] 2043.6 5635.2 8932.1 11407.2 13252.8 14789.8

ρ 8900 0 0 0 0 8900

TABLE III THE NATURAL FREQUENCIES FOR A FGM CYLINDRICAL SHELL UNDER (F-SS) BOUNDARY CONDITIONS (M = 1, H / R=0.01, L / R=20) m n ω (Hz) 1 1 0.376687 2 0.472224 3 0.496101 4 0.506079 5 0.513007 6 0.520445 7 0.530317 8 0.544065 9 0.562919 10 0.587923

To validate the present analysis, results for cylindrical shells are compared with Loy and Lam [15] and with M. R. Isvandzibaei [16]. The comparisons show that the present results agreed well with those in the literature.

n 2

ν 0.3100 0 0 0 0 0.3100

variation of the natural frequency with the circumferential wave number n for a functional graded cylindrical shell. The frequencies for the free-simply support boundary conditions increased with the circumferential wave number.

P3 are the coefficients of

temperature T ( K ) expressed in Kelvin and are unique

VI.

Nickel E 223.95×109 0 -2.794×10-4 -3.998×10-9 0 2.05098×1011

Present 2045.1 5624.6 8821.5 11437 13197.5 14790.6

Natural frequencies of the FGM cylindrical shell for (F-SS) boundary conditions are computed and plotted in Fig. 3. For this boundary conditions, the frequencies decrease first and then increase as the circumferential wave number n increases. The minimum frequency occurs in between n equals 2 and 3 for this boundary conditions. For simplicity, we actually vary the value of power law exponent whenever we need to change the volume fraction. Varying the value of power law exponent N of the FGM cylindrical shell, natural frequencies are computed for (F-SS) boundary conditions. Results are

From the comparisons presented in Table II, it can be seen that the present results agree well with those in the literature. In this paper, studies are presented for a FGM cylindrical shell with free-simply support (F-SS) boundary conditions is considered. Table III, shows the



939


also computed for pure stainless steel and pure nickel shells. All these results are plotted in Fig. 4.

[3] [4]

[5]

[6]

[7] [8] [9]

Fig. 3. Natural frequencies FGM cylindrical shell associated with Free-Simply Support boundary conditions. (m=1, h/R=0.002, L/R=20)

[10]

N=0(SS) N=0(N) N=0.5 N=0.7 N=1 N=2 N=5 N=15 N=30

[12]

70 60 50 40

f(HZ) 30 20 10

[11]

[13]

[14] [15] [16]

0 0

2

4

6

8

10

12

Review on Modelling and Simulations, Part B, Vol 3, pp: 10771086. Chung, H., 1981. Free vibration analysis of circular cylindrical shells. J. Sound vibration; 74, 331-359. Soedel, W., 1980.A new frequency formula for closed circular cylindrical shells for a large variety of boundary conditions. J. Sound vibration; 70,309-317. Vörös, G.M., 2007. Buckling and Vibration of Stiffened Plates. International Review Mechanical Engineering; Vol .1 n. 1, pp. 49 – 60. Lam, K.L., Loy, C.T., 1995. Effects of boundary conditions on frequencies characteristics for a multi- layered cylindrical shell. J. Sound vibration; 188, 363-384. Loy, C.T., Lam, K.Y., 1996.Vibration of cylindrical shells with ring support. I.Joumal of Impact Engineering; 1996; 35:455. Koizumi, M., 1993. The concept of FGM Ceramic Transactions, Functionally Gradient Materials. Makino A, Araki N, Kitajima H, Ohashi K. Transient temperature response of functionally gradient material subjected to partial, stepwise heating. Transactions of the Japan Society of Mechanical Engineers, Part B 1994; 60:4200-6(1994). Anon, 1996.FGM components: PM meets the challenge. Metal powder Report. 51:28-32. Zhang, X.D., Liu, D.Q., Ge, C.C., 1994. Thermal stress analysis of axial symmetry functionally gradient materials under steady temperature field. Journal of Functional Materials; 25:452-5. Wetherhold, R.C., Seelman, S., Wang, J.Z., 1996. Use of functionally graded materials to eliminate or control thermal deformation. Composites Science and Technology; 56:1099-104. Najafizadeh, M.M., Hedayati, B. Refined Theory for Thermoelastic Stability of Functionally Graded Circular Plates. Journal of thermal stresses; 27:857-880. Soedel, W., 1981. Vibration of shells and plates. MARCEL DEKKER, INC, New York. Loy, C.T., Lam, K.Y., Reddy, J.N., 1999.Vibration of functionally graded cylindrical shells; 41:309-324. Najafizadeh, M.M., Isvandzibaei, M.R., 2007. Vibration of functionally graded cylindrical shells based on higher order shear deformation plate theory with ring support. Acta Mechanica; 191:75-91.

n

Authors’ information Fig. 4. Natural frequencies FGM cylindrical shell associated with various power law exponents for F-SS boundary conditions

VII.

M. Setareh, Department of Mechanical Engineering, Andimeshk Branch, Islamic Azad University, Andimeshk, Iran. He has published 18 articles in journals and conferences proceeding. Pone: +98 916 6419163 Email: [email protected]

Conclusion

A study on the vibration of functionally graded (FG) Cylindrical shell composed of stainless steel and nickel has been presented. The study showed that in this boundary conditions (F-SS) the frequencies first decreases and then increases as the circumferential wave number n increases. The minimum frequency occurs in between n equals 2 and 3 for this boundary conditions. The results showed that one could easily vary the natural frequency of the FGM cylindrical shell by varying the volume fraction. The present analysis is validated by comparing results with those available in the literature.

M. R. Isvandzibaei, Department of Mechanical Engineering, Andimeshk Branch, Islamic Azad University, Andimeshk, Iran. now he is Ph.D. student all in mechanical engineering. He has published more than 43 articles in journals and conferences proceeding. Phone: +98 916 344 2982 Email: [email protected]

References [1]

[2]

Arnold, R.N., Warburton, G.B., 1948. Flexural vibrations of the walls of thin cylindrical shells. Proceedings of the Royal Society of London A; 197:238-256. Adrian Plesca, 2010. Thermal Analysis of Fuses and Busbar Connections at Different Type of Load Variations. International



940


Using Homotopy Analysis Method to Determine Profile for Disk Cam by Means of Optimization of Dissipated Energy Hamid M. Sedighi, Kourosh H. Shirazi

Abstract – Select a particular shape for cam profile can play an important role in reduction of energy in automobile motors. Engineers try to reduce this dissipated energy to improve the motor operation. This paper obtains nonlinear governing equation of disk cam follower motion and optimizes it with calculus of variation. Because of optimum cam profile, also, maximum acceleration of the follower is decreased. Finally, we solve it analytically by means of Homotopy analysis method and numerical results have been reported to prove the soundness of the analytical method. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Homotopy Analysis Method, Cam Profile, Optimization of Dissipated Energy

Cam shape optimization is typically a heavily constrained multi-objective optimization problem in which multiple, design task dependent, material strength and durability, cam manufacturing technology, geometric, kinetic and dynamic constraints should be satisfied simultaneously. For example, the minimum local concave radius of the cam shape may not be smaller than the radius of the grinding stone used for manufacturing the cam. The minimum local convex radius is constrained by the contact pressure (Hertz’s pressure) between the cam and cam follower [5]. Typical cam design objectives are maximum rise and return rates for the cam follower movements, minimum instantaneous contact force between the cam and follower simultaneously with minimum dynamic force fluctuation. Typically also one or more task dependent measures of operated device performance are among the objectives. Thus, the objectives are the target functional properties of the cam mechanism and the device operated by it, and the constraints are imposed by the restrictive design conditions. In the field of mechanical engineering, in addition to cam shape optimization discussed in this article, evolutionary shape optimization approach have been applied also for shape optimization of: a strain gauge load cell [6], a cantilever beam [7], a torque arm [7], a spherical pressure vessel [7] and a conical pivot bearing journal [8]. In this paper, at first, the second order nonlinear differential equation of follower motion is obtained with Euler-poison's equation in calculus of variation. For such a system, the governing equation is solved analytically in two separate intervals of cam rotation by means of homotopy analysis method and boundary conditions of each interval are regarded. Finally, the agreement of our results with numerical methods is illustrated.

Nomenclature Me Mt Mv Ma I r

ω µ

l1 ,l2 Ks

Tappet mass Pushrod mass Valve mass Rocker arm mass Rocker arm moment of inertia Radius of base circle of cam Cam angular velocity Friction coefficient Lengths of rocker arm Valve spring coefficient

I.

Introduction

A cam is a mechanical element, which is used to transmit a desired motion to another mechanical element by direct surface contact. Generally, a cam is a mechanism, which is composed of three different fundamental parts from a kinematic viewpoint [1], [2]. A cam, which is a driving element, a follower, which is a driven element and a fixed frame. Cam mechanisms are usually implemented in most modern applications and in particular in automatic machines and instruments, internal combustion engines and control systems [3]. This mechanism is an important place in automobile that consume a significant amount of energy and can be effective in work of motors. Shape optimization based on genetic algorithm (GA) [4], or based on evolutionary algorithms (EA) in general, is a relatively young and potential field of research. The target of the cam shape optimization is to optimize the function (movements) of the operated device without violating geometric and physical constraints of cam designing.



941

Hamid M. Sedighi, Kourosh H. Shirazi

II.

2 ⎛ ⎞ ⎛l ⎞ I B2 = ⎜ M v ⎜ 2 ⎟ + 2 + M t + M e ⎟ ω 2 ⎜ ⎟ ⎝ l1 ⎠ l2 ⎝ ⎠

The Governing Equation

The main parts of cam mechanism are shown in Figure 1.

⎛l ⎞ B3 = K s ⎜ 2 ⎟ ⎝ l1 ⎠

2

B4 = 1 − µ 2 B5 =

(5)

2µ

ξ

(6)

(7) (8)

where parameters which introduce in the above equations are defined in Appendix A. Equation (3) is the same target function which we want to optimize it with calculus of variation. So that, we find such a function y (θ ) which minimize this equation. According to calculus of variation, y (θ ) should satisfy Euler- poison's

Fig. 1. Details of cam mechanism

equation [10]:

Energy that dissipated in cam- follower for one around of cam rotation is:

∂η d ∂η d 2 ∂η − + =0 ∂y dθ ∂y ′ dθ 2 ∂y ′′

2π

E=

∫ Mdθ

(1) where η introduce the term inside integral. Put E into equation (9) instead term η get the second order nonlinear differential equation as follows:

0

where M is the total moment of forces about cam center, θ is cam rotation that varies from 0 to 2π in one revolution. The frictional moment M can be written in terms of friction and contact forces as follows: M = Fy + Ny ′

⎛ b1 y ′ + b2 yy ′ + b3 y ′2 + b4 y ′3 + ⎞ ⎟ −⎜ ⎜ +b yy ′2 + b y + b ⎟ 5 6 7 ⎝ ⎠ y ′′ = b8 + b9 y ′ + b10 y + b11 y 2 + b12 y ′2

(2)

2π

∫ 0

( B1 + B2 y ′′ + B3 y ) ( y ′ + µ y ) B4 + B5 y ′

dθ

(10)

where b1 − b12 are defined in Appendix B.

where y is a function of θ and y ′ is derivative of y with respect to θ . After writing equation of motion of any part of valve mechanism, all equations are merged and obtained the general formulation of dissipated energy. Equation (3) represents the energy consumption E of cam-follower system, which constants B1 − B5 in terms of specific characteristics of valve are given below: E=

(9)

II.1.

Boundary Conditions

The boundary conditions of equation (10) are based on available limitations in follower motion and parameters of cam-follower mechanism. So that, according to, for this object the following conditions must be satisfied:

(3)

y ( 0 ) = r, y ′ ( 0 ) = 0

(11)

y (π 2 ) = h + r, y ′ (π 2 ) = 0

(12)

where: where r is the radius of base circle of cam and h is the maximum displacement of follower. Note that, in order to solve differential equation (10) we require two boundary conditions; however equations (11) and (12) tell us that four above boundary conditions must be satisfied. In according to these conditions, the sign of y ′′

2

⎛l ⎞ l B1 = M e g − M v g 2 − K s r ⎜ 2 ⎟ + l1 ⎝ l1 ⎠ l l + K s d1 2 − Mag 2 + M t g l1 l1

(4)



942


must be changed in interval [ 0 ,π 2] . Selection of this

about q as:

point is depending on cam length, radius of base circle and etc.

∞

φ (θ ,q ) = φ (θ , 0 ) + ∑ yk (θ ) q k

(19)

k =1

III. Homotopy Analysis Method where:

III.1. Basic Ideas

yk (θ ) =

Consider the nonlinear differential equation in general form: N ⎣⎡ y (θ ) ⎦⎤ = 0

k 1 ∂ φ (θ ,q ) k ! ∂k q q =0

Assume that the series (19) converges at q = 1 . From equations (16), (17) and (19), we have the relationship:

(13)

where N is a differential operator and y (θ ) is a y (θ ) = y0 (θ ) +

solution. Applying the HAM to solve it, we first need to construct the following family of equations:

(1 − q ) {L ⎣⎡φ (θ ,q ) − y0 (θ )⎦⎤} = hqN ⎣⎡φ (θ ,q )⎦⎤

(20)

∞

∑ yk (θ )

(21)

k =1

(14)

Liao [11] provides a general approach to derive the governing equation of yk (θ ) . Substituting the series (21)

where L is a properly selected auxiliary linear operator satisfying:

into equation (14) and equating the coefficient of the like power of q , we get the kth-order deformation equations:

L ( 0) = 0

L ⎡⎣ yk (θ ) − χ k yk −1 (θ ) ⎤⎦ = h Rk (θ )

(15)

h ≠ 0 is an auxiliary parameter, and y0 (θ ) is an initial

where:

approximation. Obviously, equation (14) gives:

φ (θ , 0 ) = y0 (θ )

Rk (θ ) =

(16)

when q = 0 . Similarly, when q = 1 , equation (13) is the same as equation (15) so that we have:

φ (θ ,1) = y (θ )

∂ q

, k = 1, 2,3,...

k −1

N ⎡⎣φ (θ ,q ) ⎤⎦ dq k −1

(23) q =0

⎧0, k ≤ 1 ⎩1, k ≥ 2

χk = ⎨

(17)

(24)

It is very important to emphasize that equation (22) is linear. If the first (k-1)th-order approximations have been obtained, then the right-hand side Rk (θ ) will be

converges for all 0 ≤ q ≤ 1 , for properly selected h and the auxiliary linear operator L . Suppose further that φ (θ ,q ) is infinitely differentiable at q = 0 , that is:

k

1 d ( k − 1) !

and:

Suppose that equation (14) has solution φ (θ ,q ) that

∂ k φ (θ ,q )

(22)

obtained. So, using the selected initial approximation y0 (θ ) , we can obtain y1 (θ ) , y2 (θ ) ,... , one after the other in order. Therefore, according to equation (21), we convert the original nonlinear problem into an infinite sequence of linear sub-problems governed by equation (22). We now consider equation (10) to apply homotopy analysis method to obtain analytical solution.

(18)

q =0

exists. Thus, as q increases from 0 to 1, the solution

φ (θ ,q ) of equation (14) varies continuously from the initial approximation y0 (θ ) to the solution y (θ ) of the

III.2. Application of the HAM

original equation (13). Clearly, equations (16) and (17) give an indirect relation between the initial approximation y0 (θ ) and the general solution y (θ ) . A

According to equation (10), it is straightforward to use the set of base functions [11]:

direct relationship between the two solutions is described as follows. Consider the Maclaurin’s series of φ (θ ,q )

{θ


k

}

|k = 0 ,1, 2,3,... , 0 ≤ θ ≤ π

2

(25)


943


to represent y (θ ) , i.e.:

The results at the kth-order approximation are given by: m

y (θ ) ≈ ∑ yk (θ )

∞

y (θ ) = ∑ akθ

k

(26)

k =0

Notice that at this case Equation (22) is subjected to the initial conditions:

where ak is coefficient. This provides us with a rule, called the rule of solution expression. This rule is important in the frame of the homotopy analysis method, as shown later. We introduce 1 y (θ ) , 2 y (θ ) as solutions in first and

1 yk

(θ ) = r,

2 y0

(θ ) = r + h

1 y1

(27)

(33)

(θ ) = −

9760943 2 hθ , 10000000

Similarly, we can get

1 y2

(θ ) ,...

and

(34)

2

2 y2

(θ ) ,... .

It

should be emphasized that our results contain the auxiliary parameter h , which provides us with a simple way to ensure the convergence of our series solutions. Note that y (θ ) is a power series of h .

parameter q increases from 0 to 1, φ (θ ,q ) varies from the initial guess y0 (θ ) to the exact solution y (θ ) ,

It is found that, at the 8th-order approximation, y ′′ ( 0 ) converges to the same value in the

respectively. To ensure this under the rule of solution expression described by (26), one chooses such an auxiliary linear operator:

∂ 2θ

(0) = 0

π⎞ 16190831 ⎛ h ⎜θ − ⎟ 2 y1 (θ ) = − 10000000 ⎝ 2⎠

The homotopy analysis method is based on such continuous variations φ (θ ,q ) , that, as the embedding

∂ 2φ (θ ,q )

1 yk′

r = 0.0156 m and h = 0.006 m, we successively obtain:

is a good initial guess of 1 y (θ ) , 2 y (θ ) .

L ⎡⎣φ (θ ,q ) ⎤⎦ =

( 0 ) = 0,

For the auxiliary function H (θ ) = 1 and for values

second intervals of consideration. Based on the initial conditions (11) and (12) and the rule of solution expression described by (26), it is obvious that: 1 y0

(32)

k =0

region −0.06 ≤ h ≤ 0 , as shown in Figure 2 for the socalled y ′′ ( 0 ) - h curve. This is indeed true: y ′′ ( 0 ) converges to 0.0081 in the region −0.06 ≤ h ≤ 0 .

(28)

that: L [C1 + C2θ ] = 0

(29)

where C1 and C2 are coefficients. Note that the rule of solution expression described by (26) plays an important role while determining the initial guess and the auxiliary linear operator L . Then, due to (10), one defines the non-linear operator: N ⎡⎣φ (θ ,q ) ⎤⎦ = φθθ + +

b1φθ + b2φφθ + b3φθ2 + b4φθ3 + b5φφθ2 + b6φθ + b7 (30)

Fig. 2. The 8th-order approximation of y ′′ ( 0 ) versus h

b8 + b9φθ + b10φ + b11φ 2 + b12φθ2

The obtained results are nearly identical with the results obtained numerically using a fourth order RungeKutta method. Figure 3 shows, the comparison between the results obtained by the present solution and the numerical integration results.

where φθ = ∂φ ∂θ ,φθθ = ∂ 2φ ∂θ 2 . Assume that h is properly chosen, therefore, at q = 1 we have: ∞

y (θ ) = ∑ yk (θ )

(31)

k =0



944


control the contact forces between cam and follower. In case of high-speed cam mechanisms, minimizing the contact force to reduce dissipated energy is also one of the most important design targets, whereas, it is necessary to avoid this force to becomes zero. The value K s = 25000 N/m satisfies two above conditions and it prevents the jump of disk cam in the motions of cam follower mechanism [9].

IV.

Conclusion

In this work, we presented a reliable algorithm based on the HAM to solve the optimization equation of disk cam profile with initial conditions. Displacement, velocity and acceleration figures of follower are given to illustrate the validity and accuracy of this procedure. The series solutions obtained by HAM contain the auxiliary parameter h . The validity of the method is based on such an assumption that the series (21) converges at q = 1 . It is the auxiliary parameter h which ensures that this assumption can be satisfied. In general, by means of the so-called h -curve, it is straightforward to choose a proper value of h which ensures that the series solution is convergent. Thus, the auxiliary parameter h plays an important role within the frame of the homotopy analysis method. Unlike all previous analytic techniques, such as HPM, we can adjust and control the convergence region of the solution series by assigning h a proper value.

Fig. 3. Comparison of the homotopy analysis approximation of follower rise when h = −0.04 with the Numerical solution. Symbols: numerical solution; Solid line: 8th-order approximation

Velocity and acceleration diagram vs. angle of cam rotation indicated in Figures 4 and 5. As shown in these figures. the 8th-order approximation of (21) agrees well with the numerical results.

Appendix A M e = 0.1083 kg M t = 0.1568 kg M v = 0.2 kg

Fig. 4. Comparison the homotopy analysis approximation of follower velocity when h = −0.04 with the Numerical solution. Symbols: numerical solution; Solid line: 8th-order approximation

M a = 0.15 kg I = 0.001 kg m2 r = 0.0156 m ω = 10 rad/s µ = 0.2 l1 = 0.12, l2 = 0.0183 m K s = 25000 N/m

(A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.10)

Appendix B b1 = 3B1 B42 B5 µ

(B.1)

b2 = 4 B3 B4 B5 µ

(B.2)

2 B1 B52 µ + B3 B52 4 B3 B52 µ 2 B3 B42 µ

(B.3)

b3 = b4 =

Fig. 5. Comparison the homotopy analysis approximation of follower acceleration when h = −0.04 with the Numerical solution. Symbols: numerical solution; Solid line: 8th-order approximation

b5 = b6 =

B3 B4 B5

(B.4) (B.5) (B.6)

The most difficult problem in shape optimization is to Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


945


b7 = B1 B42 µ

b8 =

2 B1 B4 B5 + 2 B2 B42 µ

b9 = 3B2 B4 B5 µ b10 = 2 B1 B4 B5 − 2 B1 B52 µ b11 = −2 B3 B52 µ b12 = B2 B52

Authors’ information

(B.7)

Department of Mechanical Engineering, Shahid Chamran University, Ahvaz, Iran.

(B.8) (B.9)

Hamid M. Sedighi was born in 1983 in Iran. He is currently a fourth year Ph.D student working under the supervision of Dr. Kourosh H. Shirazi in mechanical engineering at the Shahid Chamran University of Ahvaz in Iran. He obtained his M.S. degree (2007) from the Shahid Chamran University of Ahvaz and his undergraduate B.S. degree (2005) from the Shiraz University. His general academic areas of interest include the Mathematics, Nonlinear Dynamical Systems, Elasticity and Machine Design. As a teacher and a teaching assistant, he has held various positions at SCU and IAU Ahvaz. The topic of PhD research is “Analysis of nonlinear dynamical behavior of multilayer beams with interlayer slip”.

(B.10) (B.11) (B.12)

References [1]

F. Y. Chen, Mechanics and Design of Cam Mechanisms (New York: Pergamon Press, 1982). [2] J. Angeles, C. S. Lopez-Cajun, Optimization of Cam Mechanisms (Dordrecht: Kluwer Academic Publishers, 1991). [3] R. Norton, Modern Kinematics: Developments in the Last Forty Years (New York: Wiley, 1993). [4] D. E. Goldberg, Genetic algorithms in search, optimization and machine learning Reading (MA: Addison-Wesley, 1989). [5] J. T. Alander, J. Lampinen, A distributed implementation of genetic algorithm for cam shape optimization (Civil-Comp Press: Edinburgh, Scotland, 1997). [6] G. M. Robinson, Genetic algorithm optimization of load cell geometry by finite element analysis, Ph.D. Thesis, Department of Electrical, Electronic and Information Engineering, City University, Measurement and Instrumentation Centre, , School of Engineering, London (UK), 1995. [7] R. A. Richards, Zeroth-order shape optimization utilizing a learning classifier system, PhD Thesis, Mechanical Engineering Department, Stanford University, 1995. [8] G. Vancsay, T. Bercsey, P. Horak, Shape optimization based on genetic algorithms, Proceedings of International Conference on Engineering Design ICED 97, Finland, Tampere, 1997, pp. 335 – 338. [9] S. Hasanifard, optimum design of disk cam profile, MS thesis, Department of mechanical engineering, Tabriz University, Iran, 2002. [10] L. Elsgolts, Differential equation and the calculus of variation (Mir publishers: Moscow, 1973). [11] S. J. Liao, Beyond perturbation: introduction to the homotopy analysis method (Boca Raton: Chapman & Hall/CRC Press, 2003).

Kourosh H. Shirazi was born in 1969 in Iran. After studying in University of Science and Technology he received a B.S. degree in mechanical engineering - solid mechanics in 1992. He pursued his study in mechanical engineering in Amirkabir University (Tehran Polytechnic) and received a M.S. degree in 1993 and PH.D. degree in 2002. He started his work as a full time faculty member in Shahid Chamran University since 2002 till present. During his work he offered 7 courses such as Linear Control Theory, Design of Chassis and Body of Vehicle and Mechanisms Design in undergraduate program and 6 courses such as Advanced Dynamics, Advanced Mathematics and Nonlinear Dynamics in graduate program. His research interest is Kinematic of Mechanisms, Vehicle Dynamics and Chaotic Dynamics. He is a member of ASME and SAE.



946


Path-Whispering in a Virtual Environment Fawaz Y. Annaz

Abstract – A mobile robot physically navigates its way to a goal in a virtual maze that was created on a computer. The maze is made up of traversable and un-traversable grids, a starting point and a goal. Based on the immediate (left, right and ahead) surrounding information revealed to it, the robot makes a decision to navigate its way to the next location, which is reported to and traced on the virtual maze. Data exchange is achieved through discrete communications between the robot and the computer via Bluetooth. Real time merging of a virtual maze to a mobile robot allows for rapid navigation algorithm development for any foreseeable/target application with minimal cost and time. To cater for realistic rescue missions, two searching algorithms were implemented; the modified wall-touching is implemented in regions outside a proximity range, where victim’s calls cannot be heard; and a modified pledge algorithm, which is activated once the robot crosses into the proximity range region. Since this paper aims for revealing the shortest rescue path to other robots or humans, the robot is capable of eliminating double treaded paths, hence increasing the efficiency of any further support missions. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Mobile Robot, Maze Solving, Virtual Maze

I.

unnecessary space, when not in use. Additionally there exists the possibility of the prototype algorithm making an erroneous judgment and causing the robot to collide with an obstacle resulting in physical damage to the maze as well as the robot. As for maze solving, algorithms have been developed for robots with varied success and efficiency levels, depending on the maze architecture and the way information is revealed about the maze and the goal location. Some of the online approaches in autonomous robots, includes: wall following method [3]; potential field methods [4], [5]; virtual target approach [6], [7]; landmark learning [8]; edge detection and graph-based methods [9]; vector field histogram methods, dynamic window approaches, neural network based methods, and fuzzy logic methods [10][15] and many others. Clearly, adopting the new concept of to navigate within a virtual maze becomes more apparent for the following reasons: • It is Cost effective. • It has a high degree of portability, and can be implemented in any open space, or the robot can be suspended from physical movement until the navigation algorithms are developed to an extent that is deemed ready for physical trials. • It permits unlimited in-depth algorithms performance assessment. • It can be effectively and efficiently modified to architecture with various degree of complexity, including those mimicking real rescue missions (i.e. beyond traditional competition architectures).

Introduction

Realistic rescue missions usually start with a search in an unknown environment, before reaching regions where the presence of a victim is detected. It is also usual for rescuers to report their present locations (to rescue mission headquarters) as they progress along a certain path and the final victim location. It is also usual to identify and report dead-end paths along the initial rescue course, to avoid delays in further support by other fellow rescuers. It is the aim of this paper to mimic such real life rescue missions in an efficient way for different maze architectures. Traditionally, researchers have developed various algorithms that were tested to different levels of success and efficiency, depending on the maze architecture, goal position, and information revealed about the maze (i.e. goal location). To implement these algorithms using mobile robots, the common factor to any of these approaches was to initially physically develop and build a maze to certain specifications. Robots are then used to test various algorithms to navigate is way to a goal. One such environment is the problem that we are trying to solve, that is rescue mission scenario, where in reality path planning is most challenging as the unknown environment can be dynamically changing [1], [2]. The problem with this approach is that a maze has to be constructed in accordance to some specific architecture, which could be costly and time consuming. Furthermore, it is also natural that such mazes has to be substantially rugged to survive the tests, which intern makes them heavy and difficult to move around, hence occupying Manuscript received and revised June 2011, accepted July 2011


947

Fawaz Y. Annaz

II.

facing; and the Robot is never given position coordinates since it is intended for the Robot to construct and memorize is own coordinate system with respect to its initial deployment position.

Path-Whispering Mobile Robot

In this paper a leading robot makes decisions (that are based on local information revealed to it) and navigates its way through the maze. The robot maps its movement to onboard memory continues to navigate and discover its goal using onboard audio, visual or other sensory systems and return to its start position without having to depend on commands from a main command centre. The navigation algorithm developed for path mapping is also capable of calculating the minimal path but omitting dead end paths, redundant loops and bridging open neighbouring gaps. This computed minimal path can be transmitted as a single burst of path data to other mobile robots once the initial robot has discovered the goal. Therefore, although the leading robot is free to navigate the maze, it whispers back a limited constrained to the other robots. To maintain this two-way-update links with the control station (PC in this case) and a final path whisper with other posse robots, Bluetooth communication is used. The virtual maze is revealed to the mobile robot on a grid by grid basis as the robot moves to a new grid. Therefore, the PC that runs the virtual maze program needs to inform the mobile robot of its surroundings.

The Virtual Maze Program will keep track of the Robot based on directional information transmitted back (by the robot) and by knowing the heading direction from previous cells.

Fig. 1. Maze with shaded Proximity region

III. The Virtual Maze Environment Figure 1 shows a maze that was developed using a Visual Basic programme and when executed it allows the user to generate a maze with an initial robot position and final goal locations. The maze can be manually or randomly generated DFS, Prims or the modified Kruskal's Algorithm. The interface also allows users to save, load, and edit mazes. The interface is also capable of setting up communications to one or more robots to discretely • unveil to the robot the maze, and • trace robot movements current position (on the maze), as the robot feeds back its location following a decision it made.

IV.

Coordinate Building

Building a coordinate system within the mobile robot is crucial in calculating minimal paths and in eliminating redundant path and loops. The robot considers its deployment point as its origin. The origin is numbered 2020, which is an (x,y) coordinates. Since the maze is 20x20 the origin is considered 2020 instead of the conventional 00. This was done for the explicit purpose of maintaining the grid numbering positive, hence, allowing for a larger addressing range. The (x,y) coordinates were built into composite numbers as shown in Figure 2(a) with respect to the Starting Position (2020). These numbers are stored as step information, which were later used in computing the minimal path and in navigating a other posse robots. Figure 2(b) depicts the movement of the robot starting at the unknown grid that the robot designates as 2020 (Start Point for the robot), before moving North, which results in the robot storing its next positions as grid 2019 and 2018. At 2018 the robot travels East and makes a Left turn till it reaches cell 2218, hence it designates and stores this cell before it heads North again. Here, it should be noted that the grid numbering here is in not related to the numbering used in the virtual maze programme; here, the robot determines its new position considering its current facing direction, as well as, its last movement. The robot has four primary step that dictates its movement; these are Move Forward, turn Left, turn Right and make a 1800 turn Back and step forward (which is essentially a move to go back).

The maze in Figure 1 shows the robots Start location marked with 'S' and its path is indicated by a solid line and the goal is marked with 'G' within a shaded 'Proximity Region'. The robot current position is marked with a circle with an arrow that depicts the robot current facing direction. Adjacent grids to the robot are marked as F, B, L, and R, indicating the Forwards, Backward, Left and Right locations to the robot current position. To navigate the maze, two main processes take place, the first is computer based, and the other is mobile robot based. The computer discretely reveals the F, L, and R locations to the robot, the robot decides on the next move (using its own algorithm) and reports its new location to the computer, and the cycle repeats, until the goal is reached. It should be noted that the adjacent grids to the robot (F,L,R,B) are based on the direction the Robot is



948

Fawaz Y. Annaz

The robot records its last movement in a buffer for later references involving step storing.

Navigation (MPN) algorithm once the Goal is within proximity. A B C D E F G H I

J K L M N O P Q R S T

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

(a)

Fig. 3. Path Simplification

Figure 3 shows a solved maze with the lead robot using a Left-MWFN algorithm for primary navigation, which starts the robots journey at grid B2 and ends at grid C16. The robot enters dead end ally at grid C4 and has to turn back at grid E2. This redundant path is removed by the redundant path checking algorithm, which stores the path. Similarly the redundant loop that is apparent by the bottle neck at grid O8 and P8 is removed by the adjacent step checking procedure of the algorithm. Once the robot enters the Proximity area (shaded around the goal) at C14, the robot switches its Left MWFN algorithm to a MPN algorithm, and the robot no longer has to follow walls. This is apparent from the robots decision to step on to grid C15 (from grid C14) instead of turning left to follow the wall. The minimal computed path (shown with dotted lines) is then transmitted to the support robot via the Bluetooth. The support robot travels this path without the need to make any movement decisions.

(b) Figs. 2. Path Coordinates construction

V.

Navigating the Maze with Hybrid Wall-Following and Pledge Algorithms

To mimic a real-life rescue operation, it was assumed that, as the robot enters a region close to the goal vicinity (shaded region in Figure 3), the goal direction is revealed to the robot. Therefore, as the robot enters this region, (Proximity Range) the goal (marked by the Chequered box), the goal directional is revealed to the Robot. Hence, the robots native search pattern will be switched to a directional based decision making algorithm. This section explains the two algorithms the lead robot uses to solve the maze, that is: • The Modified Wall-touching algorithm, which is a Primary movement algorithm that is used outside the Proximity range. • The Pledge Method, which is used once the robot enters or enters-and-leaves the Proximity range. The lead robot always stores its movement information to generate a path coordinate map to calculate the best possible minimal path. The information is then relayed to a second support robot. In this research, two 'E-Puck' robots were used utilising the built in Bluetooth for communications. When the Goal is not within the detection range, the Robot resorts to a Modified Wall Following Navigation (MWFN) and then switches over to a Modified Pledge

VI.

Proximity Region and Alert

The introduction of a Proximity region is to mimic the real life situation of a rescue where the position of the victim is initially unknown and the search progresses in a methodical manner covering grid by grid. Once the search party nears the victim within a certain range the party become aware of victims relative location by either audio or visual cue. The actual proximity awareness of a search robot will depend on its audio and video sensors as well as its ability to differentiate between cues relevant to the victim from ambient noise. The virtual maze program creates a proximity region of a predefined size. Considering the size of the maze the



949

Fawaz Y. Annaz

proximity region was set to a grid size of 5x5 which is centred about the goal as indicated in Figure 4(a). The virtual maze program monitors the robots current position. If the robot has entered the proximity region then the program sends out a corresponding number indicating the relative direction of the goal as depicted in Figure 4(b).

(a)

(a)

(b) Figs. 5. Path travelled with and without Proximity Alert (b) Figs. 4. Proximity Calculation

For example, if the robot is not within the proximity of the goal, the proximity indicator routine (in the virtual maze program) assigns the value 0 to the proximity direction. If the robot has entered the lower left corner of the proximity region (where Absolute Grid Index = index_end + 38) then the proximity indicator routine in the virtual maze program assigns the value 8 to the proximity direction. Example of paths travelled with and without proximity alerts (shown in Figure 5(a) and 5(b), respectively) prove that the Proximity Navigation Algorithm allows the robot to find a goal in less time and travel distance. The Proximity Navigation algorithm was optimised using independent prioritized set of rules, so that the robot reached the goal as soon as possible. One Primary rule dictates that the robot keeps on travelling forward within the proximity region if a possibility exists, given the exception that an immediate turn does not result in solving the maze. The advantage of this rule is depicted in Figure 6, where the path taken by the robot results in a single turn while an alternative solution contains two turns. Note that, although the travelled distance is the same, there will be time loss due to turning.

Single Turn Path

Double Turn Path

Fig. 6. Optimization of Proximity Navigation

The mentioned advantages of the Proximity Navigation algorithm do have a single downfall. That is, the algorithms decision making itself creates a Proximity Trap for the robot in a particular situation as depicted in Figures 7. Figure 7(a) shows the robot progressing in to the proximity region. It is apparent that the robot cannot reach the goal from its first proximity encounter, however, the rules of the proximity navigation algorithm do not allow the robot to leave the proximity region. Therefore, the robot dwells (within the region) unable to exit it. The solution for this situation is not to modify the proximity navigation algorithm since that would remove the advantages introduced by the algorithm. Instead, a proximity dwelling threshold should be set, so that when



950

Fawaz Y. Annaz

it is exceeded, the robot ignores the proximity alert altogether and uses the primary navigation algorithm to exit the proximity region and reset its active response status to the proximity alert once it is clear from its initial proximity encounter. The proximity threshold is set to twice the proximity region which is 25x2=50 for the maze in Figures 7. The robot exiting the first proximity encounter after a exceeding the dwelling threshold is shown in Figure 7(c). The robots optimized return path from goal to start point is shown in Figure 7(d) with a dotted line.

the robot and the computer via a Bluetooth Module. This paper successfully mimicked a robot discovering a victim using hybrid navigation modes and calculated a minimal path that is to be conveyed to support robots or personnel wirelessly or to use the minimal path for the robot itself to return back to its starting point without meaninglessly backtracking on its own redundant initial discovery path in order to provide rapid and efficient search and rescue operations.

References [1]

[2] [3] [4]

[5] (a)

(b)

[6]

[7]

[8]

[9] (c)

(d)

[10]

Figs. 7. Proximity Trap and its escape [11]

VII.

Conclusion [12]

To allow for rapid navigation algorithms development with minimal cost and time, this paper presented a real time merging of a virtual maze to a mobile robot. In the presented work, two maze solving algorithms were used to solve mazes generated by the user; a primary navigation algorithm for systematic discovery and a proximity algorithm for optimised rescue. A solution was given to preserve the advantages of proximity navigation whilst avoiding being in an unsolvable loop that was introduced as a proximity trap. Path simplification was also presented with the aim that it will be transmitted to guide a further support robot to the goal. A realistic rescue missions was taken into account, and the two presented searching MWFN and MPN algorithms were used in vicinities far away and close to goal, respectively. In both algorithms, immediate surroundings were revealed to the lead robot to enable it to decide on navigating its way to the next location. Data exchange was achieved through discrete communications between

[13] [14]

[15]

O. Motlagh, T. Hong, N. Ismail, Development of a new minimum avoidance system for a behavior-based mobile robot, Fuzzy Sets and Systems 160 (2009) 1929–1946. K. Goris, Autonomous mobile robot mechanical design, Verije University, Brussels, Retrieved January 3, 2006. Y. Owen, G. Andrew, Dynamical wall following for a wheeled robot, University Baltimore. Retrieved March 5, 2006. Y. Koren, J. Borenstein, Potential field methods and their inherent limitations for mobile robot navigation, in: Proc. IEEE Conf. on Robotics and Automation, 1991, pp. 1398–1404. M. Massari, G. Giardini, F. Bernelli-Zazzera, Autonomous navigation system for planetary exploration rover based on artificial potential fields, Retrieved January 3, 2006, from (http://naca.central.cranfield.ac.uk/dcsss/2004/B24_navigationb.p df), 2001. W. L. Xu, S. K. Tso, Sensor-based fuzzy reactive navigation for a mobile robot through local target switching, IEEE Trans. Systems, Man, Cybernet., Part C: Appl. Rev. 29 (3) (1999) 451–459. X. Yang, M. Moallem, R.V. Patel, A layered goal-oriented fuzzy motion planning strategy for mobile robot navigation, IEEE Trans. Systems, Man, Cybernet., Part B: Cybernet. 35 (6) (2005) 1214–1224. K. M. Krishna, P. K. Kalra, Perception and remembrance of the environment during real-time navigation of a mobile robot, Robot. Auton. Syst. 37 (2001) 25–51. A. V. Kelarev, Graph Algebras and Automata, Marcel Dekker, New York, 2003. M. Wang, J. N. K. Liu, Fuzzy logic based robot path planning in unknown environments, in: Proc. 2005 Internat. Conf. on Mach. Learn. and Cybernet., Vol. 2, 05, pp. 813–818. A. Zhu, S. X. Yang, A fuzzy logic approach to reactive navigation of behavior-based mobile robots, in: Proc. 2004 IEEE Internat. Conf. on Robotics and Automat., Vol. 5, 2004, pp. 5045–5050. Saylor J. Neural networks for decision tree searches. In: Proceedings of IEEE/EMBS-9 conference, vol. 2, 1987. p.373–4. Simon XY, Meng M. An efficient neural network approach to dynamic robot motion planning. Neural Networks 2000;13:143–8. Glasius R, Komoda S, Gielen SCAM. Neural network dynamics for path planning and obstacle avoidance. Neural Networks 1995;8:125–33. Nayfeh AB. Cellular automata for solving mazes. Dr Dobbs J 1993;197(February).

Authors’ information Fawaz Y. Annaz was born in Iraq in 1962. F. Y. Annaz received B.Sc. in Electrical and Electronic Engineering and M.Sc. in Engineering of Dynamic Systems from the University of East London in 1987 and 1989, respectively. He received a Ph.D. in Avionics from Queen Mary University of London in 1996. His main research interests are in electromechanical actuation design, artificial heart development, robotics and 3D Vision.



951


Modeling, Control and Analysis of a Serial And Parallelogram Lower Member Mechanism Alvaro Uribe, João Rosário, Luciano Frezzatto

Abstract – This work presents the modeling, control and analysis of a serial and parallelogram mechanisms for designing a lower member assisting motion device. Current most known developments do not cover a wide range of population due to costs, requirements and availability of these devices. The robotics impact on improving health care through rehabilitation or motion assisting devices has been increasing as the technological trends have advanced in recent years. The methodology taken for developing this work starts with the mechanism design, so mechanical properties such as masses and inertias are known. This information allows performing the kinematics and dynamics analyses for choosing the actuator and control system while improving the mechanism design for executing gait and exercise motions. The validation of both mechanisms dynamics is accomplished through the analysis of the experimental torques obtained after simulation and the linearization of the model for implementing and tuning the control system. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Assisting Motion, Lower Member, Parallelogram, Serial

The most common causes of impairment and disability include chronic diseases such as diabetes, cardiovascular disease and cancer; injuries such as those due to road traffic crashes, conflicts, falls, landmines, mental impairments, birth defects, malnutrition, HIV/AIDS and other communicable diseases. These trends are creating overwhelming demands for health and rehabilitation services, (www.who.int/disabilities/ publications/). Lower member studies have resulted in various research related to motion capture, along with mechanics, electronics and control techniques for developing humanoid robotics and orthotic devices for assisting motion and aiding rehabilitation therapies by increasing some human capabilities [1]-[4]. With the continuous advancements in mechanisms and actuators, early developments in the area of human robotics started to take place. It was in 1969 in the Mihailo Pupin Institute (Serbia) that the first exoskeletons were developed, these were based initially on pneumatic actuators, which were later replaced with electrical and then electromechanical ones. The exoskeletons also served as basis for the research in the area done by Vukobratovic [5]. Since, various mechanisms and actuation solutions have been used throughout research for reproducing human lower member motion as similar as possible. In [6], a one degree of freedom (DOF) rotational mechanism allowed the generation of suitable motion without increasing the user’s weight, while allowing the device to generate a torque similar to the human one. Using four DOFs, Kasaoka and Sankai presented in [7], a device based on a

Nomenclature L K V τ θ ζ Xref X i −1 Tri qi ci C S Mi Li Lci I g k d

Lagrangean Kinetic energy Potential energy Torque Thigh flexion/extension angle Shank flexion/extension angle Reference position vector Position vector Transformation matrix Generalized rotation coordinates Constants Cosine Sine Mass Length Center of mass length Inertia Gravity Spring elasticity constant Spring compression

I.

Introduction

An estimated 10% of the world’s population experience some form of disability or impairment. The number of people with disabilities is increasing due to population growth, ageing, emergence of chronic diseases and medical advances that preserve and prolong life.



952

A. Uribe, J. Rosário, L. Frezzatto

ballscrew and a direct current motor, where myoelectric inputs allow controlling and predicting the motion in order to better assist the user. By considering the effects of compensatory mechanisms as the main reason of exoskeletal robotic device rejection due to the high energy required for motion, Greene and Granat presented in [8] a device based on the knee’s extension and flexion without compensatory elements which allowed motion when no motor functions were present. Agrawal and Agrawal presented in [9], an alternative passive solution using a hybrid parallelogram is presented in order to balance the mechanism’s gravity and minimize torques during the motion. Whilst the number of assisting devices commercially available is growing, their production, renting and selling costs excludes a wide range of population that could benefit during their rehabilitation therapies thus, failing the objective of reaching those who need it the most. One trend shared by these solutions is to offer less invasive and portable devices. Through this work, the modeling and control of a lower member motion assisting device is presented. The proposed device goal is to be as less invasive as possible with a mechanical configuration that allows feasible development costs to widen the target population. The methodology allowed studying and analyzing the mechanism’s kinematics and dynamics model, so the control system can be proposed for assisting gait and therapy motion when applied known torques and lower member rotations. This work is organized as follows. In Section II, the system architecture to be studied is presented. In Section III, the mechanism characteristics are introduced for solving the dynamics model. In Section IV, the mathematical modeling of the three mechanisms is described. In Section V, the control system is implemented and analyzed. In Section VI, the results are presented. Finally, in Section VII the conclusions and final works are presented.

II.

kinematics thus, allowing a cooperative and collaborative flow of information for motion performing. The overall architecture is presented in Fig. 1, where the previous mentioned elements can be seen. In this diagram, X represents the positions vector for each joint.

Fig. 1. Control System architecture

The study presented in this work is centered on the dynamics of the thigh and shank highlighted with light grey on Fig. 1. The system architecture proposed considers two inputs based on stored motion for reproducing on the mechanism or user generated patterns. These sequences are sent to a set of actuators for moving the thigh and shank mechanism members, whose motion is controlled and sensed for accurate motion execution.

III. Mechanism Characteristics As the mechanism characteristics are based on the human lower member anthropometry, the biomechanics of the lower member is studied for modeling a suitable kinematics representation for analysis. This representation is composed of links resembling the thigh, the shank and the ankle. Each of the lower member articulations have 3DOF, but since the goal of the project is focused on assisting gait motion, only flexion and extension rotation are considered on the mechanism, as these are the most important movements during gait and muscle strengthen for gait, as presented in Fig. 2,[10] [11].

System Architecture

The system architecture of the mechanism assisting device is composed of an actuation and control system, the kinematics and dynamic model of the plant, and feedback blocks using sensors for guaranteeing an acceptable performance of the mechanism due to the chosen inputs. The kinematics modules allow knowing the positions and orientations during the sequences of motion, while the dynamics model allow designing and choosing the control and actuation systems. To solve the kinematics and the dynamic models, the mechanism properties and motion sequences are required. The inputs of the system can be obtained from motion data or a trajectory generator providing the positions for the mechanism to follow, along with each joint control, the system is tuned up using predefined motion data or information acquired through sensors with the forward

Fig. 2. Lower member DOF performing walking and exercises

As mass and inertia properties of each mechanism element changes the amount of required torque for



953


executing any flexion and extension rotation, two CAD models are designed to study these effects during gait and exercise motion. These mechanisms are composed of solid and hollow beams made of steel 1020, both configurations are presented in Fig. 3.

In order to find and solve the kinematics and dynamics model, an algorithm was implemented using the mechanism parameters along with the motion data. This approach allows obtaining the dynamic model for testing it with motion data from the user motion and being visually monitored through the forward kinematics model. The blocks diagram of the implemented algorithm is presented in Fig. 5.

Fig. 3. Solid and hollow mechanisms

The properties obtained from both configurations are presented in Table I.

Segment Thigh Mec1 Thigh Mec2 Shank Mec1 Shank Mec2

IV.

TABLE I MECHANISM PROPERTIES Mass Inertia 1.78 Kg 0.02 Kgm2 0.8 Kg 0.031 Kgm2 1.97 Kg 0.024 Kgm2 0.72 Kg 0.042 Kgm2

Fig. 5. Dynamic analysis diagram

Length 0.42 m 0.42 m 0.46 m 0.46 m

IV.1. Kinematics The kinematics analysis allows calculating positions given known rotations for each member using the forward kinematics. This model is calculated to serve as suitable preprogrammed input for the dynamic model as an alternative to motion generated by the user directly on the mechanism. The forward kinematics model allows calculating each joint’s position from known rotations. It is determined using the anthropomorphic measurements and the rotations of each member following the Denavit-Hartenberg (DH) method [12]. The rotation information of each mechanism segment is calculated using the transformation matrix obtained from the DH method and presented in Eq. (1):

Mathematical Modeling

Considering the anthropometric characteristics of the human lower member, a serial kinematic chain, a parallelogram and a hybrid parallelogram mechanism are used for modeling and designing the device (Fig. 4). The serial mechanism is considered because of its simplicity which can lead to reduce material and machining costs, as complex mechanism may result in expensive structures. The two parallelogram mechanism configurations are considered because of the torque and dynamics reduction offered when using parallel bars to the main links of the kinematic chain. The study in [9], presented the advantages of such configurations along with the gravitational balance of the mechanism when used in passive mode with springs.

⎡Cθi ⎢ Sθ i −1 Tri = ⎢ i ⎢ 0 ⎢ ⎣ 0

− Sα i Cθi Cα i Cθi

Sα i Sθi − Sα i Cθi

Cα i

Cα i

0

0

ai Cθi ⎤ ai Sθi ⎥⎥ di ⎥ ⎥ 1 ⎦

(1)

where, j is the number of the segment, θ its flexion or extension rotation and aj its length. It has to be noted that the mechanism entire forward solution is calculated by multiplying all the transformation matrices thus, allowing to know the position of last joint in relation to the first one.

Fig. 4. Proposed mechanisms



954


and the previous terms expresses: I for inertias, M for masses, l for lengths and g for gravity, where the indexes represent the link of the mechanism being, index 1 the thigh, 2 the shank and c information related to the center of mass With the equation Eqs. (3) and (4), the rotations can be calculated when known torques for the thigh and the shank are used as inputs.

IV.2. Dynamics The Lagrange-Euler method is used for performing the dynamic analysis; this allows studying the mechanism from an energetic point of view considering the kinetic (K) and potential (V) energies as shown in Eq. (2). With the Lagrangian expression calculated, the torque τ can be found for the thigh and the shank using the partial derivatives presented in Eq. (2). Each τ expression defines the dynamic model expressed in terms of its effective and coupling inertia, centripetal, coriolis and gravitational forces:

τ=

d ∂L ∂L − dt ∂qi ∂qi

IV.4. Parallelogram Mechanism Analysis The parallelogram is a four bar mechanism forming a closed kinematics chain as presented in Fig. 4. This configuration is often used to reproduce motion sequences by taking advantage of the main link’s rotations to induce movement through the parallel joints. The geometric relations of the parallelogram are based on the study presented by Agrawal [9] for the implementation of a passive hybrid parallelogram exoskeleton with gravity balancing. Similarly to the analysis of the serial mechanism, each link of the parallelogram is studied by calculating its COM vertical and horizontal positions. From these known positions it is possible to find the velocity expression thus, allowing the calculus of the potential and kinetics energy equation. In this case, the total Lagrangian LT is obtained by considering all kinetic and potential energies from each mechanism member, as shown in Eq. (5):

(2)

where, qi are the generalized rotation coordinates τ torque, K kinetic energy and V potential energy. IV.3. Serial Mechanism Analysis The serial configuration seeks to resemblance the lower human member, composed of two corresponding links matching the thigh and the shank rotations as presented in Fig. 4. From the kinematics analysis, the positions of the joints and the center of mass can be calculated thus, obtaining the expressions need for Lagrange and torque evaluation. Finally, with the Lagrange expression, τ1 can be calculated for the thigh and τ2 for the shank, obtaining the equations presented in Eqs. (3) and (4):

LT = K m1 − Vm1 + K m 2 − Vm 2 + + K ma1 − Vma1 + K ma 2 − Vma 2

with LT calculated and using Eq. (2), the torque expressions are defined accordingly to Eq. (5).

τ1 = θ( c1 + c2 C (ς c ) ) − ςc ( c2 + c2 C (ς c ) ) + c S (ς ) + ς 2 c S (ς ) − c S (θ − ς ) + −2θς c 2 c c 2 c c 4

(3)

IV.5. Hybrid Parallelogram

−c5 S (θ )

τ 2 = θ( c3 + c2C (ς c ) ) − ςc c3 + −θ 22 S (ς c ) + c4 S (θ − ς c )

(5)

This configuration has as main feature the gravity balance generated by the springs attached to the mechanical structure, hence the hybrid designation. The forces of the springs allow balancing the mechanism in various positions during motion. For considering the spring effects over the dynamic analysis, the potential energetic equation (Eq. (6)) is derived in relation to the corresponding rotation angle for each torque. Then, these expressions are added to the parallelogram dynamic equations LT:

(4)

where: c1 = M1 Lc12 + I l1 + M p 2 Lc12 + I l 2 + M 2 Lc 2 2 + M 2 L12 c2 = M 2 L1 Lc 2

kspring = 1 / 2kd

c3 = I l 2 + M 2 Lc 2 2

(6)

where, k is the spring constant and d is the compressed or extended distance in terms of the θ and ζc rotations.

c4 = gM 2 Lc 2

V.

c5 = gM 2 L1 + gM1 Lc1 + gM p1 L1

Control System and Analysis

The control system can be analyzed from two points of view, one considering the actuator and other the



955


mechanism for guaranteeing a suitable controller for executing lower member motions. The Proportional-Integrative-Derivative (PID) controller was selected due to the characteristics offered by its proportional, integrative and derivative gains. The uses of these gains allow configuring the controller response to reduce the state error and the overshoot by compensating the differences between real values and those estimated by the model. V.1.

have to be linearized in order to apply the controller. The Taylor series is used for linearizing the model, this method assumes a lineal approximation by supposing that the sum of all the nonlinear terms are small enough at the operation point thus, being excluded. The operation point is set to match the equilibrium point of the mechanism, which in all the cases is in its standing position without rotations in the thigh and in the shank. To linearize the dynamic equations, the Taylor series was implemented within the algorithm presented in Fig. 5. From the algorithm solution, the nonlinear terms are found and presented in Eq. (7):

Actuator Control

The dynamic analysis allowed choosing a suitable DC motor offering the needed torques for moving the mechanism. Considering the characteristics of the lower member motion during rehabilitation exercise or strengthen leg muscles, the controller design criterion is defined. As these motions are executed slowly and repeatedly, the settling time is set to 0.4 s, the overshoot to less than 5% and no steady error. It has to be noted that these parameters may be readjusted from user to user depending on how fast he can move the lower member. Then, the PID test is performed using the DC motor with a load equivalent to the member that is moving and a step input of π /2 that represents the position where the largest torque response is required. Fig. 6 presents the different position responses obtained from various PID configurations for analyzing each of the gains incidences on the motor. From this figure can be seen how the PD configuration comply with the chosen design parameters.

f1 (ς c ) = C (ς c )

(

)

S (ς ) f 2 ς c ,θ ,ςc = θς c c f3 (ς c ,ςc ) = ςc 2 S (ς c ) f 4 (θ ,ς c ) = S (θ − ς c )

(7)

f5 (θ ) = S (θ ) f 6 (θ ,ς c ) = θ 2 S (ς c )

After the linearization process is completed, the obtained model equations are simplified and the number of dynamic components reduced for both, the thigh and shank torques.

VI.

Results

The kinematics and dynamic models previously obtained were tested using the three different mechanism configurations along with gait and a rehabilitation exercise. The results are presented in the subsequent sections. VI.1. Forward Kinematics Analysis The implemented forward kinematics model was tested using the gait and exercise rotations of the thigh and shank during motion. Two outputs were obtained from the model, one quantitative allowing to know the position of each joint of the mechanism during motion, and other qualitative through a visual representation in a 3D virtual environment as presented in Fig. 7.

Fig. 6. PID response to step input with chosen actuator

V.2.

Mechanism Control

All of the calculated models have nonlinear dynamic components, these have to be processed accordingly to control strategy applied to the system. In this study, as a PID control is used, its gains are considered sufficient for satisfying the mechanism range of motions. As the PID is a linear control system, then the mechanism equations

Fig. 7. Forward kinematics positions of the three mechanisms



956


VI.2. Dynamic Torque Analysis The obtained dynamic models for the three analyzed mechanisms where tested using captured gait data and a therapeutic exercise focused on the recovery of walking capabilities. The first experiment used the mechanical configuration with the solid elements Fig. 3 and the lower member rotations obtained from motion captured during a normal gait cycle. These flexion and extension rotations are presented in Fig. 8.

Fig. 10. Calculated torques during exercise

Using normal gait data, the torques were calculated and are presented in Fig. 11, where can be seen a noticeable reduction when compared to Fig. 9.

Fig. 8. Thigh and shank rotations Fig. 11. Calculated torques for the reduced mechanism with normal gait

Using the dynamic equations with the rotational data, the torques for this mechanism on its three analyzed configurations was calculated. The obtained torques are presented in Fig. 9, from the data can be seen how for this range of motion the hybrid parallelogram requires less torque than the other two options.

With the reduced mechanism and the therapy exercise motion data, torques were calculated to visualize how the dynamic model responded. The torques in this scenario are presented in Fig. 12, where the range differences of the obtained data can be seen by comparing with Fig. 10.

Fig. 9. Thigh torques with solid bars Fig. 12. Calculated torques for the reduced mechanism when executing exercise

Continuing with the dynamic study, a second experiment used rehabilitation motion exercise from therapy. In this case, the thigh and shank flex and extends in order to strengthen lower member muscles necessary for walking. The calculated torques are presented in Fig. 10. From the acquired data, can be seen how the best option is the serial mechanism as the parallelograms performed poorly, the hybrid one because of the forces of the strings and the other because of how the torques are distributed during motion. The previous obtained torques required for moving the mechanism presented a non-viable design and implementation because the actuators needed for supporting such torque would defeat any purpose of designing a light and portable mechanism. To validate the simplified design presented in Fig. 3, the dynamic model was calculated and solved using both gait data and an exercise motion sequence.

The obtained torques during gait and the exercise with the mechanisms using the mechanical configurations, presented a similar behaviors, while the magnitudes varied. When evaluating the system with gait data, the hybrid parallelogram torque in both thigh and shank offered the smallest values, and then followed by the serial and parallelogram configurations. This result was obtained due to effects of the springs during motion as each aid the mechanism returning to its equilibrium position. A different response was obtained with the exercise data. In this case, the serial mechanism presented the most suitable torques as the given range of motion creates higher torques for both, the parallelogram and hybrid parallelogram making them not suitable for



957


implementation. The hybrid parallelogram torques are considerably increased due to the spring effects, as for each different range of motion the mechanism configuration should need to be recalculated and various spring should be available for replacement, this requirement makes it a non-viable solution. In all the previous scenarios, the parallelogram configuration presented torque values higher enough that lead to disregarded it as a viable option. The effects of adding the parallel bars increased the inertia and the force distribution, resulting in higher torques needed from the actuators.

Fig. 14. Calculated rotations given known torques with and without PID

VI.3. Rotational Dynamics Analysis

VI.4. Controller Test

Another method for validating the dynamic model is by calculating the positions when known torques are used as input. For this analysis only the serial mechanism was considered as it is the configurations that best fits gait and therapy exercises with acceptable torque ranges. The PID controller is placed between the input and the system transfer function sensing the rotation calculated, as presented in Fig. 6.

For testing the control system, a 1:3 scale robot was used and the PID gains were configured accordingly to the mechanism properties. The robot was configured to perform flexion and extension rotations at its waist and knee joints controlled by DC motors as presented in Fig, 15.

Fig. 15. Scale robot performing cyclic motion

In this scenario, because of the small mass and inertia values, the controller was able to respond better, resulting in the cyclic motions presented in Fig. 16.

Fig. 13. Implemented PID controller

In this scenario, the PID model presented in Fig. 13 with the linearized thigh and shank models was used for calculating the thigh and shank flexion and extension rotations during cyclic and repetitive motion. A sine-like input generated from the DC motor was used for generating a 15 Nm torque for the thigh and 4 Nm for the shank. Two PID controllers were configured for testing quality and robustness. The PID1 response attained stability after the first cycle in order and had an overshoot of less than 18%. A second controller, the PID2, was configured so the stability was reached at half cycle, but the overshoot raised about 33%.These results are presented in Fig. 14 along with the calculated rotations without the PID thus, showing and highlighting the controller effects over the mechanism. The motion amplitude obtained with the chosen torques and PID1 was about 124º and -124º for the thigh, and about 64º and -64º for the shank. These results confirm that the chosen control covers a wide range of motion as the maximum position would be 90º for the thigh and the shank when flexing. The PID2 resulted in a wider range of motion due to the chosen PID gains.

Fig. 16. Calculated rotations given with adjusted PID using the scale robot

VI.5. Mechanical Design Analysis From the dynamic analysis it was possible to choose the actuators for satisfying and fulfilling the chosen motion sequences studied in this work. With the actuator information the mechanic design can be optimized for guaranteeing smaller torques by reducing mass properties and inertias of the elements. The mechanism used was presented in Fig. 3 and was validated using the Finite Element Method. For running the simulation, the mechanism was fixed at the waist thus, holding the actuator and lower member weight during motion. For validating the mechanism, the VonMises failure criterion was used in order to test if the structure could withstand the chosen motion [13].



958


The obtained results are presented in Fig. 17, from them, it can be seen that the maximum stress calculated value was of 1.03e8 Pa and of 5.16e-4 Pa for the strain; these values did not exceed the 2e11 Pa elastic modulus and 7.7e10 Pa guaranteeing that the structure supports the actuator and the lower member during motion.

the motion sequences to be tested. The FEM analysis with the1020 steel proved to withstand the actuators weight during the motion sequences. Future works will be centered on prototyping the mechanism for making any required mechanical and control system adjustments. The full architecture of the system will be analyzed including an inverse kinematics module.

Acknowledgements The authors would like acknowledge the support of FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo), under process 2009/05396-4, the Virtual Reality Center of the Universidad Militar Nueva Granada, the Instrumentation Laboratory for Biomechanics (LIB), and the Automation and Robotics Laboratory of the UNICAMP.

Fig. 17. Calculated VonMises stress, strain and deformation

VII.

References

Conclusion [1]

Through the dynamic analysis, the calculated torques for three mechanism configurations allowed selecting which is the most suitable as a lower member assisting device. When evaluating the mechanisms with gait data, the lowest torques were obtained using the hybrid parallelogram. Due to the small range of rotations during gait, the mechanism springs aid recovers its equilibrium position. The serial mechanism torque results were close to the hybrid configuration presenting variations smaller than 10 Nm. When evaluating the mechanisms with exercise data, the most suitable torques were calculated using the serial mechanism. The hybrid parallelogram proved not to be suitable due to reactions of the spring forces when wide motions take place. With these results, the selection process was reduced to the serial mechanism as better option and in order for improving it, hollow parts were considered for reducing the mass and inertias of the parts thus, reducing the needed torques for moving the mechanical structure. This analysis allowed to select and adequate actuator which weight and size makes it a viable solution. The PID controller proved to behave adequately as it can be configured to obtain a desired response from the system accordingly to the needs of motion. Although separately each actuator reaches the desired parameters, when integrated with the system models the PID offered a suitable solution for small torques and slow motion. This lead to conclude that there is room for further improvement for reaching better overshoot and stability responses. While the obtained steady error with PID1 persists for a cycle and a half due to slow execution of motion during rehabilitation it is not larger enough to affect motion performance. Finally, the dynamic analysis allowed choosing the actuation system, hence completing the mechanism design by considering the weights of the actuators and validating the mechanism accordingly to

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10] [11] [12] [13]

S. Kajita, H. Hirukawa, K. Harada, and K. Yokoi, Introduction a la commande des Robots Humanoides (Springer, 2009). A. Grabowski and H. Herr, Leg exoskeleton reduces the metabolic cost of human hopping, Journal of Applied Physiology, vol. 107, pp. 670–678, 2009. S. Banala and S. Agrawal, Gait rehabilitation with an active leg orthosis, Proceedings of IDETC/CIE 2005 ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2005. G. Sawicki, K. Gordon and D. Ferris, Powered lower limb orthoses: Applications in motor adaptation and rehabilitation, Proceedings of the 2005 IEEE 9th International Conference on Rehabilitation Robotics2005, pp. 206–211, 2005. M. Vukobratovic and B. Borovac, Note on the article ”zeromoment point - thirty five years of its life, I. J. Humanoid Robotics, vol. 2, no. 2, pp. 225–227, 2005. B. Ruthenberg, N. Wasylewski, and J. Beard, An experimental device for investigating the force and power requirements of a powered gait orthosis, Journal of Rehabilitation Research and Development, vol. 4, pp. 203–213, 1997. K. Kasaoka and Y. Sankai, Predictive control estimating operator’s intention for stepping-up motion by exoskeleton type power assist system hal, International Conference on Intelligent Robots and Systems, 2001. Proceedings. 2001 IEEE/RSJ, pp. 1578–1583, 2001. G. P. J. and G. M. H., A knee and ankle flexing hybrid orthosis for paraplegic ambulation, Medical engineering & physics, vol. 25, pp. 539–545, 2003. A. Agrawal and S. K. Agrawal, Design of gravity balancing leg orthosis using non-zero free length springs, Mechanism and machine theory, vol. 40, pp. 693–709, 2005. [V. Inman, H. Ralston, and F. Todd, Human Walking (Williams & Wilkins, 1981). J. Hamill and K. Knutzen, Bases Biomecânicas do Movimento Humano, (Manole, Ed. Manole, 2008). J. Denavit and R. Hartenberg, Kinematic Synthesis of Linkages, (Mc- Graw Hill, 1964). S. M. A. Kazami, Solid Mechanics, (Mc Graw Hill, 2004)

Authors’ information UNICAMP, Universidade Estadual de Campinas, DPM, Brazil.



959


Alvaro J. Uribe, has a Bachelor degree in Mechatronics Engineering from the Militar Nueva Granada University in Bogotá, Colombia in 2003, and a M.Sc. in Mechanical Engineering at the State University of Campinas (UNICAMP), Campinas – São Paulo, Brazil in 2008, currently is enrolled in Ph.D. studies.

Luciano A. Frezzatto Santos, has a Bachelor degree in Computer Engineering from University of Campinas (UNICAMP), Campinas – São Paulo, Brazil in 2009, currently is enrolled in M. Sc. Studies. He has work in the implementation of control techniques for CNC machine tools. His research interests are on control techniques and robotics.

He has work in the development of Virtual Reality tools for architecture, engineering and medical applications. His research interest is on applied Virtual Reality. João M. Rosário, was educated at Campinas State University, São Paulo, Brazil receiving the B Sc. degree in mechanical engineering in 1981 and the M.Sc. degree in systems and control in 1983 and Specialization Degree in Production and Automation Systems in 1986 at Nancy University, France. He was awarded the Ph.D. degree in 1990 by Ecole Centrale – Paris, France, for research into Automation and Robotics. He worked briefly as a control engineer and robotics in the Hispano Suiza, France and underwater robotics at GKSS, Germany. Actually, he’s invited professor at Automation and Control department, at SUPELEC, France. Currently he is an associated professor at Faculty of Mechanical Engineering at the University of Campinas, UNICAMP, responsible of the Automation and Robotics Laboratory, and coordinator of Robotics and Automation in the Brazilian Manufacturing Network. From 1998-2002 was the head of the graduate course of Automation and Control Engineering (Mechatronics) Department. Currently develops various industrial research projects at national and international level in different areas such as: Industrial Automation, Control Design of Mechatronics Systems and Biomechanics.



960


Parametric Study of Electro-Hydraulic Servo Valve Using a Piezo-Electric Actuator S. F. Rezeka, A. Khalil, A. Abdellatif Abstract – This paper deals with the mechatronics approach for the design of a piezoelectric actuator and its integration into a servo valve. A piezo-element is introduced as an actuator, instead of conventional electrically operated torque motor, to operate the spool valve. The non dimensional equations describing the system were derived. PD controller was also synthesized. Parametric simulation was conducted to study the effects of the system parameters on its performance. The simulation results are presented and discussed. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Electro, Hydraulics, Servo Valve, Piezo Actuator, PD Control, Parametric Study

x

Nomenclature A As C Cd C

m

D

F

z

Ktl kmp kp M m

eff

ms P Ps T U(t) Uz V1 ,V2 Vout vm .

w

m

Effective area of the piston [m2] Spool area [m2] Hydraulic cylinder damping coefficient [Ns/m] Coefficient of discharge [dimensionless] Piezo-element mechanical stiffness [N/m]

xp Z ∆ ρ

Displacement of the iron core w.r.t. the transformer Spool damping coefficient [N s/m] External load force [N]

Linear movement of piezo-element ending [m] Displacement of the piston [m] Valve opening [m] Effective bulk modulus of oil [N/m2] Difference in pressure in piston chambers [m2] Oil density [kg/m3]

I.

Introduction

In most electro hydraulic control loops the hydraulic servo valves are widely used. Their usage allows obtaining both good dynamic and static performances. Many works were reported in the literature in this area [1]-[6]. First development on servo valves was by J. C. Jones and Hons Monash [1]. More Hydraulic control attempts were made on the bases of modeling equations and discussed by H.Merritt [2]. Development of a MATLAB /Simulink hydraulic servo control design experiment was presented by Charles Birdsong [3]. While hydraulic amplification allows for low power electrical command signals, the response time of this amplification limits the bandwidth of the resultant actuation systems. To overcome the bandwidth limitation of hydraulic amplification, one possible solution is electrical amplification of the command signal. With electrical amplification of the command signal, an electronic amplifier and electromagnetic motor directly control the position of the spool [6]. However, the weak force and energy density of electromagnetic motors limit the force and bandwidth of these direct drive systems for a given device size. High energy density piezoelectric materials present a possible alternative to electromagnetic actuation to further improve the response time of the spool in direct drive systems [5]. Also the main advantage of these piezo elements actuators used in

Equivalent stiffness of the system [N/m] Piezo element damping coefficient [N s/m] Constant of proportionality [N/m3] Stiffness of hydraulic cylinder [N/m] Sensitivity of the transformer [dimensionless] Effective mass of piezo element [kg] Mass of the spool and the working fluid [kg] Oil pressure [N/m2] Supply pressure [N/m2] Time [s] Voltage on transducer plates [V] Voltage from controller to the piezo element [V] Volume of first and second cylinder chambers Output voltage of transducer [V] Linear velocity of piezo-element end [m/s] Velocity of the piston [m/s] Acceleration of the spool [m/s2] Area gradient of the spool [m] Displacement of the spool [m] Velocity of the spool [m/s]



961

S. F. Rezeka, A. Khalil, A. Abdellatif

is relatively big displacement (> 1 mm) and low supply voltage (< 50 V), but such transducer produces forces only till 0.5 N [8]. Piezo actuators are characterized by low power consumption, low manufacturing costs and good dynamic properties. The first introduction of future actuators and Microsystems, including the usage of piezo electric material as an actuator, was by Hiroyuki Fujita [4]. A single-stage servo-valve using direct piezoelectric actuator drive was described by Jason E. Lindler and Eric H. Anderson [5]. More investigations of electro hydraulic servo valve with piezo-element, mathematical modeling and simulation were conducted by Andrzej Milecki [6]. Further research and experimental characterization of a high dynamic servo valve integrated with piezoelectric actuator were done by Karunanidhi and M Singaperumal [7]. In this paper a nonlinear mathematical model of electro-hydraulic servo valve that implement piezo actuators, instead of torque motors, is derived in a nondimensional form. A proportional differential controller (PD) is synthesized in order to minimize the drive power and improve the dynamic properties. The effects of the non dimensional parameters on system were studied.

II.

Fig. 1. Electro-hydraulic Servo Valve With Piezo actuator: 1–Flapper, 2–Nozzles, 3-Piezoelement displacement, 4– Spool valve, 5Hydraulic Piston, 6- LVDT Transducer

Fig. 2. Piezoelectric Element

The piezo bender transducer type PL 112.10 [8] applied here is characterized by displacement equal to ±0.08 (± 20%) µm after putting the maximum voltage equal to ±60 V. It produces the maximum output force equal to 2 N. In produced nowadays servo-valves, torque motors with maximum torque from 20 to 50 mN.m are applied [6]. Defining [2]:

Electro-Hydraulic Servo Valve Model

The servo used in this system is an electro-hydraulic valve where the torque motor is replaced by piezo bender actuator (Fig. 1). Its plate acts as a flapper placed between two nozzles. LVDT transducer [9] is used to measure hydraulic piston displacement. The output signal from the transducer is compared with the desired one and the controller produces the voltage required for the piezo element. Piezoelectric actuator: Piezo electric elements make use of the deformation of electro-active piezo-ceramics when they are exposed to electrical fields. This deformation can be used to produce motions with suitable force. The generated forces (depending on the construction) range from mN to many kN obtaining movements from μm to mm. They require application of high voltage, which can reach 1000 v [6]. There are two types of action displacement in piezoelectric actuators: longitudinal in which the lengthen direction is the same as electric filed direction, and shear where lengthen direction is perpendicular to polarization direction. The second effect is used in bender piezo actuators. Bender transducer is build similarly to bimetal. It consists of two plates (Fig. 2). Applying electrical voltage, lengthens one plate and shortens the second one resulting in a bending moment [11], [13], [14].

xr  tr 

V , A mV Ps A2

vr  Cd

,

wr zr A

2



Ps

Dimensionless quantities can be introduced as ,

=

,

=

,

, U*

, t 

t , tr

. The resultant voltage is considered from the initial voltage input to the system: (1) where:

The resultant voltage starts the deformation of the piezo element leading to a displacement of flapper at a non dimensional velocity of: (2)



962


=

1

(8)

where:

2

3

Hydraulic Cylinder: Hence, from both pressures chamber 1 & 2 the spool starts to move back and forth simultaneously moving the connected load with velocity and a displacement :

The resultant non-dimensional displacement of the piezo element is : (3)

(9)

where:

(10) where: The resultant linear displacement of the piezoelectric material creates a difference in pressure in orifices of the flapper ( : (4) Spool Valve: As the pressure value at the flappers starts to change by the value these results in a change in the pressure of valve chambers and a movement of the spool valve with velocity and a displacement :

LVDT Transducer: The displacement is measured by a LVDT transducer to convert the resultant displacement of the load into a corresponding value of voltage Vout. The type of LVDT is open wiring LVDT, where the number of coil windings is uniformly distributed along the transformer; the voltage output is proportional to the iron core displacement when the core slides through the transformer [9]. The transducer equation is:

(5) (6) where:

(11) where D is the iron core displacement, M is the LVDT sensitivity, and is the output voltage. The load displacement moves proportionally with D resulting in LVDT output voltage which is compared with the value of the voltage that corresponds to the desired piston position.

By considering the continuity equation and the resultant flapper pressure difference, the pressure at the first chamber of the valve is:

III. Controlled System Results A numerical simulation was conducted to evaluate the performance of the electro-hydraulic servo valve integrated with a peizo actuator. The nominal values for the dimensionless system parameters are listed in Table I The controller used for this system is a proportional differential controller whose parameters are Kp=3.1 and Kd=85.5. Figure 3 includes the controlled input voltage

(7) At the second chamber the pressure

is:



963


that is fed to the piezo actuator and the response of servo valve states variables at desired non-dimensional piston displacement ( ) of 0.5. These states are: velocity and displacement of piezo actuator, velocity and displacement of the spool valve, pressures in the chambers of the hydraulic cylinders as well as its piston velocity and displacement. The responses are stable and the dimensionless controlled input maximum value is 0.95 as shown in the figure. Similar results are illustrated in Figure 4 at different desired piston displacement.

dimensionless piezo element displacements are illustrated in Figure 5. It can be noticed from the figure that as 1 increases from 1.0×10-6 to 1.65×10-6 , the dimensionless settling time decreases from 315 to185. The peak value of increases from 2×10-4 to 3.3×10-4, respectively to move the piston displacement to its desired value. At steady state, the piezo element returns to its neutral position ( =0) to stop the piston motion. As 1 is reduced to 0.1×10-6, the peak and final value of is 0.28×10-4 which means that the piston will not stop after reaching the desired position as indicated in Figure 6.

TABLE I VALUES OF NOMINAL NON-DIMENSIONAL PARAMETERS 135.082

20.378

0.152

12.350

1.4901×10-6

0.050

1.976

10.943

109.435

0.174

0.135

0.3623

Fig. 5. Responses of piezoelectric displacement at different

1

Fig. 3. Non-dimensional response at desired piston displacement = 0.5 Fig. 6. Settling time steady state error in piston position at different values of 1

The value of the dimensionless force due to piezo element stiffness ( 2) is 0.0496. It is changed from 0.040 to 0.5 and the resulted dimensionless piezo element displacements are plotted in Figure 7.

Fig. 4. Response of Piston Velocity & Displacement at different desired piston displacement

IV.

Parametric Study

For desired non-dimensional piston displacement of 0.5, and control gains of Kp=3.1 and Kd=85.5, the system dimensionless parameters are changed around their nominal values. The effects of these changes on the system performance are investigated. Effects of piezo element parameters : The nominal value of the dimensionless force generated due to the maximum applied voltage ( 1) is 1.4901×10-6. It is changed from 0.1×10-6 to 1.65×10-6 and the resulted

Fig. 7. Responses of Piezoelectric Displacement at different 2 values

The figure shows that as 2 increases from 0.04 to 0.06, the peak value of decreases from 3.7×10-5 to -5 2.5×10 , respectively. The piezo element returns to its neutral position ( =0) to stop the piston movement. As



964


is 2 is increased to 0.5, the peak and final value of 3.6×10-4 which results in unstoppable motion of the piston in the hydraulic cylinder as shown in Figure 8.

dimensionless spool valve displacements and decrease in the settling time. For β1 =50, the response is slow and the steady state error in the spool displacement is 0.003. The settling time and the steady state error in the displacement of the hydraulic cylinder piston are illustrated in Figure 11.

Fig. 8. Settling time steady state error in piston position at different values of 2

The dimensionless force due to piezo element damping ( 3) is changed from 0.05 to 6 (nominal value equals to 1.976) and the resulted responses of piezo element displacement are presented in Figure 9. The decrease of 3 to 0.5 renders an oscillatory response. The increase in 3 above the nominal value results in slow response and both the rise and the settling time increase. Meanwhile the maximum piezo element displacement will not change.

Fig. 11. Settling time steady state error in piston position at different values of β1

The nominal value of the dimensionless damping force on the spool valve (β2) is 20.3775 and it is changed from 0.1 to 40 and the results are plotted in Figure 12. The increase in the spool damping reduces the peak value of the dimensionless spool valve displacements and increases both the rise and settling time. An under damped oscillatory response is observed when β2 is 0.1. The piston of the hydraulic cylinder reaches the desired value for all values of β2 . The value of the dimensionless force due load stiffness ( 2) is 0.1349 and is changed from 0.05 to 1 and the results are plotted in Figure 15. As 2 is set lower than the nominal value (0.1349), the piston can be adjusted to reach the desired position with steady state error equals to zero. For 2 = 0.5 and 1(higher than nominal value), the steady state error in the dimensionless piston displacement are 0.15 and 0.32, respectively.

Fig. 9. Responses of Piezoelectric Displacement at different 3 values

Effects of spool valve parameters: The value of the dimensionless force that drives the spool valve β1 (nominal value is 135.082) is changed from 25 to 150 and the results are shown in Figure 10.

Fig. 12. Responses of spool valve Displacement at different (β2) values

Effects of hydraulic cylinder parameters: The value of the dimensionless pressure force driving the load ( 1) is changed from 0.05 to 0.5 and the resulted dimensionless piston displacements are illustrated in Figure 13. It

Fig. 10. Responses of spool valve Displacement at different β1 values

The increase in the value of β1 results in steady state error equals to zero, an increase in the peak value of the Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


965


The nominal value of the dimensionless damping force on the load ( 3) is 0.3623 and it is changed from 0.001 to 500 and the resulted dimensionless piston displacement ( ) are shown in Figure 17. It can be noticed from the figure that the increase in 3 will not affect the steady state error which is equal to zero. Meanwhile the dimensionless settling time increases from 192 to 270 as 3 is increased from 0.001 to 300, respectively.

can be noticed from the figure that as 1 increases from 0.1 to 0.5, the steady state error is zero and the dimensionless settling time decreases. As 1 is reduced to 0.050, the desired position ( =0.5) will not be attained and the steady state error is 0.129 as depicted in Figure 14.

Fig. 13. Responses of piston displacement at different

1

values

Fig. 17. Responses of piston displacement at different values of

V.

Conclusion

A piezo-element is integrated to the electro-hydraulic servo valve to act as an actuator, instead of conventional electrically operated torque motor. The non dimensional equations describing the system were derived. Eight dimensionless groups were introduced to describe the system design. Three groups are related to the piezo actuator to describe the force generated due to the applied voltage, stiffness, and damping with respect to inertia force of the piezo element. The design of the spool valve is described by two groups; the driving pressure force and the damping force as a ratio of its inertia force. The hydraulic cylinder dimensionless groups are the driving pressure force, the stiffness and the damping forces as ratio of the inertia force of the piston. The synthesized PD control was able to adjust the position of the hydraulic piston at any desired value. To have more insight into the dynamic behavior of valves of similar configuration but with different geometry, parametric simulation was conducted. The results of the numerical simulation indicate that the increase in the driving forces decreases both the settling time and the steady state error in the displacement of the hydraulic cylinder. An opposite effect is observed for the increase in the stiffness force of both the piezo element and the hydraulic cylinder. The decrease in the damping force renders an oscillatory response without affecting either the steady state error or the settling time. The obtained results agree with those reported in [6].

Fig. 14. Settling time steady state error in piston position at different values of 1

Fig. 15. Responses of piston displacement at different values of

3

2

References [1] Fig. 16. Settling time steady state error in piston position at different values of 2


J. C. Jones, H. Monash, Developments in design of electrohydraulic control valves from their initial design concept to present day design and applications, Moog Australia PTY LTD. Workshop on Proportional and Servovalves , Monash University,


966


[2] [3]

[4]

[5]

[6]

[7]

[8] [9] [10] [11]

[12]

[13]

[14]


Melbourne, Australia. November 1997. H. Merritt, Hydraulic Control Systems (John Wiley & Sons, New York NY, 1967). C. Birdsong, Development of a MALAB / Simulink hydraulic servo control design experiment, California Polytechnic State University, American Society for Engineering Education, 2007. Hiroyuki Fujita, Future of actuators and Microsystems, Institute of Industrial Science, the University of Tokyo. 7-22-1 Roppongi. Minatolm. Tokyo 106. Japan, Sensors and Actuators, 1996. Jason E. Lindler, Eric H. Anderson, Piezoelectric Direct Drive Servo valve, SPIE Paper 4698-53, Industrial and Commercial Applications of Smart Structures Technologies, San Diego, March 2002. A. Milecki, Modeling and investigations of electro hydraulic servo valve with piezoelement, Institute of Mechanical Engineering, Posnan University of technology, Vol. 26 nr 2, 2006. S. Karunanidhi, M. Singaperumal, Mathematical modeling and experimental characterization of a high dynamic servo valve integrated with piezoelectric actuator, Proc. IMechE Vol. 224 Part I: J. Systems and Control Engineering, 5 February 2010. http://www.piceramic.de/products.html. http://www.efunda.com/DesignStandards/sensors/ lvdt/lvdt_theory. cfm. MathWorks Inc., MATLAB, version 2008, Natick, MA: Math Works, Inc., 2008. R. H. Bishop, The Mechatronics Handbook (The University of Texas at Austin, Texas, The University of Texas at Austin, CRC PRESS,2002). Yannis Koveos, Anthony Tzes, Demosthenes Tsahalis, Poppet Valve Parameter Optimization of a High Frequency Operating Pump System+B438, International Review of Mechanical Engineering, IREME, Vol.4 No 7, pp 71-79, November 2010. M. Mailah, Gigih Priyandoko, Mechatronic Implementation of An Intelligent Active Force Control Scheme Via A Hardware-in-theLoop Configuration, International Review of Mechanical Engineering, IREME, Vol.4 No 7, pp 899-907, Nov 2010. M. Abdi, A. Karami Mohammadi, Numerical Simulation and Active Vibration Control of Piezoelectric Smart Structures, International Review of Mechanical Engineering, IREME, Vol.3 No 2, pp 175-181, March 2009.

Sohair F. Rezeka received the B.Sc. degree from Alex. Univ., Egypt in 1976 and the M.Sc. and the PhD degrees in Mechanical Engineering from Wayne State University, Detroit, Michigan in 1980, and 1984, respectively. She is a professor of Mechanical Engineering in Alex. Univ., Egypt and now she is on leave at Arab Academy for Science and Technology, Alex., Egypt. Her professional interests include intelligent control systems for autonomous vehicles and HVAC systems, fault diagnostics and identification, modeling and simulation of dynamic systems and mechatronics. Alaa Khalil received the B.Sc. degree from Alex. Univ., Egypt in 1991, M.Sc. degree from Alex. Univ., Egypt in 1994, . and the PhD degree in Electrical and power Engineering Dept. from Ain Shams Univ. in 1999. He is professor of Automatic Control Engineering in Arab Academy for Science and Technology, Alex., Egypt. His professional interests include Control applications in power systems an electrical drives, artificial intelligent control, identification and adaptive control. A. Abdellatif received the B.Sc. degree from Arab Academy for Science and Technology and Maritime Transport (AAST), Alexandria, Egypt in 2008. He is a graduate teacher assistant in Arab Academy for Science and Technology, Cairo, Egypt. His professional interests include Mechatronics, Automatic control and Hydraulic systems.



967


The Effect of the Fibre Orientation on the Failure Load of Face Sheets Composite Sandwich Beams F. Bourouis, F. Mili Abstract – Sandwich beams subjected to three points bending may fail in several ways, including tension or compression failure of the facing; face yielding and face wrinkling. In this paper three face sheets materials were used to study the effect of the fibre orientation on static failure of composite sandwich beams carbon/epoxy, kevlar/epoxy, glass/epoxy of stacking sequence [+θ/-θ] 3s, [0°/90°] 3s and [45°/-45°]3s. The stresses in the face were calculated using maximum stress criterion and the simple beam theory, including transverse shear effect. The obtained different results show that the sandwich beams with carbon/epoxy, and glass/epoxy face sheets are the best materials, in return the kevlar /epoxy facing characterised by a low mechanical resistance in compressive and tensile. The critical failure loads, depends on the properties of the face sheets materials and the fibre orientation. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Beam Theory, Fibre Orientation, Transverse Shear, Wrinkling, Yielding

Nomenclature l b h t x y z zk

σf

Beam length, (mm) Width of the beam, (mm) Core thickness, (mm) Face thickness, (mm) Direction x Direction y Direction z Distance from the middle plane to the top of the kth layer, (mm) Thickness of the layer k, (mm) Number of layers in the upper skins Normal stress in the face, (N/mm2)

σ yf

Normal tensile stress in the face, (N/mm2)

σ wy

Normal stress in the compression face, (N/mm2) Normal stress, (N/mm2) of the kth layer

ek n2

k σ xx

(Q ) Dij

Plane reduced stiffness constants of the kth layer Bending stiffness

Cij2

The stiffness coefficients in the upper skins

∆

Dij*

Determinant of the matrix ⎡⎣ Dij ⎤⎦ The inverse matrix of ⎡⎣ Dij ⎤⎦

Vf

Fibre volume fraction

El Et

Longitudinal young’s modulus , (Gpa) Transverse young’s modulus, (Gpa)

ij k

Glt

ν lt Xt , Xc Yt ,Yc S

Longitudinal shear modulus, (Gpa) Poisson ratio The tensile and compressive strengths in the longitudinal direction, respectively, (Mpa) The tensile and compressive strengths in the transverse direction, respectively, (Mpa) The in –plane shear strength of the layer, (Mpa)

I.

Introduction

Sandwich structured composites are a special class of composite materials ,which have become very popular due to high specific strength and bending stiffness. Sandwich beams subjected to bending may fail in several ways; face yielding, face wrinkling, bond decohesion, core shear, core tension and core compression [1]. Core failure by shear has been studied by Allen [2]. Debonding of facing from the core may occur during the fabrication or due to overloads, this failure mode was studied in terms of the critical strain energy release rate of the interface by Triantafillon and Gibson [3]. Operative failure mechanisms are identified and failure maps are constructed by Craig A, Steeves, Norman A. Fleck [4], the geometry of sandwich beams is optimized to minimise the mass for a required load bearing capacity in three-points bending. Face wrinkling failure loads were measured by [5] and compared favourably with an early expression of Hoff and Mautner for the case of the columns and long beams, where the core is in the linear elastic range. An experimental study in plane compressive failure mechanism of foam cored sandwich specimens with an implanted through with



968

F. Bourouis, F. Mili

Dij = hCij2

face/core is presented by [6]; the observation of the response of specimens during testing showed that failure occurred by buckling of the debonded face sheet. S. Ellagoune, L. Fatmi and A. Rouili [7] working on damage mechanism of a sandwich material tested under static and cyclic loadings in a three points bending experiment. A number of failure modes were recorded and studied in the composite sandwich beams subjected to three points bending, they include wrinkling of the compression, and yielding of the tensile facing, these failure modes are discussed in the present work. The main goal of this work is to study the effect of fibre orientation on the failure loads of symmetric sandwich beams with two identical skins, including transverse shear deformation in laminate theory; the sandwich beams where fabricated from 6 ply unidirectional fibre with various materials, carbon/epoxy, kevlar/epoxy, glass/epoxy of stacking sequence [+θ/-θ] 3s, [0°/90°] 3s and [45°/-45°] 3s.

II.

( )

k

φx =

pl * ⎡ ⎛ x⎞ D11 ⎢1 − 4 ⎜ ⎟ ⎥ 16b ⎝ l ⎠ ⎦⎥ ⎣⎢

p=

* D11

(

D12 D22 D26

D16 ⎤ D26 ⎥⎥ D66 ⎥⎦

1 2 = D22 D66 − D26 ∆

)

k

x

(7)

k 4bσ xx

( )

* hD11 Q11

(8) x

k

The mechanical characteristics of the skins composites materials are listed in the Table I, [8]. TABLE I MECHANICAL CHARACTERISTICS MEASURED ON VARIOUS UNIDIRECTIONAL FIBRE EPOXY COMPOSITE Mechanical Glass E/ Carbon HR Kevlar 49/ characteristics epoxy /epoxy epoxy 0.6 0.6 0.6 V f

El (Gpa) Et (Gpa)

ν

(Gpa)

46

159

84

10

14.3

5.6

4.6

4.8

2.1

ν lt

0.31

0.32

0.34

X t (Mpa) X c (Mpa) Yt (Mpa) Yc (Mpa) S (Mpa)

1400

1380

1400

910

1430

280

35

40

15

110

240

50

70

70

35

III. Analytical Formulation of Failure Mode It is well known that sandwich beams fail by one of several competing failure modes [9]; the operative mode is determined by the beam geometry, material properties, and the loading configuration. In this studies we describe two failure modes face yielding and face wrinkling. A common failure mode of composite sandwich members is the yielding of the face material due to high normal stresses generated by bending, when the normal tensile stresses equal or exceeds the yield stress of the face material (Fig. 1):

(2)

where p is the total load applied at middle of beam: ⎡ D11 Dij = ⎢⎢ D12 ⎢⎣ D16

( )

ph * D11 Q11 4b

The symmetry of the problem leads to consider only half of the beam; in our study we consider upper skin:

(1)

2⎤

(6)

k =1

k σ xx =±

A plus sign being associated to the upper skin and maximum sign with the lower. In the case of a three points bending, we have:

2

∑ ( Qij )k ek zk

We obtain the normal stress as:

Consider a simply supported sandwich beam loaded in three points bending. Let l be the beam length between the supports, b the width of the beam, h the core thickness, and t the face thickness. l = 180 mm, b = 50 mm, t = 6 mm, h = 10 mm. We consider the case of the symmetric sandwich consisted of two identical skins with orthotropy directions, parallel to the direction x and y of the beam, and core with material direction parallel to the directions x and y. The normal stress written as [8]: k σ xx = ± Q11

n2

Cij2 =

Three Points Bending of Sandwich Beams

h dφ x 2 dx

(5)

(3)

k σ yf = σ f = σ xx

(4)


(9)


969


load is high in compression than in tensile. From 6° to 35° the loads are the same for two failure modes. From 36°, the sandwich beams loaded in three points bending with carbon/epoxy face sheets resist better in compression than in tensile. Fig. 4 shows that from 0° to 7° we have very significant values of failure loads in face yielding than in face wrinkling. From 8° the sandwich beams with kevlar/epoxy face sheets have a low mechanical resistance in tensile and compression and the load is almost similar for the two failure modes.

Fig. 1. Schematic of face yielding failure

The compression face of sandwich beam subject to bending is likely to fail by a particular kind of local instability described as wrinkling (Fig. 2).

Failure load P/b N/mm

1000

Fig. 2. Schematic of face wrinkling failure

Face wrinkling occurs when the compressive stress in the face is: k σ wf = σ f = σ xx

(10)

400

0 0

IV.1. The Effect of the Fibre Orientation on Failure Load in Sandwich Beams with Various Face Sheets Materials Oriented at [+θ /-θ]3s

10

20

30

40 50 Angle(°)

60

70

80

90

Fig. 4. Failure load in sandwich beams with kevlar/epoxy face sheet oriented at [+θ/-θ]3s on three points bending

Generally the fibre orientation constituting the coatings sandwich beams materials have various values 0° to 90° thus, it is necessary to study the influence of the fibre orientation on the variation of failure loads of sandwich beams loaded in three points bending. In order to demonstrate the effect of the choice of the fibre angle we considered x = l/2, which the stresses are maximum.


1000

1000 Failure load P/b N/mm

600

200

IV. Results and Discussion

face wrinkling face yielding

800

(face wrinkling) (face yielding)

800

(face wrinkling) (face yielding)

800

600

400

200

0 0

600

10

20

30

40

50

60

70

80

90

Angle(°)

400

Fig. 5. Failure load in sandwich beams with glass/epoxy face sheet oriented at [+θ/-θ]3s on three points bending

200

Fig. 5 shows, that the failure load is very important for the small values of the fibre orientation. From 0° to 4° the sandwich beams with glass/epoxy face sheets resist better in tensile than in compression. From 5° to 26°, the failure loads are similar for two failure modes. From 27°, the face yielding appears before face wrinkling failure mode. Fig. 6 shows the influence of face sheets materials in face wrinkling failure mode.

0 0

10

20

30

40 50 Angle(°)

60

70

80

90

Fig. 3. Failure load in sandwich beams with carbon/epoxy face sheet oriented at [+θ/-θ]3s on three points bending

In Fig. 3, we observe that for small values of the fibre orientation the failure load is very significant. We note that fibre orientation from 0° to 5°; the failure Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


970


IV.2. Failure Load in Sandwich Beams with Various Face Sheets Materials Oriented at [0° /90°] 3s on Three Points Bending at Different Span Lengths

We notice that for small fibre orientation values the sandwich beams with carbon/epoxy facing resist better than sandwich beams with glass/epoxy and kevlar/epoxy face sheets. From 5° to 26° the sandwich beams with glass/epoxy and carbon /epoxy face sheets will fail at the same load. From 27° to 63° the sandwich material with glass/epoxy face sheets is better than the two other materials. From 64° the carbon/epoxy is the best skins material in sandwich beam material loaded in three points bending. The sandwich beam with kevlar/epoxy face sheet is always the less material resistant.

Fig. 8 shows that the failure load in compression is very important for short spans. The sandwich beam with carbon/epoxy face sheets oriented at [0°/90°]3s is the best material in compression. Fig. 9 shows that the failure load in tensile is very significant for short spans, and the three beams sandwich materials will fail at the same load. 4500


4000


1000 (carbon/epoxy) (kevlar/epoxy) (glass/epoxy)

800

600

3500

(carbon/epoxy) (kevlar/epoxy) (glass/epoxy)

3000 2500 2000 1500 1000

400

500 200

0

0 100 200 300 400 500 600 700 800 900 1000 span mm 0 0

10

20

30

40 50 Angle(°)

60

70

80

90

Fig. 8. Face wrinkling failure mode in sandwich beams with various face sheets materials oriented at [0°/90°]3s on three points bending at different span lengths

Fig. 6. Face wrinkling failure mode in sandwich beams with various face sheets materials oriented at [+θ/-θ]3s on three points bending

5000 4500

(carbon/epoxy) (kevlar/epoxy) (glass/epoxy)

4000

(carbon/epoxy) (Kevlar/epoxy) (glass/epoxy)

800



1000

600

400

3500 3000 2500 2000 1500 1000 500

200

0 0

0 0

10

20

30

40 50 Angle(°)

60

70

80

90

100 200 300 400 500 600 700 800 900 1000 span mm

Fig. 9. Face yielding failure mode in sandwich beams with various face sheets materials oriented at [0°/90°] 3s on three points bending at different span lengths

Fig. 7. Face yielding failure mode in sandwich beams with various face sheets materials oriented at [+θ/-θ]3s on three points bending

IV.3. Failure Load in Sandwich Beams with Various Face Sheets Materials Oriented at [45° /-45°]3s on Three Points Bending at Different Span Lengths

Fig. 7 shows the influence of face sheets materials in face yielding failure mode. The sandwich beams with carbon/epoxy and glass/epoxy face sheets will fail at the same load from 0° to 45° ,and the kevlar/epoxy is the less material resistant in tensile. From 46°the value of failure load is the same for the three materials.

Figs. 10 and 11 show that the beam with glass/epoxy face sheets oriented at [+45/-45]3s resist better in tensile and compression than sandwich material with carbon/epoxy and kevlar/epoxy facing.



971



600 550 500 450 400 350 300 250 200 150 100 50 0

References (carbon/epoxy) (kevlar/epoxy) (glass/epoxy)

[1]

[2] [3] [4]

[5]

0

[6]

100 200 300 400 500 600 700 800 900 1000 span mm

[7] Fig. 10. Face wrinkling failure mode in sandwich beams with various face sheets materials oriented at [+45°/-45°] 3s on three points bending at different span lengths


[8] 300 275 250 225 200 175 150 125 100 75 50 25 0

[9] (carbon/epoxy) (kevlar/epoxy) (glass/epoxy)

Ashby MF, Evans AG, Fleck NA, Gibson LJ, Hutchinson JW, Wadley HNG, Metal foams a design guide (Butter worth Heinemann, 2000). H. G Allen, Analysis and design of structural sandwich panel (Pergamon, London, 1969). TC. Triantafillon, LJ. Gibson, Debonding in foam core sandwich panels, Mat. Struc, Vol 22, pp 64-69, 1989. Craig A. Steeves, Norman A. Fleck, Material selection in sandwich beam construction, Scripta Materialia, Vol. 50, pp 1335-1339, 2004. EE. Gdoutos, I. M. Daniel, K. A.Wang, Compression facing wrinkling of composite sandwich structures, Mechanics of Materials, Vol 35, pp 511-522, 2003. V. Vadakke, Leif A. Carlsson, Experimental investigation of compression failure of sandwich specimen with face/core debond, Composites, Vol 35, n. B, pp 583-590, 2004. S. Ellagoune, L. Fatmi, A. Rouili, Damage Mechanism of a Sandwich Material Tested Under Static and Cyclic Loading in a Three Points Bending Experiment, International Review of Mechanical Engineering, Vol. 2, n. 1, pp. 56-61,2008. J. Marie Berthelot, Mechanical behaviour of composite materials and structures (Masson Paris, 1996). D. Lukkassen, A. Meidell, Advanced Materials and structures and their fabrication (processes, Book manuscript, Navic University college, Hin, 2007).


0

Fairouz Bourouis (Constantine, Algeria 24 / 3 / 1972) Magister in Mechanical Engineering University Mentouri Constantine Algeria 2005. Her current interests include studies of; Mechanical behaviour of laminates and sandwich beams. Cylindrical bending and buckling of sandwich plates. Failure modes of composite sandwich beams. Design optimization of composites using genetic algorithms. E-mail: [email protected]

100 200 300 400 500 600 700 800 900 1000 span mm

Fig. 11. Face yielding failure mode in sandwich beams with various face sheets materials oriented at [+45°/-45°] 3s on three points bending at different span lengths

V.

Conclusion

In most application sandwich construction is used to reduce the failure of structural components. In this studies we describe the effect of fibre orientation in face yielding and face wrinkling failure modes. The sandwich beam with glass/epoxy and carbon/epoxy face sheets oriented at [+θ/-θ]3s resist better in tensile and compression than the material with kevlar/epoxy skins. When the laminates oriented at [0°/90°]3s the three types of sandwich beams will fail at the same load in tensile, on the other hand the sandwich beam with carbon /epoxy face sheet is the best in compression. The material with glass/epoxy facing oriented at [45/45]3s have high mechanical resistance in tensile and compression than the two others materials. The good choice of the fibre orientation, composite face sheets materials, span lengths allow to maximized the failure loads and the performance of the sandwich beams.



972


An Overview on Thermal Barrier Coating (TBC) Materials and its Effect on Engine Performance and Emission Pankaj N. Shrirao1, Anand N. Pawar2, Atul B. Borade1 Abstract – Ceramic based thermal barrier coatings are considered as candidate materials for coating of engineering components subjected to elevated temperatures in operating conditions. In this study efforts are taken to gather the information regarding the TBC materials, Bond coat materials, coating methods and effect of TBC materials on engine performance and emission. Ceramics, in contrast to metals are often more resistance to oxidation, corrosion, wear as well as being better thermal insulator. Many researchers have carried out large number of studies on LHRE concept. In case of LHR engine almost all theoretical studies predict improved performance but many experimental studies shows different picture. It is concluded that much more research is needed to overcome the practical problems before LHR engines can be put into production. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Diesel Engine, TBC, LHRE

I.

basic requirements: (1) high melting point, (2) no phase transformation between room temperature and operation temperature, (3) low thermal conductivity, (4) chemical inertness, (5) thermal expansion match with the metallic substrate, (6) good adherence to the metallic substrate and (7) low sintering rate of the porous microstructure. Properties of some ceramics that can be used as TBC material are summarized in Table I[1]. Among those properties, thermal expansion coefficient and thermal conductivity seem to be the most important.

Introduction

Energy conservation and efficiency have always been the quest of engineers concerned with internal combustion engines. The diesel engine generally offers better fuel economy than its counterpart petrol engine. Even the diesel engine rejects about two thirds of the heat energy of the fuel, one-third to the coolant, and one third to the exhaust, leaving only about one-third as useful power output. Theoretically if the heat rejected could be reduced, then the thermal efficiency would be improved, at least up to the limit set by the second law of thermodynamics. Low Heat Rejection engines aim to do this by reducing the heat lost to the coolant. The diesel engine with its combustion chamber walls insulated by ceramics is referred to as Low Heat- Rejection (LHR) engine. The LHR engine has been conceived basically to improve fuel economy by eliminating the conventional cooling system and converting part of the increased exhaust energy into shaft work using the turbocharged system. Most of researchers have concluded that insulation reduces heat transfer, improves thermal efficiency, and increases energy availability in the exhaust. However contrary to the above expectations some experimental studies have indicated almost no improvement in thermal efficiency and claim that exhaust emissions deteriorated as compared to those of the conventional water-cooled engines. An attempt will be made here to review the previous studies to look into future possibilities of the LHR engine from the viewpoint of combustion, heat transfer and emission.

II.

II.1.

Materials Used as Thermal Barrier Coating II.1.1. Zirconia

Zirconia, a ceramic material that has very low thermal conductivity values, good strength, and thermal expansion coefficients similar to metals and is able to withstand much higher temperatures than metals.Silicon Nitride Syalon (Si-Al-O-N) ceramics is one derivative of this classification. The major advantage of the material is the low creep characteristics at high temperatures. The material also has low density and low coefficient of friction. This will be good for reciprocating parts such valves and bearings. Ceramic materials are brittle in general and which develops crack due to stress concentration. II.1.2. Mullite Mullite is an important ceramic material because of its low density, high thermal stability, stability in severe chemical environments, low thermal conductivity and favorable strength and creep behavior. It is a compound of SiO2 and Al2O3 with composition 3Al2O3. 2SiO2.

Literature Review

The selection of TBC materials is restricted by some Manuscript received and revised June 2011, accepted July 2011


973

Pankaj N. Shrirao, Anand N. Pawar, Atul B. Borade

compared with YSZ, mullite has a much lower thermal expansion coefficient and higher thermal conductivity, and is much more oxygen-resistant than YSZ. For the applications such as diesel engines where the surface temperatures are lower than those encountered in gas turbines and where the temperature variations across the coating are large, mullity is an excellent alternative to zirconia as a TBC material. Life of Mullite coating is more than the zirconia coating tested under same condition. Mullite is most promising coating material for the SiC substrate because their thermal expansion coefficients are similar.

hardness and stoichiometry change in coating due to the vaporization of CeO2 reduction of CeO2, reduction of CeO2 into CeO2 into Ce2O3 and accelerated sintering rate of coating. II.1.4. Pyrochlore Oxides Pyrochlore oxides of general composition, A2B2O7, where A is a 3+ cation (lanthanum to lutetium) and B is a 4+ cation (zirconium, hafnium, titanium, etc.), are one such class of ceramic materials. These oxides have a high melting point, a relatively high coefficient of thermal expansion, and low thermal conductivity, which make them suitable for applications as high-temperature thermal barrier coatings. The primary objective of this study at the NASA Glenn Research Center was to devise approaches to further lower the thermal conductivity of pyrochlore oxide compounds. An oxide-doping approach was used where part of cation A was substituted by other cations (e.g., A1-xMxB2O7, where x = 0 – 0.5 and M = rare earth or other cations) in the pyrochlore materials. Powders of various compositions were synthesized by the sol-gel method and hot pressed into dense 1-in.diameter disks. The thermal conductivity of these disks was measured at temperatures up to 1550 °C using a steady-state laser-heat-flux technique. As an example, results for La2-xMxZr2O7 systems (where M = gadolinium (Gd), ytterbium (Yb), or Gd + Yb) are shown in the graph. The rare-earth-oxide-doped pyrochlores (La,Gd)2Zr2O7, (La,Yb)2Zr2O7, and (La,Gd,Yb)2Zr2O7 have lower thermal conductivity than the undoped La2Zr2O7. The thermal conductivity of material codoped with Gd + Yb is ~30 percent lower than that of the undoped oxide. These results clearly demonstrate that the thermal conductivity of pyrochlore oxides can be reduced greatly by doping, especially through codoping with other cations.

TABLE I TBC MATERIALS AND THEIR CHARACTERISTICS Materials Advantages Disadvantages YSZ

(1) high thermal expansion coefficient (2) low thermal conductivity (3) high thermal shock resistance (4) oxygen-transparent

(1) sintering above 1473 K (2) phase transformation (1443 K) (3) corrosion

Mullite

(1) high corrosion-resistance (2) low thermal conductivity (3) good thermal-shock resistance below 1273 K (4) not oxygen-transparent

(1) crystallization (1023-1273 K) (2) very low thermal expansion coefficient

Alumina

(1) high corrosion-resistance (2) high hardness (3) not oxygen-transparent

(1) phase transformation (1273 K) (2) high thermal conductivity (3) very low thermal expansion coefficient

YSZ+CeO2

(1) high thermal expansion coefficient (2) low thermal conductivity (3) high corrosion-resistance (4) high thermal-shock resistance

(1) increased sintering rate (2) CeO2 precipitation (> 1373 K) (3) CeO2-loss during spraying

La2Zr2O7

(1) very high thermal stability (2) low thermal conductivity (3) low sintering (4) not oxygen-transparent

(1) relatively low thermal expansion coefficient

(1) Cheap, readily available (2) high corrosion-resistance

(1) decomposition into ZrO2 and SiO2 during thermal spraying (2) very low thermal expansion coefficient

Silicates

III. Coating Deposition Methods General methods of producing thermal barrier coatings are mentioned below: - Electron Beam Physical Vapor Deposition: - EBPVD - Air Plasma Spray: APS - Electrostatic Spray Assisted Vapour Deposition: - ESAVD - Direct Vapor Deposition

IV.

II.1.3. CeO2+YSZ CeO2 has higher thermal effiency & lower thermal conductivity than YSZ is effective for the improvement of thermal cycling life. Stress generated by bond coat oxidation is smaller in smaller in CeO2+YSZ coating due to better thermal insulation , thermal expansion coefficient is larger in CeO2+YSZ coating.Decrease in

Effect of Insulation on Engine Performance IV.1.

Volumetric Efficiency

Volumetric efficiency is an indication of breathing ability of the engine. It depends on the ambient conditions and operating conditions of the engine. Reducing heat rejection with the addition of ceramic



974


such as volumetric efficiency, air-fuel mixing etc, which in turn affect fuel consumption. Hence it is felt that, comparison between the two engines should be made at same engine operating conditions and same engine operating parameters. In general, it has been reported that fuel consumption of, naturally aspirated LHR engine is in the range of 0 to 10% higher, turbocharged LHR engine in the order of 0 to 10% lower and turbocompounded LHR engine in the order of 0 to 15% lower, when compared with the conventional cooled engine.

insulation causes an increase in the temperature of the combustion chamber walls of an LHR engine. The volumetric efficiency should drop, as the hotter walls and residual gas decrease the density of the inducted air. References [2], [3], [4], [5] and [6] on LHR engine show decreased volumetric efficiency. The deterioration in volumetric efficiency of the LHR engine can be prevented by turbocharging, and that there can be more effective utilization of the exhaust gas energy. IV.2. Thermal Efficiency

V.

Thermal efficiency is the true indication of the efficiency with which the chemical energy input in the form of fuel is converted into useful work. Improvement in engine thermal efficiency by reduction of in-cylinder heat transfer is the key objective of LHR engine research. Much work has been done at many research institutes to examine the potential of LHR engines for reducing heat rejection and achieving high thermal efficiency. References [7], [8], [9], [10], and many others have reported improvement in thermal efficiency with LHR engine. They attribute this to incylinder heat transfer reduction and lower heat flux. However investigations of others such as [11], [12], [13], [14] and some others report that thermal efficiency reduces with insulation. They all attribute this to an increase in the convective heat transfer coefficient, higher heat flux (increase in incylinder heat transfer) and deteriorated combustion. The in-cylinder heat transfer characteristics of LHR engine are still not clearly understood. Thus the effect of combustion chamber insulation on reducing heat rejection and hence on thermal efficiency is not clearly understood as on date.

Effect of Insulation on Emission V.1.

Unburned Hydrocarbon

The emission of unburned Hydrocarbon from the LHR engines is more likely to be reduced because of the decreased quenching distance and the increased lean flammability limit. The higher temperatures both in the gases and at the combustion chamber walls of the LHR engine assist in permitting the oxidation reactions to proceed close to completion. Most of the investigations show reduction in HC level. However References [2], [18] indicate increased level of HC emissions. They attribute this to deterioration in diffusion combustion. The burning of lubricating oil due to high wall temperature is believed to be the other reason for increased UBHC level. V.2.

Carbon Monoxide

It might be expected that LHR engines would produce less Carbon monoxides, for reasons similar to those for unburned Hydrocarbon. In fact many investigations indicate lower level of CO emissions. They attribute this to high gas temperature and combustion chamber walls. The reduced level of premixed combustion in the insulated engine decreases the initial production of CO and the higher temperatures during diffusion combustion accelerate the oxidation of CO.

IV.3. Fuel Consumption Numerous investigators have modeled and analyzed the effects of in-cylinder thermal insulation on fuel consumption. References [15], [16], [17], [18] and [19] have reported improvement in the reduction of fuel consumption in LHR engine. The level of improvement that has been predicted ranged from 2 to 12 %. They attribute this to insulation of in-cylinder components. It has been predicted that insulation of in-cylinder components is a more effective means of reducing heat rejection and reducing fuel consumption. The reference [20] indicates reduction in fuel consumption, and attributes this to reduced friction due to increased wall temperature. He also states that there is no measurable improvement in fuel consumption based on the thermodynamics involved. Reference [5] indicates higher fuel consumption of LHR engine. They attribute this to the increase in reciprocating mass. Reference [3] shows that LHR engine has roughly 25% greater fuel consumption than the baseline engine. Reference [4] indicates comparison of SFC between baseline and LHR engine should be done carefully, because reducing the heat rejection affects other engine operating parameters

V.3.

Nitrogen Oxides

NOx is formed by chain reactions involving Nitrogen and Oxygen in the air. These reactions are highly temperature dependent. Since diesel engines always operate with excess air, NOx emissions are mainly a function of gas temperature and residence time. Most of the earlier investigations show that NOx emission from LHR engines is generally higher than that in watercooled engines. They say this is due to higher combustion temperature and longer combustion duration. Reference [9] reports an increase in the LHR engine NOx emissions and concluded that diffusion burning is the controlling factor for the production of NOx. Almost equal number of investigations report declining trend in the level of emission of NOx. Reference [18] indicates reduction in NOx level. They reason this to the



975


shortening of the ignition delay that decreases the proportion of the premixed combustion. V.4.

barrier coatings under thermo mechanical conditions has gained increasing interest in recent years. Plasma spread yttria (6-8 %) stabilized zirconia (PSZ) is currently the most advanced system envisaged for TBC applications on super alloy turbo engine components. Although this system is known for its excellent thermo mechanical resistance and elevated temperatures, the relationship between the performance and microstructure of such complex coatings is not fully understood [21]. Yet TBCs represent an attractive material to enhance the high temperature limits of the super alloy engine components now with standing its limitations to spilling by oxidation of the bond coat. At high temperatures the interdiffusion processes between the bond coat and the substrate could be very important and in extreme cases, lead to a rapid degradation of coating properties, like the Co based coating and rapid diffusion of Al from Co based coatings on Ni based super alloy at high temperatures with formation of an intermediate brittle intermetallic layer. The coating could play its protective role only when it could withstand erosion and impact in addition to induced thermo-mechanical stresses during on and off periods of the engine. In studying the life of TBCs in recent times more efforts have been made in studying the mechanism of crack initiation and propagation under controlled experimental conditions [22]. Studies undertaken at NML Jamshedpur [23] reveal the following facts. • Ni base (Inconel 617) alloy was taken as substrate material on which NiCrAlY type metallic undercoat or bond coat was first applied by plasma spraying. Zirconia stabilized with 8 wt % Yttria thermal barrier coating was applied over the bond coat by plasma spraying under reduce pressure and in an inert gas atmosphere. • During the Metallographic studies the micrographs of plasma sprayed to the TBC substrate revealed the presence of porous effects arising out of the processing. It was also evident that the bond coat also possessed a large quantity of porosities. Carbides precipitation at the grain boundaries and within the grains were also seen. An Al depleted zone just below the TBC along the TBC/bond coat interface as well as the bond coat substrates interface was also observed. The Al depletion was primarily due to its oxidation to Al2O3 and Al diffusion into the substrate. • The crack propagation behavior of the TBC revealed that the cracks propagated linearly in the TBC with increase in bending loads till the yield point of the substrate was reached thereafter at the interface of the TBC and bond coat a high threshold load was required for the crack to propagate further in the bond coat. Once the threshold was reached the crack propagated faster in the bond coat without any appreciable increase in the load. It is noteworthy that the crack path trajectory in the TBC often showed crack branching and deflection. At temperatures of 800oC the crack was found to propagate only in the

Smoke and Particulates

It might be expected that LHR engines would produce less smoke and particulates than standard engines for reasons such as high temperature gas and high temperature combustion chamber wall. Earlier investigations show that smoke and particulates emission level increased in some cases and decreased in a few others. The results obtained by [18] show significant reduction in smoke emission. They attribute this to enhanced soot oxidation, which was made possible by both the high combustion temperature and the intense turbulence created by the reversed squish. However investigations carried out at SwRI show increased level of smoke. They attribute it to increased oil consumption resulting from the loss of oil control at the higher temperatures. Factors such as short ignition delay, poor air-fuel mixing are also responsible for the formation of smoke and particulates.

VI. Drawbacks with TBC During operation TBCs are exposed to various thermal and mechanical loads such as thermal cycling, high and low cycle fatigue, hot corrosion and high temperature erosion. Currently, because of reliability problems, the thickness of TBCs is limited, in most applications, to 500µm. Increasing coating thickness increases the risk of coating failure and leads to a reduced coating lifetime. The failure mechanisms that cause TTBC coating spallation differ in some degree from that of the traditional thinner coatings. A major reason for traditional TBC failure and coating spallation in gas turbines is typically bond coat oxidation. When the thickness of the thermally grown oxide (TGO) exceeds a certain limit, it induces the critical stress for coating failure. Thicker coatings have higher temperature gradients through the coating and thus have higher internal stresses. Although the coefficient of thermal expansion (CTE) of 8Y2O3-ZrO2 is close to that of the substrate material, the CTE difference between the substrate and coating induces stresses at high temperatures at the coating interface. The strain tolerance of TTBC has to be managed by controlling the coating microstructure. Use of thicker coatings generally leads to higher coating surface temperatures that can be detrimental if certain limits are exceeded. In the long run, the phase structure of yttria stabilised zirconia (8Y2O3ZrO2) is not stable above 1250°C. Also the strain tolerance of the coating can be lost rapidly by sintering if too high surface temperatures are allowed.

VII.

Failure Analysis

Research on evaluating lifetime of candidate super alloy turbine blade materials with ceramic-coated thermal Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved


976


improved fuel economy and reduced emissions are attainable, but much more investigations under proper operating constraints with improved engine design are required to explore the full potential of Low Heat Rejection engines.

TBC as the coating having ductility at these temperatures offered an attractive means of protecting the substrate material from thermal loads. Much of the failures of such coatings has been attributed to the thermal expansion mismatch between the ceramic coat and the ductile substrate primarily arising during cooling cycle. The crack in the TBC during cooling was monitored and showed no increase in length. This could be due the compressive stresses in the ceramic layer placed by larger expansion of the metallic substrates. In crack propagation studies it was clear that for the oxidized samples in the TBC crack growth was observed even at constant load. The porosities played a definitive role in acting as stress raisers and initiating cracks. Higher temperature heat treatments at 1000 oC for 200 h densified the TBC coating thereby reducing the probability of crack source, yet permitting the crack to propagate unhindered by the reduced pores acting as sinks. Consequently crack growth was noticed within the TBC at constant load in the case of oxidized specimens. It was found that the crack length varies linearly with deflection in the ceramic layer of the TBC as well as the bond coat.

References [1] [2]

[3] [4] [5]

[6]

[7] [8]

[9]

VIII. Applications

[10]

The thermal barrier coats find vast applications in following areas: • Cutting Tool. • Gas Turbine Blades. • I.C. Engines. • Space Shuttles.

[11]

[12]

[13]

IX. Conclusion

[14]

The TBC system is outlined and discussed. Selection of proper TBC material depends upon the nature of the substrate and a proper deposition coating process results in the increase in the life of the substrate. The failure of the TBC material in the severe operating conditions is its major draw back. The failure analysis reveals that the porosities play a definite role in acting as stress raisers and igniting crack. Oxidation leads to crack growth at constant loads. Mismatch in the properties of the coating and the substrate alloy are responsible for exacerbating the failure mechanism. The mismatch may arise due to differences in the thermal expansion coefficients, ductility, strength and elastic module and can significantly reduce the fatigue life. Further research in development of smart coats with thermal expansion coefficients closed to that of substrate coupled with other mechanical properties and proper deposition methods may reduce the crack formation and result in increased TBC and substrate life. The objectives of improved thermal efficiency,

[15] [16] [17] [18]

[19] [20]

[21]

[22]

[23]


X. Q. Cao, R. Vassen, D. Stoever, 2004. Journal of the European Society 24, pp.1-10. D. Assanis, K. Wiese, E. Schwarz, W. Bryzik, “The Effects of Ceramics Coatings on Diesel Engine Performance and Exhaust Emissions”, SAE Paper No.910460. J. A. Gatowski, “Evaluation of a Selectively-Cooled Single Cylindered 0.5-L Diesel Engine”, SAE Paper No.900693. R. H. Thring, “Low Heat Rejection Engines”, SAE Paper No.860314. Y. Miyairi, T. Matsuhisa, T. Ozawa, H. Oikawa, N. Nakashima,” Selective Heat Insulation of Combustion Chamber Walls for a DI Diesel Engine with Monolithic Ceramics”, SAE Paper No.890141. T. Suzuki, M. Tsujita, Y. Mori “An Observation of Combustion of Phenomenon on Heat Insulated Turbocharged and Inter-cooled DI Diesel Engines”, SAE Paper No.861187. P. H Havstad, I. J. Gervin, W. R Wade, “A Ceramic Insert Uncooled Diesel Engine”, SAE paper No.860447. C. H. Moore, J. L.Hoehne, “Combustion Chamber Insulation Effect on the Performance of a Low Heat Rejection Cummins V903 Engine”, SAE Paper No.860317. A. C. Alkidas, "Performance and Emissions Achievements an Uncooled Heavy Duty Single Cylinder Diesel Engine”, SAE Paper No.890144. T. Morel, E. F. Fort, P. N.Blumberg, “Effect of Insulation Strategy and Design Parameters on Diesel Engine Heat Rejection and Performance”, SAE Paper No.850506 W. K. Cheng, V. W. Wong, F. Gao, “Heat Transfer Measurement Comparisons in Insulated and Non-Insulated Diesel Engines, “SAE Paper No.890570. G. Woschni, W. Spindler, K. Kolesa, ”Heat Insulation of Combustion Chamber Walls – A Measure to decrease the Fuel Consumption of I.C.Engines?” SAE Paper No.870339 S. Furuhama, Y. Enomoto, “Heat Transfer into Ceramic Combustion Wall of Internal Combustion Engines” SAE Paper No.870153. D. W. Dickey, “The Effect of Insulated Combustion Chamber Surfaces On Direct-Injected Diesel Engine Performance, Emissions and Combustion”, SAE Paper No.890292. R. Kamo, W. Bryzik. “Adiabatic Turbocompound Engine Performance Prediction”, SAE Paper No.780068. V. Sudhakar, “Performance Analysis of Adiabati Engine” SAE Paper No.820431. Yoshimitsu et al, “capabilities of Heat Insulated Diesel Engine”, SAE Paper No.840431. W. R. Wade, P. H. Havstad, E. J. Ounsted, F. H. Trinker, I. J. Garwin, “Fuel Economy Opportunities with an Uncooled DI Diesel Engine”, SAE Paper No.841286. K. L. Hoag, M. C. Brands, W. Bryzik, “Cummins/ TACOM Adiabatic Engine Program”, SAE Paper No 850356. S. Henningsen, “Evaluation of Emissions and Heat Release Characteristics from a Simulated Low Heat Rejection Engines”, SAE Paper No.871616. U. Schulz et al, “Thermocyclic Behavior of Microstructurally Modified EB-PVD Thermal Barrier coatings,” Mater. Sci. Forum, 251-254 (1997) 957-964. A. S. James, A. Matthews, “Thermal Stability of Practically – Ytrria-stabilized Zirconia Thermal Barrier Coatigs deposited by r. f. plasma-assisted Physical vapor deposition,” Surf. Coat. Technol., 41(3) (1990) 305-313. A. K. Ray, N. Roy, K. M. Godiwalla, “Crack propagation studies and bond coat properties in thermal barrier coatings under bending”, Bull. Mater. Sci., Vol. 24, No 2, April 2001, pp. 203209.


977


Dr. Atul B. Borade is working as Professor and Head of Mechanical Engineering Department at Jawaharlal Darda Institute of Engineering and Technology, Yavatmal, India. He owes around 15 years of experience in teaching. He is basically a Mechanical Engineer. He then persuaded M .E. and PhD in Production Engineering. He also holds MBA as an additional qualification. He has published 15 papers in National and International Journals. His few publications have appeared in prestigious Springer, Emerald, and Inderscience journals. He is regular reviewer with Omega, Journal of Manufacturing Engineers, Journal of Information, Information Technology and Organizations, Journal of Information System and Technology Management, Journal of Industrial Engineering and Management, International Journal of Industrial Engineering - Theory, Applications and Practice. Chinese Journal of Industrial Engineering, Journal of Manufacturing Technology and management, Computers and Mathematics with Applications He has worked as Editorial Board Member with International Journal of Information Technology and Knowledge Management, International Journal of Manufacturing Science and Manufacturing Management. Currently he is working as Editorial Board Member with Contemporary Management Research, Asian Journal of Industrial Engineering, and International Journal of Business Management and Research. He is working as an Editor in Chief with International Journal of Manufacturing Systems.

Authors’ information 1 J D College of Engineering and Technology, Yavatmal, India 445001. 2 Govt Polytechnic College, Yavatmal, India 445001.

Pankaj N. Shrirao is working as Assistant Professor of Mechanical Engineering Department at Jawaharlal Darda Institute of Engineering and Technology, Yavatmal, India. He owes around 07 years of experience in teaching.He has completed his Bachelor degree in mechanical engineering from Govt. college of engg., Amravati, India. He then persuaded M.E in Thermal Power engg. From Govt. college of engg., Amravati, India. Now he is pursuing PhD in Thermal Engineering from Amravati University India. He has published 05 papers in National and International Journals. Also he has presented 03 papers in international conferences. Dr. Anand N. Pawar is working as Professor of Mechanical Engineering Department at Govt. Polytechnic Amravati India. He owes around 17 years of experience in teaching. He has completed his Bachelor degree in mechanical engineering. He then persuaded M.E in Thermal Power engg. From Govt. college of engg., Amravati, India. He has persuaded PhD in Thermal Engineering from Amravati University India. He has published number papers in National and International Journals. Also he has presented number papers in international conferences. His few publications have appeared in prestigious Springer, Emerald, and Inderscience journals. His area of interest is Low Heat Rejection engine.



978


Process Robustness in a Dimensional Testing Laboratory Caterina Poustourli1, Vrassidas I. Leopoulos2 Abstract – In this study we consider a method to make robust the process of high-precision length measurements. Quality improvement efforts in many instances have been directed at reducing the variation of a particular characteristic around a nominal design specification. In the included case study we organized and executed controlled experiments for the comparative measurement of the inside diameter of plain ring gages in the infrastructures of an accredited Laboratory. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Process Robustness, Design of Experiments, Taguchi, Quality Improvement, Length Measurement

I.

of Quality Engineering’s literature, Section 2 describes the steps of the proposed method, Section 3 contains a case study used to demonstrate the applicability of the proposed method. Finally, Section 4 presents conclusions from the usage of the system and directions for future research Continuous improvement is a requirement included in all management system standards (ISO 17025, ISO 9001, ISO 14001, OHSAS 18001, etc). Efforts towards integrated management systems include process continuous improvement as a common requirement [6], [8]. Quality engineering is considered as an appropriate improvement methodology. Taguchi, the father of Quality Engineering, has been very successful in integrating powerful applied statistical methods into engineering processes for achieving greater capability and stability [1], [9]. Aim of the measurement processes are accuracy and precision along with repeatability and reproducibility [12]. Three steps are required to achieve robustness in a process such as dimensional measurement [1]-[2]-[3]: 1. System design (the concept developmental stage), 2. Parameter design (enhancement of the system design so that the process consistently performs as intended or better), 3. Tolerance design (determining the tolerances and grades of materials and nominal values defined in the parameter design stage). Robust Engineering (Taguchi-DOE) methods focus on the second phase, parameter design [5]. Taguchi's parameter design leads to the maximization of performance and quality at minimum cost. This is fundamentally achieved by determining the best settings of those design or process parameters which influence the performance variation and by fine tuning those design or process parameters which influence the average performance [2]-[3]-[5]. Amongst the several experimental designs related in

Introduction

In this paper we present a method to make robust the measurement process for high-precision length measurements [1]-[14]. The method is illustrated through a case study in a laboratory accredited for length measurements according to the standard ISO17025. The method has been tested and included in the quality management system of the laboratory. Aim of the method is the comparison of the effect of a predefined number of control factors with interactions and presence of controllable noise factors. Series of length measurement experiments are executed using inexpensive Quality/Robust Engineering (Taguchi-DOE) techniques to set each factor on different levels. The measurement results are compared against the calibrated values of the gages. The set of levels that minimizes the quality loss function is further checked for robustness. If using this setting of factors the process is robust, the measurement process and the relevant quality procedures are accordingly updated. In the case study the inside diameter of plain ring gages have been measured. Four control factors (environmental conditions and the operator) and one noise factor have been chosen. Analysis of variance was employed to quantify the effect of each factor considered. The application of the method resulted in a less variable measurement process. The experiments have been conducted using a length measuring instrument (LMI-comparator model with horizontal base) largely used in the industrial and laboratory conditions. The instrument is used for highprecision length measurements on precision parts such as gears, journals, ring gages, gage blocks, gear shafts etc, in accordance to and DIN 2250:1989 Part 1. The measuring instrument that we use for our experiments correspond to measuring range of 14mm to 150mm (internal measurement). The paper is organized as follows: The remainder of this Section provides a brief review Manuscript received and revised June 2011, accepted July 2011


979

Caterina Poustourli, Vrassidas I. Leopoulos

The method necessitates five primary tools for the process improvement, as generally used in robustness strategy [2]-[3]: 1. P-Diagram is used to classify the parameters associated with the measurement process into noise, control, signal (input), and response (output) factors. 2. Objective Function is used to mathematically specify the ideal form of the signal-response relationship as embodied by the design concept for making the higher-level system work perfectly. 3. Quadratic Loss Function (also known as Quality Loss Function) is used to quantify the loss incurred by the user due to deviation from target performance. 4. Signal-to-Noise Ratio is used for predicting the field quality through laboratory experiments. 5. Orthogonal Arrays are used for gathering dependable information about control factors (design parameters) with a small number of experiments. The method starts with the experimental variables selection (step 1 – 2). Thus, the control variables and their levels are selected to investigate the desirable response variable behavior (step 3). The next step concerns the choice of Taguchi experimental array, in order to conduct the experiment and minimize the total number of variable level combinations (runs) (step 4). There are several experimental arrays proposed by Taguchi, presented in a catalog in order to simplify the selection and facilitate the user [2]-[3]-[5].

literature, the Taguchi experimental designs are suitable to use in problems where there are many variables to investigate and it is necessary to reduce drastically the total number of experimental runs [5]-[13]-[14]. Taguchi arrays are equivalent to fractional factorial designs, but the codification applied by Taguchi facilitates the application even for the inexperienced user [5].

II.

Steps of the Proposed Method

The idea of this work is to make the measurement process as robust as possible at the lowest possible cost, without emphasizing cost more than quality [2]. Quality is improved at minimal cost by optimizing these parameters (factors) that are the least expensive to optimize. Instead of eliminating all causes of variation, the process is designed to minimize the effect of the variation in parameters that are expensive to control [2][3]. The method includes nine steps as the following diagram (Figure 1) reflects [6]-[7].

III. Case Study III.1. Problem Statement and Objective of the Experiment (System Design / Step 1) Objective of the case study is the robustness of measurement process and further the improvement of measurement practice for reducing measurement variability: “The value of measurement result as closed as possible to the calibrated value “. The robust engineering techniques include functions which model the amount the measurement variation in plain ring gages measurement as functions of environmental conditions. The infrastructure (Measurement Room) of the Laboratory operates in conformance with the VDI/VDE 2627 requirements [10]-[11]. The part to be measured is a plain ring gage with calibrated value of 54,9990mm and the reference part is a plain ring gage with calibrated value of 14,0000mm. III.2. Identification of Factors and Interactions (System Design / Step 2) A structured brainstorming session, in which participated the director of the laboratory, the technical director, the quality manager and the operators, identified firstly the parameters (factors) which presumably influence the performance and the variability of the

Fig. 1. The proposed method



980


Reference Part (ring): a plain ring of 14.00000 mm internal diameter. Objective (Ideal) Function: Nominal the Best. The control factors identified are: - Factor A - Ambient temperature of measurement room - Factor B - Part (plain ring gage) temperature - Factor C - Operator (level of metrological proficiency) - Factor D – Relative humidity of measurement room. - Possible interactions between factors are: - AXB - AXC - BXC The noise factor identified and selected is the temperature of reference part (plain ring).

measurement process (Figure 2).

III.3. Choice of Factors Levels (Parameter Design / Step 3)

Fig. 2. Factors influencing the measurement process

In the following table are presented the critical control and noise factors that have been selected for the experiment as well as their selected levels and interactions.

The quality characteristic of the process is presented in the following Table I. TABLE I QUALITY CHARACTERISTIC (RESPONSE) Q.C.

Description

Measure

dimension

Internal diameter

mm

TABLE II SELECTED FACTORS AND THEIR LEVELS

Secondly, the critical factors as well as signal factor, control factors, noise factors and interactions affecting the comparative measuring process of internal diameter plain rings (Figure 3), has been identified. P-diagram Process of Comparative Dimensional Measurement of Internal Diameter Plain Ring in Mahr Opal ULM 600

a/a

Factor Description

Level

Factor

0

Internal diameter of the part to be measured

54,99900

1

Ambient Temperature of Measurement Room (Cooling System of Liebert Hiross-HPM for temperature and humidity control ) Temperature of Part (plain ring) (of 54,9990 mm internal diameter plain ring gage) Operator (level of proficiency)

1=19.00-19.70 oC, 2=19.71-20.50 oC

M Signal Factor Z1=A Control Factor

Relative Humidity of Measurement Room (Cooling System of Liebert Hiross-HPM for temperature and humidity control ) Interaction “Part- Operator” Interaction “Ambient Temperature Operator” Interaction “Ambient Temperature Part” Temperature of reference part (plain ring) (of 14,0000 mm internal diameter reference plain ring)

1=38.00-43.00 %, 2=43.01-48.00 %

2

Noise Factors-NF = ο C of reference plain ring

Nominal Value Internal Diameter of Plain Ring gage (54,9900mm)

Process of Comparative Dimensional Measurement

Result of Internal Diameter Measurement

3

Control Factors-CF = Ambient Temperature (o C), 55mm Plain Ring Temperature, Humidity, Operator Interactions

4

Fig. 3. P-diagram for Internal Diameter Plain Ring Measurement process

5 6

The P-diagram of the plain ring measurement process gives: Input/Signal Factor: the middle calibrated value of the part to be measured. Output/Response: the result of measurement (true value). Type of problem: Static (one signal factor always, the middle calibrated value of the part to be measured). Part to be measured: a plain ring of 54.99900 mm internal diameter.

7

8


1=19.00-19.70 oC, 2=19.71-20.50 oC

Z2=B Control Factor

1=Operator, 2=Expertise

Z3=C Control Factor Z4=D Control Factor

2

Z5=BXC

2

Z6=AXC

2

Z7=AXB

1=19.00-19.70 oC, 2=19.71-20.50 oC

X1=N Noise Factor


981


called outer array.

Hence, based on Taguchi’s quality loss function [2] the above quality characteristic is classified as Nominalthe-Best and thus all statistical analysis that follows makes usage of the equations developed for this case. III.4. Selection of Appropriate Orthogonal Array (OA), (Parameter Design / Step 4) The selection of OA is based on the number of factors and the number of levels of factors [2]-[5]. This selection should satisfy the following conditions:

Fig. 5. Linear Graph for the L8(27) Experiment TABLE III INFORMATIVE TABLE FOR THE FACTORS ASSIGNMENT

νLN ≥ νrequired for factors and interactions where νLN are the degrees of freedom available in an OA νLN - N-1, where N is the number of trials and ν (for a factor) = number of levels -1. The selection of a suitable Orthogonal Array (OA) depends on the total degrees of freedom required to study the main and interactions effect [3]-[5]. In accordance to Table II and their degrees of freedom, an L8 OA is selected based on the inequality. The international symbolism for the orthogonal arrays is Ln(XY), [3]-[5]. In the experiment, the L8 (27) OA symbolism, means that we have an experiment with:

a/a

Factor

d.o.f.

1 2 3 4 5 6 7 8

A B C D BXC AXC AXB N

2-1=1 2-1=1 2-1=1 2-1=1 (2-1)X(2-1)=1 (2-1)X(2-1)=1 (2-1)X(2-1)=1 2-1=1

OA Column Assignment 1st 2nd 4th 7th 6th 5th 3rd Outer array

TABLE IV ASSIGNMENT OF FACTORS AND THEIR INTERACTIONS IN L8 OA N N1 N2 N1 N2 N1 N2 1 2 3 4 5 6 7 1 2 1 2 1 2 Exp A B AXB C AXC BXC D R1 R2 R3 R4 R5 R6 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 3 1 2 2 1 1 2 2 4 1 2 2 2 2 1 1 5 2 1 2 1 2 1 2 6 2 1 2 2 1 2 1 7 2 2 1 1 2 2 1 8 2 2 1 2 1 1 2

8 = number of runs 2 = number of levels 7 = number of factors. The optimum assignment of factors and their interactions in orthogonal array achieved with the usage of the Taguchi’s triangular tables and linear graphs tools [2]-[4]. The interactions will be assigned in the following columns, in accordance with the standard Triangular Table of Taguchi: the AXB in column 3, the AXC in column 5 and the BXC in column 6.

III.5. Preparation of the Experiment (Parameter Design / Step 5) For the experimental conduction, were prepared the measurement equipment (length measuring instrument), the parts (plain ring and reference plain ring) and the cooling system in accordance to the measurement method and the adjustments as seen in the Table II. The measurements conducted by two operators with different metrological experience. TABLE V ADJUSTMENT FACTORS IN COOLING SYSTEM & ALMEMO OF LMI Ambient Relative Parts Temperature Moisture Temperature Adjustment Adjustment Adjustment

Fig. 4. Triangular Table of Taguchi for interactions in L8 Orthogonal Array

Similarly, the Linear Graph of Taguchi, verifies the above assignment of the seven factors as in the Fig. 5. The third column of the Table III reflects the degree of freedom for each factor and the forth column reflects the OA column assignment. In the columns 1 to 7 of the L8 OA (Table IV) are assigned the factors and their interactions (this part of OA called inner array). In the columns R1 to R6 will be placed the measurement results of the experiments conduction (horizontal lines combination of conditions = runs). This part of OA

1=19.00-19.70 oC, 2=19.71-20.50 oC

1=38.00-43.00%, 2=43.01-48.00 %

1=19.00-19.70 oC, 2=19.71-20.50 oC

III.6. Implementation of the Experiment (Parameter Design / Step 6) The experiments were conducted with appropriate



982


C D A*B A*C B*C

sequence and timing in order to avoid errors of repeatability and reproducibility (used by the proposed R&R provision of Minitab v.15). Six sets are conducted the results of which were recorded on the relative Minitab worksheet.

Analysis of Variance for SN ratios Source DF Seq SS Adj SS Adj MS F P A 1 2,6285 2,6285 2,6285 * * B 1 4,2029 4,2029 4,2029 * * C 1 7,0718 7,0718 7,0718 * * D 1 1,6714 1,6714 1,6714 * * AXB 1 8,6388 8,6388 8,6388 * * AXC 1 18,4936 18,4936 18,4936 * * BXC 1 13,1941 13,1941 13,1941 * * Residual Error 0 * * * Total 7 55,9012 Response Table for Signal to Noise Ratios Nominal is best 10*Log10 Ybar**2/s**2 Level A B C D 1 104,3 104,4 102,8 103,3 2 103,2 103,0 104,7 104,2 Delta 1,1 1,4 1,9 0,9 Rank 3 2 1 4 Response Table for Means Level A B C D 1 55,00 55,00 55,00 55,00 2 55,00 55,00 55,00 55,00 Delta 0,00 0,00 0,00 0,00 Rank 1 2 3 4 Response Table for Standard Deviations Level A B C D 1 0,000351 0,000351 0,000413 0,000392 2 0,000401 0,000401 0,000338 0,000360 Delta 0,000050 0,000050 0,000075 0,000032 Rank 3 2 1 4

Fig. 6. Minitab Worksheet for the L8(27) Experiment

III.7. Experiment Results Analysis with Minitab v.15 (Parameter Design / Step 7) With a range of appropriate options from the menu of Minitab, recording all the necessary parameters of the experiment such as: Type of design (2-Level Design), number of runs (L8), Assign Factors, Interactions, Measurement Results, Columns of Responses, Type of results (S/N, Means, Standard deviations, Anova), Type of the objective function (Nominal is best), Predicted values. The overall picture and results of the L8(27) experiment project by Minitab, appears in the following worksheet and the session of data.

As the Analysis of Variance for S/N ratios shows, we have strong interactions, in order of importance, between AXC (Ambient Temperature of Measurement Room X Operator) factors, between BXC (Temperature of Part to be measured X Operator) and AXB (Ambient Temperature of Measurement Room X Temperature of Part to be measured) factors. As the Response Tables for S/N ratio shows the most important factor in our measurement process is C (the Operator) and follows the B (Temperature of plain ring), A (temperature of measurement room) and negligible D (relative moisture). The levels that minimize the variation (or optimize the process) are the level 2 for the operator (expertise) the level 1 for the factor B, the level 1 for the factor A and the level 2 for the factor D [6]-[7].

Fig. 7. Minitab Worksheet for the L8(27) Results Experiment

III.8. Export Optimal Design: Optimum Factors Levels (Interpretation and Conclusion of Experiment / Step 8)

Session of L8(27) experiment (Minitab data analysis): Welcome to Minitab, press F1 for help, Taguchi Design / Taguchi Analysis: Term A B

Minitab v.15 used for the exploitation of predicted values: Predicted values : S/N Ratio Mean StDev Ln(StDev)



983


54,9995 0,0002395

A takes a value of 2.

-8,33716

Factor levels for predictions: A B C 1 1 2 19.00 oC 19.01 oC Expertise 19.70oC 19.70oC

Interaction Plot for SN ratios

D 2 43.01% 48.00%

Data Means 106

C Χειριστής) 1 2

105 104 SN ratios

107,223

Best settings / adjustments: A1B1C2D2 The S/N ratios under the optimum and initial conditions, denoted by ηoptimum and ηpresent respectively, are predicted by:

103 102 101 100 1

ηoptimum - ηpresent = 107,22 - 101,12 = 6,1 dΒ

2 A (oC Α ΕΣ)

Signal-to-noise: Nominal is best (10*Log10(Ybar**2/s**2))

where:

Fig. 9. Interaction Plot for SN ratios (AXC)

ηopt = 104,3+104,4+104,7+104,2-3Χ103,725=107,22 dΒ Interaction Plot for SN ratios

and:

Data Means 107

C Χειριστής) 1 2

ηpresent = 103,2+103,0+102,8+103,3-3x103,72=101,12 dΒ 106

SN ratios

In addition Minitab v.15 used for the exploitation of Main Effects Plot. The Main Effects Plot for SN ratios below shows that the greatest impact on S/N ratio is the factor C-operator taking the value of Level 2. Unlike the factors A and B give higher yield (S/N) values in level 1. Finally, the factor D affects less than the other three factors (less verticality) and more regulation (level 2).

105 104

103 102 1

2 B (oC ∆ακτυλίου)

Signal-to-noise: Nominal is best (10* Log10(Ybar* * 2/s* * 2))

Fig. 10. Interaction Plot for SN ratios (BXC)

Main Effects Plot for SN ratios Data Means A (oC ΑΕΣ)

B (oC ∆ακτυλίου)

Interaction Plot for SN ratios Data Means

104,0

106,5

103,5

106,0

103,0

B (oC ∆ακτυλ ίου) 1 2

105,5 1

2

1

C Χειριστής)

2

SN ratios

Mean of SN ratios

104,5

D (Υγρασία)

104,5 104,0

105,0 104,5 104,0 103,5

103,5

103,0

103,0 1

2

1

102,5

2

1


2 A (oC ΑΕΣ)


Fig. 8. Main Effects Plot for Means

Fig. 11. Interaction Plot for SN ratios (AXB)

Strong interaction between A and C factors (AXC): The level 2 of C increases as the A takes a value of 2, while the level 1 of C decreases as the A factor takes a value of 2, [6]. Strong interaction between B and C (BXC): The Level 2 of C decreases as the B takes a value of 2, while the level 1of C increases as B take a value of 2. Strong interaction between A and B (AXB): The Level 2 of B increases slightly as the A factor takes a value of 2, while the level 1 of B factor decreases as the

III.9. Running of Confirmation (Experiments and Final Conclusions / Step 9) Three confirmation experiments running in best settings of A1B1C2D2. Minitab v.15 used for the calculations: Taguchi Analysis: Y1; Y2; ... versus A; Β; C; D



984


measurement process. A procedure of recovery and reuse of this experimental array at regular intervals for the quality improvement has been included in the Quality Management System of the laboratory. In summary, the proposed method achieved for the first year of operation of the laboratory, improved process measurement, through identifying easily controlled factors and their arrangements, which minimize variability. These factors were adjusted to optimal levels and the process is robust against possible changes in environmental conditions. In addition the number of lengthy trials has been greatly reduced at 8 instead of 128 that would be required if traditional (factorial) design of experiments (DOE) had been used. Cost in money and time has thus been reduced.

Predicted values S/N Ratio Optimum conditions: 107,223 dB. 1st Confirmation Experiment 106,913 dB. 2nd Confirmation Experiment 107,012 dB. 3rd Confirmation Experiment 106,998 dB. TABLE VI PERFORMANCE OF COMPARATIVE INTERNAL DIAMETER PLAIN RING PROCESS MEASUREMENT (DB) 1 2 3 Average Expi ηi 106,913 107,012 106,998 106,974

The above verification process showed that the confirmation experiments are within acceptable limits. Therefore, these optimum settings are adequate to describe the achievement of measurement process robustness for the current time period.

A 1 19.00 oC 19.70oC

TABLE VII BEST SETTINGS OF FACTORS B C 1 2 19.01 oC Expertise 19.70oC

IV.2. Strength and Weakness Points The arguments over Taguchi's methods could indeed prove counterproductive. The statistically fearful may now be afraid of easy-to-use methods and, consequently, not use any of the statistical methods. Even though there are areas where Taguchi's methods do not apply, there are other areas they have proven themselves to be valid. It is important that reason and objectivity not be lost. In general, however, we think DOE when we have no idea about the fundamental mechanisms governing our process, when we have no idea about interactions between our inputs, and when experiments take so long that we absolutely must get it right the first time. We think Taguchi and Robust Engineering when we have a fairly firm grasp of the underlying processes and are just trying to optimize an additional notch, when robustness or consistency of output is just as important as maximizing (or minimizing) our output, when we cannot afford to test all possible combinations of inputs, and when we have the luxury of going back and doing a final confirming test later. As always, we try to understand the principles, and then we do what makes sense.

D 2 43.01% 48.00%

or in letters form, Best Settings: A1B1C2D2. III.10.

Improvement of the Quality Management System / Step 10

A new infrastructure and equipment has been installed in the laboratory under consideration. The laboratory has been accredited for length measurements according to the standard ISO17025:2005. In order to satisfy the requirement of continuous improvement, decisions had to be taken that concerned: - Further investment in equipment. - Investment in retraining operators. The highly automated equipment raised expectation that even non experienced operators could perform accurate and precise measurements. The results of the experiments led to the conclusion that further investment in the operators training is important. In addition the ambient temperature stability is very costly as the system must work for several hours (actually days) to stabilize the temperature. The experiments led to the best combination of the levels of the controlled parameters, including the targeted temperature, in order to arrive at robust results. When the removal of the cause of bad influence is impossible, or as in this case costly, the method seeks the elimination of the bad influence of the cause.

IV.

IV.3. Generally – Robust Engineering Although Robust Engineering has an important role in process improvement the successful application of this powerful methodology in testing laboratories and manufacturing sectors is still limited. A number of key points could increase the chances for making the application of Robust Engineering successful in organizations and are the object of further research: Regarding the quality management system: • Procedures of continuous improvement that lead to a clear understanding of the scope, feasibility and boundaries of the experiment. • Formation of a team involving the direction, quality managers, technical managers, operators, and all people involved in process improvements. • Conducting a number of exhaustive and detailed brainstorming sessions to ensure that all the factors which presumably influence the performance and the

Conclusion

IV.1. For the Laboratory The proposed method is an important tool for continuous improvement leading to a robust



985


• • • • • • •

Engineering, IREME, Vol. 2 n. 2, pp. 296 – 303. [14] G. A. Ibrahim, C. H. Che Haron, J. A. Ghani, H. Arshad, March 2010 , ‘Taguchi Optimization Method for Surface Roughness and Material Removal Rate in Turning of Ti-6Al-4V ELI’, International Review of Mechanical Engineering, IREME, Vol. 4. n. 3, pp. 216-221.

variability of the process have been taken into consideration. Regarding the application of the method: Determine what to measure and how. Re-consider the measurement procedure for the measurement of the quality characteristic. Choose the experimental parameters to be studied. Select the most suitable orthogonal array for the experiment. Randomize the trials if necessary. Replicate the trials if noise factors are suspected but cannot be studied for the experiment. Regarding the validation of the results: Perform confirmatory runs or experiments to validate the results. Participation in inter-laboratory comparison scheme when possible. Back to the quality management system: Review of the continuous improvement procedure in order to include the achieved results.


Technological Educational Institution of Serres, Greece. 2 National Technical University of Athens, Greece.

Caterina Poustourli 2011 PhD in Quality/Robust Engineering, National Technical University of Athens, School of Mechanical Engineering, Section of Industrial Management and Operational Research, Greece. 1992 Dipl. Production & Management Engineer, Technical University of Chania, Crete, Greece. 2006 till now: Studies & Construction Department/Directorate of Technical Services & Information Systems, Responsible for Health & Safety/Energy/Quality/Maintenance topics, Technological Educational Institution of Serres, Greece. 2000 till now: Certification Division, Hellenic Organization of Standardization, ISO Management Systems Auditor (freelance), Greece 2001-2006: Carrier Services Office - Head Office, Technological Educational Institution of Serres, Greece 1998-2006: Business Administration Dept. School of Business Administration - Invited lecturer, Technological Educational Institution of Serres, Greece 1996-2001: Dromeas SA, Office Furniture Industry, ISO Management Standards Systems Manager, Inventory Management, Serres, Greece 1992-2006: ISO Management Standards Systems Consultant, Greece 1992 till now: Accredited Instructor in Continuing Vocational Training Programmes in topics of TQM, Health & Safety, Energy, Environment, Production Systems etc. She specializes in ISO Management Systems (Management Standards Systems and Integration Management Systems).

References [1]

[2] [3] [4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12] [13]

Lisenkov a. n., Measuring Techniques, Methodology of the Analysis of Variability and Robust Optimization for Quality Control Systems and Measurement Problems, Springer, Vol. 45, n. 7, pp. 679-688, 2002. G. Taguchi, S. Chowdhury, Y. Wu, Taguchi’s Quality Engineering Handbook (1st edition, J.Wiley & Sons, 2005). G. Taguchi, S. Chowdhury, S. Taguchi, Robust Engineering (1st edition, McGraw Hill, 2000). Veronica Czitrom, Communications in Statistics-Theory and Methods, Taguchi methods: linear graphs of high resolution (24th edition, Taylor & Francis, 2011). Ranjit K. Roy, Design of Experiments Using the Taguchi Approach: s: 16 Steps to Product and Process Improvement (1st edition, Wiley, 2001). Caterina Poustourli, Vrassidas Leopoulos, “Integration Management System of a Dimensional Laboratory”, 6th International Conference “Standardization, Protypes and Quality: A Means of Balkan Countries’ Collaboration”, Thessaloniki Greece, 9-10/10/2009, ISBN: 978-960-87973-9-0, pp.129-141. Caterina Poustourli, Vrassidas Leopoulos, “Taguchi methods to minimize variation: an example with plain ring gages”, 6th International Conference “Standardization, Protypes and Quality: A Means of Balkan Countries’ Collaboration”, Thessaloniki Greece, 9-10/10/2009, ISBN: 978-960-87973-9-0, pp.195-209. Leopoulos V., Voulgaridou D., Bellos E., Kirytopoulos K., 2010, ‘Integrated management systems: Moving from function to decision view’ The TQM Journal, Emerald, vol. 22 no. 6, pp. 594628. Cudney E., Fargher J., (2005) “Lean and Six Sigma: Integrating Lean and Six Sigma in a Systematic Approach”, and Cudney, E. “Applying Robust Engineering to Six Sigma”, Proceedings of the Institute of Industrial Engineers International Conference. VDI/VDE 2627 Blatt 1. Meßräume - Klassifizierung und Kenngrößen - Planung und Ausführung (Measuring rooms Classification and characteristics), Ausgabe: 1998. VDI/VDE 2627 Blatt 2. Meßräume – Leitfaden zur Planung, Erstellung and zum Betrieb (Measuring rooms, Guide for planning, construction and operation), Ausgabe: 2005. NPL, Measurement Good Practice Guide No. 80, Fundamental Good Practice in Dimensional Metrology, pp. 26-27. I. Belaidi, K. Mohammedi, B. Brachemi, March 2008, ‘An Efficient Tolerances Optimization Algorithm Based in Taguchi Gradients Method’, International Review of Mechanical

Vrasidas I. N. Leopoulos Associate Professor. National Technical University of Athens, Heroon Polytechniou 9, 15780 Zografou, Athens, Greece, School of Mechanical Engineering, Section of Industrial Management and Operational Research 1985 Phd Université PARIS IX Dauphine 1983 Diplôme d’Etudes Approfondies (DEA), Ecole des Mines, ENSAE, Université PARIS IX Dauphine 1980 Diploma, Mechanical and Electrical Engineer, National Technical University of Athens 1988 Invited lecturer in the Technical University of Crete 1995 Lecturer in the National Technical University of Athens 2001 Assistant Professor in the National Technical University of Athens 2008 Associate Professor in the National Technical University of Athens He specializes in Quality Management Systems, Project Management, Industrial Facilities Planning and Industrial Information Technology. He is the director of the Metrotechnics Laboratory of National Technical University of Athens. Prof. Leopoulos is a Member of the Technical Chamber of Greece, Member of the Greek Association of Mechanical and Electrical Engineers, Member of the Hellenic Management Association - Institute of Production and Operations Management and Member of the ΡΜΙ Project Management Institute.



986


Analytical Solution to Transient Temperature Field in Semi-Infinite Body Caused by Moving Ellipsoidal Heat Source Aniruddha Ghosh1, N. K. Singh2, Somnath Chattopadhyaya2 Abstract – Submerged Arc Welding process (SAW) provides high quality voluminous deposition process. It has lot of social and economical implications. Lot of critical input variables are involved in this process which are needed to control to get quality weld. Main input variables of this process are function of temperature distribution on the welded plates. Critical investigation of the transient temperature distribution is important for maintaining quality of the Submerged Arc welded plates. This paper makes an attempt to uncover an important domain of the studies of temperature distribution during submerged arc welding process. This analysis may pave the way for the application of micro structural modeling, thermal stress analysis, residual stress/distribution and simulation in welding process. Prediction of temperature variations of entire plates during welding is done through an analytical solution. It is derived from the transient multi dimensional heat conduction of semi infinite plate. The heat input that is applied on the plate is considered to be same amount of heat lost form electric arc. It is assumed that a moving double ellipsoidal heat source with Gaussian distribution is responsible for Submerged Arc Welding process for some input parametric settings. In the analysis it has been observed that the predicted temperature distribution values are in good agreement with the experimental results for a particular input parametric setting. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Submerged Arc Welding, 3-D Gaussian Heat Distribution, Multi Dimensional Heat Flow Equation, Ellipsoidal Heat Source

fabrication processes in industries because of its inherent advantages, including deep penetration and a stable bead geometry. It is used to join the plates of higher thickness in load bearing components. This joining process is critically important for fabricating structures, bridges, ships, boilers, etc. This method of arc welding provides a purer and cleaner high volume weldment that has relatively higher material deposition rate in comparison to the traditional welding methods. In a country like India, in the context of infrastructural development, the SAW process has much more useful applications in welding of critical components and equipment. Use of this technology has huge economic and social implications in the national perspective. A common issue in the application of SAW process raises a concern about the uncertainties involved with the bead geometries. The most concerned query is about the formation of the bead of undesired shape and sizes and that imparts some uncertainties in the welded quality. The output response, bead geometry depends on the input variables current, voltage, travel speed etc. These input variables influence the heat input into the welding. So the heat input is the key factor for quality of welding process and for the shape of weld bead. Lot of research works have been continuing on effects of heat input on welding quality. Generally moving heat source is applied in welding process.

Nomenclature α Q Q0 T T0 v x,y,z x’,y’,z’ t t’ a’,a”,b,c ρ L B P ff, fb

Thermal diffusivity Heat distribution Net heat input per unit time Temperature of body Initial Temperature of body Travel speed Position where temperature is measured Position heat source at time t’ Time of temperature measurement at a chosen point Time when position of heat source is at (x’,y’,z’) Ellipsoidal Heat source parameters Density of the welded plates Specific heat at solid phase Reinforcement height Bead width Penetration Proportion coefficient representing heat appointment in front and back of the heat source respectively

I.

Introduction

Submerged arc welding (SAW) is one of the major Manuscript received and revised June 2011, accepted July 2011


987

Aniruddha Ghosh, N. K. Singh, Somnath Chattopadhyaya

modified Rosenthal’s theory to include a two dimensional (2-D) surface Gaussian distributed heat source with a constant distribution parameter (which can be considered an effective solution of arc radius) and found an analytical solution for the temperature of a semi-infinite body subjected to this moving heat source. Their solution is a significant step for the improvement of temperature prediction in the near heat source regions. Jeong and Cho [12] introduced an analytical solution for the transient temperature field of fillet welded joint based on the similar Gaussian heat source but with different parameters (in two directions x and y). Using conformal mapping technique, they have successfully transformed the solution for temperature field in the plate of finite thickness to the fillet welded joint. Even though the available solutions using the Gaussian heat sources could predict the temperature at regions closed to the heat source. They are still limited by the shortcoming of the 2-D heat source itself with no effect of penetration. Goldak, et al.[13] (3-D), first introduced the three dimensional double ellipsoidal moving heat sources. Finite element modeling (FEM) was used to calculate the temperature field of a bead-on-plate and showed that this 3-D heat source could overcome the shortcoming of the previous 2-D Gaussian model to predict the temperature of the welded joints with much deeper penetration. However, up to now, an analytical solution for this kind of 3-D heat source was not yet available [13], and hence, researchers must rely on FEM for transient temperature calculation or other simulation purposes, which requires the thermal history of the components. Therefore, if any analytical solution for a temperature field from a 3-D heat source is available, a lot of CPU time could be saved and the thermal-stress analysis or related simulations could be carried out much more rapidly and conveniently. Nyguyen, et al. [14] derived analytical solution for the transient temperature field of the semi infinite body subjected to 3-D power density of a dynamic heat source (such as semi-ellipsoidal and double ellipsoidal heat source). However, the results are not satisfactory with the single semi-ellipsoidal 3-D heat source with respect to the double ellipsoidal one. In present paper double ellipsoidal heat source model has been utilized to predict the temperature distribution. Basic deference of this paper w.r.t other related papers is consideration of equal Proportion coefficient representing heat appointment in front and back of the heat sources. Good agreement between predicted and experimental results for a particular input parametric setting has been achieved.

More than forty years ago the first attempt was reported to model a travelling point heat source for the case of welding process. The stationary and dynamic heat sources are frequently modeled for many popular manufacturing processes. In recent years, application of localized heat source has been employed for laser and electron beam based methods in material processing, such as welding, cutting, heat treatment of metals and manufacturing of electronic components [1]-[2]. Analytical and numerical models for the prediction of the thermal fields induced by the stationary or dynamic heat sources are useful tools for investigating the specific cases problems as mentioned [2]. In some laser beam applications, such as surface heat treatment, the contribution of convective heat transfer is also considered for in-depth analysis [3]. Quasi-steady state thermal fields induced by moving localized heat sources have been extensively investigated [3], [4]. However, further attention seems to be devoted to the critical analysis of temperature distribution in the transient heat conduction domain due to the significant role of the temperature distribution on the residual stresses, distortion and the fatigue behavior of the welded structures. From the existing literature, a significant number of closely related works are found. Classical solutions for the transient temperature field such as Rosenthal’s solutions [5] are dealt with the semi infinite body subjected to an instant point heat source, line heat source or surface heat source. These solutions can be adroitly employed to predict the temperature field at a distance far away from the heat source. Rykalin [6] pointed out that a heat flow model needs to be considered for the factors mainly variable thermal properties, temperatures of phase transformation, the magnitude of heat and characteristics of heat distribution, plate geometry, convection and surface depression in the weld pool [7]. The work of Grosh [8] showed that that the effect of nonconstant thermal properties can only make a 10-15 % difference in the weld pool geometry and Malmuth [9] has shown that the effect of latent heat can only make a 5-10 % difference. Obviously, the heat input magnitude solution of Rosenthal modified for non constant thermal properties and latent heat, cannot explain unambiguously the difference between theoretically predicted values and experimentally measured observations related to the heat distribution of the plate surface. Fortunately several investigators have been measured for actual heat distributions in arcs on a water cooled copper anode [10]. Using these results it is possible to determine the presence of a distributed heat source rather than a point or linear source of heat. The assumptions used by Rosenthal are not pragmatic as they have not been included for heat convection in the weld pool and the latent heat of phase transformation. The only development from the previous work is that a distributed heat source is considered instead of a point source. Rosenthal fail to predict the temperature in the vicinity of the heat source [5]. Eager and Tsai [11]

II.

Experimental Procedure

The experiments were conducted as per the design matrix randomly to avoid errors due to noise factors. The job (300x150x20 mm - 2 pieces) is cut and V groove of angle 60o as per the standards are prepared. The job was firmly fixed to a base plate by means of tack welding and then the submerged arc welding was finally carried out.



988


The welding parameters were recorded during actual welding to determine their fluctuations, if any. The slag was removed and the job was allowed to cool down. Welding is carried out for the square butt joint configuration. The job was cut at three sections by a hacksaw cutter and the average values of the penetration, reinforcement height and width were recorded using digital venire caliper of least count 0.02mm. Fig. 2(b). Spherical heat source in which heat is distributed in a Gaussian manner throughout the heat source’s volume

Fig. 1. Bead geometry, P-Penetration, L-Reinforcement height, B-Bead Width

Job No. A1 A2 A3 A4 B1 B2 B3 B4

TABLE I OBSERVED YIELD PARAMETERS OF SAW PROCESS Heat Penetration Reinforcement Bead Width Input (mm) Height(mm) (mm) (kJ/mm) 3.09 6.8 2.4 18.0 4.32 3.8 2.4 22.0 3.97 6.7 3.2 21.0 5.56 8.3 2.8 31.0 1.75 5.4 1.1 14.1 2.45 4.5 1.8 20.2 2.25 6.7 2.3 16.0 3.09 7.8 2.0 23.0

Fig. 2(c). Conical heat source in which heat is distributed in a Gaussian manner throughout the heat source’s volume

IV.

Gaussian Heat Distribution

Let us consider a fixed Cartesian reference frame x, y, z. Initially proposed a semi-Central Conicoidal heat source in which heat is distributed in a Gaussian manner throughout the heat source’s volume. The heat density q(x, y, z) at a point (x, y, z) with in semi ellipsoidal is given by the following equation:

III. Prediction of Temperature Distribution III.1. Graphical Representation of Gaussian Heat Distribution Figs. 2 show heat source volume/shape in which heat density is distributed in a Gaussian manner throughout the heat source’s volume. Major process control parameters (i.e. current, voltage, travel speed, stick out, electrode wire diameter, polarity etc.) are involve in Submerged Arc Welding operation which forms the heat input function and the shape of heat source is varied with change of the major process control parameters of SAW process. This is the main motivation for considering an ellipsoidal heat source. Parameters controlling the heat source are obtained through the measurement of weld pool geometry.

, ,

0

(1)

where q (0) is Gaussian heat distribution parameter and a, b, c are central Conicoidal heat source parameters. If Q0 is the total heat input, then: , ,

2Q0= q(0) =

√

×Q0

dxdydz (2)

Here, Q0=I×V×arc efficiency; V, I welding voltage, current and respectively. Arc efficiency is taken 1 for submerged arc welding process. It has been observed that the calculated temperature distribution on different points of submerged arc welded plates with the help of aforesaid heat distribution equation is not agreed with experimentally observed temperature values. To overcome this, combined two semi-Central Conicoids and a new heat source called “double central Conicoidal heat source” has been proposed. Double ellipsoidal heat source: Since two different semi Central Conicoidal are combined to give a new heat

Fig. 2(a). Sketch of the work piece (semi-infinite body) for submerged Arc Welding process



989


source, the heat density within the each semi Central Conicoidal heat source are described by different equations. For a point (x, y, z) with in the first semiCentral Conicoid located in front of welding arc, the heat density equation is described as: √

q(x,y,z) =

×Q0 ff×

source applied at point (x’, y’, z’) at instant t’, assuming the body to be infinite with an initial homogeneous temperature. Then, due to the linearity of equation (19), the temperature variation induced at point(x, y, z) at time t by instantaneous heat source of magnitude is q(x’,y’,z’,t’) applied at (x’, y’, z’) at time t’ is

(3)

q (x’, y’, z’, t’) G (x-x’, y-y’, z-z’, t-t’) Assuming that heat has been continuously generated at point (x’, y’, z’) from t’=0, the temperature increment at point(x, y, z) at time t is:

and for point (x, y, z) with in the semi- ellipsoid covering rear section of the arc, as: √ "

q(x,y,z) =

"

×Q0 ff×

(4)

’, ’, ’, ’ G (x-x’, y-y’, z-z’, t-t’) dt’

where a’, a”, b, c are ellipsoidal heat source parameters; ff,, fb = proportion coefficient representing heat appointment in front and rare of the heat source, respectively, where ff + fb=2. Let, ff=1; fb=1.

V.

If we assume that the heat has been continuously generated from t’=0 throughout an infinite medium, the temperature increment at any point (x, y, z) and at any instant t takes the form: ∆ (x, y, z, t) = = .

Moving Double Ellipsoidal Heat Source Problem

Let us consider a heat source located at x=0 at time t=0 moves with constant velocity v along the x axis and heat emitted at a point (x, y, z) at instant t by the double ellipsoidal heat source. Mathematical expression of double ellipsoidal heat source is (5):

∆

=

√ √

(x, y, z, t)= ′

= ′

[ ′, [√ " [ x ′,

{

′

"

′

[(Ib ×Ic) × (Ia’ + Ia”)]dt’

, FOR x ′

, FOR

(9)

or

}

T(x, y, z, t) –T0= =

V.1.

(8)

Then the temperature induced by the double Central Conicoidal heat source defined by equation is:

Q(x, y, z)= √

q x’, y’, z’, t’ G (x-x’, y-y’, z-z’, t-t’) ′ ′ ′dt’

′

[(Ib ×Ic) × (Ia’ + Ia”)]dt’

Induced Temperature Field

Heat conduction in a homogeneous solid is governed by the linear partial differential equation:

where, T(x, y, z, t), T0 are the temperature at point (x,y, z) any time t, and initial temperature of test specimen respectively, and:

(6) T+q= (18) where T=T(x, y, z, t) is the temperature at point (x, y, z) at point (x, y, z) at time t, q the heat source, is the density, c is the heat capacity and k is the thermal conductivity of the plates of welded plates. The fundamental solution of equation (18) is the Green function:

Ia’ =

Ia” =

′

′

"

"

Ib = G (x-x’, y-y’, z-z’, t-t’) = ′

=

′

(7)

Ic =

/

It is worth noting this double Central Conicoidal distribution heat source is described by four unknown parameters: a’, a”, b, c. Goldak, et al. [13] implies equivalence between the source dimensions and those of

where α =k/ ) is the thermal diffusivity. Equation (6) gives the temperature increment at point (x, y, z) and instant t due to an instantaneous unit heat



990


the weld pool and suggested that appropriate values of a’, a”, b, c could be obtained by measurement of weld pool geometry as shown in Goldak et al.,2005 [23], where a’= , a”= , b= , c= . ′ " Values of (m’), (m”) are taken in front and rare portion of weld bead along x-axis for the two ellipsoids having equations: "

For example, after one second from the starting of welding process, predicted temperature is 1074.7K and measured temperature is 1073K. This result proves that shape of moving heat source on the welded plates is double ellipsoidal for particular input parametric setting (for heat input 3.09kJ/mm of Table I).

VII.

1 1

Heat distribution on welded plate is ellipsoidal shape for Submerged Arc Welding process for particular input parametric setting and parameters of this heat source can be measured from the dimension of bead geometry. Transient temperature distribution on welded plate can be calculated with the help of Gaussian ellipsoidal Heat distribution technique. In this study, analytical solutions for the transient temperature field of a semi infinite body subjected to 3-D power density moving heat source (such as double ellipsoidal heat source, which is first time attempted in this work) were found and experimentally validated. The analytical solution for double ellipsoidal heat source was used to calculate transient temperatures at selected points on a mild steel plates which are welded by taking x- axis along welding line, origin is starting point of welding, yaxis is perpendicular to welding line and z-axis towards plate thickness. Both numerical and experimental results from this study have showed that the present analytical solution could offer a very good prediction for transient temperatures near the weld pool, as well as simulate the complicated welding path. Furthermore, very good agreement between the calculated and measured temperature data indeed shows the creditability of the newly found solution and potential application for various simulation purposes, such as thermal stress, residual stress calculations and microstructure modeling etc.

and, half of bead width= n , penetration=o. It has been found from experiment for submerged arc welding process that 3.8m’=m”, and: n = 0.6m’ (for heat input 3.09kJ/mm of Table I).

VI.

Conclusion

Comparison of Predicted and Experimental Data

Fig. 3 shows a comparison of Predicted (from equation (9)) and Experimental Temperature distribution at the position of 15 mm away from weld line (at the point x=0,y=15mm,z=0) on the plate top surface for square butt welding of 20 mm thick plate (for heat input 3.09kJ/mm of Table I; heat input-2.84kJ/mm, k=54W/m o C, ρ=7833kg/m3 ,i.e. α=k/ cp)=1.48×10-5 m2/s ,cp=465 J/kg 0C). Lot of critical set of input variables i.e. current, voltage, electrode diameter, travel speed, wire feed rate, stick out etc. are involved in submerged arc welding process. Temperature distribution patterns depend on these welding process parameters because these are the functions of heat input. So, shape of heat source will be changed with the change of input parameters of SAW process. From Figure 3, it has been revealed that very good agreement exists between predicted and measured temperature values on welded plates.

References [1] [2]

[3]

[4]

[5]

[6] [7] Fig. 3. Comparison of Predicted(from equation (9)) and Experimental Temperature distribution at the position of 15 mm away from weld line (at the point x=0,y=15mm,z=0) on the plate top surface for square butt welding of 20 mm thick plate (for heat input 3.09kJ/mm of Table I; heat input-2.84kJ/mm, k=54W/m oC, ρ=7833kg/m3 ,i.e. α=k/ cp)=1.48×10-5 m2/s ,cp=465 J/kg 0C)

[8] [9]


Tanasawa, I. and Lior, N., 1992, Heat and Mass Transfer in Material Processing, Hemisphere, Washington, D.C. Viskanta, R. and Bergman, T. L., 1998, Heat Transfer in Material Processing, in Handbook of Heat Transfer, Chap. 18, McGrawHill, New York. Shuja, S. Z., Yilbas, B. S., and Budair, M. O., 1998, Modeling of Laser Heating of Solid Substance Including Assisting Gas Impingement, Numer. Heat Transfer A, 33, pp. 315-339. Bianco, N., Manca, O. and Nardini, S., 2001, Comparison between Thermal Conductive Models for Moving Heat Sources in Material Processing, ASME HTD, 369-6, pp. 11-22. Rosenthal, D. 1941, Mathematical theory of heat distribution during welding and cut-ting, Welding Journal20 (5),pp. 220-s 234-s. Rykalin, N.N., and Nikolaev, A.V., Welding Arc heat Flow, welding in World, 9(3/4), 1971, pp.112-132. Malmuth,N.D.,Hall,W.F.,Davis,B.I.,and Rosen,C.D.,Transent Thermal Phenomena and Weld Geometry in GTAW,Welding Journal 53(9),1974,pp.388s. Grosh,R.J.,Trabant,E.A.,Arc Welding Temperature, Welding Journal,35(3),1956, pp. 396-s – 400-s. Lin, M.L., Influence of Surface Depression and Convection on weld Pool Geometry, Maters Thesis, MIT, Cambridge, MA, 1982.


991


[10] Nestor, O.H., Heat Intensity and Current Density Distribution at Anode of High Current Inert Gas Arcs, Journal of Applied Physics, 33(50,1967,pp.1638 – 1648. [11] Eager, T. W., and Tsai, N. S. 1983, Temperature fields produced by traveling distributed heat sources, Welding Journal 62(12), pp.346-s - 355-s. [12] Jeong, S. K., and Cho, H. S. 1997, An analytical solution to predict the transient temperature distribution in fillet arc welds. Welding Journal 76 (6), pp. 223-s - 232-s. [13] Goldak, J., Chakravarti, A., and Bibby, M. 1985. A Double Ellipsoid Finite Element Model for Welding Heat Sources, IIW Doc.No. 212-603-85. [14] Nguyen,N.T.,Ohta,A.,Suzuki,N.,Maeda,Y.,Analytical Solutions for Transient Temperature of Semi-Infinite Body Subjected to 3D Moving Heat Source, Welding Journal,August,1999,pp. 265-s – 274-s. [15] Painter, M. J., Davies, M. H., Batters by, S., Jarvis, L., and Wahab, M. A. 1996. A literature review on numerical modeling the gas metal arc welding process. Australian Welding Research, CRC. No. 15, Welding Technology. [16] Pillai,K.R,Ghosh,A.,Chattopadhyaya,S.,Sarkar,P.K., (2007) Some investigation on the Interaction of the Process Parameters of Submerged Arc welding, Manufacturing Technology & Research (An International Journal),Vol. 3No.1&2,June-July. [17] Gunaraj, G., Murugan, N., Prediction of Heat-Affected Zone Characteristics in Submerged Arc Welding of Structural Steel Pipes, Welding Research, January 2002,pp.94-s – 98-s. [18] Wang, Y., Tsai, H.L.,Impingement of filler droplets and weld pool dynamics during gas metal welding process, International Journal of Heat and Mass Transfer,Vol.44,isue11,June 2001,pp. 2067-2080. [19] Carraugo et.al, Microstructural change in high temperature heataffected zone of low carbon weldable “13% Cr” martensitic stainless steels, Stainless Steel World, October 2002, pp.16-23. [20] Murugan, N., Gunaraj, V., Prediction of Heat-Affected Zone Characteristics in Submerged Arc Welding of Structural Steel Plates, Welding Research,January,2002,pp.94-s – 98-s. [21] Dutta, P., Weld Pool Dynamics during Submerged Arc Welding Process, North Bengal University Review (Science & Technology), Vol.9.No.1 (June 1997),pp-83-88. [22] Sreedhar, U., Krishnamurthy, C.V., Balasubramaniam, K., Raghupathy, V.D. and Ravishankar, S., Modeling and Simulation for Temperature Prediction in Welding Using Infrared Thermography, Proceedings of the National Seminar & Exhibitionon Non-Destructive Evaluation,NDE 2009, December 10-12, 2009,pp.396-400. [23] Goldak, J.A., Akhlaghi, M., Computational Welding Mechanics, Springer, pp.31. [24] Kumar, A., Deb Roy, T., Calculation of Three dimensional electromagnetic force field during arc welding, Journal of Applied Physics, (July, 2003), Vol-94, No.-2, pp.1267-1277.


Govt College ofEngg. & TextileTechnology, Berhampore,WB,India. 2 ME &MME Dept, ISM, Dhanbad, India.

Aniruddha Ghosh is Assistant professor in Mechanical Engineering of Govt. College of Engg & Textile Technology, Berhampore, WB, India. He received B.E. degree from North Bengal University. His research interests, apart form Welding Technology, include Heat and Mass Transfer, Optimal System Design. He has several papers in the International Journals as well as National Journals. Dr N. K. Singh isAssistant Professor(Work Shop) of ISM,Dhanbada,India.His highest qualification is PhD(Engg).He has twenty years industrial experiences (two years & two months in M/c shop (production), eleven years and four months in Inspection and Quality Assurance, six years and nine months training experience in Metrology, SQC & ISO 9000 including workshop). Now he is working as faculty in deptt. of ME &MME,ISM,Dhanbad,India since 15.01.2008. Somnath Chattopadhyaya is Associate Professor in the department of Mechanical Engineering and Mining Machinery Engineering of Indian School of Mines, Dhanbad, India. He received his bachelor of Engineering (Production) degree from Jadavpur University, Kolkata, M.Tech (Mechanical) from Indian School of Mines, Dhanbad and PhD (Production) from Birla Institute of Technology, Mesra, Ranchi . His area of Specialization is Advanced Manufacturing Systems. He has near about 10 technical papers in the impacted International Journals and about 10 technical papers in the reputed National Journals. He has delivered several key note Lectures the international conference of East European countries. He is the author of a book “Transfer of Transfer of Agile Manufacturing Technology to Small Scale Industries” - LAP Lambert Academic Publishing, March 2011.



992


Research Methodology of an Integrated Approach for Thermal Mapping of Hot Section Components of Gas Turbine Engines Sachin V. Bhalerao1, A. N. Pawar2, Atul B. Borade1 Abstract – High levels of thrust demanded from modern gas turbines have led to higher turbine entry temperatures with concomitant detrimental effect on the hot section components such as turbine rotor disc, blade and nozzle guide vane.In a quest to increase the efficiency and power of a gas turbine engine designers are continuously trying to raise the maximum turbine inlet temperature .Further economical and todays environmental concerns continue to provide impetus for operating the aeroengines at ever increasing temperatures thereby improving the thermodynamic efficiency and reducing pollutant emissions. But on the other hand, the maximum allowable metal temperature needs to be limited to 1400oC even in the case of the most advanced super alloys. Accurate full-field assessment of metal temperatures of hot section components is essential for gas turbine designers not only to prevent hot spots but also to produce reliable and durable engines. Gas turbine engineer would have an affluence of useful data if a permanent fullfield record of actual hardware temperatures, i.e., metal temperatures, could be made from an operating gas turbine. Conventional thermometry falls short to record the thermal gradient across the various hot section components due to certain limitations. Temperature sensing thermal paints provide a better alternative but the data interpretation issue using the thermal paints is still under research. In this article an inline effort taken to develop a proper research methodology for implementation of a thermal mapping technique of gas turbines using thermal paints is discussed. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Thermal Mapping ,Gas Turbines, Thermal Paints, Automated Thermal Interpretation

I.

Gas turbine plant owners are, for obvious reasons, very interested in keeping the intervals between overhauls as long as possible. Temperature measurements are needed to complement the calculations in order to reach a high level of confidence in life expectancy. The measurements also provide the possibility to detect any life issues at an early stage, or to identify potentials to reduce the cooling air consumption, improving the overall engine performance. While spacecrafts experience temperatures from -120 to 110 °C on the orbit, their surface reaches extremely high temperatures, well above 1000 °C, during descent into the atmosphere due to aerodynamic heating. Sophisticated insulation systems are designed for thermal protection. One of the steps in designing a protection system is experimental temperature measurements. New efficient engine and thermal protection systems design is the major tasks for the space industry. The elevation in operating temperatures increases the engine efficiency but in turn reduces the material life. New hightemperature materials are being developed, and include high-temperature alloys, composites, and ceramic materials. Since materials strength decreases with temperature, it is essential to develop adequate cooling systems. As the maximum operating temperature is

Background and Justification: Need of Thermal Mapping

Modern gas turbine engines are designed with higher and higher turbine inlet temperature in order to increase the efficiency. To achieve a high benefit from increased temperature level it is necessary to minimize the amount of cooling air to increase the engine performance factor. The difficulty in turbine design is to find the optimal path to increase the efficiency without sacrificing the component lifetimes. Modern gas turbine materials usually suffer a steep decrease in material properties when a certain temperature is exceeded. It is extremely important to know the component temperatures in real engine conditions with good accuracy in order to predict the component lifetimes. For the heavily cooled components, the main damage mechanism is often due to thermo-mechanical fatigue, TMF caused by the thermal gradients within the component The Availability factor (AF), Reliability factor (RF) and Mean time before failure (MTBF) are very important parameters for gas turbines and every effort should be made to increase it further and hence extensive thermal mapping is done to understand the nature of stresses and their effects on engine components for all operating conditions.



993

Sachin V. Bhalerao, A. N. Pawar, Atul B. Borade

design. As a result, cylinder head tends to fail in operation (Distortions, Fatigue cracking) due to over heating in regions of limited cooling..About 50 % of the heat to the engine coolant is through the engine head and valve seats, 30 % through the cylinders sleeve or walls and remaining 20 % through exhaust port area. Proper thermal mapping of I. C .Engines is needed for designing the tailored cooling to yield maximum efficiency and safe running.

normally just under a hundred degrees below the maximum allowed for a particular material, it is extremely important to gain knowledge of temperatures reached in operating devices. While predictive thermal models exist, the trusted temperature data can be obtained only experimentally. Accurate experimental temperature measurement techniques become especially useful during the engine fine tuning, although measuring high temperatures in operating engine is challenging, since: 1) The operating temperatures are high 2) Normally, large temperature gradients are present, and need to be accounted for; 3) The geometry of engine elements is complex; 4) New generation engine elements become smaller and smaller; 5) Engine elements should stay intact during testing, i.e. the thermal sensor should not cause engine parts damage; 6) Sensor should not affect tested material thermal properties; 7) Getting readings from sensors installed in an operating engine is complicated. When designing the cooling system for cooled components in modern gas turbines it is necessary to spend a minimum of cooling air, in order to minimize the negative effect on the efficiency from the cooling air injection. This means the cooling has to be tailor-made for the gas temperature distributions of the particular turbine. Regarding I. C. Engines as an appreciable amount of heat is transferred through the I. C. engine which effects the engine performance, it is therefore essential to look forward to analyze the modes of heat transfer and keep record of temperature variations in the engine components. Temperatures within the combustion of an engine reach values on the order 2700 K & up. Materials in the engine cannot tolerate this kind of temperature & would quickly fail if proper heat transfer did not occur. Removing heat is highly critical in keeping an engine & engine lubricant away from thermal failure. On the other hand it is desirable to operate an engine as hot as possible to maximise thermal efficiency. Continuous monitoring of temperature in various critical areas is a important design parameter in I. C. Engines. Satisfactory engine thermal mapping is required for a number of important reasons, including material temperatures limits, emissions & knock. The temperatures of certain critical areas need to be kept below material design limits. Aluminum alloy begin to melt at temperatures greater than 775 K & the melting point & iron is about 1800 K. Differing temperatures around the cylinder bore will cause bore distortion & subsequent increased blow-by, oil consumption & piston wear. Cylinder head is one of the most complicated parts of the internal combustion engine. It is directly exposed to high combustion pressures and temperatures. In addition, it need to house intake and exhaust valve ports, fuel injector and complex cooling passages. Compliance of all these requirements leads to many compromises in

II.

Need of Innovative Thermal Sensors

A number of temperature measuring techniques are in use today in the industry and their advantages and limitations are well known to experts. These techniques include thermocouples (Slip-Ring Assembly or Radio Telemetry System), Optical Pyrometers, Thermo Plugs etc. Thermal paint temperature Sensor is an addition to this list with a distinctively different principle of operation and technical characteristics. All experimental temperature measurements are based on some sort of properties or matter state change (i.e. fluid volume or pressure, electric force induced by two dissimilar materials contact, etc.). There are over 30 different phenomena and materials properties used for measuring temperature, with even a larger number of temperature measurement tools available today. There is a vast variety of applications, which necessitate a need for new methods, enhanced measurement range, accuracy, reduced sensor size, etc. Measuring temperature in hardto-access moving parts, such as rotating turbine disks, or jet nozzles, is even more challenging. These kinds of parts carry most of the mechanical, as well as thermal loads, which necessitates precise thermometry. In most cases, “conventional” thermometry techniques are not adequate for such applications. For example, signal from thermocouples needs to be somehow transferred from the measured moving parts to a readout unit. While these kinds of mechanisms exist, they cause additional errors due to contact friction, and are not usable at all at high rotating speeds. Moreover, they change the thermal transfer conditions, and the “true” part temperature can not be obtained. Both Temp.-Plug and thermal pyrometry methods are less accurate, then the thermal paint technique due to numerous technological constrains, subjectivity in data interpretation, survivability issues, and tendency, to modify thermal behavior of a component during testing, will make it difficult to get accurate data. In thermal paint technique a temperature sensitive coating used to measure peak surface temperatures irreversibly change its color and obtains a visual record of the temperature variations over the surface of components (parts). With more traditional instrumentation using thermocouples it is not possible to install enough measuring points on the component to really catch the gradients. Thermal paints show the gradients and enables measurement of the temperature with good accuracy in many points on the same component.



994


•

Wide temperature range (150-1450 °C), “no-lead” installation, and exceptional accuracy make thermal paints very attractive for use in small, rotating and “hardto-access” parts, including, but not limited to gas turbine blades, space shuttle ceramic tiles, automobile engines, etc.

Establish the DIP set up required for automatic data acquisition from isothermal fringes. Capturing of luminance and colour coordinates corresponding to transformation fringes. Develop the DIP procedures for establishing the color space for fitment of experimental data. Analyse the application of Rapid Prototyping as a valid tool for prototype manufacturing. Conducting of FEM based validation and establishing the correlation coefficients for various mulit-change thermal paints.

•

III. Objective and Specific Aim of the Research Work

•

Specific and primary aim of the research work is to develop a research methodology for assessing the fullfield surface temperature plots of high speed compressor and turbine modules through thermal paint technique.To achieve this objective several secondary tasks need to be completed such as: • Design and development of calibration rig for determining the isothermal transformation points of single and multi change paints. The rig caters to different thermal paints that have operational range of 110 – 1290 degree Centigrades. • Design and development of a Spinning test rig for experimental evaluation of rotational effects over the thermal paint contours. • Design and development of a test rig for observing the effect of heating time on the thermal paint behaviour. • Application of rapid prototyping as a tool for hot section component prototype manufacturing. • Design and development of a Digital Image Processing technique for automatic interpretation and data acquisition from thermal paint fringe pattern. • Carrying out validation studies through Finite Element Analysis.

IV.

V.

Methodology and Proposed Plan Work

V.1.

Design and Development of a Calibration Test Rig for Calibration of Single and Multi Change Paints

Single-change paints operate in the temperature range of 1200 C to 5900 C and each of the paints has got only one transformation temperature.

Research Methodology

The proposed research addresses these important functional issues and specifically aims at: • Evaluation of paint adherence to high speed components – Study the substrate materials of high speed compressor and turbines. Examine the surface topology parameters and develop surface suitable preparation procedures. • Design and development of the calibration rig incorporating the features for thermocouple based validation. Generation of isothermal data for single change paints through calibration coupons. Application of “bow tie” coupons for multi change paints. • Study of the influence of peak temperature duration over transformation temperatures. Conducting of paint calibration tests for various durations in the range of 5 to 60 minutes. • Analyse the effect of rotational speeds on isothermal data – Design, manufacture and installation of rig for spinning a painted disc at various rotational speeds and observe the effect of rotation on thermal paint behavior.

Calibration test rig

Furnace for heating test coupons

Fig. 1. Schematic of calibration facility

Multi-change paints operate in the temperature range of 5000 C to 12900 C and each paint has got 7 to 10 transformation temperatures. Calibration coupons are prepared by coating them in the thermal paint to be calibrated and then heating them. The coupons are heated at different temperatures in a 15 degree centigrade step for a heating time of 5 minutes in each temperature zone. Alternately, where a small temperature change produces rapid colour changes, coupons may be provided at narrower temperature bands.



995


a fixed speed and allowed to heat. Now after every interval of 60 seconds the image of the color pattern on the disc is taken and observed using DIP technique for a detailed analysis of the effect of heating time on paint behaviour.

The images of the coupons are further acquisited for preparing the digital calibration data bank. The calibration rig consists of electrical heating system provisioned with a facility for passing of high amperage current through a test coupon for stipulated duration. Varying cross section of the bow tie test coupon gives rise to differential heating levels at different cross sections leading to capturing various transformation points. The bow tie test coupons have multiple numbers of thermocouples or Infrared sensors attached to them and temperature levels are recorded from digital display systems. A timer is provisioned for facilitating the studies on peak temperature durations via-a-vis transformation temperatures. V.2.

V.3.

Thermochromic paint, as its name suggests, is paint that changes colour with temperature. This change is permanent and allows a rough temperature measurement to be made by observation of the colours on the sample; a task currently performed by a human operator. After heating, the colour profile is such that at certain temperatures, the colour changes quite prominently as perceived by the eye. At these points the temperature is calibrated empirically; it is by locating these colour boundaries, and then classifying the thermal contours done by the human operator who performs his analysis. When trying to locate colour boundaries, the problem is one of defining a colour edge, or more specifically, how to combine the gradient measures from the three monochromatic gradients into a salient measure of color gradient. In the present trends used the colour boundary detection methods fails to give satisfactory results. The main reason is that for same temperature band transitions the contrast is very weak and leaves much room for human error. This nesseciates a need of an automatic interpretation system which may avoid the manual error. The proposed research activity involves setting up a DIP set up consisting of polarizing filter, image grabber, cross-polariser and broad optical band width light source of constant optical intensity (xenon) source. The colour information, obtained through image grabber, is filtered using a smoothing function. Necessary corrections are incorporated to compensate for brightness variations caused by the shape of the part or by camera distortions. A method of analysing a thermal paint includes the step of defining the colour space in at least two dimensions, the dimensions representing different luminance and/or pixel values. The location of reference points, representing calibration data, is defined within the colour space for comparison with colour information relating to the part to be analysed. Each pixel of an image of the part to be analysed is given a location in colour space and the nearest calibration point is determined, leading to approximate establishment of the temperature value. Automatic temperature data assignment is done through a custom developed algorithm. Full-field temperature plots are generated corresponding to compressor and turbine components and they are superposed over the corresponding 3D CAD models. Comparison with FEM plots leads to defining of transformation coefficients. Based on the data extracted from various spin tests, temperature plots are generated delineating the influence of rotational effect over the temperature plots. The

Evaluation of the Effect of Rotational Speed and Heating Time on the Thermal Paint Behaviour

The activity proposes to carry out an experimental study to analyse the effect of heating time and rotational speed on the paint behaviour. The test rig consists of a circular disc mounted on a motor shaft. The motor receives the electric supply through a rheostat which is used to regulate its speed. The disc is painted and heated using a propane torch. The flame of the torch is allowed to strike the disc through a hole in the insulating slab placed in front of the disc increasing the heating intensity. Infrared sensors are mounted on the test rig to record the local temperatures. The disc with the thermal paint applied on it is rotated at different speeds for fixed time intervals say 5 minutes for each speed. LED Camera

Power plug Motor

Disc

Design and Development of a DIP Facility for Automatic Interpretation of Thermal Paint Test Data

Insulator

Rheostat

IR Sensor

Fig. 2. Schematic view

It should be observed that for every set of rotational speed the heating temperature should be the same. Now the thermal paint color contours will be observed for different rotational speeds and its effect will be noted. Images of the contour patterns will be taken and processed using the DIP algorithm for detail analysis. Similarly for observing the effect of heating time on the paint behavior the painted disc should be kept rotating at



996


Time can be saved by reducing the amount of time required to produce the prototype parts and tooling. Production Costs are lowered as the mistakes regarding tooling are identified in the early stages and hence the modification cost can be cut down in actual production. Since visualization capabilities are enhanced various factors like manufacturability, robustness and functionality of the design can be properly checked before sending them for production. These multiobjective advantages of Rapid Prototyping are continuously motivating to use it as a tool for prototype manufacturing for evaluation and testing.

context of this study is the development of an accurate color measurement device including a digital color camera connected to an image acquisition and processing system. The industrial application is the measurement of temperature via thermal paints which are used by gas turbine manufacturers to determine the maximum temperature reached in a combustion chamber or on high-speed rotating components under combustion test. These mechanical parts are examined off-line, once cooled down and dismantled. The color-to temperature mapping is established from painted samples, separately brought to different temperatures in the range of interest. At present, restitution of temperature maps on mechanical components is still a human process. Its automation will reduce interpretation time, improve accuracy and make temperature data available as direct input into numerical models. A particular attention is to be paid on lighting and viewing conditions.

VII.

The painted components are assembled in a rig/engine prior to any testing and handled using clean nylon hand gloves to prevent contamination of the painted surfaces. If the test part is part of a gas turbine engine, there should be no compressor wash or wet motor before the test run so that the painted surfaces remain uncontaminated. Whenever possible, engine light-up should be made using natural gas fuel. The fuel flow should be kept to a minimum during ignition to avoid turbine torching, and light-up should preferably occur on the first start. When a test run encompasses turbine nozzle and blade paint tests, the running time at full-load conditions should be achieved as quickly as possible and time spent at full load should not exceed 5 minutes to avoid paint removal by the scrubbing action of the exhaust gases. When paint testing heavy structures such as disks, diaphragms or casings, the running time should be 5 hours. Combustor liners can undergo much longer exposure times without adverse effects. Color change calibrations for thermal paint are available up to some 50 hours exposure time although 30 minutes is the test norm. Estimates of temperature contours after a series of short "runs" even at nominally the same conditions can be misleading. At the conclusion of a test run, the engine fuel should be cut-off at the maximum load conditions in order to reduce any effect of transient temperature exposure. The engine should then be motored at cranking speed for 15 minutes to cool the hot section parts. This procedure ensures that the heat lost by cooling at all locations is greater than the heat gained by transfer from the hotter to the cooler parts of any test section. If this precaution is disregarded, it may lead to a certain amount of heat-soaking and parts attaining a higher temperature during the cooling phase than they had reached during the test itself. The hot section components are then disassembled and the their images are taken to capture the isothermal contours for further detailed temperature analysis using the Digital Image Processing set up.

Fig. 3. Schematic of DIP facility

VI.

Thermal Paint Test on Gas Turbines

Prototype Manufacturing

During product evaluation and testing experimentations are done on geometrically representative test models. In context of aero engines the components have complex geometries and forms. Manufacturing the replicas of these intricate components by conventional methods put forth enormous challenges involving significant time effort and skill. Rapid prototyping enables realization of metallic parts directly from CAD models, without the use of tools or fixtures.Some of the RP techniques like metal sintering and laser engineered net shaping can help in rapid production of engineering parts. This research work propose to use RP for production of test parts for thermal paint experimentation.Also Rapid Prototyping has significant advantage in cost and time savings. Savings of development time and process cost are some of the greatest advantages of Rapid Prototyping.



997


VIII. Validation Studies through Finite Element Analysis

[5]

Computer Aided Engineering is extensively sed in design and simulation of engineering components. The CAE software are nowadays widely used for simulation as the result achieved are quite close and approximate to the real condition. Various software are available for modeling and analysis purpose. Using these high end Software ,modeling the complete prototype geometry precisely and analyzing with proper boundary conditions yields results which are quite close to the actual performing conditions. These software based on Finite element analysis technique can be used for Validation of the prototype experimental work.

[6]

[7]

[8] [9]

[10] [11]

IX.

Conclusion

[12] [13]

Thermal paints also known as temperature indicating paints provide an experimental way to record the temperature variations over the surface of the test components. Thermal paints are characterised by “isothermal” or “transformation” temperatures at which they undergo irreversible colour change. Out of numerous temperature measurement techniques like pyrometry, thermo couples, liquid crystals etc., thermal painting has got the innate ability to provide full-field temperature data even from the rotating and hard to access components, upto 1200 degree Centigrades. As reported by engine developers thermal paint tests on static components such as combustor, nozzle guide vane, fuel injector and exhaust nozzle have consistently provided wealth of information to the designers. But the published data pertinent to the application of thermal paints on rotating components such as high pressure compressor and turbine rotors and aspects related to paint adhesion to rotating components, rotational and heating time effects, surface preparation, isothermal data acquisition and interpretation are scanty. A systematic orderly approach right from application to interpretation of the thermal paints needs to be generated. The research approach presented in this article is expected to develop a proper methodology for utilization of thermal paints as an effectve tool for thermal mapping of gas turbine engines.

[14] [15]

[16]

[17]


Jawaharlal Darda Institute of Engineering and Technology, Yavatmal. 2 Goverenment Polytechnic, Amravati. Sachin V. Bhalerao is working as Assistant Professor in Mechanical Engineering Department at Jawaharlal Darda Institute of Engineering and Technology, Yavatmal, India. He owes around 13 years of experience in teaching. He has completed his Bachelor’s and Master’s degree in Mechanical engineering and is pursuing his Ph. D. in Thermal Engineering from Amravati University India. His thrust area of research is thermal mapping of hot section components of gas turbines and space vehicles. He has published about 12 papers in various National, International conferences and Journals and guided many research projects in thermal engineering.

References [1] [2] [3]

[4]

U. Chandrasekhar and Rajeev Jain “Manufacturing paradigm for gas turbine engine test components through laser net shaping and allied layered technologies”, Proc. ISABE – International Society for Air Breathing Engines, 2005. “Effect of rotation on leading edge region film cooling of a gas turbine blade with three rows of film cooling holes”, International Journal of Heat and Mass Transfer, Volume 50, Issues 1-2, January 2007, pp. 15-25. Z.hi Taoa, Xiaojun Yanga, Shuiting Dinga, Guoqiang Xua, Hongwei Wua “Experimental study of rotation effect on film cooling over the flat wall with a single hole, Experimental Thermal and Fluid Science.”, Volume 32, Issue 5, April 2008, pp. 1081-1089 “Advances in Techniques for engine applications”, R. Dunker, pp. 51-122, John Wiley & Sons, 1995 M.K. Douglas Smith and D. Marriott “Apparatus and method for calibration of thermal paint”, Patent Application Note No – 553 892, European Gas Turbines Ltd., 1995 M. K.Douglas Smith “Interpretation of thermal paint”, Rolls Royce plc, US Patent 6434267 – B1, 2002. Andrew J Neely and Philip J.Tracy. “Transient response of thermal paints for use on short duration hypersonic flight test.” “Appratus and method for calibration of thermal paints.” U.S.Patent 5720554. Surface temperature measurements on Engine Components by means of irreversible thermal coating.” C Lempereur, R Andral and J. Y. Prudhomine SciTechnol 19 October 2008. Advantages of Rapid Prototyping.” Thomas E Endres , SAE Publication /1991-01 3433 /Oct 1999. “Rapid Prototyping Technology Application and Benefits for rapid product development.” S.O. Onuh, Y.Y. Yusuf. Journal of Intelligent Manufacturing, Springer Netherlands Volumn 10, Sept. 1999 , Page 301 – 311. “New Developments and Trends in Rapid and High Performance Tooling.” Terry Wohlers at World Wide Advances in Rapid and High Performance Tooling. Conference Dec. 2002 Prototype Advantages and Rapid Prototyping Benefits,Free Press Release, PR log July 7,2008, Hobart , Washington. www.prototypezone.com

Campbell, B. T., Liu, T., and Sullivan “Temperature Sensitive Fluorescent Paint Systems”, J. P., AIAA Paper 94-2483, 1994. Mike Connolly “Surface temperature measurements through thermal paints”, Rolls Royce GmbH, Deutschland Note RR-2005. Kesavan, V., Suresh, B. and Kaushal, M., “Experimental validation of thermal model of cooled nozzle guide vanes”, Proceedings of National Conference on Air Breathing Engines, NCABE, 2002. Moon H. K. and Glezer “Application of Advanced Experimental Techniques in the Development of Cooled Turbine Nozzle”, , B, ASME Paper 96-GT-233, 1996.

Dr. Anand N. Pawar is working as Professor of Mechanical Engineering Department at Govt. Polytechnic Amravati India. He owes around 17 years of experience in teaching. He has completed his Bachelor degree in mechanical engineering. He then persuaded M.E in Thermal Power engg. From Govt. college of engg., Amravati, India. He has persuaded PhD in Thermal Engineering from Amravati University India. He has published number papers in National and International Journals. Also he has presented number papers in international conferences. His few publications have appeared in prestigious Springer, Emerald, and



998


Inderscience journals. His area of interest is Low Heat Rejection engine. Dr. Atul B. Borade is working as Professor and Head of Mechanical Engineering Department at Jawaharlal Darda Institute of Engineering and Technology, Yavatmal, India. He owes around 15 years of experience in teaching. He is basically a Mechanical Engineer. He then persuaded M .E. and PhD in Production Engineering. He also holds MBA as an additional qualification. He has published 15 papers in National and International Journals. His few publications have appeared in prestigious Springer, Emerald, and Inderscience journals. He is regular reviewer with Omega, Journal of Manufacturing Engineers, Journal of Information, Information Technology and Organizations, Journal of Information System and Technology Management, Journal of Industrial Engineering and Management, International Journal of Industrial Engineering - Theory, Applications and Practice. Chinese Journal of Industrial Engineering, Journal of Manufacturing Technology and management, Computers and Mathematics with Applications He has worked as Editorial Board Member with International Journal of Information Technology and Knowledge Management, International Journal of Manufacturing Science and Manufacturing Management. Currently he is working as Editorial Board Member with Contemporary Management Research, Asian Journal of Industrial Engineering, and International Journal of Business Management and Research. He is working as an Editor in Chief with International Journal of Manufacturing Systems.



999


The Pseudo Radiation Energy Amplifier (PREA) A. Boucenna

Abstract – In this paper we show that a gray body which separated from vacuum by a material interface and submitted to outside incident radiation can behave like a Pseudo Radiation Energy Amplifier. The Earth (Earth + atmosphere) is not a simple isolated gray body but it is in fact a Pseudo Radiation Energy Amplifier with adequate reflection coefficients. The balance of the energy exchanged between Earth and outer space is reconsidered and the estimated Earth’s ground temperature mean value 15 °C is then derived. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Gray Body Radiation, Reflection Coefficient, Material Interface

In this paper we develop a new concept of the Pseudo Radiation Energy Amplifier (PREA) which is a gray body separated from vacuum by an interface and submitted to outside incident radiation, and we show that the Earth, which is a gray body separated from the space by the atmosphere, does not behave like a simple isolated gray body but like a Pseudo Radiation Energy Amplifier gray body. The balance of the energy exchanged between Earth and outer space is reconsidered and the 15 °C Earth’s ground temperature mean value is then derived.

Nomenclature t T T0 T∞ Tsun ∆T

σ

Rsun Rearth A Asun Aearth V

ρ

cv ∆Q

orbit

orbit

The time The temperature The temperature at t = 0 The saturation temperature ( t = ∞ ) 5780 °K The temperature change The Stefan constant The sun radius The earth orbit radius The surface of the body The sun area The earth orbit area The body volume The density The specific heat of the material The calorific energy

I.

II.

The Isolated Gray Body Radiation Exchange II.1.

Introduction

From the radiation balance diagram [1] illustrating the IPCC reports [2] one can estimate the Earth received power from the sun at Pin = 342 W/m2. The consumed, reemitted and reflected power by the earth and its atmosphere can be estimated at Pout = 599 W/m2. It seems that the earth emits more power than it receives. The earth’s ground mean temperature is estimated at 15 °C. A calculation based on the black body radiation theory [3] gives an earth’s ground mean temperature of the order of -18 °C which is much lower than 15 °C. The important gap, between these calculated and estimated temperature mean values, is often explained by the atmosphere gas greenhouse effect [4]-[15]. This explanation is in contradiction with the thermodynamics second law [3].

Isolated Gray Body Radiation

If an isolated gray body receives a radiation power P0 from an outside source and if R is the total mean reflection coefficient at the AB interface between this gray body and the vacuum (Fig. 1), then the reflected power is given by the relation: Pr = RP0

(1)

and the absorbed power by the gray body is given by : Pa = (1 − R ) P0

II.2.

(2)

The Gray Body Temperature

The absorbed power Pa gives the gray body temperature which satisfies the Stefan Boltzmann law [4]: σ T 4 = Pa (3) The gray body temperature is given by:



1000

A. Boucenna

T=

4

(1 − R ) σ

(4)

P0

If the gray body temperature is known, one can deduce the radiated power and vice versa, if a gray body absorbs a Pa power one can deduce its temperature.

If the external radiation source is off. The isolated gray body can lose its energy only by radiation. The gray body cooling through radiation, from the temperature T0 at t = 0, is governed by the following differential equation: V ρ cv

RP0

P0

dT + A σT 4 = 0 dt

(10)

The solution of the above differential equation gives:

VACUUM A

B

T (t ) =

(1-R)P0 GRAY BODY

(11)

3 A σ T03 3 1+ t V ρ cv

where the temperature T0 is given by Eq. (4). The Eq. (11) gives the evolution of an isolated gray body instantaneous temperature. It describes an isolated gray body cooling by radiation.

Fig. 1. Isolated Gray Body Radiation Exchange

II.3.

T0

Gray Body Warming and Cooling

The calorific energy transferred between a sample of material with mass m and its surrounding is related to its temperature change by the following relation [16]:

III. The Pseudo Radiation Energy Amplifier (PREA)

cv m ∆T = ∆Q

III.1. Pseudo Radiation Energy Amplifier Gray Body Radiation Exchange

(5)

The temperature of an isolated gray body submitted to a radiation of power P0 is governed by the following differential equation:

V ρ cv

dT + A σ T 4 = (1 − R ) AP0 dt

(6)

A particular solution of the Eq. (6) is given by the Eq. (4) and the general solution is given by: 1

T (t ) = 3

⎛ 3 Aσ ⎞ B ⎜1 + t⎟ ⎝ BV ρ cv ⎠

+4

(1 − R ) σ

P0

(7)

B is a constant. Eq. (7) gives the instantaneous temperature of an isolated gray body. It describes the warming of an isolated gray body submitted to a radiation. Let us put: T∞ = 4

(1 − R ) σ

A Pseudo Radiation Energy Amplifier consists of a gray body which separated from the vacuum by a material interface (Fig. 2). If a gray body receives a radiation power P0 from an outside source and if R is the total mean reflection coefficient by all AB interfaces separating the gray body from the vacuum, then the power Pr = RP0 is reflected and the power Pa = (1 − R ) P0 is absorbed by the gray body. The gray

body radiates towards the vacuum the absorbed power. This latter tries to cross all AB interfaces which characterized by a total mean reflection coefficient r All AB interfaces can have variable reflection coefficients. In the case of the earth’s ground, the AB interface is constituted by the Earth atmosphere. Only the power (1 − r )(1 − R ) P0 crosses all AB interfaces, whereas the power P1 = r (1 − R ) P0 is reflected by all AB interfaces to the gray body itself which absorbs and reemits this power to the vacuum. P0

P0

RP0 VACUUM

A

That Corresponds to the saturation temperature ( t = ∞ ) into Eq. (7). We can get: T∞

T (t ) = − 3 1+

3 Aσ t V ρ cv T∞3

(1-r) (1-R)P0

(8)

+ T∞

(9)

(1-R)P0

(1-R)P0

B

Material interface

r (1-R)P0

Pseudo Radiation Energy Amplifier GRAY BODY

Fig. 2. The Pseudo Radiation Energy Amplifier (PREA)



1001

A. Boucenna

All AB interfaces will then reflect to the gray body the power P2 = r P1 = r 2 (1 − R ) P0 , and so on. At the end of the nth reflection, the power reflected by all AB interfaces is Pn = rPn −1 = r n (1 − R ) P0 . Therefore, the gray body did not absorb solely the power (1 − R ) P0 but it absorbed in addition a part of the power that is itself radiated and that is reflected back by all AB interfaces to the gray body. Then, the effective power absorbed by the gray body is: Pa = (1 − R ) P0 + P1 + P2 + P3 + … + Pn + ... =

(

P = RP0 + (1 − r )(1 − R ) P0 + (1 − r ) r (1 − R ) P0 + + (1 − r ) r 2 (1 − R ) P0 +…+ + (1 − r ) r n (1 − R ) P0 + ... = = RP0 + (1 − r )

)

= (1 − R ) P0 + r1 P0 + r 2 P0 + . . . + r n P0 + ...

(12)

The effectively absorbed power Pa by the Pseudo Radiation Energy Amplifier gray body, reemitted as IR radiations, is equal to the sum of the geometric progression terms, where the first term is (1 − R ) P0 and the reason is r, thus: Pa =

The Pseudo Radiation Energy Amplifier is not a Radiation Energy Amplifier. The total power reflected and reemitted by the Pseudo Radiation Energy Amplifier is equal to the received power from the outside source and is given by the relation:

1− R P0 1− r


1− R P0 = 1− r

= P0

III.2. The Pseudo Radiation Energy Amplifier Gray Body Temperature The absorbed power Pa gives the Pseudo Radiation Energy Amplifier gray body its temperature which satisfies the Stefan Boltzmann law: Pa =

(13)

The effectively absorbed power by the Pseudo Radiation Energy Amplifier gray body, reemitted as IR, is not given by relation (1) but is given by relation (13). The total mean values of the reflection coefficients R and r depend on the wavelengths of incident radiation and the one reemitted by the gray body once it absorbed the power Pa. If the whole power absorbed is reemitted, the Pseudo Radiation Energy Amplifier gray body must radiate towards the vacuum the absorbed power Pa. Depending on the mean values of the R and r reflection coefficients, the effective absorbed power Pa by the Pseudo Radiation Energy Amplifier gray body can be equal, lower or larger than the received power P0 from the outside source. Thus: - if R = r , then Pa = P0 , the effective absorbed and reemitted power by the Pseudo Radiation Energy Amplifier gray body is equal to the received power from the outside source. The system behaves like a pseudo black body. The particular case of a black body corresponds to R = r = 0 ; - if R > r , then Pa < P0 , the effective absorbed and reemitted power by the Pseudo Radiation Energy Amplifier gray body is lower than the received power from the outside source, one has a loss of power ; - if R < r , then Pa > P0 the effective absorbed and reemitted power by the Pseudo Radiation Energy Amplifier gray body is larger than the received power from the outside source, one has a gain of power. The system (gray body + material interface) behaves like a Pseudo Radiation Energy Amplifier (PREA).

(14)

1− R P0 = σ T 4 1− r

(15)

If the temperature of a gray body is known one can deduce the absorbed radiated power and if a Pseudo Radiation Energy Amplifier gray body absorbs a power Pa one can deduce its temperature T. Then: T=4

1 (1 − R ) P σ (1 − r ) 0

(16)

Eqs. (4) and (16) show that for the same received power P0 from an outside source, a Pseudo Radiation Energy Amplifier gray body and an isolated gray body constituted with the same material, will have different temperatures. This difference is due to the effect of the material interface introduced between the gray body and the vacuum. In the case R < r , the Pseudo Radiation Energy Amplifier gray body temperature is superior to the one of an isolated gray body. III.3. The Pseudo Radiation Energy Amplifier Gray Body Warming and Cooling The temperature of a Pseudo Radiation Energy Amplifier gray body submitted to a radiation of power P0 is governed by the following differential equation: V ρ cv

dT + (1 − r ) A σ T 4 = (1 − R ) AP0 dt

(17)

The relation (16) gives a particular solution of this equation. The general solution is given by the relation:


1002

A. Boucenna

T= 3

1 + ⎛ 3 A (1 − r ) σ ⎞ B ⎜1 + t⎟ BV ρ cv ⎝ ⎠

4

1 (1 − R ) P σ (1 − r ) 0

(18)

P = S cos α

where B is a constant. The solution of the differential equation (17) gives the variation of the temperature of the Pseudo Radiation Energy Amplifier gray body in function of time. Putting: T∞ =

4

1 (1 − R ) P σ 1− r 0

T∞ 3 A (1 − r ) σ T∞3 3 1+ t V ρ cv

+ T∞

(20)

Eq. (20) gives the temperature of a Pseudo Radiation Energy Amplifier gray body as a function of time. It describes the warming of a Pseudo Radiation Energy Amplifier gray body submitted to a radiation of power P0. If the external radiation source is off, the cooling by radiation of the Pseudo Radiation Energy Amplifier gray body, having the temperature T0 at t = 0 and given by (19), is governed by the differential equation: V ρ cv

dT + A (1 − r ) σ T 4 = 0 dt

(21)

The solution of this differential equation gives: T=

T0

(22)

3 A (1 − r ) σ T03 3 1+ t V ρ cv

The temperature T0 is given by the relation (19). The factor (1-r) makes the difference between the relations (11) and (22). Eq. (22) gives the evolution of a Pseudo Radiation Energy Amplifier gray body temperature with time. It describes the cooling of a Pseudo Radiation Energy Amplifier gray body having the temperature T0 at t = 0.

IV.

Earth Case

The received power S by the earth from the sun at the equator is given by: 4 S = σ Tsun

Asun Aearth _ orbit

4 = σ Tsun

2 Rsun 2 Rearth _ orbit

(23)


(24)

The mean power P0 coming from the sun on the whole Earth’s ground, is then: P0 =

1 S cos α dA = A exp osed _ ∫face _ area π

(19)

Corresponding to the temperature of saturation ( t = ∞ ), one gets: T =−

At any α latitude point of the Earth exposed face, the power coming from the sun (Fig. 3) is given by:

S = A

(25)

2π 2

∫ 0

S ∫ cos α sin α dα dϕ = 4 0

where A is the Earth area. For S = 1368 W/m2, one has P0 = 342 W/m 2 , which is the power value Pin received from the sun and given in the references [1],[2]. Assuming that the interface [(Earth+Atmosphere) – space] presents the R and r total mean reflection coefficients to solar radiations and terrestrial IR radiations respectively, the effectively absorbed power by the Earth is given by relation (13). This absorbed power is equal to the consumed, reemitted and reflected power Pout by Earth and its atmosphere, therefore: Pout =

1− R P0 1− r

(26)

The power values mentioned in references [1],[2] give us a value of the total mean reflection coefficient R. One 107 has R = ≈ 0.31 . 342 Using the relation (26) one can deduce the reflection coefficient total mean value r ≈ 0.61 of the reflection coefficient. One can see that for the Earth R < r , the effectively absorbed power by the Earth and reemitted is greater than the received power from the sun. We have a gain of power. The Earth behaves like a Pseudo Radiation Energy Amplifier. The power effectively absorbed by the Earth’s ground and reemitted as IR radiations must satisfy the Stefan Boltzmann law: 4 σ Tmean =

1− R P0 1− r

(27)

In fact, a fraction Su of this power is consumed by Earth. The power used in the ocean water evaporation is estimated at 78 W/m2 s and the one used to heat the air is 24 W/m2 s. If one takes into account this power loss, relation (27) becomes: 4 Tmean =

1 ⎡ 1− R

⎤

P −S σ ⎢⎣ 1 − r 0 u ⎥⎦

(28)


1003

A. Boucenna

0.31. The total reflection coefficient r for earth’s IR radiations that results from the contribution of the interface earth-atmosphere with reflection coefficient re and of the interfaces: vacuum-gas, gas-cloud and all others sprays with reflection coefficient rc, is:

Ox

SUN RADIATIONS

r ≈ re + rc

ϕ α

EARTH

Assuming that the reflection coefficient rc is practically equal to the cloud reflection coefficient, one can deduce an approximate value of rc: rc ≈ r − re ≈ 0.21

Oy Fig. 3. Exposed Earth face. The α latitude is the angle between the Oz axis (passing by the Earth center and crossing the equator) and the vector locating the solar radiation impact point (on the Earth’s ground)

For R = 0.31, r = 0.61, P0 = 342 W/m2 and Su = 102 W/m2, one can obtain: Tmean = 306.89K = 33.89 °C . This mean temperature value is much higher than the 15 °C Earth’s ground temperature mean value. To have this temperature, one must take r = 0.521. This reflection coefficient verifies the relation R < r which means that the Earth behaves like a Pseudo Radiation Energy Amplifier. Using this value and relation (28), one can recalculate the Pout power. One finds Pout ≈ 492 W/m 2 , instead of Pout ≈ 599 W/m 2 given by the references [1], [2]. This latter value is probably overestimated. The reflection coefficient r for a radiation at the interface between two media of refractive indexes n1 and n2 is given by the relation [17]: ⎛n −n ⎞ r =⎜ 1 2 ⎟ ⎝ n1 + n2 ⎠

(30)

Oz

2

(29)

The reflection at the interfaces gas-vacuum and gasgas is negligible: n2 − n1 ≈ 0 since n2 ≈ n1 ≈ 1 . Gases have a very weak reflection coefficient because of their refractive index value n ~ 1. They do not reflect light since the molecule sizes are too small compared to the radiation wave length. The Tmean = 288 K = 15°C earth’s ground mean temperature has been obtained while considering an earth IR reflection coefficient r equal to 0.521. This reflection coefficient cannot come from the atmosphere gases (O2, N2, A, H2O (steam), CO2, Kr, H, N2O, Xe, O3 (ozone), CH4). These are the sprays, that are microscopic particles hanging on the air, and the clouds composed of water droplets (liquid) or water solid (ice) that can reflect and diffuse the radiations coming from the earth or from the sun. At the interface between the earth’s ground and the atmosphere, the IR radiation emitted by the earth is reflected with a reflection coefficient re estimated at


(31)

The reflection coefficient rc is linked to the atmosphere composition (clouds, smoke, dust …). The clouds having a strong reflection coefficient reflect a good part of earth’s IR radiations toward the earth’s ground and that is what keeps the night earth’s ground temperatures moderate in the presence of clouds. On the other hand when the sky is cleared, the reflection coefficient is weak and the reflection of earth’s IR radiations is less important giving colder nights.

V.

Conclusion

The solar power absorbed by the Earth does not explain the gap noted between the estimated Earth’s ground temperature mean value and the one calculated while considering the earth as an isolated gray body. The Earth (Earth-atmosphere) is in fact a Pseudo Radiation Energy Amplifier with adequate reflection coefficients. The effectively absorbed power by a Pseudo Radiation Energy Amplifier is the sum of the power provided by an external source, and the power emitted by the Pseudo Radiation Energy Amplifier gray body itself and reflected back at the interface separating the Pseudo Radiation Energy Amplifier gray body from the vacuum. In this scheme, the solar and terrestrial radiation reflection coefficients by the earth and its atmosphere acquire a privileged role. Thus, the balance of the energy exchanged between Earth and outer space is reconsidered and the 15 °C estimated Earth’s ground temperature mean value is then derived. Our result revives the discussion on the parameters that control the Earth climate. The Pseudo Radiation Energy Amplifier can be a starting point to improve the energy use. It can find its application for a better use of the energy resources. The materials having the adequate reflection coefficients will be indicated for specific applications.

Acknowledgements Thanks are due to Professor A. Layadi and Professor J. Slimani for their help.


1004

A. Boucenna

References [1]

[2] [3]

[4] [5]

[6]

[7]

[8]

[9]

[10]

[11] [12] [13]

[14]

[15]

[16]

[17]


J. T. Kiehl, and K. E.Trenberth, 1997. Earth's Annual Global Mean Energy Budget, Bull. Amer. Meteor. Soc. , 78, 197-208 (1997). Intergovernmental Panel on Climate Change, IPCC, Rapports d'évaluation, (1990, 1995, 2001, 2007). G. Gerlich, and R. D. Tscheuschner, 2009. Falsification of the Atmospheric CO2 Greenhouse Effects Within the Frame of Physics, International Journal of Modern Physics B, Vol. 23, No. 3, 275-364 (2009). Intergovernmental Panel on Climate Change, IPCC, Rapports d'évaluation, (1990, 1995, 2001, 2007). V. K. Saxena and J. D. Grovenstein, The role of clouds in the enhancement of cloud condensation nuclei concentrations, Atmospheric Research 31(1994), 71-89. J. Servant, Carbon cycle variations on the Earth's surface during the last two glacial cycles, Atmospheric Research, 30(2003), 223232. E. Kaas and P. Frich, Diurnal temperature range and cloud cover in the Nordic countries: observed trends and estimates for the future, Atmospheric Research, 37(1995), 211-228. Y. S. Chung, The variations of atmospheric carbon dioxide at Alert and Sable Island, Canada, Atmospheric Environment (1967), 22(1988), 383-394 Chris D. Jones and Peter M. Cox, Constraints on the temperature sensitivity of global soil respiration from the observed interannual variability in atmospheric CO2 Atmospheric Science Letters, 2(2001), 166-172 O. Kärner, S. Keevallik and P. Post, A two-parameter approximation in cloudiness variability studies, Atmospheric Research, 27(1992) 231-252 E. K. Bigg, Measurement of concentrations of natural ice nuclei, Atmospheric Research, 25(1990), 397-408 A. Boucenna, The Great Season Climatic Oscillation, International Review of PHYSICS 1(2007) 53-55. A. A. Pawar, R. R. Kulkarni, J. B. Sankpal, Prediction of CO2 in the Exhaust of C.I. Engine Using Emission Model, International Review of Mechanical Engineering 1(2007) 127-130. N. Tadj, T. Bartzanas, B. Draoui, C. Kittas, Experimental and Numerical Investigation of Convective Heat Transfer in a Tunnel Greenhouse, International Review of Mechanical Engineering 2(2008) 872-878. N. Tadj, T. Bartzanas, B. Draoui, C. Kittas, Experimental and Numerical Investigation of Convective Heat Transfer in a Tunnel Greenhouse, International Review of Mechanical Engineering 2(2008) 872-878. R. A. Serway and W. Jr Jewett, 2006, Physics for Scientists and Engineers with modern physics, 6th edition, Tomson Brooks/Cole edition, p.608, 628. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interaction and Propagation of Light. 6th Edition (Cambridge University Press, Cambridge, UK, 1997).


Laboratoire DAC, Département de physique, Faculté des Sciences, Université Ferhat Abbas, 19000 Sétif, Algeria. E-mails: [email protected] [email protected] A. Boucenna, Place and date of birth Sétif 1956 Doctorat d’Etat en Physique, USTHB, Algiers, Algeria, 1988. Field of study: Nuclear Physics. Research interests: Nuclear Physics, Climate, History and Philosophy of Sciences Prof. Boucenna was a regular Associate at the ICTP (1994-2002), he is the President of the ANPMC (Associaion National des Professeurs et Maître de Conferences) in Algeria.


1005

International Review of Mechanical Engineering (I.RE.M.E.), Vol. 5, N. 5 July 2011 Extracted by ICOME 2011

A Simplified Method for Thermally-Induced Volumetric Error Compensation Yuxia Lu, M. N. Islam

Abstract – The thermally-induced volumetric error of a machine tool has been recognised as a major contributor to dimensional and geometric errors of component parts produced on it. Consequently, considerable research has focused on compensating for this type of error. The traditional model of compensation for thermally-induced volumetric error of a three-axis machine tool requires measuring 21 geometric error components, which are difficult and time consuming to gather. This paper describes the development of a simplified method of compensating for thermally-induced error based on only three axial positioning errors, which are assumed to be functions of ball-screw nut temperature and travel distance. It is a more efficient and comparatively cheaper method of compensating for thermally-induced error. The results indicate that only a negligible amount of the total dimensional accuracy is sacrificed by adopting the proposed model instead of the traditional model. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Error Compensation, Dimensional Error, Machine Condition, Thermally-Induced Positioning Error

PC S IT

Nomenclature δx(x) δy(y) δz(z) δy(x) δz(x) δx(y) δz(y) δx(z) δy(z) εx(x) εy(x) εz(x) εy(y) εz(y) εx(y) εz(z) εy(z) εx(z) εx(r) εy(r) εz(r) x, y, z a Txnut Tynut Tznut βx, βy, βz x0(Txnut) y0(Tynut) z0(Tznut)

x-axis positioning error y-axis positioning error z-axis positioning error Horizontal straightness error in x-axis Vertical straightness error in x-axis Horizontal straightness error in y-axis Vertical straightness error in y-axis Horizontal straightness error in z-axis Vertical straightness error in z-axis x-axis roll error x-axis yaw error x-axis pitch error y-axis roll error y-axis yaw error y-axis pitch error z-axis roll error z-axis yaw error z-axis pitch errors Squareness error in yoz plane Squareness error in xoz plane Squareness error in xoy plane Axial travel distance Coefficient of linear thermal expansion x-axis nut temperature y-axis nut temperature z-axis nut temperature Multiplication factors in x-, y- and z-axis x-axis origin offsets (varies with nut temp.) y-axis origin offsets (varies with nut temp.) z-axis origin offsets (varies with nut temp.)

Process capability (mm) Magnitude of size dimension (mm) Standard tolerance grade number

I.

Introduction

Thermal effects on machining accuracy have been researched extensively for the last 40 years and various models have been proposed for error reduction [1]-[9]. One of the common approaches for thermal error reduction has been volumetric error compensation. The general compensation model for calculating the volumetric error of a machine tool is based on the homogeneous coordinate transformation [10]. The main difficulty with this traditional method is that the model includes 21 pre-calibrated geometric errors. In practise, as the machine tool’s running time increases, thermally-induced errors start to play a major role in machined workpiece accuracy. Some research reported that if only compensated by the geometric error model, the results become even worse than with no compensation [11]. As such, it is imperative to include the thermal error components in any volumetric error model. A few researchers [8], [12] have attempted to insert the thermal errors into the geometric error model and then to use this synthesized model to predict and compensate for the positioning error at any point within the working zone. The most accurate method is the on-line measurement of thermal errors followed by instantaneous compensation. This method is not practical, however, because it requires having a laser interferometer located beside the working


1006

Yuxia Lu, M. N. Islam

machine at all times during the production process. A substitute method which addresses this shortcoming is to predict the thermal errors from a group of temperature points related to the increase of thermal errors. Chen et al.[8] developed a real time error compensation system to enhance the time-variant volumetric accuracy of a 3-axis machining centre through sensing, metrology and computer control techniques. A synthesized volumetric error model combining 21 geometric errors and 11 additional thermal errors was proposed. Each error component was modelled using regression methods. The main difficulties with this system are that the regression equation for each error component is very complicated and that the large number of thermocouples required to monitor each error component can obstruct the machine tool’s routine work. In our previous work [11], a new approach for thermally-induced volumetric error compensation was suggested. Instead of predicting each error component using a regression equation which requires several thermocouples to detect the temperatures for one single error component, this model demonstrated that the thermally-induced positioning errors are functions of the ball-screw nut temperature and the travel distance, which involves only one nut temperature for each positioning error. The other 18 geometric errors are assumed to remain constant at their values which were pre-calibrated at the cold start temperature of close to 20°C. This model still requires using a laser interferometer to check regularly for basic linear and angular geometric errors at a micron level accuracy, This model encounters difficulties in the industry because a laser interferometer is an expensive piece of equipment and difficult to use; even an experienced and skilled operator needs a considerable amount of time for each measurement. In this paper, a further simplified thermally-induced volumetric error compensation model is proposed [11] that permit more accurate parts to be manufactured while avoiding high costs in terms of both time and money. The proposed simplified model ignores angular errors, straightness errors, and their variations with temperature and instead compensates for volumetric error by considering only the thermally-induced positioning errors. This proposition is justified by the fact that temperature has a significant effect on positioning errors, and that such positioning errors dominate the volumetric error among 21 geometric errors [9], [11].

II.

Volumetric Error Model II.1.

General Model

The geometric error of a three-axis machine tool consists of 21 error components, of which six are along each axis and three are squareness errors between axes. The six error components along each axis consist of three translation errors (one positioning error and two straightness errors) and three rotational errors (pitch, yaw


and roll) (Fig. 1). For a typical linear carriage, three translational and three rotational errors can be represented using a 4 x 4 transformation matrix [Tx], [Ty] and [Tz] for the x-, y- and z-axis as follows: ⎛ 1 −ε z ( x ) ε y ( x ) δ x ( x ) ⎞ ⎜ ⎟ 1 −ε x ( x ) δ y ( x ) ⎟ ⎜ ε z ( x) [Tx ] = ⎜ ⎟ 1 δ z ( x) ⎟ ⎜ −ε y ( x ) ε x ( x ) ⎜ 0 0 0 1 ⎟⎠ ⎝

(1)

⎛ 1 −ε z ( y ) ε y ( y ) δ x ( y ) ⎞ ⎜ ⎟ 1 −ε x ( y ) δ y ( y ) ⎟ ⎜ εz ( y) ⎡⎣Ty ⎤⎦ = ⎜ ⎟ 1 δz ( y) ⎟ ⎜ −ε y ( y ) ε x ( y ) ⎜ 0 0 0 1 ⎟⎠ ⎝

(2)

⎛ 1 −ε z ( z ) ε y ( z ) δ x ( z ) ⎞ ⎜ ⎟ 1 −ε x ( z ) δ y ( z ) ⎟ ⎜ εz ( z) [Tz ] = ⎜ ⎟ 1 δz ( z) ⎟ ⎜ −ε y ( z ) ε x ( z ) ⎜ 0 0 0 1 ⎟⎠ ⎝

(3)

Similarly, the squareness error can be represented by using a 4 x 4 transformation matrix as follows: −ε z ( r ) ε y ( r ) ⎛ 1 ⎜ ε (r ) 1 −ε x ( r ) [Tr ] = ⎜⎜ z −ε ( r ) ε x ( r ) 1 ⎜ y ⎜ 0 0 0 ⎝

0⎞ ⎟ 0⎟ 0⎟ ⎟ 1 ⎟⎠

(4)

The error vector of all the combined errors in the work volume (space) is defined as the volumetric error. The G volumetric error vector v (Vx, Vy, Vz) can be expressed using a homogeneous transformation matrix [T] as Eq. (5): G G G v = [T ] ∗ pd − pd

(5)

[T ] = [Tx ] ∗ ⎡⎣Ty ⎤⎦ ∗ [Tz ] ∗ [Tr ]

(6)

G where, pd (Xd, Yd, Zd) is the designed position vector. Substituting Eqs. (1) - (4) in Eqs. (5) and (6) and neglecting second order terms, we get the following relationships (7): ⎧Vx = ( −Yd ) ∗ ⎡⎣ ΣZ + ε z ( r ) ⎤⎦ + Z d ∗ ⎡ΣY + ε y ( r ) ⎤ + ∆X ⎣ ⎦ ⎪ ⎪ ⎨V y = ( − Z d ) * ⎣⎡ ΣX + ε x ( r ) ⎦⎤ + X d * ⎣⎡ΣZ + ε z ( r ) ⎦⎤ + ∆Y ⎪ ⎡ ⎤ ⎩⎪Vz = ( − X d ) * ⎣ ΣY + ε y ( r ) ⎦ + Yd * ⎡⎣ΣX + ε x ( r ) ⎤⎦ + ∆Z

International Review of Mechanical Engineering, Vol. 5, N. 5 Extracted by ICOME 2011

1007


⎧ ΣX = ε x ( x ) + ε x ( y ) + ε x ( z ) ⎪ ⎪ ΣY = ε y ( x ) + ε y ( y ) + ε y ( z ) ⎪ ⎪ ΣZ = ε z ( x ) + ε z ( y ) + ε z ( z ) ⎨ ⎪∆X = δ x ( x ) + δ x ( y ) + δ x ( z ) ⎪∆Y = δ x + δ y + δ z y( ) y( ) y( ) ⎪ ⎪∆Z = δ ( x ) + δ ( y ) + δ ( z ) z z z ⎩

II.3.

(8)

Fig. 1. A typical linear carriage with six degree of freedom

II.2.

Thermally-Induced Volumetric Error Model (TIVEM)

Based on the above stated mathematical model and the fact that temperature has a significant effect on three positioning errors [9], a new thermally-induced volumetric error model was developed [11], which is expressed by Eq. (9). The model assumes that thermally-induced positioning errors are functions of the ball-screw nut temperature and the travel distance. It also assumes that the other 18 error components remain constant at their pre-calibrated cold start values: ⎧Vx ⎪ ⎪ ⎪ ⎪⎪V y ⎨ ⎪ ⎪ ⎪Vz ⎪ ⎪⎩

= ( −Yd ) * ⎡⎣ΣZ + ε z ( r ) ⎤⎦ + Z d * ⎡⎣ΣY + ε y ( r ) ⎤⎦ + + E x ( x,Txnut ) + δ x ( y ) + δ z ( z )

= ( − Z d ) * ⎡⎣ΣX + ε x ( r ) ⎤⎦ + X d * ⎡⎣ΣZ + ε z ( r ) ⎤⎦ +

(

)

+δ y ( x ) + E y y,Tynut + δ y ( z )

Simplified Thermally-Induced Volumetric Error Model

For the purposes of this paper, the thermally-induced volumetric error model described in the previous sub-section is termed the ‘full-error model.’ Although the full-error model is a significant improvement from the traditional model, it still requires measuring 21 error components with a high resolution laser interferometer, which makes the process costly, complicated, and time consuming. Therefore, in this paper an attempt has been made to further simplify the model by including only the three thermally-induced positioning error components. The remaining error components, such as the straightness, squareness and angular error components are ignored because they introduce only small contributions to the total volumetric error [9],[11]. The proposed simplified model will require measuring only the temperature of three nuts, axis origin offsets and their variations, for which a simple Laser Doppler Displacement Meter (LDDM) will be sufficiently accurate. The simplified model can be expressed as Eq. (11): ⎧Vx ⎪ ⎪ ⎪ ⎪V y ⎨ ⎪ ⎪ ⎪Vz ⎪ ⎩

= Ex ( x,Txnut ) = = x0 (Txnut ) + β x * α * x* (Txnut − 20 )

( ) = y0 (Tynut ) + β y * α * y* (Tynut − 20 ) = E y y,Tynut =

(11)

= Ez ( z,Tznut ) = = z0 (Tznut ) + β z * α * z* (Tznut − 20 )

III. Calculation Methods To demonstrate the effectiveness of the simplified model, two types of machining jobs are simulated on a three-axis horizontal CNC machining centre (Fig. 3): end-milling faces A0B0C0D0 and A1B1C1D1, and drilling holes 1 through 8 (Fig. 2).

(9)

= ( − X d ) * ⎣⎡ΣY + ε y ( r ) ⎦⎤ + Yd * ⎡⎣ΣX + ε x ( r ) ⎤⎦ + +δ z ( x ) + δ z ( y ) + Ez ( z,Tznut )

where, Ex(x, Txnut), Ey(y, Tynut), and Ez(z, Tznut) are thermally-induced positioning errors along axes. They are defined by the following equations: ⎧ E x ( x,Txnut ) = x0 (Txnut ) + β x * α * x* (Txnut − 20 ) ⎪⎪ ⎨ E y y,Tynut = y0 Tynut + β y * α * y* Tynut − 20 (10) ⎪ ⎪⎩ E z ( z,Tznut ) = z0 (Tznut ) + β z * α * z* (Tznut − 20 )

(

)

(

)

(

)


Fig. 2. Simulated machining components


1008


The simulations were performed for machining three cuboids sized 280 x 50 x 50 mm, 150 x 25 x 25 mm, and 30 x 10 x 10 mm. For any nominal tool position Pd set by the CNC controller, its compensation point Pc and error-remains after compensation can be calculated by applying the proposed model in conjunction with the algorithm proposed in our previous paper [11] and Lee et al. [13]. Testing of the full-error model for a three-axis machine tool requires the difficult and time consuming task of measuring 21 geometric error components and the positioning error variation data at different temperatures, which are difficult and time consuming to gather.

This paper uses data presented in [9] to demonstrate the effectiveness of the simplified model.

y

x

z Fig. 3. Test machine structure

Fig. 4. Thermally-induced volumetric errors of plane A1B1C1D1 at different machine conditions Plot 1: high temperature without compensation; Plot 2: compensated by proposed simplified compensation model; Plot 3: compensated by the full-error model [11] including all 21 errors; Plot 4: cold start

30

60

25.667

cold start 40

Linear dim. err (µm)


48.302

full err model simplified model

20

no compensation 0 -5.174 -20

cold start 20

ful err model simplified model 10

no compensation 0 -2.675 -4.847

-8.418 -8.855

-10

-4.954

Compensation methods

Compensation methods

Fig. 6. Linear dimension error of a 150 x 25 x 25 mm workpiece (length = 150 mm)

Fig. 5. Linear dimension error of a 280 x 50 x 50 mm workpiece (length = 280 mm)



1009


IV. Results and Analysis

6


cold start

IV.1. Volumetric Errors

3.809

4

Fig. 4 illustrates the thermally-induced volumetric errors of plane A1B1C1D1 machined under different machine conditions. The top layer shows that the average volumetric error is calculated as 192 microns if machined at high temperature without compensation. The second layer shows that the average volumetric error is calculated as 82 microns if machined at high temperature and compensated by the proposed model presented in this paper, which is a 57.3% reduction from the uncompensated error. The third layer shows that the average volumetric error is calculated as 77 microns if machined at high temperature and compensated by the full-error model, which is a 59.9% reduction from the uncompensated error. The bottom layer depicts that the average volumetric error is calculated as 11 microns if machined at cold start state, which does not require any thermal compensation. Therefore, the difference between the proposed model and the full-error model [11] is 5 microns or 2.6% of the uncompensated thermal error, which is negligible.

full err model simplified model

2

no compensation 0 -1.011 -2 -2.312

-2.316

-4

Compensation methods Fig. 7. Linear dimension error of a 30 x 10 x 10 mm workpiece (length = 30 mm)

60

Linear dim.err (µm)

50 40

280x50x50mm

30

150x25x25mm

20

30x10x10mm

IV.2. Linear Dimension Error The linear dimension, the distance between planes A0B0C0D0 and A1B1C1D1, was calculated to evaluate the effectiveness of the simplified compensation model on linear dimension compensation. The linear dimension error is defined as the measured value minus the designed value. Fig. 5 shows that the linear dimension error for a large workpiece increased by 0.437 microns (|(-8.855)-(-8.418)|) if compensated by the simplified model rather than the full-error model. Similarly, for medium and small sized workpieces the errors are increased by 0.107 and 0.004 microns, respectively, using the simplified model (Fig. 6 – Fig. 7). A comparison of linear dimension errors between different size workpieces (Fig. 8) confirms that the linear dimension error is proportional to the size of the workpiece.

10 0

1

-10

2

3

4

-20

Compensation methods 1 -cold start 2 -full err model 3 -simplified model 4 -no compensation Fig. 8. Comparison of linear dimension error of different workpiece sizes

cold start(main Y) no compensation(main Y) simplified model(minus Y) full err model(minus Y)

60

150

55

100 50 50 45

0 0

1

2

3

4

5

6

7

8

-50

IV.3. Positioning Error of Holes Eight holes were drilled on plane B0C0C1B1 of the test part (Fig. 2). Each hole’s positioning error along the x-axis (Ex) and the z-axis (Ez) is defined as measured value minus designed value. The holes’ positioning errors in the xoz plane are calculated using Eq. (12) and the results are illustrated in Fig. 9:

Holes' pos. err.(um)

Holes' pos. err (um)

200

E xz = E x2 + E z2

9

(12)

40 Holes number in order

Four types of machine conditions are shown. It is observed that at the cold start stage, the positioning errors are close to zero, whereas at high temperature without compensation, the errors reach as much as 154 microns.

Fig. 9. Positioning errors of holes in xoz plane



1010


Figs. 10. The percentage of thermally-induced dimensional error with and without compensation: (a) thermally-induced dimensional error comprises 75.94% without compensation (b) thermal error is reduced by 70.23% if compensated by the full-error model [11] (c) thermal error is reduced by 69.46% if compensated by the proposed simplified model

The figure also reveals that differences between the holes’ positioning errors estimated by the full-error model or by the simplified model are insignificant.

IV.4. Assessment of Compensation Results by Process Capability Data The effectiveness of the simplified compensation model is evaluated against process capability data to compare the error improvement against real machining error. Process capability is the smallest tolerance that can be maintained economically by a particular process; as such it represents the precision of a process. The smaller the process capability value is, the more precise the process is, yielding a higher quality product with smaller variability in dimensions. Process capability data can be estimated by applying the following equation [14]:

(

)

PC = 0.45 3 S + 0.001S 10

IT −16 5

The results show that thermally-induced dimensional error comprises 75.94% of total dimensional errors caused by machine conditions (Fig. 10(a)). When compensated by the full-error model, 70.23% of total dimensional errors can be eliminated, which leaves 5.71% thermally-induced dimensional error remaining (Fig. 10(b)). When compensated by the simplified model, 69.46% of total dimensional errors can be eliminated, which leaves 6.48% thermally-induced dimensional error remaining (Fig. 10(c)). This suggests that only 0.77% of total dimension compensation accuracy will be sacrificed by adopting the simple and economical model proposed in this paper instead of the complicated and costly full- error model.

V. z

(13)

There are six main factors affecting process capability [14]: (i) type of material machined, (ii) shape of the part, (iii) surface area, (iv) number of operations, (v) machine conditions, and (vi) skill of the operator. The change in PC value caused by machine conditions can be estimated by considering the other five process condition factors as fixed values and following the method described by Farmer [14]. Machine condition is affected by several factors, such as the built-in geometric and kinematic errors, thermally-induced errors, errors caused by the cutting process, and other machine system errors. Considering other aspects unchanged, the dimensional error caused by the machine’s temperature rise can be calculated from measured data. From this baseline, the percentage of error reduced by using the simplified compensation model out of total dimensional error caused by machine conditions can be obtained. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

z

z

Conclusion

A simplified thermally-induced volumetric error model which takes into account any time and thermal variations has been proposed to compensate for the volumetric error at any point within the work volume of a 3-axis horizontal CNC machining centre. The simplified compensation model requires only measurement of the three axis origin offsets and their variations with temperature, therefore a simple Laser Doppler Displacement Meter (LDDM) will be enough to meet the accuracy requirement. The simulated machining results indicate that only a negligible amount of total dimensional accuracy will be sacrificed by adopting the simplified compensation model instead of the full-error model comprised of 21 error components.

References [1] [2]

Bryan, J., International status of thermal error research, Annals of the CIRP Vol. 39, n. 2, pp. 645-656, 1990. Nasri, M. Ben Said, W. Bouzid, and O. Tsoumarev, A steady state thermal behavior study of 3D ball end milling model by using finite element method, International Review of Mechanical Engineering (I.RE.M.E.), Vol. 2, n. 6, pp. 845-855, 2008.


1011


[3]

[4]

[5]

[6]

[7]

[8]

[9] [10]

[11] [12]

[13]

[14]

S. Meguellati, and D. Bouzid, Measurement of thermal expansion by moiré interferometry, International Review of Mechanical Engineering (I.RE.M.E.), Vol. 1, n. 2, pp.190-194, 2007. Kim, J.-J., Y.H. Jeong, and D.-W. Cho, Thermal behavior of a machine tool equipped with linear motors, International Journal of Machine Tools and Manufacture, Vol. 44 (7-8), pp. 749-758, 2004. Wu, C.-H., Y.-T. Kung, Thermal analysis for the feed drive system of a CNC machine center, International Journal of Machine Tools and Manufacture, Vol. 43, n. 15, pp. 1521-1528, 2003. Ramesh, R., M.A. Mannan, and A.N. Poo, Thermal error measurement and modelling in machine tools: Part i. Influence of varying operating conditions, International Journal of Machine Tools and Manufacture, Vol. 43, n. 4, pp. 1235-1256, 2003. Yang, J., J. Yuan, and J. Ni, Thermal error mode analysis and robust modeling for error compensation on a cnc turning center, International Journal of Machine Tools and Manufacture, Vol. 39, pp. 1367-1381, 1999. Chen, J.S., Fast calibration and modeling of thermally-induced machine tool errors in real machining, International Journal of Machine Tools and Manufacture, Vol. 37 (2), pp. 159-169, 1997. R. Venugopal, Thermal effects on the accuracy of numerically controlled machine tools, Thesis, Purdue University. 1985. P.M. Ferreira, and C.R. Liu, A contribution to the analysis and compensation of the geometric error of a machining center, Annals of CIRP, Vol. 35, pp. 259-262, 1986. Yuxia Lu, M.N. Islam, A new approach to thermally induced volumetric error compensation, under review. J.-P. Choi, S.-J. Lee, and H.-D. Kwon, Roundness error prediction with a volumetric error model including spindle error motions of a machine tool, International Journal of Advanced Manufacture Technology, Vol. 21, pp. 923-928, 2003. E.-S. Lee, S.-H. Suh, and J.-W. Shon, A comprehensive method for calibration of volumetric positioning accuracy of CNC-machines, International Journal of Advanced Manufacture Technology, Vol. 14, pp. 43-49, 1998. Leonard E. Farmer, Dimensioning and tolerancing for function and economic manufacture (Blueprint Publications, 1999).


Authors’ information Yuxia Lu earned her bachelor’s and master’s degree in thermodynamics engineering from Northwestern Polytechnical University, Xian, China in 1987 and 1993, respectively. She worked as a mechanical engineer in China’s Department of Defense and at Beijing University of Aeronautics and Astronautics. She has been interested in doing research on the thermal behavior of machine tools, and she is currently enrolled as a PhD candidate at the Department of Mechanical Engineering, Curtin University, Australia. M. N. Islam (Corresponding A.) obtained his first degree in engineering (a combined bachelor’s and master’s degree in Mechanical Engineering) from the Technical University of Varna, Bulgaria (1978). He obtained his M.E. (Hons.) in Mechanical Engineering from the University of Wollongong, Australia (1990) and his PhD in Mechanical and Manufacturing Engineering from the University of New South Wales, Australia (2000). He is currently working as a senior lecturer at the Department of Mechanical Engineering, Curtin University, Australia. His research interests include dimensioning and tolerancing, dimensional analysis of manufacturing processes/process capability, design for manufacture, and precision machining.


1012


Fatigue and Brinelling Evaluation of ASME Extraction Pressure Vessel Closure with Locking Ring A. M. Senthil Anbazhagan, M. Dev Anand

Abstract – Closure used to close and open the pressure vessel comes in special type of vessels instead of the fixed dish heads. This type of vessel design purely depends on the process and maintenance requirements. Closures may be in different thicknesses, dimensions and materials according to the requirements. The design of closure in the vessel head area is subjected to mechanical and hydrostatic loads. Need of this work accounts to the often failures of closures in the oil, gas, natural chemical product production industries due to the insufficient design, over pressure loading during hydro test and over mechanical loading during operation. Estimates of fatigue and brinelling were obtained from the finite element analysis from the range of work hardening plasticity. Comparisons were made with the outputs taken from the finite element analysis for the range of loading conditions, vessel and closure geometry and its shape. Generally it is found that although there are significant variations between the different finite element solutions, satisfactory estimates of fatigue and brinelling that are most conservative are obtained when the reference stress produced is adopted. In this paper the evaluation of fatigue and brinelling for 300 Liter High Strength Steel TYPE-17-4PH pressure vessel closure with locking ring is discussed. The reasons for the failures were studied. New design methodology and recommendations are suggested based on the analysis with respect to the requirements of ASME. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Brinelling, Finite Element Method, Fatigue, Failures Vessels

I.

Introduction

This paper explains the design and developments of ASME based extraction pressure vessel closure and its locking ring using finite element technology. The 300L high strength ASME Section VIII DIV-I [1] extraction vessel has been taken for this study. Since the vessel and closure are axis symmetry, the 2-D axis symmetry FEA model along with closure and locking ring has been generated for analysis using ANSYS software. Different types of locking ring design with various dimensions have been generated and checked under various required design and operating loading conditions. Fatigue and Brinelling effects of the closure and locking ring are evaluated with respect to ASME [1] [4] requirements. To counsel the safe design and development, the effects of changes in dimensions with respect to Increase Vessel Groove Depth and Height, Decrease Vessel Groove Depth and Height, Increase Locking Ring Width and Decrease Locking Ring Height, Decrease Locking Ring Width and Increase Locking Ring Height have been made and analyzed and its design advantages and disadvantages also have been discussed and listed in the below study. Finally suitable locking ring design with various design recommendations has been suggested for the taken 300L extraction vessel in lie you with the recommendation of ASME [1].

The finite element based software ANSYS is used to find out the solution.

II.

Geometry and Modeling

The finite-element modeling of the 300L extraction vessel has been generated for this analysis. The baseline dimensions of the vessel solid geometry are shown in Figure 1 and the resulting 2-D axisymmetric FEA model developed from this geometry is shown in Figure 2. This model used high quality ANSYS mid-side nodded elements for the purpose of calculating the general and local membrane and bending stresses necessary for evaluating the structural adequacy of the design in accordance with ASME Code requirements [1]. However, local contact effects were not investigated analytically in the original analysis. The contact interfaces along with the gap element locations are identified in the Figure 3. There are two types of gap elements according to the FEA modeling are represented in the below model Figure 3. The round cloud markings in the messed model in Figure 3 are showing those areas. The surfaces between top of the vessel closure and bottom portion of the top locking ring, and the surfaces between locking ring top and vessel head groove bottom are represented as axial gaps.


1013

A. M. Senthil Anbazhagan, M. Dev Anand

The surfaces inside the groove vertical and locking ring vertical contact areas are represented as radial gaps. However the FEA model has been generated very carefully to extract the seriousness of load and stress transformations through these critical meshed surfaces and gaps of these regions.

III. Material Properties The head, vessel, and locking ring for the 300L extraction vessel are all fabricated from forged bar, Type 17-4PH, Condition H1150. The material properties were taken directly from the Tables TE-1, TM-1, and NF-1 in Part D of the ASME SECTION-II [2],[3] Material Specification Code are summarized in the below Table I. TABLE I SUMMARY OF MATERIAL PROPERTIES FOR SA-564, TYPE

Fig. 1. Geometry and Pertinent Dimensions Used to Develop ANSYS FEA Model of Extraction Pressure Vessel

Material Properties Elastic Modulus, E (106 psi) Thermal Expansion Coefficient, α (10-6 in/ºF) Poisson’s Ratio, ν

IV.

TEMPERATURES DEGREE F 70 28.5

200 27.8

400 26.6

600 25.5

700 24.9

5.9

5.9

5.9

5.9

5.9

0.31

0.31

0.31

0.31

0.31

Loads and Boundary Conditions

The vessel was designed for a maximum temperature of 200 degree F and pressure cycles ranging from 0 to 10000 psi. It was understood that there were no thermal transient stresses imposed upon the cleaning vessel; therefore, our analysis was an isothermal evaluation to design the closure and lock at a temperature of 200 degrees F and at the maximum rated pressure of 10000 psi. Figure 2 shows the regions where we have applied pressure loading on the ANSYS finite element model. To extract the thrust load of the vessel under pressure, we restrained the model in the axial direction near the bottom of the model, where the local restraint would not affect the areas of interest near the top of the model. One modification that we made to the FEA model was to replace the coupled interface at the postulated contact regions with analytical contact elements. This allowed us more flexibility in generating the closure model, and improved the accuracy of the solution. However, the contact elements are non linear in nature, resulting in longer run times to obtain a solution, and in some cases resulted in non converged solutions. To assess the effects of the ends of the locking ring on the vessel groove, we developed several 3-D models of the extraction vessel: a one-sixty symmetry model of the upper portion of the vessel and closure components, and a sub-model of this 3D model to obtain more details about the bearing stress at the end of the locking ring.

Fig. 2. Plot of Boundary Conditions Applied to the 2-D Axisymmetric Model of the Extraction Vessel

Fig. 3. Close-Up Head, Extraction Vessel, and Locking Ring Interfaces with Gap Element Locations Identified



1014


Fig. 4(a). Parametric 2-D Axisymmetric Results for Baseline (2” Wide by 2” High) Locking Ring with 0.500” Groove-Depth

\ Fig. 4(d). Parametric 2-D Axisymmetric Results for Baseline (2” Wide by 2” High) Locking Ring with 1.25” Groove-Depth

Fig. 4(b). Parametric 2-D Axisymmetric Results for Baseline (2” Wide by 2” High) Locking Ring with 0.750” Groove-Depth

Fig. 5(a). Parametric 2-D Axisymmetric Results for Baseline (2” Wide by 1.5” High) Locking Ring with 0.500” Groove-Depth

Fig. 4(c). Parametric 2-D Axisymmetric Results for Baseline (2” Wide by 2” High) Locking Ring with 1.00” Groove-Depth

Fig. 5(b). Parametric 2-D Axisymmetric Results for Baseline (2” Wide by 1.5” High) Locking Ring with 0.750” Groove-Depth



1015


Fig. 5(c). Parametric 2-D Axisymmetric Results for Baseline (2” Wide by 1.5” High) Locking Ring with 1.00” Groove-Depth

Fig. 6(b). Parametric 2-D Axisymmetric Results for Baseline (1.5” Wide by 2” High) Locking Ring with 0.750” Groove-Depth

Fig. 5(d). Parametric 2-D Axisymmetric Results for Baseline (2” Wide by 1.5” High) Locking Ring with 1.25” Groove-Depth

Fig. 7(a). Parametric 2-D Axisymmetric Results for Baseline (2.5” Wide by 2.5” High) Locking Ring with 1.25” Groove-Depth

Fig. 6(a). Parametric 2-D Axisymmetric Results for Baseline (1.5” Wide by 2” High) Locking Ring with 0.500” Groove-Depth

Fig. 7(b). Parametric 2-D Axisymmetric Results for Baseline (2.5” Wide by 2.5” High) Locking Ring with 1.5” Groove-Depth



1016


Fig. 8. Close-Up of 2-D Axisymmetric Model of Extraction Vessel Assembly Showing Linearized Stress Cutline Definitions

Fig. 11. Plot of 3-D Stress Intensity Results for Extraction Vessel with 0.30” Clearance

V.

Fig. 9. Plot of 2-D Axisymmetric Stress Intensity Results for 2.5” x 2.5” Locking Ring with 1.500” Groove-Depth

Fig. 10. Plot of 3-D Stress Intensity Results for Extraction Vessel with 0.050” Clearance


Preliminary Assessment and Observations

From this FEM study, we concluded that the taken 300L extraction vessel met the ASME code stress limits [1], [2] and [3]. Also we understood from the result photographs of the brinelling in the vessel groove and on the locking ring indicated that the failures are not only due to inadequate sizing, but it also with the result of very narrow line contact of the components in the closure path. A review of assembly tolerances indicated that as much as 0.060inch of clearance could exist at worst case and is normally 0.40inch.This allows the locking rings to rotate through the clearance when they are loaded. Logic suggests that shear keys (i.e., locking rings) require very tight clearances to ensure that pure shear transfer occurs and that rotation of the locking ring within the groove or slot is minimized. Because of the requirement for rapid and frequent opening and closing of the vessel, it is apparent that a tight toleranceing scheme, which makes assembly and disassembly of the extraction vessel difficult, is undesirable. So we decided that a minimal clearance of 0.020 inch will still allow disassembly of the vessel. The design fatigue curve for 17-4 PH Type 630 Condition H1150 according to ASME [1], [4] we referred to compute the number of cycles is as shown in Figure 12. The design fatigue life for Type 630 Condition H1150 material has been interpolated from the values taken from this chart Figure 12 [1], [4], [10]. So the estimated fatigue number of cycles for the enlarged locking ring groove depth and height is also extracted from this chart in accordance with the obtained finite element result output Figure 9.


1017


the edges of the locking ring, on the locking ring bottom surface coincident with the OD of the closure head, and on the locking ring upper surface coincident with the ID of the vessel flange.

Fig. 12. Design of Fatigue Curve for 17-4 PH Type 630 Condition H1150

VI.

FEA Analysis and Results

A parametric study of the locking ring sizes and groove geometries was performed using the 2D axisymmetric FEA model. We evaluated the effect of varying groove depth and locking ring width and height on the general stresses in the vessel flange as well as the bearing stress patterns on the locking ring. Results of the 2D axsymmetric analyses are shown in Figures 4 through 7. Note that the (unmodified) initial key size is 2 inches x 2 inches. Based on the parametric study and references [5], [6], [7], [8], [9], and [10] we established the below summary Table II of results. From our study, it was concluded that the membrane and bending stresses in the closure head and the locking ring are acceptable so long as the ID of the locking ring is maintained at 20.05 inches and the locking ring height is no less than two inches. To minimize impact on the current hardware, we retained these parameters. Therefore, only changes in groove depth and height in the vessel flange were considered. As noted above, increasing the groove depth results in more uneven bearing stresses on the locking ring. To offset this, any increase in groove depth is accompanied by a similar increase in locking ring height. It should also be noted that from a stability standpoint, both a wider and a taller locking ring improve stability. However both adversely impact the bending stress on the vessel flange. Therefore, we re-evaluated the fatigue capability of vessel flange [4], [6]. Taking a vertical linearized stress cut across the flange Figure 8, we noted that the stress intensity is 32,400 psi in Figure 9. This results in an estimated fatigue life of 70000 cycles with the enlarged groove depth and height. The results from the 3D stress analysis of the region at the ends of the keys are shown in the Figures 10 and 11. Based on the comparisons with 3D stresses away from the ends of the keys, the estimated peaking factor is 1.7. This is less than the value of 2.0 used in the previous fatigue findings and study [5], [6], and [10] of the locking ring. Therefore, it can be concluded that the previous fatigue assessment of the locking ring is conservative. However we noted that significant peaking occurs on the vessel coincident with


TABLE II EFFECTS OF CHANGES IN CLOSURE DIMENSIONS Parameter Variation Disadvantage Advantage ¾ Higher Bending ¾ More Stable1. Increased Vessel Stress in the Vessel Less Rotation as Groove Flange .Fatigue life Tolerances Depth/Height must be evaluated. Increase. ¾ Lower Bearing Loads under line contact ¾ Less Stable-More ¾ Lower 2. Decrease Vessel Rotation as Bending Stresses Groove Tolerances Increase on Vessel Depth/Height ¾ Higher Bending Flange. Loads under line contact ¾ Uneven Bearing ¾ More Stable3. Increase Locking on Locking Ring Less Rotation as Ring ¾ More Bending in tolerance Width/Decreases Locking Ring Increase. Locking Ring ¾ Lower Height Bearing Loads under Line Contact. ¾ Less Stable-More ¾ More Even 4. Decrease Locking Rotation as Bearing on Ring Tolerances Increase Locking Ring. Width/Increase ¾ Higher Bearing ¾ Less Bending Locking Ring Loads under line in Locking Ring. Height. contact

These local, high stress regions are identical to the regions where brinelling was observed on the vessel and the locking ring. As a result, we concluded that measures be taken to minimize line loading at these locations. Also we concluded rounding the edges of the following locations on the model: 1. The edges of the locking rings (both top and bottom surfaces to protect against upside-down installation of the ring segments). 2. The OD of the closure head along the top bearing surface, and 3. The vessel flange ID along the underside of the vessel groove upper lip. In order to better determine the minimum round edge size, we created an FEA model with round edges at the above locations. However, a converged analytical solution was not reached with this model. Nevertheless, we observed that minimum ¼ inches round edge required for these surfaces. In addition to performing a parametric study on various groove depths and locking ring heights and a 3D analysis of the vessel, we also assessed the impact on the locking ring load distribution of increasing the groove clearances. It can be seen from Figures 10 and 11 that the peak stresses intensity due to contact of the locking ring is at least 11% higher when the clearance is 0.050 versus an assumed line to line contact. When the clearance is


1018


reduced to 0.030, the difference in the peak stress intensity value is less than 1%, which is judged to be negligible. TABLE III SUMMARY OF STRESS RESULTS FOR EXTRACTION VESSEL LOCATION VESSEL AT LOCKING RING SLOT Stress Intensity, SINT (ksi) 32,357 PSI Stress Concentration Factor 3.0 3D Peaking Factor 1.0 Alternation Stress, Sa (Ksi) 48,536 KSI Cycles 70,000

Notes: 1. For 3D results, stress values in indicate stresses at 50o location. 2. Alternating stress is calculated as the stress intensity times the SCF times the peaking factor (if any) divided by 2.

VII.

Recommendation and Conclusions

Based on the results of our study and FEM evaluations, we concluded that the taken vessel design will require the following design changes according to the fatigue and brineling evaluation of the vessel closure and locking ring: Enlarging the locking ring cross section in both the height and width by ½ inches from the original dimension of the vessel. Tightening the axial clearance between the locking ring and the groove in the vessel flange to 0.030 inches maximum, and Rounding of the edges of the locking ring as well as the mating edges on the closure head OD and the upper lip ID of the vessel flange, using a ¼ inch radius. These changes will reduce, but may not completely eliminate brinelling of the locking ring, head and vessel flange.

References [1] [2]

[3] [4] [5]

[6]

[7]

ASME Code, Rules for construction of Pressure Vessel, Section VIII-Div-I, Subsection-A, General Requirements, 2007, pp.8-104. ASME Code, Specification for Steel, Sheet, and Strip, Hot Rolled, Carbon, Structural, High Strength Low Alloy, and High Strength low Alloy with Improved Formality, Section II, 2007 pp.16071615 ASTM Code, Specification for Steel Bars, Carbon and Alloy, Hot Wrought, General Requirements, 2007. ASME Code, Improvement of ASME NH Creep and Creep Fatigue, 2008. Atluri, S.N and Kathireson, K., Outer and Inner Surface Flaws in Thick Walled Pressure Vessels, Transcript of the fourth International Conference on Structural Mechanics and Reactor Technology, San Francisco,1977. Chopra, O.K and Shack, W.J, Review of Margins for ASME Code Fatigue Design Curve Effects of Surface Roughness and Material Validity, US Nuclear Regulatory Commission, Washington, DC,20555-0001, 2003. Kirkhope, K.J.Bell, R. and Kirkhope,J. ,Stress Intensity Factor Equations for Single and Multiple Cracked Pressurized Thick


Wall Cylinders, International Journal of Pressure Vessels and Pipings,Vol.41,pp 103-111,1990 [8] Mackerle. J, Finite Elements in the Analysis of Pressure Vessels and Piping’s, a bibliography (1998-2001) Int J, Pressure Vessels Piping’s 2002; 79(1):1-26. [9] Newman,J.C and Raju,I.S., Stress Intensity Factors for Internal Surface Cracks in Cylindrical Pressure Vessels, Journal of Pressure Vessel Technology-Transaction of the ASME, Vol. 102, pp 342-346,1980. [10] Zahavi. E, A Finite Element Analysis of Flange Connections, Journal of Pressure Vessel Technology, ASME, 115-1993, pp. 327-330. [11] B. Kenmeugne, B. D. Soh Fotsing, G. F. Anago, M. Fogue, J-L. Robert, Application of Multiaxial Fatigue Criteria to Mechanical Design, International Review of Mechanical Engineering (IREME), Vol. 5 n. 3, pp. 426-435, March 2011. [12] A. M. Alshoaibi, A. K. Ariffin, Finite Element Modeling of Fatigue Crack Propagation Using a Self Adaptive Mesh Strategy, International Review of Mechanical Engineering (IREME), Vol. 2 n. 4, pp. 537-544, July 2008.

Authors’ information Er. A. M. Senthil Anbazhagan, is a Research Scholar in the Department of Mechanical Engineering, Nooral Islam Center for Highr Education (NICHE), Kumarakoil under the control of Nooral Islam University (NIU), Tamil Nadu, INDIA. Born in 20th of March 1978 in Tamil Nadu, INDIA. He has completed his Masters of Engineering (ME) Degree in Design Engineering Disiplince from Government College of Technology (GCT), Coimbature, under the control of Anna Unitversity Chennai, Tamil Nadu, INDIA. He has completed his Bachlor of Engineering (BE) Degree in the disipline of Mechanical Engineering from National Engineering College (NEC), Kovilpatti, under the control of Manonmaniyam Sundaranar University, Tamil Nadu, INDIA. The major area of interest of the scholor is the Research Based Design and Development of Offshore and Onshore Oil, Gas, Nuclear, Chemical and Powerplant Industries Static Equipments like Pressure Vessels, Heat Exchangers, Columns, Reactors, Silos and Storge Tanks using Finite Element Calucations. Dr. M. Dev Anand is a Professor and Deputy Director Academic Affairs in the Department of Mechanical Engineering, NICHE, Kumaracoil, India. He completed his BE (Mechanical) from NEC, Kovilpatti in the year 1998, ME (Production) from Annamalai University, Chidambaram in the year 2000, his PhD from the NIT, Tiruchirapalli in the year 2008. He worked as R&D Engineer in biomedical instrumentation company, Karnataka, served as Faculty and Administrator in Engineering Colleges. He has published more than 37 papers in national and international conferences, 11 papers in international journals and one in national journal. He is a member of the IE and Life Member of ISTE. His research interests are intelligent manufacturing and mechatronics.


1019


The Modeling of the 2D Continuum with Non-Linearities Jiří Podešva

Abstract – The modeling of the textile fabric is an interesting area of mechanical problems. The subject of modeling can be a several types of fabric bags, either prismatic or flat. The reason for modeling is to investigate the deformation of the bag, the stress state in the material and the total volume of the full bag. The modeling bears two problems. First is the geometric non-linearity. Because the stiffness is consequent on the deformation, the mechanical behavior depends on the deformation. The solution must be performed in iterative cycles, during which the stiffness matrix is updated in every solution step with respect to the calculated deformation. The geometric nonlinearity is one of the typical problems of non-linear static and the solvers have tools for iterative solution. The second problem is “the first step problem”. The stiffness is consequent on the curved shape of the fabric. In the first step, when the model of the fabric is flat, it gives the zero stiffness. For this reason the solution of the first step can not be found. The paper demonstrates a kind of trick for “the first step solution”. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Finite Element Method, Fabric Bag, Discretization, Non-Linearity, Large Deformation, Newton-Raphson Approach

The “stiffness” is consequent on the longitudinal tension and, of course, on the deformation of the initially flat material. Consider the fabric material of no bending stiffness, exposed to the 2 axis in plane tension τx and τy and perpendicular pressure p (see Fig. 3). After deformation, displacement u perpendicular to the x-y plane, the tension τx and τy is not really perpendicular to the pressure. Conversely the component of the tension, perpendicular to the x-y plane, is in equilibrium with pressure. This gives the mathematical description in the form of partial differential equation:

Nomenclature x, y τ x, τ y u p

λ

R V

The coordinate system axis The tension force per unit width [N/m] The deflection [m] The normal pressure [Pa] The relative loading [m-1] Radius [m] Volume [m3]

I.

Introduction

The fabric material is used for flat (see Fig. 1) or prismatic (see Fig. 2) bag. The bag contains either bulk material (such as sand) or liquid. Under internal pressure the bag sides are deformed. For the bag design both deformation and stress state are important. It can be the subject of the computer modeling (Eischen [2]), using finite element method. The results of such modeling are deformation and stress state and, for instance, the final volume of the bag - the storage capacity. But the process of modeling brings specific problems, which makes it complicated, not routine work.

II.

τx ⋅

∂ 2u ∂x 2

+τ y ⋅

∂ 2u ∂y 2

=p

The Exact Solution

The problem of the 2D continuum mechanics has the mathematical description in the form of partial differential equation (1). The fabric is the kind of material with no bending capability.

Fig. 1. The flat fabric bag


1020

(1)

Jiří Podešva

tension

y

pressure Ω the fabric covering the hole

the supported fabric

x tension

Fig. 4. The 2D continuum discretized y

x

Fig. 2. The prismatic fabric bag

τy

y Fig. 5. The parabolic deflection

p

The results of the computer modeling (see Fig. 4 and Fig. 5) can be compared with the exact solution.

x τx

u

III. The Material Properties The fabric material can be characterized as the structure of perpendicular threads (see Fig. 6, King [3], Zhang [5], [6]). The threads in the direction of the long stripe of fabric are called warp, while these perpendicular to the warp threads are weft threads (Peng [7]). The material stiffness was measured in both warp and weft direction.

x

τx

p τx

Fig. 3. The 2D continuum with pressure and tension

If the tension in the x and y direction is equal (τx=τy=τ), then the equation (1) is simplified: ∂ 2u

the long stripe of fabric

u Ey

y

warp

weft x

∂ 2u

p

pressure + 2 =λ= = 2 tension τ ∂x ∂y

(2) Ex

The solution of the equation (1) or (2) then depends on the area Ω, on which it is solved. If the fabric covers the hole of certain shape, this shape represent the area. On the hole edges the displacement is zero - the boundary conditions. For the circular area of the radius R the solution of (2) is:

(

u = 14 ⋅ λ ⋅ x 2 + y 2 − R 2

)

(3)


the stripe width

Fig. 6. The stripe of fabric, the warp-weft structure


1021

load

linear approximation

specified load

sought deflection

tension

Jiří Podešva

deformation

deflection Fig. 7. The tension test record

The behavior under loading is not linear, but not strongly. The tension-deformation curve was linearized (see Fig. 7) and the Young modulus Ex and Ey for both directions was determined.

IV.

The Geometric Non-Linearity and the “First Step Trick”

It is clear from the description in chapter 2 that the resistance against pressure, the stiffness, depends on the deformation. Such behavior is called “the geometric nonlinearity” (see Fig. 8). the non-linear case tension

tension

the flat membrane - zero stiffness

Fig. 9. The non-linear load - deflection behavior

The commercial software has the tools for the iterative solution of non-linear problems (Crisfield [1]). The common approach is the Newton-Raphson method (see Fig. 9). The idea is to solve the non-linear task in subsequent iterations during which the stiffness matrix is re-calculated in every solution step. But in the solution process to solve the first iteration becomes to be a problem. The described procedure fails in the case of planar membrane. If the area is initially flat, then the structure has zero stiffness in the first iteration and no first approximation can be found. The additional stiffness can be added to every node of the structure just to solve the first solution step. In that case the first iteration solution is found, which is the gate for subsequent steps. Depending on the value of the additional stiffness the first approximation can be near to or far from the final deformed shape. In subsequent solution steps this additional stiffness is not taken in account and correct final deformed shape is found. tension

tension

tension

tension the slightly curved membrane - small stiffness the flat membrane with additional stiffness

tension

Fig. 10. The additional stiffness for the first step

Once, when even the first approximation of the deformed shape is found, the subsequent solution can be performed using standard non-linear algorithms. tension

V. the deeply curved membrane - large stiffness

The Real Task

The above described methodology was used to investigate the deformation and planar stress state of the real fabric bag (see Fig. 1 and Fig. 2 in the introduction).

Fig. 8. The stiffness resulting from deformation



1022

Jiří Podešva

The bag is exposed to the internal hydrostatic pressure due to the weight of the enclosed material. To create the finite element model the membrane triangular elements were used for the bag walls and the spring elements were used for the additional stiffness (Bijian [8], Shahkaramia [9]). The model of the prismatic bag (see Fig. 11) has the 3D shape, while the model of the flat bag is initially 2D simply two coincident walls. The walls of the bag are exposed to the internal pressure, which increases downward. In case of prismatic bag it is rather easy to define the pressure (see Fig. 12, for the bag strength testing the weight is 5x increased, that is why the pressure on the top edge is not zero).

Finally, if the iteration process converge to the correct solution, the deformed shape (see Fig. 13) and the stress state (see Fig. 14) is found. The stress state is expressed in terms of N/mm (the tension per unit width of the cloth), not in N/mm2 (the tension per unit area) as usual. On the figures the tension in the warp threads direction (vertical in this case) and the weft threads direction (horizontal in this case) is displayed.

the additional springs

Fig. 13. The prismatic bag deformation

The weft threads tension (horizontal)

Fig. 11. The finite element model

The warp threads tension (vertical)

the internal pressure increasing downward

Fig. 14. The prismatic bag stress state

In case of flat bag it is more complicated. The two horizontal walls are initially on the same height, so it has the same pressure across the whole area. In the process of deformation the initially flat walls are warped, the bag begins to have the 3D shape and the pressure varies across the wall (see Fig. 15). The pressure than must be redefined in each iteration cycle to be higher on element moving down and to be lower on elements moving up (see Fig. 16). The results of the solution are the deformation of the bag (see Fig. 17) and the stress state (see Fig. 18). Fig. 12. The bag exposed to the internal pressure



1023

Jiří Podešva

The walls has the longitudinal stripes of higher thickness and subsequently higher stiffness. This is the reason of irregularity in the stress distribution. Finally the resulting volume of the full bag was calculated. This is the important question because the bag - initially two coincident walls of zero volume, gain its volume capacity directly from the deformation. The final volume calculation is not supported by the software tools. The macro was created for this reason. From the nodal displacements the volume V1 below the top wall and the volume V2 below the bottom wall were calculated. The resulting volume V is the difference between these two (see Fig. 19).

p=0 h p Fig. 15. The pressure, increasing downward

the lower pressure on the top fabric wall

the top wall

V1

the bottom wall

V2

-

the resulting volume

=

V

Fig. 19. The volume calculation

VI.

Conclusion

The modeling of the fabric bags, the deformation and the stress state, is the case of the finite element method modeling. We use the generally defined element types, such as the shell element (the membrane element) and spring (or spar) element. The modeling brings the problems. Some of them (the geometric non-linearity) can be solved using the routines, built in the software (Newton-Raphson method). But some of them (the first step problem) are specific and the software has not the built in tools. To solve this the specific methodology was developed. Also the post-processing has some specific features, we mean the volume calculation. To solve this the deep insight into the software is necessary and the special macros have to be developed.

the higher pressure on the bottom fabric wall Fig. 16. The pressure distribution across the top and bottom bag wall

Acknowledgements The work was done with support of the MSM 6198910027 project.

References

Fig. 17. The flat bag deformation [1]

[2]

[3]

[4]

Fig. 18. The warp threads tension stress


Crisfield, M.A. Non-linear finite element analysis of solid and structures. John Wiley & Sons, Baffins Lane, Chichester, 1997. ISBN 0-471-92956-5. Eischen, J.W.; Shigan Deng; Clapp, T.G. : Finite-element modelling and control of flexible fabric parts. Computer Graphics and Applications, IEEE, Volume: 16 Issue:5, 1996, ISSN: 02721716. King, M.J., Jearanaisilawong, P., Socrate, S. : A continuum constitutive model for the mechanical behaviour of woven fabrics. International Journal of Solids and Structures, Volume 42, Issue 13, 2005, ISSN: 0020-7683. Duana, Y., Keefeb, M., Bogettic, T.A., Cheesemanc, B.A. Modelling friction effects on the ballistic impact behaviour of a single-ply high-strength fabric. International Journal of Impact Engineering. Volume 31, Issue 8, September 2005, Pages 9961012, ISSN: 0734-743X.


1024

Jiří Podešva

[5]

Zhang, Y.T., Fub, Y.B. A micromechanical model of woven fabric and its application to the analysis of buckling under uniaxial tension: Part 1: The micromechanical model. International Journal of Engineering Science, Volume 38, Issue 17, November 2000, Pages 1895-1906, ISSN: 0020-7225. [6] Zhang, Y.T., Fub, Y.B. A micro-mechanical model of woven fabric and its application to the analysis of buckling under uniaxial tension. Part 2: buckling analysis. International Journal of Engineering Science, Volume 39, Issue 1, January 2001, Pages 1-13, ISSN: 0020-7225. [7] Peng, X.Q., Cao, J. A continuum mechanics-based nonorthogonal constitutive model for woven composite fabrics. Composites Part A: Applied Science and Manufacturing, Volume 36, Issue 6, June 2005, Pages 859-874, ISSN: 1359-835X. [8] Bijian Chen, Muthu Govindaraj. A Physically Based Model of Fabric Drape Using Flexible Shell Theory. Textile Research Journal, June 1995, vol. 65, no. 6 324-330, ISSN: 0040-5175. [9] Shahkaramia, A.,Vaziri, R. A continuum shell finite element model for impact simulation of woven fabrics. International Journal of Impact Engineering. Volume 34, Issue 1, January 2007, Pages 104-119, ISSN: 0734-743X. [10] Breen, D. E., House, D. H., Wozny, M. J. A Particle-Based Model for Simulating the Draping Behavior of Woven Cloth. Textile Research Journal, November 1994, vol. 64, no. 11 663-685, ISSN: 0040-5175. [11] Ansys - Structural Nonlinearities, User‘s Guide. SAS IP, Inc. 1999.


Authors’ information Jiří Podešva Born 1. Dec. 1959 in Ostrava, Czech Republic, still alive. The graduate in 1982, VSB - Technical University of Ostrava, Ostrava, Czech Republic. Ph.D. in 1999 in applied mechanics, nowadays associated prof. on the VSB - Technical University of Ostrava, dep. of mechanics. Interested in the finite element method modeling, the practice, non-linear static and dynamic.


1025


Flying Qualities Estimation Methods for Small Unconventional Aircraft P. Hospodář, P. Vrchota, A. Drábek Abstract – This article describes a methodology of the aerodynamic calculation and experimental measurement, which leads to determination of flight conditions within design of small unconventional aircraft. The first part contains the application of various types of computational methods to determine the aerodynamic characteristics. The results serve as the source material for the flight qualities determination. The second part describes validation and evaluation process. The data obtained from calculation, wind tunnel testing and flight tests were compared with special focus on flying qualities determination. Individual types of methods were step by step applied on the development of the small unconventional aircraft. The various stadium of the aircraft design were considered. The handbooks methods used in the preliminary design through the “simple” panel methods up to the advanced CFD methods. These advanced methods can be used in the final part of the geometrical design. This procedure reduces time and financial demands on the development. Second part is devoted to the validation of the computational results. The wind tunnel results and data from flights tests were used as a baseline. Two aspects were considered during evaluation of obtained results. One aspect takes into account flying qualities where the frequency and the damping are determined by the calculation of eigenvalues of the nonlinear model. Another aspect allows obtaining the complex view and the possibilities of the various methods which can be applied in the individual phases of the aircraft design from the point of view of the aerodynamics and the flight dynamics. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Flying Qualities, Aerodynamic Derivatives, CFD, Flight Testing, Wind Tunnel

φθψ δ eδ c δ a δ r

Nomenclature α β S c b V q ρ u, v, w p,q,r cD,cL,cY cl,cm,cn X,Y,Z L,M,N I g G Ω m G F G M

Angle of attack [rad] Sideslip angle [rad] Wing area [m2] Mean aerodynamic chord [m] Wing span [m] Flight airspeed [m·s-1] Dynamic pressure [kg m-1 s-2] q = 1 2 ⋅ ρ ⋅V 2 Air density [kg m-3] Linear velocities [m·s-1] Angular velocities [rad·s-1] Dimensionless force aerodynamic coefficient (drag, lift, side) Dimensionless moment aerodynamic coefficients (roll, pitch, you) Aerodynamic forces (body axis system) Aerodynamic moments Moment of inertia[kg s-2] Gravity constant [m s-2] Angular velocity vector [rad s-1] Mass [kg] Force vector [N] Moment vector [N m]

M φ M φ

Mφ cmα cmq clp

cnr cnβ

x(t) z(t) y(t) u(t) R Θ

Euler angles; roll, pitch, yaw angle [rad] Control surface deflection (elevator, canard, aileron, ruder) Moment generated by mass of inertia and angular acceleration Moment generated by friction and angular rates Moment generated by spring and deflection Pitch moment coefficient derivative by 1st derivative of α' Pitch moment coefficient due to pitch rate derivative Roll moment coefficient due to roll rate Yaw moment coefficient due to yaw rate Yaw moment coefficient due to 1st derivative of sideslip angle derivative State vector Response of the system Computed response Input vector Covariance matrix Vector of unknown parameter


1026

P. Hospodář, P. Vrchota, A. Drábek

I.

Nowadays is possible to obtain data for the flight qualities identification and properties by a few manners with a usage a various empirical computational methods, wind tunnel tests and flight measurements. At the present time is possible to observe the trend of the displacement of the major part wind tunnel testing by the computations by the help of the advanced CFD methods. Wind tunnel tests are performed beyond the final geometry, which was designed or optimized by the CFD methods which reduce time and the financial demand on the development. Individual methods of aerodynamics coefficients determination are compared from standpoint of speed (prize) and the accuracy of the obtained results. Fast results are more likely needed in the preliminary design of aircraft, mostly with the less accuracy for the reason to suggest basic conception and estimate loads in a real time. These results are gradually specified (refined) by the using of the more complexity and more demanding methods. Defining the stability and control characteristics of an airplane are probably one of the most difficult and expensive aspects of an aircraft development. The difficulties are partially due to the fact that the stability and control phase of design is extended to the very end of the development process and sometimes even beyond it, causing occasionally unexpected and expensive twists along the project paths requiring changes on the aircraft, Since these can occur in the very late stages of the project, they are very expensive and often comes with detrimental effect to the expected performance. It is therefore of utmost importance to be able to predict Stability and control characteristics of the aircraft in the early stages of its development.

II.

G G ∂ I ⋅Ω G G M = + Ω× I ⋅Ω ∂t

(

)

(1)

(2)

Fig. 1. Body axis components; definition of moments L, M, N; forces X, Y, Z; velocities u, v, w; angular rates p, q, r; angle of attack α and sideslip angle β

Expressing the vectors as the sum of their components with respect to the body-fixed reference frame (Fig. 1) and with addition of gravity components and engine forces Ft gives: 1 ( X + FT ) m 1 v = p ⋅ w − r ⋅ u + g ⋅ sin φ ⋅ cos θ + Y m 1 w = q ⋅ u − p ⋅ v + g ⋅ cos φ ⋅ cos θ + Z m

u = r ⋅ v − q ⋅ w − g ⋅ cos θ +

(3)

The same procedure can be used for obtaining moment equations in component form:

Mathematical Model of an Aircraft

A general formulation of aircraft flight dynamics is derived from Newton’s second law of motion. These can be investigated from the viewpoint of stability or flying qualities. There are used following simplifying assumptions: the aircraft is rigid body with fixed mass, gravitational acceleration is constant and the aircraft is without structural deformation. Following equation and variables are relative to the body axes system (see Fig. 1). II.1.

G G ⎛ ∂v G G ⎞ F = m⎜ + Ω×v ⎟ ⎝ ∂t ⎠

Introduction

p = ( c1 ⋅ r + c2 ⋅ p ) ⋅ q + c3 ⋅ L + c4 ⋅ N

(

)

q = c5 ⋅ p ⋅ r − c5 ⋅ p 2 − r 2 + c7 ⋅ M

(4)

r = ( c8 ⋅ p − c2 ⋅ r ) ⋅ q + c4 ⋅ L + c9 ⋅ N

where the moment of inertia components are given by [4]:

( ) Γ ⋅ c2 = ( I x − I y + I z ) ⋅ I xz

Γ ⋅ c1 = I y − I z ⋅ I z − I xz 2

Six-Degree of Freedom Equation of Motion

The general forces and moments equations can be described by Newton’s second law of motion in translational and rotational form [1]. Next we consider rotating axis system, where the derivate operator applied to vectors has two parts: one for the rate of change of the vector, and one for axis system rotation:


Γ ⋅ c3 = I z , Γ ⋅ c4 = I xz

c5 = ( I z − I x ) I y , c6 = I xz I y , c7 = 1 I y

(

(5)

)

Γ ⋅ c8 = I x ⋅ I x − I y + I xz 2 Γ ⋅ c9 = I x , Γ = I x ⋅ I z − I xz 2


1027


Last three differential equations describe the relation between vector of Euler angle and angular velocity vector: φ = p + tan θ ⋅ ( q ⋅ sin φ + r ⋅ cos φ ) (6) θ = q ⋅ cos φ − r ⋅ sin φ

ψ =

in the body XB axis. The most significant components are aerodynamic moment and force. Aerodynamic moment and force can be described: X = q ⋅ S ⋅ cD Z = q ⋅ S ⋅ cL

q ⋅ sin φ + r ⋅ cos φ cos θ

Y = q ⋅ S ⋅ cY M = q ⋅ S ⋅ c ⋅ cm

Main aerodynamic variables are total air speed, angle of attack and sideslip angle (Fig. 1): V=

(u

2

+ v 2 + w2

)

⎛ w⎞ ⎟ ⎝u⎠ ⎛v⎞ β = sin −1 ⎜ ⎟ ⎝V ⎠

α = tan −1 ⎜

(7)

Definition of air speed V, angle of attack α and sideslip angle β shown in Fig. 1 and it is described in (eq. (7)). Body-axis velocity components are related to V, α and β by: u = V cos α cos β v = V sin β (8) w = V sin α cos β For many purposes, they are aerodynamic values measured in wind axes. In addition, the dimensionless aerodynamic force and moment coefficients are generally characterized as a function of angle of attack, sideslip angle and Mach number. Therefore, it is often useful write the force equations in terms of V, α and β instead u, v and w. Differentiating (eq.(7)) with respect to time gives: 1 V = ( uu + vv + ww ) V ( uw − wu ) α = 2 u + w2 ⎡ ⎤ (9) ⎛ Vv − vV ⎞ ⎢ 1 ⎥ = β = ⎜ ⎟⎢ ⎥ 2 ⎝ V ⎠ ⎢ 1− v / V 2 ⎥ ⎣ ⎦ =

(u

2

N = q ⋅ S ⋅ b ⋅ cn

Each dimensionless aerodynamic coefficient can be described as a function of other flight parameters:

(

(10)

L = q ⋅ S ⋅ b ⋅ cl

cL = cL (α ,q,α ,δ e ,M ,Re )

(11)

This description is too detailed for mathematical modeling and too difficult for measurement. Therefore, it is used a simplified description where the effect of each flight parameters is separated. Aerodynamic lift coefficient then given by: cL = cL0 + cLα ⋅ α + cLq

+ cLα

α ⋅ c 2 ⋅V

q⋅c + 2 ⋅V

(12)

+ cLδ ⋅ δ e + cLδ ⋅ δ c e

c

Total aerodynamic coefficient equals summation of initial value (value on lift line at zero angle of attack), product of lift line derivatives and angle of attack, product of non-dimensional pitch rate derivatives and non-dimensional pitch rate, product of derivation of nondimensional angle of attack change and non-dimensional angle of attack change and product of control derivatives and control deviation (elevator).

)

)

+ w2 v − v ( uu + ww ) V 2 u 2 + w2 Fig. 2. Definition of control surface sign deflection

II.2.

Aerodynamic Forces and Moments

We consider two components of external moment and three components of external force. The gyroscopic moment from the rotating machinery in the engine is neglected in this case. First component of external force is gravity, which affects a force along earth Z axis (eq. (3)). Second component engine thrust, which we assume


It can be used more comprehensive aerodynamic model which describes flow separation on lift curve [2] or dynamic component as a function of angle of attack and rate of change of it. The mathematical model of dimensionless aerodynamic coefficient (eq. (12)) is used in this paper for flight parameter estimation, computation, wind tunnel testing and for stability and flying qualities analyses. International Review of Mechanical Engineering, Vol. 5, N. 5 Extracted by ICOME 2011

1028


III. Computational Methods

III.2. Panel Methods

Among currently used methods for estimation (calculation) of aerodynamic characteristics focused on stability and controllability in design stage are Hand book, panel and CFD methods. The choices which methods will be used depend on design level, time and requirement of accuracy results. The hand book and panel methods will be used in early stage of the design because of a very fast estimation of variables. The advance CFD method will be used for the detail analysis of the design object in the latest stage of the design process.

Panel method is computational method for solutions of the airflow around the external surfaces airplane. Applicability of the panel method is essentially limited to the flow of potential characteristics - thus solution of the wing surfaces at small angles of attack and sideslip with the some influence of compressibility. There are many programs which solves potential flow, it this paper is used program AVL and Tornado.

III.1. Hand Book Methods This method is focused on prediction of stability and controllability of airplane in state of preliminary design. The method arose from connection of experimental data with empirical, semi - empirical and theoretical terms. There are two methods for computation of aerodynamic characteristic: Datcom and AAA. Datcom was issued in 1960s in cooperation of Aircraft Division of McDonnell Douglas co. in California and Flight Dynamic Laboratory of Airforce in Ohio. The creation of detailed structure of airplane model in Datcom paper version can be sometimes difficult. Therefore the Digital Datcom [3] was developed. Digital Datcom program is an electronic form of the USAF Stability and Control Datcom intended for preliminary aerodynamic design of an aircraft preferably in terms of stability and control. Advanced Aircraft Analysis (AAA) [4] method is a method developed and distributed by DAR (Design, Analysis, Research) Corporation, Kansas, USA. The AAA is software designated to computations of different aspects of aircraft design, it is based on the method published by Jan Roskam. In principle, Roskam´s method is a handbook method that compiles different methods and sources in comprehensive manner. Experience with AAA method was gained on several low-speed aircraft of general aviation category in the Aerodynamics, Performance, Geometry, Stability & Control and Dynamics modules, so the following remarks should be seen as not necessarily concerning all available scope of the method. Comprehensive system includes all fundamental aspects of aircraft aerodynamics, flight performance and flight dynamics. Very quick input data of aircraft are entered, the productivity is several orders over the productivity of manual use of Roskam´s books. There are also some drawbacks. Insertion of the wing of more complex geometry is difficult or even impossible (even kinked wing of two trapezes in plan is automatically replaced by substitute wing of single trapeze, no possibility to work with winglets).


The AVL (Vortex Lattice method) has a very simple implementation of solutions for control surface and wing mechanization effects and uses basic aerodynamic derivatives for longitudinal and lateral movement. Compressibility is included in the AVL over Prandtl Glauert correction, so the applicability is limited. The AVL provides reasonable results in terms of static factors (forces and moments) for small angles of attack and sideslip angle. Tornado [5] is a 3D-vortex lattice program. Stability derivatives with respect to angle of attack, angle of sideslip, angular rates and rudder deflections are provided. In order to be easily ported, Tornado is written in Matlab. Tornado is based on standard vortex lattice theory, stemming from potential flow theory. The classical “horse-shoe” arrangement of other vortex-lattice programs has been replaced with a “vortex-sling” arrangement. There is same limit as in the AVL: the vortex lattice theory is assumed the small angle of attack. No consideration is taken to fuselage effects or to any friction drag. III.3. Computational Fluid Dynamics Nowadays there are several approaches how to solve governing equations in CFD [6] solvers. These approaches could be from the simplest invisced to the most time-consuming Direct Numerical Simulation (DNS). An average CFD model for DNS, such as airplane, would contain billions of grid cells and it would take many months if not years to solve a single model. That is the reason why RANS is still in use for CFD calculation in engineering application. III.3.1. Grid creation One of the significant steps in computation process is to create sufficiently quality grid for CFD users due to regard for time consumption, simplicity of grid and amount of cells. Generating a good-quality grid is currently the single most important user-controlled aspect of CFD, and is the most difficult aspect of CFD modeling. For very complex geometry, it can take days to weeks (or even months) to create a quality grid. The tetrahedral grid has some advantages and on the other hand some disadvantages in comparison with the hexahedral grid. Each of them needs specific procedure and the time consumption, rate of user-controlled time International Review of Mechanical Engineering, Vol. 5, N. 5 Extracted by ICOME 2011

1029


specifically, is different. Nowadays there are several possibilities how to create a grid. The user can decide to use structured or unstructured grid with some specific shape of the elements (it usually depends on the solver). Grids, used in this study, were generated in ANSYS ICEM CFD and directly in STAR CCM+. STAR-CCM+ contains tools which can be used to generate a volume mesh starting from a surface. The surface can be read directly from either a surface mesh file, a neutral format file or translated from native CAD data. These tools can be used instead of or in addition to the volume mesh import options in order to obtain the volume mesh that will be used for the simulation. In other words, a part of the volume mesh can be imported while the rest can be generated directly within the software if desired. Alternatively, the entire mesh can be generated directly within STAR-CCM+. The core volume mesh contained a polyhedral type of cells. Additionally, prism layers were automatically included next to all wall boundaries. The mesh has nominally 10 prism layers with the stretching ratio of 1.2. To total number of cells is approximately 7 million.

CCM+. EDGE [7],[8] uses a finite volume approach to solve governing equations for unstructured meshes. It employs local time-stepping, local low speed preconditioning, multigrid and dual stepping for steady state and time dependent problems. The data structure of the code is edge based so that the code is constructed as cell vertex. It can be run in a parallel on a number of processors to efficiently solve large flow cases. It is equipped with a number of turbulence models based both on the eddy-viscosity and an explicit algebraic Reynolds stress model (EARMS) assumption. The model that was used during this study was Menter SST k-ω. The code has a number of boundary conditions for inlet and outlet boundaries. The pressure farfield boundary condition was used on the input boundary and pressure output on the output boundary of the computational domain.

Fig. 4. STAR CCM+ result for sideslip angle 20° - front view Fig. 3. Result for angle of attack 12°- flow separation can be seen on the canard and start of the separation on the wing, it means that that when crossing the critical angle of attack, the first flow separate canard and nose of the aircraft begins to fall

III.3.2. ICEM CFD This ANSYS ICEM CFD semi-automated meshing module presents rapid generation of multi-block structured or unstructured hexahedral volume meshes. Blocks can be built and interactively adjusted to the underlying CAD geometry. This blocking can be used as a template for other similar geometries for full parametric capabilities. Complex topologies, such as internal or external O-grids can be generated. The total number of element is about 10 millions depending on the calculated configurations (necessity of more amounts of cells if the aileron is deflected etc.). The height of the first layer was adequately small to satisfy the requirements of the turbulence model on wall function y+. III.3.3. Solver The CFD flow solvers used for this study were EDGE developed at FOI and commercial CFD code STAR


The STAR CCM+ is commercial software code based on a finite volume approach, also. The same boundary conditions, as in EDGE, were used. The k- ω model of turbulence was used [9]. The initial conditions were the same as in the test section of the wind tunnel, for both CFD codes (Re, M). The Reynolds number corresponded to flight Reynolds number and during tunnel campaign and his value was 0,65x106. The CFD calculations were stopped after the changes of residuals and aerodynamic forces were negligible.

IV.

Wind Tunnel Testing

Experimental test in a wind tunnel leads to measurement of aerodynamic characteristic of an airplane. The measurement was executed in 3mLSWT. The 3mLSWT is an atmospheric, return circuit, open test section low-speed wind tunnel. The test section is of circular cross section, of 3 m diameter and 3 m length. Model for measuring the aerodynamic characteristic is of the 1:3 to the real airplane. The measurements were performed at Re= 0,65x106 for referential cSAT w = 0,1713 m, wing area S = 0,2591 m2 and wing span b = 1,533 m.


1030


This is 2nd order standard differential equation with damping. The parameters ks, kd and I are mechanical constants of the system and Φ is a variable of the equation and it is time dependent variable. Now we can imagine a situation: we have some rigid body (model of airplane) on the torque spring, we start oscillation and we want to estimate the constants of the dynamic system. We can estimate the period of oscillation T, frequency of oscillation f, angular velocity of oscillation ω and logarithmic damping parameter δ. By the help of the Laplace transformation we rewrite (eq. (13)) as a: Fig. 5. Return circuit, open test section low-speed wind tunnel

IV.1. Static Wind Tunnel Testing

s 2φ − s

kd k φ+ s =0 I I

(14)

Measurement of aerodynamic forces and moments is realized by the help of strain gauge balance. Static wind tunnel tests program was aimed at acquisition of static aerodynamic characteristics during zero sideslip angle with variable AOA as well as during a combination of AOA and sideslip angle. The program included a measurement of impact of deflection of elevator and canard on longitudinal characteristics, deflection of the rudder on side characteristics and ailerons deflection on transverse and directional characteristics. An influence of vortex generators and Re on canards surfaces was verified during the measurement of the symmetrical cases. Peaks and drops of lift distribution were determined. IV.2. Dynamic Wind Tunnel Testing

2nd order dynamic system has following form:

(

)

s2 − 2 ⋅δ ⋅ s + ω 2 + δ 2 = 0

(15)

If compare the (eq. (14)) and (eq. (15)) we get results: I=

ks

ω +δ 2 2

=0

(16)

kd = 2 ⋅ I ⋅ s

Now we can see: if we know the spring constant, we can make measurement and from time response of the system we can estimate ω and δ and next we can estimate the other mechanical constants I and kd. IV.2.1. Yaw and Pitch Rate Oscillation Balance

Aerodynamic damping derivatives have significant impact during quick and short movements of an aircraft and their components present important factor for the solution of flying qualities [9]. The most common determination of these characteristics comes from hand book methods but the results have only indicative character. Wind tunnel measurements can accurately determine the dynamic effects of the analyzed model and verify the calculations and flying qualities. A forced oscillation principle by the help of the oscillation balance was used. The oscillation balance is equipment which is inserted into the aircraft model and the entire assembly is placed on a sting and support in the wind tunnel. It is used to deflect the aircraft model from stable position thus inducing angular oscillations which are monitored by signal sensors. From the signal distribution, after conversion from previous calibration of the oscillation balance sensor, frequency and damping of the oscillations are estimated. Aerodynamic derivatives can be estimated from these physical values. We assumed that the mechanical oscillation is physically described as mass oscillation on a spring or as mass momentum on a spring. A summation of moments must be zero: M φ + M φ + M φ + M = 0 (13) − I φ + kd φ − k sφ + M = 0


This oscillation balance can be used for measurement of pitch and yaw moment damping derivatives. Principle of this kind of balance is following: after reaching swipe end position by the trigger mechanism the swipe and also the oscillation balance head are deflected into the oscillation beginning position (+5 deg or -5 deg when moving backwards). By the next movement in given direction the swipe and the head are released and the whole assembly „swipe - oscillation balance head – aircraft model“ begins to oscillate. Hall sensor is placed in the oscillation balance axis and firmly connected to its head. The sensor processes the measured signal corresponding to the actual balance head deflection in time.

Fig. 6. Trigger mechanism in position closely before swipe release together with oscillation balance head


1031


From the sinusoidal signal the oscillation frequency ω and damping δ are estimated. The measurements are taken for given airspeed in wind tunnel. These two parameters so far have been as unknown values in the equations for the aerodynamic derivatives estimation.

moment part of a dynamic motion (forces and cross moments are neglected). We assume (eq. 13) for rotation about each axis. Next we describe relationship between measured values (damping and frequency) and pitch moment coefficient due to pitch rate derivative. The motion equation is for the wind tunnel model: I yyα = −kd ⋅ α − k s ⋅ α + M α

(17)

where Iyy is a moment of inertia about side axis. From (eq. (17)) we can see that the aerodynamic moments operate in the movement direction which is only a matter of definition. These can be divided into four parts:

Fig. 7. The oscillation balance is connected to the aircraft model and a stand according to the illustration (placing for measurement of pitch moment derivatives)

IV.2.2. Roll Rate Oscillation Balance This torsional oscillation balance consists of a main body attached to a column, rotating parts and a drive. The rotating part is stored in the main body with ball bearings and a cross torsion spring. The airplane model is attached to it through the console and reinforced frame also known as the box.

(

M α = q0 ⋅ S ⋅ c ⋅ cm q ⋅ q + cmα ⋅ α

)

(18)

These equations for the aerodynamic moments remain in this form only under presumption of one degree of freedom for oscillation otherwise it is necessary to differentiate between the angle of attack and angular velocity. Both values are equal in this mathematical model. Whenever possible, there are non dimensional values established in aerodynamics. Appointing of (eq. (17)) into (eq. (18)) gives equation in same form as (eq. (13)). After rearrangement and application Laplace transformation we get a final form for aerodynamic damping derivatives estimation: ⎛

( cmq + cmα ) = ⎜⎜ 2 ⋅ δ − Ikd ⎝

yy

⎞ 2 ⋅ V ⋅ I yy ⎟ ⎟ q ⋅S ⋅c 2 ⎠ 0

(19)

There is possible derive aerodynamic damping derivatives for a lateral motion, by the same procedure. For yaw derivatives:

(c

nr

⎛ k ⎞ 2 ⋅ V ⋅ I zz − cnβ = ⎜ 2 ⋅ δ − d ⎟ I zz ⎠ q0 ⋅ S ⋅ b 2 ⎝

)

(20)

and for roll derivatives: ⎛ k ⎞ 2 ⋅ V ⋅ I xx clp = ⎜ 2 ⋅ δ − d ⎟ I xx ⎠ q0 ⋅ S ⋅ b 2 ⎝

Fig. 8. Detail of oscillation balance with damping component

The torsional stiffness of the balance is determined by the cross spring. The balance uses a strain gauge pad to measure the deflections. This pad is inserted into the bottom part of the main body. The drive – stepper motor is located on the leeward side of the pillar and is connected to the main body by a centering pin only. The transmission from the stepper motor is mediated by a cam. The cam symmetrically rests on the teeth attached to the box which consequently results in the required motion. IV.2.3. Damping Derivatives Measurement Theory

V.


Flight Testing

Flight test is a branch of aeronautical engineering that develops and gathers data during flight of an aircraft and then analyzes the data to evaluate the flight characteristics of the aircraft and validate its design. Therefore we used flying laboratory called Vilík. It is RC model with measuring equipment for estimation of aerodynamic parameters. V.1.

This oscillation model in the wind tunnel is considered only with 1 degree of freedom i.e. we can estimate only

(21)

Instrumentation

Control system is modeler radio, where a receiver in the model by the help of servo-motors controls an International Review of Mechanical Engineering, Vol. 5, N. 5 Extracted by ICOME 2011

1032


aileron, elevator, rudder, canard and electrical engine motors. Drive is secured by an electric motor connected to an appropriately dimensioned three-phase regulator. Power and control of the controller comes from receiver’s battery. Power circuit itself is powered by the propulsion batteries galvanically separated from receiver circuit. Fuel cells are battery A123 (sometimes marked as a LiFe), 30 cells linked as: 10 cells in the series, 3 parallel series (10s3p). For measurement of performances and features of the Vilík model it is necessary to measure and record whole set of parameters with relatively high frequency of recording. The elementary principle of sampling rate is influenced by Shannon’s sampling theorem. That say the sample rate muss be two-times higher then maximal frequency of measured signal. To obtain good results in practice, it is necessary to sample at a rate much higher than the theoretical minimum rate. Typical frequency of the rigid body dynamic model is below 2 Hz. Therefore we choice sampling rate 50 Hz. Flight test data necessary for flight parameter estimation are obtained by measurement of following variables: airspeed, angle of attack, sideslip angle, control deflection, angular rates, accelerations and Euler angles. Airspeed is measured by the pitot-static probe and differential manometers. Angle of attack and sideslip angle are measured by directional flag with Hall sensor and magnet1. Air temperature (density is function of temp.) in the boundary layer is measured by temperature sensor. Deflections of all control surfaces are measured by Hall sensor and magnet. Acceleration of translational, rotational and Euler angels in all 3 axes is measured equipment Crossbow AHRS400. AHRS (Attitude and Heading Reference Systems) from the company Crossbow is nine-axes measuring system, which is composed of linear accelerometers, angular rate sensor and magnetometer. Collection of analog data from Hall sensors, temperature sensors and pressure sensors are read by National Instruments A/D cards. Connection to PC is done via USB. Vilík has an on-board computer, which is used for collecting and recording data from sensors. AHRS base Crossbow communicates via COM1. Vilík is connected to the ground WiFi data transmission, through which it is possible to monitor the process of measurement. V.2.

Parameter Identification

Flight test was focused on estimation of aerodynamic parameters and determination of flying qualities. For this reason, the towing model was used to lifts the test model to the measured height without losing engine battery power on test model. Test model break away from the towing model of sufficient height to and implements appropriate maneuvers to identify the flight data without engine thrust. During the maneuver, the airplane lost 1

This equipments were calibrated in wind tunnel


altitude. The airplane gains a certain height after the engine starts and continues with identifying maneuvers. This process is repeated few times during one flight.

Fig. 9. Model Vilík during flight test

Main goal of the flight testing is an estimation of aerodynamic characteristics [11],[12],[13]. There are many methods for aircraft parameter identification and here are presented some generalized method for identification of dynamic model. The basic principle of dynamic system identification is based on transfer function estimation by the help of method of least squares. This method derived Carl Friedrich Gauss, and it describes the linear relationship between variables x (input) and y (output) as overdetermined system. To identify the dynamics of the aircraft, essentially nonlinear, it may include a priori information knowledge of the mathematical description of motion equations of the airplane. Then it is more appropriate and more accurate to estimate parameters of the system rather than transfer function. This procedure can be used for estimation of aerodynamic derivatives for each aerodynamic coefficient separately. It would be better to estimate all longitudinal or lateral aerodynamic coefficients together depending of type of flight maneuver. There is not possible estimate measurement error and error of sensors (offset, bias). Therefore is preferred a method which uses whole nonlinear mathematical model of aircraft motion, where we estimate the aerodynamic derivatives and other parameters. The name of this method is output error method (OEM) and Fig. 10 shows a block schematic. We briefly describe the form of the output error parameter estimation method. The aircraft is a continuous-time dynamic system. Measurements are made at discrete time intervals for analysis. The system equations, in general form are: x ( t ) = f ( x ( t ) ,u ( t ) , Θ ) (22) z ( t ) = g ( x ( t ) ,u ( t ) , Θ ) + ε We assume that the measurement noise ε is a sequence of independent Gaussian random vectors with zero mean and covariance R.


1033


simplification a substitution is introduced:

Measurement noise

Input (control signal) u

Measured output

+

Aircraft Aircraft Dynamic Dynamic system system

N ⎡ ∂y t ⎤ ( k ) −1 G = −∑ ⎢ ⎥ R ⎡⎣ z ( tk ) − y ( tk ) ⎤⎦ k =1 ⎣ ∂Θ ⎦ T

+ z

Mathematical Mathematical model model

y

+ -

Integration of state and observation equations

T

Resulting equation defining the parameters vector for system response definition and residua definition in the next step of optimization is:

Response error z- y

Parameter Parameter update update

ˆ ˆ Θ i +1 = Θi + ∆Θ ˆ = −G F ∆Θ

Fig. 10. Schema of Output Error Method

∑

The basic optimization problem is to find the value of the vector Θ that gives the smallest or largest value of the scalar-valued function J(Θ). The necessary condition for minimization of the likelihood function with respect to the unknown parameters is given by:

∂Θ

=0

(24)

There are many optimization techniques which can be applied for optimization of nonlinear case. GaussNewton method has convenient rapid tendency of convergence. This method requires first and second order of gradient of cost function from Taylor’s series. Necessary condition for determination of the function minimum is a presumption, which relates to the previous term: ∂ ⎡ J ( Θ0 + ∆Θ ) ⎤⎦ = 0 (25) ∂Θ ⎣ Substitution (eq. (23)) into (eq. (25)) and arrangement we get final form of Gauss-Newton method with second order of gradient (Hess matrix) of cost function. Solution of Hess matrix in the final form is harder and more time consuming than solution of the gradient of the first order. That’s why a following approximation of Hess matrix is used: ⎡ ∂y ( tk ) ⎤ −1 ⎡ ∂y ( tk ) ⎤ ≈ ∑⎢ ⎥ R ⎢ ⎥ 2 ∂Θ ⎣ ∂Θ ⎦ k =1 ⎣ ∂Θ ⎦ ∂2 J

N

T

(26)

This simplified version is called modified GaussNewton (referred to as Newton-Raphson) method. For Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

40 pitch rate [deg.s -1]

T 1 N ⎡⎣ z ( tk ) − y ( tk ) ⎤⎦ R −1 ⎡⎣ z ( tk ) − y ( tk ) ⎤⎦ (23) J (Θ) = 2 k =1

(28)

That concludes the entire optimization procedure based on maximum likelihood function.

20 0 -20 -40 -60

0

2

4

6

8

10

12

14

16

18

8 10 time [s]

12

14

16

18

measured estimated 1st iteration estimated last iteration

100 pitch acceleration [deg.s -2]

The parameter estimation task is to estimate the value of the unknown parameter vector Θ. The output-error estimate of the unknown parameter vector is the value that minimizes the maximum likelihood cost function:

∂ J (Θ)

(27)

⎡ ∂y ( tk ) ⎤ −1 ⎡ ∂y ( tk ) ⎤ F = ∑⎢ ⎥ R ⎢ ⎥ ⎣ ∂Θ ⎦ k =1 ⎣ ∂Θ ⎦ N

50 0 -50 -100

0

2

4

6

Fig. 11. Comparison of flight measured and estimated parameters, longitudinal maneuver

V.2.1.

Estimation Procedure

First it is necessary to specify the model that shall be estimated (longitudinal / lateral / fully nonlinear model / aerodynamic coefficients). Based on the model a vector of the starting parameters is selected. Consequently movement of the model is defined using Runge-Kutta integral calculating of the output values of the model. Based on simulated and measured signals maximum likelihood matrix with covariance matrix are defined. Next step initiates optimization phase described in the previous part. Particular vector elements with searched parameters are perturbed and each perturbed value is complemented with state and output equations. Based on particular perturbed outputs a sensitivity equation is derived which is consequently used for the determination of the matrixes F a G using the Newton-Raphson method. Finally a change of parameters vectors is calculated for next iteration step.


1034

-0.03

-0.2

-0.04

-0.4 Cma

Cm0


-0.05 -0.06

-0.6 -0.8

0

5 iteration #

-1

10

-0.3

0.55

-0.35

0.5 Cmdd

Cmde

-0.07

-0.4 -0.45 -0.5

changes during short period maneuver on short period dynamics is small. The Phugoid mode is lightly damped low-frequency oscillation. The aircraft responses in Phugoid mode are very slow compared to the changes in the motion parameters in short period mode. This mode describes the translator motions of the vehicle center of mass with practically no change in the angle of attack. Frequency and damping ratio are obtained from state space model using investigation of an eigenvalues.

0

5 iteration #

10

0.45

VI.2. Lateral Motion

0.4

0

5 iteration #

10

0.35

0

5 iteration #

Lateral motion uses side force, roll and yaw moment. Solving lateral motion eigenvalues we get two real roots corresponding to the spiral and roll mode and pair of complex roots defines the dutch roll mode:

10

Fig. 12. Distribution of aerodynamic derivations vs. particular steps of optimization (iterations), longitudinal maneuver

VI.

Flying Qualities

Flying qualities describe static and dynamic stability of the aircraft motion [14]. They are also used for aircraft determination MIL category. Six-degrees-of freedom equation of motion (see II. Mathematical model of an aircraft) is very complex. A simplified model for investigation of flying qualities is used. Some of the reasons why flight analysts prefer the simplified models are that full complex models are difficult to interpret and analyze. It is possible to separate the model equations into independent subsets without much loss accuracy and linear model could be created. The nonlinear 6DOF equation of motion can be linearized using the small disturbance theory2 and neglecting small terms. Results of linearization are separated longitudinal and lateral model [15]. VI.1. Longitudinal Motion Longitudinal motion considers only pitch moment, lift and drag force equation: F qS V = − cD + T cos α + g ⋅ sin (α −θ ) m m F qS g cL + q − T sin α + cos (α − θ ) α = − mV mV V qS c q = cm Iy

g qS sin φ cos θ + p sin α − r cos α + cY V m 1 p = ( qSbI z cl + qSbI xz cn ) 2 I x I z − I xz

β =

(29)

θ = q Longitudinal motion has two distinct modes: one lower frequency and another at relatively higher frequency. The short period is a relatively well-damped, high frequency oscillation mode. The effect of pitch angle 2 Disturbance around working point (flight level, airspeed, angle of attack,…)


r =

1 2 I x I z − I xz

(30)

( qSbI x cn + qSbI xz cl )

φ = p + r tan θ cos φ Dutch roll is a relatively lightly damped oscillatory mode that consists of primarily the sideslip and yawing moment. One of the real roots with a small value (relatively long time-period) indicates the spiral mode. The root can have a negative or positive value, making the model convergent or divergent. This mode is dominated by rolling and yawing motions. Roll mode is highly damped mode with a relatively short time period.

VII.

Results and Graphs

Here are presented and compared the final result of flying qualities. The flying qualities and dynamics of an aircraft motion are described base on linear model in the state space form. Results based on each methods of aerodynamic derivatives estimation are presented. Eigenvalues positions of the longitudinal motion are shown in following figure. There are two complex conjugated pairs of short period and phugoid mode. From previous figure is evident that phugoid mode (smallest frequency that short period – Fig.13 right side) has complex eigenvalues very close. But a small difference is seen on a peak of frequency characteristic at lower frequency (Fig. 14). This peak means that the slower mode (phugoid) isn’t more damped.


1035


Lateral motion eigenvalues are displayed on following figure. Eigenvalues for lateral motion is displayed in (Fig. 16).

8 0.6

0.46

0.34

6 0.76 4 0.92

0.24 CFD 0.15 0.07 7 6 panel method flight test 5 hand book method4 wind tunnel 3

Imaginary Axis

4 0.95

2

2

0.9

3 0.978

1

2

1

0.994 Imaginary Axis

2 0.92

3

-4

4 5

-6 0.76

6 0.6

-8 -7

0.46

-6

-5

0.34

-4 -3 Real Axis

0.24

0.15

-2

1 10

8

6

4

2

-2 -3 0.978

X: -0.7417 Y: -3.734

0.95 -4 -14

Bode Diagram

-12

0.9 -10

20

0.82

0.7

-8 -6 Real Axis

40

Magnitude (dB)

0.3

0.994

0

Fig. 13. Pole map in complex plane of the longitudinal motion -4

0.52 -2

0.3 0

Fig. 16. Eigenvalues of lateral motion

0

There are shown complex conjugated pair (dutch roll), one quick real pole (roll mode) and one slow real pole (spiral mode). The biggest difference between the different methods of estimating the aerodynamic derivative shows a panel method. The frequency characteristic that shows lightly damped mode (dutch roll) at frequency about 3 rad/s is depicted in (Fig. 17). This mode is shown as a step response on following figure.

-20 -40 -60 90 CFD panel method flight test hand book method wind tunnel

45 Phase (deg)

0.52

-1

0.07 7

-1

12

0

0.7

CFD panel method flight test hand book method wind tunnel

0 -2

0.82

0 -45 -90 -135 -180 -3 10

-2

10

-1

10

0

10

1

10

2

10

Bode Diagram 30

Frequency (rad/sec)

10 0 -10 -20 -30 -40 90 45

Phase (deg)

On the following figure is an impulse characteristic (time response on Dirac impulse) of longitudinal motion for short period mode. There can be shown that frequency of short period is similar, only panel method has a minor deviation from damping and frequency.

Magnitude (dB)

20

Fig. 14. Frequency response of longitudinal motion (canard deflection input – pitch angle output)

CFD panel method flight test hand book method wind tunnel

0 -45 -90 -4 10

-3

10

-2

10

-1

10

0

10

1

10

2

10

Frequency (rad/sec)

Fig. 17. Frequency response of lateral motion (rudder deflection input – yaw rate output)

VIII. Conclusion In this paper numerical, wind tunnel and flight tests results of flying laboratory Vilík were compared. In the previous chapter there are tables with aerodynamic derivations for all methods and their comparison. The results are closely matching. Particular computational methods show following features.

Fig. 15. Impulse response of longitudinal motion (canard deflection input – pitch rate output)



1036


and computation is shorter than in case of EDGE, where mesh must be created in ICEM CFD software. At the other side EDGE enables usage of unlimited amount of licence, possibility of the cooperation with EDGE developers at FOI on various modes of computation. Both solvers were used for the next computation in this research project. Sideslip regime was computed for a different angles of attack, and analysis of the various deflection of the ailerons and vertical tails were done. Influence of the canard surfaces and the horizontal tails were treated by STAR CCM+. These results were provided for simulation of fully non linear aircraft motion.

Step Response 1.2 CFD panel method flight test hand book method wind tunnel

1

Amplitude

0.8

0.6

0.4

0.2

0

-0.2

0

1

2

3

4

5

6

7

8

9

Wind tunnel measurement

10

Time (sec)

Fig. 18. Step response of lateral motion – dutch roll (rudder deflection input – yaw rate output)

Panel methods From comparison of CFD and wind tunnel results it is apparent that panel methods (AVL and TORNADO) produce credible results for static, control and damping aerodynamic derivations in linear area for small values of AOA and yaw. Panel methods are suitable for quick estimation of flight characteristics. These methods however are not able to include the factor of viscosity into the calculations. The drag characteristics depending on AOA are only calculated as induced drag (without friction and compressibility). Empirical methods AAA provides fast results of aerodynamic analysis. They are suitable for standard conceptions of aircraft (wings and horizontal tail), therefore only side characteristics are preferably compared form flying model Vilík. CFD Methods This method covers even non-linear issues and includes viscous calculations (flow separation, wingfuselage interaction, wing wake…). CFD is however, depending on the complexity of the mesh (every point on polar has to be calculated separately) very demanding on time. It still provides the most accurate results of all numerical methods. Computed results showed very good agreement in case of characteristics of forces, where were differences at the most to 5 per cent. (In case of drag force coefficient the difference was little bit higher. The very similar behavior of the flow at important angles of attack at various regimes was treated by the visualization of the fluid flow. Reached results shows that both solvers can be used for low speed aerodynamic computations and brought comparison of commercial and non-commercial software. Computation process revealed an advantage of the STAR CCM+ in such things as that the mesh generator is included and usage of polyhedral grid (less amount of cells). Time consumption for grid generation


This method provides the most accurate results; however it is influenced by the wind tunnel characteristics, model fastening and model accuracy. Wind tunnel, numerical a flight test results of damping derivatives are shown in tables 1. A measured longitudinal damping derivative differs to computed values because the chosen numerical methods do not accurately calculate aerodynamic influence of a fuselage (the programme Tornado doesn’t calculate with a fuselage). The fuselage and the wing take a main part on a yaw aerodynamic moment. This is main difference between longitudinal measured and calculated values. Lateral measured aerodynamic derivatives relatively good agreement with the calculated and estimated values. Flight testing Flight tests are the most accurate way to determine the aerodynamic characteristics of aircraft. The aerodynamic derivatives have a very good agreement. In final part of this article some summarization of the results from economical aspect point of view is done. The aspect is namely the time needed for obtaining of the results. The beginning of the working process is usually geometry acceptance. The data needed for the design are at the end of this process. Time needed for analyses is increased about one order of magnitude if CFD methods are compared with panel methods (3D analyses of the aircraft by panel methods can takes approximately one week, but the same analyses by CFD can take months or later, for example). The most time consuming parts are wind tunnel testing and flight measurement. Wind tunnel campaign can take several months (it depends on the complication of the model, how quick a manufacturing is and the number of testing cases). The almost same time requirements as for wind tunnel testing are for flight measurement (it is mainly cause by the manufacturing of the model). It also very time consuming even for flight model. If the flight measurement are done on the full scale airplane, the time and financial requirements are much higher. So at the end we would like to emphasize that it the decision what methods will be used depend on the designer and on the stage of the project. The designer needs very quick stability and control and performance


1037


assessments, at the expanse of lower accuracy of the results, in the preliminary parts of the project. The advance CFD method and especially wind tunnel testing and flight testing should be used for detail analyses and tuning of specific parts of the airplane.

Acknowledgements The research was supported by the Ministry of Education, Youth and Sports of the Czech Republic, MDM0001066901 Development of the applied external aerodynamics project.

Authors’ information P. Hospodář (Czech Republic, 1983) 2008 graduated at Czech Technical University in Prague, Czech Republic, Faculty of electrical engineering. Ph.D. student at Department of Control Engineering at Czech Technical University in Prague Research engineer - Aerodynamic department of VZLU Aeronautical Research and Test Institute in Prague. P. Vrchota (Czech Republic, 1981) 2005 graduated at Czech Technical University in Prague, Aerospace Engineering Department Ph.D. student at Aerospace Engineering Department at Czech Technical University in Prague Research engineer - Aerodynamic department of VZLU aeronautical research and test institute in

References [1] [2] [3]

[4]

[5]

[6] [7]

[8] [9]

[10]

[11] [12] [13] [14] [15]

Etkin, B., Dynamics of atmospheric flight (Wiley and Sons, Inc., N.Y. 1972) Jategaonkar R. V, D. Fischenberg Identification of Aircraft stall Behavior from Flight Test Data RTO-MP-11, 1999, Paper No. 17 W. B. Blake, Prediction of Fighter Aircraft Dynamic Derivatives Using Digital Datcom, US. Air Force Wright Aeronautical Labs., Wright-Patterson AFB, AIAA-85-4070 J. Roskam, Airplane Design, Part VI: Preliminary Calculation of Aerodynamic, Thrust and Power Characteristics, (Roskam Aviation and Engineering Corporation 1987) T. Melin, A Vortex Lattice MATLAB Implementation forLinear Aerodynamic Wing Applications, Master Thesis, Royal Institute of Technology, 2000 J.D. Jr. Anderson, Computational Fluid Dynamics The basics with applications (McGraw-Hill international edition, 1995) P. Eliasson, Edge, a Navier-Stokes solver for unstructured grids, Proceedings. To Finite Volumes for Complex Applications III, ISBN 1 9039 9634 1, pp.527-534, 2002 A. Jirasek, R.M. Cummings: Assessment of Sting Effect on X-31 Aircraft Model Using CFD, AIAA 2010-104. A. Drábek, P. Hospodář, P. Vrchota, Comparison of CFD methods applied on Vilik flying model, Czech Aerospace proceedings, 4/2010 P. Hospodář, J. Pleskač, Wind-tunnel measurement of unsteady aerodynamic characteristics; Czech Aerospace proceedings, ISSN 1211—877X, 2/2011 V. Klain, E. A. Morelli: Aircraft System Identification, AIAA 2006 R. V. Jategaonkar, Flight Vehicle System Indetification – A Time Domain Methodology, AIAA 2006 J. R. Raol, G. Girirja, J. Singh, Modelling and Parameter Estimation of Dynamic Systems, IEE, 2004 J. R. Raol, J. Singh, Flight mechanics modeling and analysis, CRC Press, 2009 B. N. Pamadi, Performance, stability dynamics, and control of airplanes, (2nd edition, AIAA, 2004)


Prague. A. Drábek (Czech Republic, 1981) 2006 graduated at Brno University of Technology, Faculty of mechanical engineering, Czech Republic Ph.D. student at Faculty of mechanical engineering at Brno University of Technology Research engineer - Aerodynamic department of VZLU aeronautical research and test institute in Prague.


1038

1970-8742(201107)5:5;1-R Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved