Effect of Elastic Modulus and Density Ratio on

A dissertation Report on

“Effect of Elastic Modulus and Density Ratio on Vibration Characteristics of Rotating Turbine Blade” Submitted to

VISVESVARAYA TECHNOLOGICAL UNIVERSITY BELGAUM In partial fulfillment of the requirements for the award of degree of

Master of Technology In

MACHINE DESIGN By

MADHU E USN: 1OX12MMD10 Under the guidance of

Mr. Nandish R V Assistant Professor Dept. of Mechanical Engineering The Oxford College of Engineering Bommanahalli, Bangalore-560068.

DEPARTMENT OF MECHANICAL ENGINEERING THE OXFORD COLLEGE OF ENGINEERING HOSUR ROAD, BOMMANAHALLI, BANGALORE -560068 2013-14

ACKNOWLEDGEMENT I have taken efforts in this project. However, it would not have been possible without the kind support and help of many individuals and organizations. I would like to extend my sincere thanks to all of them. I have a great pleasure in expressing my deep sense of gratitude to founder Chairman Sri S. NarasaRaju and to our Executive Director Sri S.N.V.L. NarasimhaRaju for proving me a great infrastructure and well-furnished labs. I take this opportunity to express my profound gratitude to Principal Dr. R Nagaraj for his constant support and encouragement. I would like to express my special gratitude and thanks to my guide Mr. Nandish R V, Asst. Professor Department of Mechanical Engineering, The Oxford College of Engineering, Bommanahalli, Bangalore, for the constant support and encouragement for throughout the course of my academic studies and M.Tech project work and providing necessary information regarding the project & also for their support in completing the project. Your advice on both project as well as on my career have been priceless. I wish to express my sense of gratitude to Dr. S. Ramachandra, Scientist ‘F’, GTRE, Bangalore, for his constant support throughout the project.

I am also grateful to the Head of the department Prof. NageswaraRao, HOD, Department of mechanical engineering, for his unfailing encouragement and suggestion given to me for completing my project. I am thankful to ADA Librarian and IISc Main and Mech. Dept Librarian for allowing me to use the Library and Journal facilities. A special thanks to my family. Words cannot express how grateful I am to my Parents Shri. Eswarappa and Smt. Puttamma, for all of the sacrifices that you’ve made on my behalf. Your prayer for me was what sustained me thus far. Also my Friends and Relatives for their encouragement help and support for this project. At the end I would like to express my appreciation to the Department of Mechanical Engineering. MADHU E (1OX12MMD10) ii

ABSTRACT Accurate prediction of vibration characteristics is crucial in the design stage of turbo machinery because prototyping and testing costs are exceptionally high and failure is generally disastrous in the practical applications of these systems. As the vibratory failures generally occur in the blades, the researchers are mostly focused on the blade vibrations. This project presents an in-depth study of Euler-Bernoulli beam vibration by varying young modulus and density ratio in flat, tapered and parabolic blades. Vibration is generally recognized as one of the most significant causes of high cycle fatigue failure in turbine. Fatigue failure is caused by repeated cyclic loads on a structural member. It is important to determine characteristics of the vibration of blade for various ratios of Young modulus to density of the material and its cross section over the length of the blade. Finite Element Method is used to solve the problems and compared by developing codes in MATLAB in which the Stiffness and Mass matrices of structure are constructed and natural frequencies are computed. Also compared the theoretical results with ANSYS and MATLAB results they are well matched.

The finite element software ANSYS is used to simulate the free vibrations. A variety of parametric studies are carried out to see the effects of various changes in the parameters of turbine blade on the natural frequencies. The parameters investigated include the effects of cracks relative to the restricted end, depth of cracks, and position of cracks in length of beam for inertia or rotating effect. The study shows that the highest difference in frequencies occur when the value of the notch radius increases. An increase of the depth of the cracks leads to a decrease in the values of natural frequencies.

iii

List of Figures

Figure No

Title of the Figure

Page No.

Figure 1.1

A Simple Gas Turbine

2

Figure 1.2

Example of Nodal Diameter m

5

Figure 2.1

An SGT-800 Gas Turbine.

9

Figure 2.2

A Ni-Based Single-Crystal Superalloy Gas Turbine Blade.

10

Figure 2.3

Causes of Failures to Aircraft Turbine Engines during Service

11

Figure 2.4

Comparisons of Different Material Properties.

19

Figure 2.5

Process Flow of FEA

24

Figure 3.1

A Cantilever Beam

35

Figure 3.2

The Beam under Free Vibration

35

Figure 3.3

The First Three Undamped Natural Frequencies and Mode Shape of Cantilever Beam

37

Figure 3.4

Uniform Elements Undergoing Transverse Deflection

38

Figure 4.1

The Location And Positive Directions Of These Displacements in a Typical Linearly Tapered Beam Element.

Figure 4.2

40

Plan and Elevation View of Cantilever Tapered Beam with Linearly Varying Width and Depth.

43

Figure 5.1

Frequency v/s Mode Number for Flat Beam

52

Figure 5.2

Frequency v/s Mode Number for Taper Beam

52

Figure 5.3

Frequency v/s Mode Number for Flat and Tapered Beam

53

iv

Figure No

Figure 5.4

Title of the Figure

Page No.

Frequency v/s Mode Number for Parabolic, Flat and Tapered Beam

Figure 5.5

54

Frequency V/S Mode Number for Flat Beam and Cantilever Taper Ratio Beams

Figure 5.6

55

Frequency V/S Mode Number for Flat Beam and Cantilever Parabolic Ratio Beams.

Figure 5.7

56

Frequency V/S Mode Number for Cantilever Parabolic Ratio Beams and Taper Beam.

57

Figure 5.8

Frequency v/s Mode Number for Different E/ρ Ratios.

57

Figure 6.1

Turbine Blade without Notch

59

Figure 6.2

Turbine Blade with Notch

59

Figure 6.3

Semicircular Notches at 20mm from the Base

60

Figure 6.4


61

Figure 6.5


61

Figure 6.6


62

Figure 6.7

U-notch for a depth b=1mm at 20mm from Base

62

Figure 6.8


63

Figure 6.9


63

Figure 6.10


64

Figure 6.11


64

Figure 6.12


65

v

Figure No

Title of the Figure

Page No.

Figure 6.13


65

Figure 6.14


66

Figure 6.15

Semicircular Notch at R=2mm at Different Positions

66

Figure 6.16


67

Figure 6.17


67

Figure 6.18


68

Figure 6.19


68

Figure 6.20

U-notch a depth b=1mm & Radius R=2mm at Different Positions 69

Figure 6.21

U-notch for a depth b=1mm and Radius R=4mm at Different Positions

Figure 6.22

70


Figure 6.23

70


Figure 6.24

71


Figure 6.25

71


Figure 6.26

72


72

vi

Figure No

Figure 6.27

Title of the Figure

Page No.


Figure 6.28

73


Figure 6.29

73


74

Figure A1

Flat Beam Model

81

Figure A2

First Mode Shape for Flat Beam

81

Figure A3

Second Mode Shape for Flat Beam

81

Figure A4

Third Mode Shape for Flat Beam

81

Figure A5

Fourth Mode Shape for Flat Beam

82

Figure A6

Fifth Mode Shape for Flat Beam

82

Figure A7

Taper Beam Model

82

Figure A8

First Mode Shape for Taper Beam

82

Figure A9

Second Mode Shape for Taper Beam

83

Figure A10

Third Mode Shape for Taper Beam

83

Figure A11

Fourth Mode Shape for Taper Beam

83

Figure A12

Fifth Mode Shape for Taper Beam

83

Figure B.1

Mode Shape for Cantilever Rectangular Beam with Five Elements 84

Figure B.2

Mode Shape for Cantilever Rectangular Beam with Ten Elements 85

vii

Figure No

Figure B.3

Title of the Figure

Page No.

Mode Shape for Cantilever Rectangular Beam with Fifteen Elements

Figure B.4

85

Mode Shape for Cantilever Rectangular Beam with Twenty Elements

Figure B.5

86

Mode Shape for Cantilever Rectangular Beam with Twenty Five Elements

86

Figure B.6

Mode Shape for Cantilever Taper Beam with Five Elements

87

Figure B.7

Mode Shape for Cantilever Taper Beam with Ten Elements

88

Figure B.8

Mode Shape for Cantilever Taper Beam with Fifteen Elements

88

Figure B.9

Mode Shape for Cantilever Taper Beam with Twenty Elements

89

Figure B.10

Mode Shape for Cantilever Taper Beam with Twenty Five Elements

89

viii

List of Tables

Table No

Title of the Table

Page No.

Table 2.1

Mechanical Properties of the Different Materials.

Table 4.1

Different Cross-Sectional Shapes of Tapered Beam with Shape Factors

19

42

ix

CONTENTS •

DECLARATION

i

•

ACKNOWLEDGEMENT

ii

•

ABSTRACT

iii

•

LIST OF FIGURES

iv

•

LIST OF TABLES

ix

CHAPTER – 1 INTRODUCTION

1

1.1. Introduction to Gas Turbines

1

1.2. Working of Gas Turbine

1

1.3. Gas Turbine Engine Failure Mechanisms

2

1.4. Types of Engine Blades and Defects

3

1.4.1. Compressor Blades and Turbine Blades

3

1.4.2. Blade Defects

3

1.5. Turbine Blade Vibration

3

1.6. Vibration Analysis of Turbine Blades

4

CHAPTER – 2 LITERATURE REVIEW

6

2.1. The History of the Gas Turbine

6

2.2. Gas Turbines

8

2.2.1 General Description

8

2.2.2 The Gas Turbine Blade

9

2. 3. Description of Failures to Gas Turbine Vanes and Blades

11

2.4 .Dynamic Analysis

12

2.4.1 Vibrations

12

2.4.2 Classification of Vibration

13

2.5 Crack 2.5.1 Classification of Crack

14 14

2.6. Flow Chart of Program

15

2.7. Physical Properties of Materials

16

2.7.1 Density

16

2.7.2 Young’s Modulus (Elastic modulus)

16

2.8. Mechanical Properties of the Different Materials

16

2.8.1 Steel Properties

16

2.8.2 Titanium Properties

17

2.8.3 Nickel Properties

18

2.9. Literature Survey on Finite Element Method

20

2.9.1 Definition of Finite Element Method

20

2.9.2 Advantages of FEA

21

2.9.3 Assumptions and Limitations

22

2.9.4 Basic Steps Followed in FEA

22

2.9.5 Engineering Applications of the Finite Element Method

24

2.9.6 Commercial Finite Element Analysis Packages

26

2.9.7 Structural Analysis

27

2.9.8 Modal Analysis of Beams using Matlab

28

CHAPTER – 3 CANTILEVER RECTANGULAR BEAM

35

3.1. Mathematical Analysis Cantilever Rectangular Beam

35

3.2. Finite Element Formulation of Cantilever Rectangular Beam Theorem

37

CHAPTER – 4 CANTILEVER TAPER BEAM

40

4.1. Theoretical Modeling of Tapered Beam

40

4.1.1 Linearly Tapered Beam Element

40

4.1.2 Formulation of Governing Differential Equation of Tapered Beam

42

4.2. Finite Element Formulation

44

4.2.1 Calculation of Shape Function

45

4.2.2 Stiffness Calculation of Tapered Beam

47

4.2.3 Mass Matrix of Tapered Beam

49

CHAPTER –5 VIBRATION ANALYSIS STUDIES

51

5.1. Modeling in ANSYS

51

5.2. Validations for Frequency

51

5.2. 1 Calculation of Frequency of Rectangular and Taper Beam Using Different Methods

51

5.3. Effect of Different Cross Section on Resonant Frequency

53

5.4. Effect of Different Cross Section ratios on Frequency

55

5.5. Effect of Variation of E/ρ Ratio on the Frequency

57

CHAPTER –6 TURBINE BLADE VIBRATION ANALYSIS

59

6.1 Different Notches at Constant Positions for Different Radius

60

6.2 Different Notches at Constant Radius for Different Positions

66

CONCLUSION

75

SCOPE FOR FUTURE WORK

76

BIBLIOGRAPHY

77

APPENDIX –A MODE SHAPES FOR FLAT AND TAPER BEAMS USING ANSYS

81

A.1 Flat Beam

81

A.2 Taper Beam

82

APPENDIX-B MODE SHAPES FOR FLAT AND TAPER BEAMS USING MATLAB

84

B.1 Mode Shapes for Flat Beam with Varying Elements

84

B.2 Mode Shapes for Taper Beam with Varying Elements

87

APPENDIX –C MATLAB PROGRAMS

90

C.1 Taper Beam

90

C.2 Cantilever Rectangular Beam

105

C.3 Parabolic Beam

105

APPENDIX-D TECHNICAL PAPER PUBLISHED

107

Effect of Elastic Modulus and Density Ratio on Vibration Characteristics of Rotating Turbine Blade

CHAPTER-I

INTRODUCTION

Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068


1. INTRODUCTION

1.1. Introduction to Gas Turbines The gas turbine is an internal combustion engine that uses air as the working fluid. The chemical energy is extracted by the engine and converts it to mechanical energy using the gaseous energy of the working fluid (air) to drive the engine and propeller. The basic principle of the airplane turbine engine is identical to any and all engines that extract energy from chemical fuel. The basic 4 stages of an internal combustion engine are: i. Intake of air ii. Compression of the air iii. Combustion iv. Expansion and exhaust The turbine section of the gas turbine engine has the task of producing usable output shaft power to drive the propeller. It must also provide power to drive the compressor and all engine accessories. It is done by expanding the high temperature, pressure, and velocity gas which in turn convert the gaseous energy to mechanical energy in the form of shaft power.

1.2. Working of Gas Turbine Gas Turbine Working Principle a schematic drawing of a simple gas turbine is shown in figure 1.1. The working principle behind the gas turbine is as follows. Ambient air is extracted from atmosphere and it is compressed in the compressor. Compressed air is directed to the combustion chamber. Compressed air in the combustion chamber is mixed with vaporized fuel and burned under constant pressure. Burning results in releasing of hot gas with high energy content. Hot gas is allowed to expand through the turbine here the energy in the gas is converted to a rotation of the turbine shaft.


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The turbine shaft over both the compressor and a generator used to obtain electrical power from the gas turbine. [1]

Figure1.1.A Simple Gas Turbine.

1.3. Gas Turbine Engine Failure Mechanisms Gas Turbine Engine failure mechanisms include: •

High cycle fatigue

•

Low cycle fatigue

•

Thermo-mechanical fatigue

•

Creep

•

Overstress

•

Corrosion

•

Erosion

•

Fretting and wear

•

And Combination of above

The ability of a component to avoid these failure mechanisms depends on the component design, material properties, and the operating environment. The component design and the material used are fixed. The operating environment may


2


change and have a detrimental effect on the life of the component. The inter relation of all these factors impacts the rate of component life consumption. Other factors affecting life consumption include: •

Manufacturing and material defects

•

Build and maintenance errors

•

Foreign object damage (FOD)

•

Limit exceedances (i.e. over speeds, over temperatures).

1.4. Types of Engine Blades and Defects 1.4.1. Compressor blades and turbine blades The blade families can be categorized as compressor blade and turbine blade according to their use in unheated and heated areas. They can also be classified as high, intermediate and low pressure blades (HP, IP and LP blade) according to their use in the three stages. Compressor blades normally use titanium alloys, and turbine blades often use nickel alloys, including expensive and difficult to produce single crystal nickel alloys.

1.4.2. Blade defects Blade failures in gas turbine engines often lead to loss of all downstream stages and can have a dramatic effect on the availability of the turbine engines. In general blade failures can be grouped into two categories: (a) fatigue, including both high (HCF) and low cycle fatigue (LCF) and (b) creep ruptures. Blade fatigue failures are often related to anomalies in mechanical behavior and manufacturing defects. Owing to normal wear or foreign object damage, defects will appear on both compressor and turbine blades

1.5. Turbine Blade Vibration Turbine blades are subjected to very strenuous environments inside a gas turbine. They are subjected to high temperatures, high stresses, and high speed. All these factors can lead to blade failure like Cracking, bulging, twisting, bending and Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

3


breaking of blades. The damage or failure of turbine blades will affect the performance of the gas turbine engine hence early detection of blade problems is necessary to ensure the availability, reliability and good performance of gas turbine engines or other machines with blades. Due to the harsh environment the gas turbine blades are operating in, it is very difficult to directly monitor the blades working conditions. Some studies have shown that rotating blades produce mechanical vibration and there is a coupling between blades and rotor vibration [2]. Therefore, blade vibration and problem can be indirectly detected from the vibration of the rotordisk-blade system.

1.6. Vibration Analysis of Turbine Blades It is very important that turbomachinery be designed in a way to assure trouble-free operation. Jet Engine failures or in-flight shutdowns put lives at risk and can be very costly for both the user and the original equipment manufacturer. Turbomachinary components, especially blades, are exposed to loads that can cause failure, designing reliable components requires in-depth vibration and stress analysis. One of the main causes of turbine blade failure is high cycle fatigue or HCF. Fatigue failure is caused by repeated cyclic loads on a structural member. The fatigue life of a part is defined by the number of load cycles it can survive. The fatigue life depends on the stress cycles magnitude and the part’s material properties. In most cases the higher the stress the shorter the fatigue life. Fatigue failure occurs as follows. A crack initiates after a number of stress cycles. This happens at a location of relatively high stress concentration. Cooling holes, sharp fillets or internal core features of a turbine blade are typical high stress concentration spots. By applying stress cycles the initiated crack grows. Final failure occurs very rapidly after the crack reaches some critical length. HCF failure corresponds to fracture due to a relatively large number of stress cycles caused by vibrations.[3] Natural frequency is the frequency at which an object vibrates when excited by force. At this frequency, the structure offers the least resistance to a force and if left uncontrolled, failure can occur. Mode shape is deflection of object at a given natural frequency. A guitar string is a good example of natural frequency and mode shapes. When struck, the string vibrates at a certain frequency and attains a deflected shape. Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

4


The Eigen value (natural frequency) and the accompanying eigenvector (mode shape) are calculated to define the dynamics of a structure. A turbine bladed disk has many natural frequencies and associated mode shapes. In the case of a bladed disk, the mode shapes have been described as nodal diameters. The term nodal diameter is derived from the appearance of a circular geometry, like a disk, vibrating in a certain mode. Mode shapes contain lines of zero out-of-plane displacement which cross the entire disk. In other words, a node line is a line of zero displacement and the displacement is out of phase on the sides of the line represented by white and gray shades in Figure 1.2. These are commonly called nodal diameters. Hence the natural frequency and nodal diameter are required to describe a bladed disk mode.

Figure 1.2 Example of Nodal Diameter m

After establishing natural frequency and mode shapes, alternating forces must exist to excite a structure and make it vibrate. These forces have inherent frequencies and shapes just as bladed disks do. In a gas turbine, the most common sources of excitation are running speed harmonics and vane passing frequencies [4].


5


CHAPTER-II

LITERATURE REVIEW



2. LITERATURE REVIEW

2.1. The History of the Gas Turbine The development of the gas turbine took place in several countries. Several different schools of thought and contributory designs led up to Frank Whittle’s 1941 gas turbine flight. Despite the fact that NASA’s development budget now trickles down to feed the improvement of flight, land based and marine engines, the world’s first jet engine owed much too early private aircraft engine pioneers and some lower profile land-based developments. The development of the gas turbine is a source of great pride to many engineers worldwide and, in some cases takes on either industry sector fervor (for instance the aviation versus land based groups) or claims that are tinged with pride with one’s national roots. People from these various sectors and subsectors can therefore get selective in their reporting. So for understanding the history of the gas turbine, one would have to read several different papers and select material written by personnel from the aviation, and landbased sectors. At that point, one can “fill in the gaps”. What follows therefore are two different accounts of the gas turbine’s development. Neither of them is wrong. The first of these presents an aircraft engine development perspective. Attempts to develop gas turbines were first undertaken in the early 1900’s, with pioneering work done in Germany. The most successful early gas turbines were built by Holzwarth, who developed a series of models between 1908 and 1933. The first industrial application of a gas turbine was installed in a steel works in Hamborn, Germany, in 1933. In 1939 a gas turbine was installed in a power plant in Neuchâtel. 1. 1931 U.S. army awards GE a turbine-powered turbo supercharger development contract 2. 1935 U.S. Army, Northrop, TWA, and GE combine to test fly a Northrop Gamma at 37,000 feet from Kansas City to Dayton. This led to a production contract for GE to build 230 units of the “Type B” supercharger and led to establishment of


6


the GE Supercharger Department in Lynn, Massachusetts (later the site of the I-A development based on the Whittle engine). 3. 1938 Wright Aeronautical Corporation designs its own vaned superchargers for its own engines, although the superchargers were manufactured for Wright by GE. 4. 1940 NACA joins with Wright, Allison and P&W to standardize turbo supercharger testing techniques. 5. 1925 R.E. Lasley of Allis-Chalmers receives the first of several patents on gas turbines. 6. Around 1930 he forms the Lasley Turbine Motor Company in Waukegan, IL. With the goal of producing a gas turbine for aircraft propulsion. 7. 1934 U.S. Army personnel from Wright Field visit Lasley’s shop and inspected his hardware and the engine which he had filmed in operation earlier that year. However, neither the Army nor Navy would fund Lasley. 8. 1939 GE studies gas turbine aircraft propulsion options and concludes the turbojet is preferable to the turboprop. Note, however, that two years later they changed their minds and proposed a turboprop to the Durand Committee. 9. 1941 GE Steam Turbine Division (Schenectady) participates in the Durand Special Committee on Jet Propulsion and proposes a turboprop, designated the TG-100 (later the T31), which ran successfully in May 1943 under Army sponsorship. 10. 1941 GE Turbo Supercharger Division (Lynn, Massachusetts) receives the Whittle W.1.X engine and drawings for the W.2.B improved version. A top secret effort begins to build an improved version, known as the I-A, for flight test in the Bell P59. 11. 1941 Durand Committee also awards Navy contracts to Allis-Chalmers and Westinghouse. The Westinghouse W19, a small booster turbojet, resulted from this but Allis-Chalmers dropped out of the “gas turbine race” in 1943. 12. 1942 In April, the GE I-A runs for the first time in a Lynn test cell. In October, it powers the Bell P-59 on its first flight at Muroc Dry Lake, CA. 13. 1929 Haynes Stellite develops Hastelloy alloy for turbine buckets, allowing operation up to gas temperatures of over 1800 F. This superior alloy was later crucial to the successful operation of the I-A and it gave U.S. turbine manufacturers the ability to use uncooled designs rather than include the complexity of blade cooling. Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

7


By the latter part of 1942, the following “native” aircraft gas turbine efforts were proceeding. These projects included: 1. Northrop Turbodyne turboprop 2 .P&W PT-1 turboprop 3. GE/Schenectady TG-100 turboprop 4. Allis-Chalmers turbine-driven ducted fan 5. NACA piston-driven ducted fan 6. Westinghouse 19A turbojets 7. Turbo Engineering Corporation’s booster-sized turbojet.[5]

2.2. Gas Turbines 2.2.1 General description Gas turbines are mainly used for power generation. The general idea behind a gas turbine is that it extracts mechanical energy from a hot gas stream, which is produced from combusting fuel. Gas turbines consist of three main parts: the compressor, the combustor and the turbine. In Figure 2.1 the Siemens gas turbine SGT-800 is shown, and the function of the gas turbine is as follows: 1. Air inlet: Air is taken in through the air inlet. 2. Compressor: The air enters the compressor. By use of compressor discs and blades, the air is compressed and its temperature is therefore increased. 3. Combustor: The compressed hot air now enters the combustor. In the combustor, the hot air is mixed with fuel, and ignited. 4. Turbine: When the hot gas is ignited, the temperature increases and the air desire to expand. Hence, the air expands through the turbine, causing a mass flow from where mechanical energy is extracted by the gas turbine blades which start to rotate.


8


5. Shaft: The rotating turbine blades are coupled to a shaft. The shaft transfers the mechanical work from the turbine blades to a generator, which in its turn generates electrical work. It should be said that part of the mechanical work from the turbine stage is also needed to drive the compressor. Therefore, not all the energy generated by the turbine can be converted into electrical work.

Figure 2.1: An SGT-800 Gas Turbine.

The function of an aero engine is very similar to that of a landbased gas turbine. However, an aero engine works at maximum capacity only during take-off and landing, while a landbased gas turbine works at maximum capacity over longer times. Another difference between the two applications is safety. An aero engine has very high safety precautions, and here, failure of the most critical components cannot be tolerated since it can have terrible consequences. However, for a landbased gas turbine, the failure of a critical component will not have the same terrible consequences. Of course, failure in a landbased gas turbine is not desirable, but is easier to accept. This means that the components in landbased gas turbines can have much longer inspection intervals and service life than aero engine components.

2.2.2 The gas turbine blade Gas turbine blades are positioned in the turbine stage after the combustor, see Figure 2.1. For a landbased gas turbine, it is common to have three or four rows of turbine blades, where each row consists of around 60-100 turbine blades. Figure 2.2 displays a gas turbine blade. When the hot gas expands through the turbine stage, the Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

9


hot gas first hits the first row of turbine blades. All the turbine blades are shaped in such a way, that the resulting force from the hot gas stream on the blade becomes perpendicular to the gas stream. Hence, the turbine blades start to rotate. The turbine blades are attached to a disc, which in turn is attached to the shaft. When the blades start to rotate, the disc and shaft also rotate. During service, the turbine blades rotate with a rotational speed of up to 10000 rpm at temperatures up to 1000 °C. Hence, the gas turbine blades are subjected to significant centrifugal forces and high temperatures at the same time, which put extreme requirements on the turbine blade material.

Figure 2.2: A Ni-Based Single-Crystal Superalloy Gas Turbine Blade.

As mentioned, there are three or four rows of turbine blades in the turbine stage. The first row is subjected to the most severe conditions; since it is here the hot gas first enters and has the highest temperature. By the time the air reaches the second, third and fourth rows of turbine blades, the temperature has gradually decreased. First stage turbine blades are most commonly coated with a thermal barrier coating (TBC) to protect the blade material from the high temperature. At the same time, the blade is continuously cooled by air from the compressor. The efficiency of the gas turbine is very much dependent on the gas temperature; the higher temperatures of the gas in the turbine stage the higher efficiency for the turbine. Further, the gas temperature can only be as high as what the first row turbine blades can withstand. This implies that it is on the performance of the first row of turbine blades that the whole turbine engine efficiency is determined.[2]


10


2. 3. Description of Failures to Gas Turbine Vanes and Blades The process of gas turbine operation is associated with various failures to structural components of gas turbines, in particular blades. Condition of the blades is of crucial importance to reliability and life time of the entire turbine, and the ‘parent’ subassembly where it is installed. This is why the blades are subject to scrupulous checks, both during the manufacture and at the stage of assembly, when any deviations from the specification are detected and eliminated. Analysis of the literature show that only a small portion of damages/failures to turbine vanes and blades are caused by material defects, structural and/or engineering process attributable defects; most damages/failures are service attributable (Fig.2.3).

Figure.2.3.Causes of Failures to Aircraft Turbine Engines during Service (In Percentage Terms)

Durability of turbine vanes and blades is a sum of a number of factors, where material quality is the matter of crucial importance. With respect to materials, durability can be defined as time of item operation when alloy properties developed during the manufacturing process remain steady (unchanged). Stability of the properties (the assumed service time) is defined at the design stage by selection of the desired characteristics (as compared to the expected loads and with account taken of the fact that the properties are subject to changes with time). High and stable strength properties of superalloys offer suitable microstructures that are resistant to any deterioration during the service. These structural features have been assumed a durability criterion.


11


During the service, gas turbine components may be subject to failures resulting from the following processes 1. Creeping; 2. Overheating and melting; 3. Low-cycle and high-cycle fatigue due to thermal and thermo mechanical factors, 4. Corrosion and fatigue cracking 5. Chemical and intercrystalline corrosion, 6. Erosion 7. Other factors of less importance. [6]

2.4 Dynamic Analysis 2.4.1 Vibrations Vibration is time dependent displacements of a particle or a system of particles w.r.t an equilibrium position. If these displacements are repetitive and their repetitions are executed at equal interval of time w.r.t equilibrium position the resulting motion is said to be periodic. One of the most important parameters associated with engineering vibration is the natural frequency. Each structure has its own natural frequency for a series of different modes which control its dynamic behavior. Whenever the natural frequency of a mode of vibration of a structure coincides with the frequency of the external dynamic loading, this leads to excessive deflections and potential catastrophic failures. This is the phenomenon of resonance. An example of a structural failure under dynamic loading was the well-known Tacoma Narrows Bridge during wind induced vibration. In practical application the vibration analysis assumes great importance. For example, vehicle-induced vibration of bridges and other structures that can be simulated as beams and the effect of various parameters, such as suspension design, vehicle weight


12


and velocity, damping, matching between bridge and vehicle natural frequencies, deck roughness etc., on the dynamic behavior of such structures have been extensively investigated by a great number of researchers . The whole matter will undoubtedly remain a major topic for future scientific research, due to the fact that continuing developments in design technology and application of new materials with improved quality enable the construction of lighter and more slender structures, vulnerable to dynamic and especially moving loads. Every structure which is having some mass and elasticity is said to vibrate. When the amplitude of these vibrations exceeds the permissible limit, failure of the structure occurs. To avoid such a condition one must be aware of the operating frequencies of the materials under various conditions like simply supported, fixed or when in cantilever conditions.

2.4.2 Classification of vibration Vibration can be classified in several ways. Some of the important classifications are as follows

i) Free and forced vibration If a system, after an internal disturbance, is left to vibrate on its own, the ensuing vibration is known as free vibration. No external force acts on the system. The oscillation of the simple pendulum is an example of free vibration. If a system is subjected to an external force (often, a repeating type of force), the resulting vibration is known as forced vibration. The oscillation that arises in machineries such as diesel engines is an example of forced vibration. If the frequency of the external force coincides with one of the natural frequencies of the system, a condition known as resonance occurs, and the system undergoes dangerously large oscillations. Failures of such structures as buildings, bridges, turbines and airplane have been associated with the occurrence of resonance.

ii) Undamped and damped vibration If no energy is lost or dissipated in friction or other resistance during oscillation, the vibration is known as undamped vibration. If any energy lost in this way, however, it is called damped vibration. In many physical systems, the amount of damping is so Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

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small that it can be disregarded for most engineering purposes. However, consideration of damping becomes extremely important in analyzing vibratory system near resonance.

iii) Linear and nonlinear vibration If all the basic components of vibratory system the spring, the mass and the damper behave linearly, the resulting vibration is known as linear vibration. If however, any of the basic components behave nonlinearly, the vibration is called nonlinear vibration. [7]

2.5 Crack A crack in a structural member introduces local flexibility that would affect vibration response of the structure. This property may be used to detect existence of a crack together its location and depth in the structural member. The presence of a crack in a structural member alters the local compliance that would affect the vibration response under external loads.

2.5.1 Classification of Crack Based on geometries, cracks can be broadly classified as follows Transverse Crack-These are cracks perpendicular to beam axis. These are the most common and most serious as they reduce the cross sections as by weaken the beam. They introduce a local flexibility in the stiffness of the beam due to strain energy concentration in the vicinity or crack tip. Longitudinal Cracks-These are cracks parallel to beam axis. They are not that common but they pose danger when the tensile load is applied at right angles to the crack direction i.e. perpendicular to beam axis. Open Cracks- These cracks always remain open. They are more correctly called “notches”. Open cracks are easy to do in laboratory environment and hence most experimental work is focused on this type of crack. Breathing crack- These are cracks those open when the affected part of material is subjected to tensile stress and close when the stress is reversed. The component is Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

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most influenced when under tension. The breathing of crack results in non-linearity in the vibration behavior of the beam. Most theoretical research efforts are concentrated on transverse breathing cracks due to their direct practical relevance. Slant Cracks-These are cracks at an angle to the beam axis, but are not very common. Their effect on lateral vibration is less than that of transverse cracks of comparable severity. Surface Cracks-These are the cracks that open on the surface. They can normally be detected by dye‐penetrates or visual inspection. Subsurface Cracks-Cracks that do not show on the surface are called subsurface cracks. Special techniques such as ultrasonic, magnetic particle, radiography or shaft voltage drop are needed to detect them.

2.6. Flow Chart of Program Read Beam Geometry Material Properties Boundary Conditions Expression for Standard Procedure: Element stiffness matrix [Ke] mass matrix [M]

Boundary coditions: [ Ke], [M]

Free Vibration Givenλ, Find ω

Determine Non-dimensional Parameter as: ωn Determined Vibration Modes


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2.7. Physical Properties of Materials These properties concerned with such properties as melting, temperature, electrical conductivity, thermal conductivity, density, corrosion resistance, magnetic properties, etc. and the more important of these properties will be considered as follows

2.7.1 Density Density is defined as mass per unit volume for a material. The derived unit usually used by engineers is the kg/m3. Relative density is the density of the material compared with the density of the water at 4C. [9] The formulae of density and relative density are: Density ρ =

Relative density d =

mass m

volume V

Density of the material Density of pure water at 4 c

2.7.2 Young’s Modulus (Elastic modulus) It is the slope of the initial, linear-elastic part of the stress-strain curve in tension or compression. But accurate moduli are measured dynamically. It is a measure of rigidity of the material. Young’s Modulus (or Elastic Modulus) is the proportionality constant of solids between elastic stress and elastic strain and describes the inherent stiffness of material. It can be expressed in the following equation where, E is Young’s Modulus; [10] E= (Elastic stress / Elastic strain)

2.8. Mechanical properties of the different materials 2.8.1 Steel Properties When selecting a material for a particular application, engineers must be confident that it will be suitable for the loading conditions and environmental challenges it will be subjected to while in service. Understanding and control of a material’s properties Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

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is therefore essential. The mechanical properties of steel can be carefully controlled through the selection of an appropriate chemical composition, processing and heat treatment, which lead to its final microstructure. The alloys and the heat treatment used in the production of steel result in different property values and strengths and testing must be performed to determine the final properties of steel and to ensure adherence to the respective standards. There are many measurement systems used to define the properties of given steel. For example, Yield strength, ductility and stiffness are determined using tensile testing. Toughness is measured by impact testing; and hardness is determined by measuring resistance to the penetration of the surface by a hard object. Tensile testing is a method of evaluating the structural response of steel to applied loads, with the results expressed as a relationship between stress and strain. The relationship between stress and strain is a measure of the elasticity of the material, and this ratio is referred to as Young's modulus. A high value of Young's modulus is one of steel’s most differentiating properties; it is in the range 190-210 GPa, which is approximately three times the value for aluminum. The physical properties of steel are related to the physics of the material, such as density, thermal conductivity, elastic modulus, Poison’s ratio etc. Some typical values for physical properties of steel are: density ρ = 7.7 - 8.1 [kg/dm3] elastic modulus E=190 -210 [GPa] Poisson’s ratio ν = 0.27 - 0.30 Thermal conductivity κ = 11.2 - 48.3 [W/mK] Thermal expansion α = 9 - 27 [10-6 / K]

2.8.2 Titanium Properties Titanium is light weight, strong, corrosion resistant and abundant in nature. Titanium and its alloys possess tensile strengths from 30,000 psi to 200,000 psi (210-1380 MPa), which are equivalent to the strengths found in most of alloy steels. Titanium is a low-density element (approximately 60% of the density of iron) that can be strengthened by alloying and deformation processing. Titanium is nonmagnetic and Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

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has good heat-transfer properties. Its coefficient of thermal expansion is somewhat lower than that of steels and less than half that of aluminum. One of titanium’s useful properties is a high melting point of 3135°F (1725°C). This melting point is approximately 400°F above the melting point of steel and approximately 2000°F above that of aluminum. Titanium can be passivated, and thereby exhibit a high degree of immunity to attack by most mineral acids and chlorides. Titanium is nontoxic and generally biologically compatible with human tissues and bones. The excellent corrosion resistance and biocompatibility coupled with strength make titanium and its alloys useful in chemical and petrochemical applications, marine environments, and biomaterial applications. density ρ = 4.45 - 4.71 [kg/dm3] elastic modulus E= 90-110[GPa] Poisson’s ratio ν = 0.28 - 0.36 Thermal conductivity κ = 21.9 [W/mK] Thermal expansion α = 8-10 [10-6 / C]

2.8.3 Nickel Properties Ni-based superalloys have some remarkable properties which make them suitable for high temperature applications. The fact that the yield strength of superalloys increases with increased temperature is particular and together with the good fatigue and creep properties makes them a good choice for turbine blade material. [11] The physical properties of nickel are related to the physics of the material, such as density, thermal conductivity, elastic modulus, Poison’s ratio etc. Some typical values for physical properties of nickel are: density ρ = 8.1 -8.9 [kg/dm3] elastic modulus E= 200-210 [GPa] Poisson’s ratio ν = 0.31


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Table 2.1 Mechanical Properties of the Different Materials.

Sl

Properties

No

Elastic

Density(ρ)

(E/ρ)

Modulus(E)

(1000

(×106

(GPa)

kg/m3)

Materials

Nm/kg)

1.

Titanium Alloy

96

4.620

20.77

2.

Nickel Alloys

210

8.890

21.18

3.

Structural Steel

200

7.85

25.47

250 210

200

200 150 100

96

50 20.77 4.6

8.89

21.18

25.47 7.85

0 Titanium Alloy Nickel Alloys Structural Steel Elastic Modulus(E) in (GPa) Density(ρ) in 10^3kg/m^3 E/ρ in 10^6Nm/kg

Figure 2.4 Comparisons of Different Material Properties.

Nω Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

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2.9. Literature Survey on Finite Element Method In the early 1940s in the field of solid mechanics Mc Henry, Hrenikoff and Newmark shown that a reasonably good solutions to continuum problem can be obtained by substituting small portions of the continuum by an arrangement of simple elastic bars. Later in the same context Argyris and Turner etal showed that a more direct, but less intuitive, substitution of properties can be made much more directly by considering that small proportions or elements in a continuum behave in a simplified manner. It is from the engineering ‘direct analogy’ view that the term ‘Finite element ‘has been born. Clough appears to be first to use this term, which implies in it a direct use of standard methodology applicable to discrete systems. Since in early 1960s much progress has been made, and today the pure mathematical and analogy approaches are fully reconciled [12]. By the early 1970’s, finite element analysis had become established as a general numerical technique for the solution of any system of differential equations. The method was still only used at this time on a limited basis, however, because of the availability of powerful computers in industry. The use was primarily in the aerospace, automotive, defense, and nuclear industries. With the advent of microcomputers (pc’s and workstations) in the 1980’s, however, the methods have become more widely used. It is now possible for engineers in virtually every industry to take advantage of this powerful tool.

2.9.1 Definition of Finite Element Method Finite element method is a method of approximation to continuum problems such that (a) the continuum is divided into a finite number of parts (elements), the behavior of which is specified by a finite number of parameters, and (b) The solution of the complete system as an assembly of its elements. •

The elements have a finite number of unknowns, hence the name finite elements. Finite Element Analysis is a way to simulate loading conditions on a design and determine the design’s response to those conditions. The FEA can be FEA is applicable to any field problem: stress analysis, heat transfer, and magnetic field and soon.


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•

There is no geometric restriction. The body or region analyzed may have any shape.

•

Boundary conditions and loading conditions are not restricted. For example, in stress analysis, any portion of a body may be supported, while distributed or concentrated forces may be applied to any other portion.

•

Material properties are not restricted to isotropy and may change from one element to another or even within an element.

•

Components that have different behaviors, and different mathematical descriptions, can be combined. Thus a single FE model might contain bar, beam, plate, cable and friction elements.

•

An FE structure closely resembles the actual body or region to be analyzed.

Explained through the physical concept and it is amenable to systematic computer programming. Finite element analysis (FEA) also called as finite element method (FEM), is a method for numerical solution of field problems. Application range from deformation and stress analysis of automotive, aircraft, building and bridge structures to field analysis of heat flux, fluid flow, magnetic flux, seepage, and other flow problems[13].

2.9.2 Advantages of FEA FEA has advantage over most other numerical analysis method, including versatile and physical appeal •

The approximation is easily improved by grading the mesh so that more elements appear where field gradients are high and more resolution is required.

•

FEA reduce the amount of prototype testing and simulate designs that are not suitable for prototype testing

•

Cost savings, Time savings and create more reliable and better quality designs are some bottom lines.

Other numerical methods have arisen since FEA appeared, but at present only FEA can confidently claim all these attributes. Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

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2.9.3 Assumptions and Limitations A number of important assumptions and limitations are inherent in linear static analysis. •

Linear elastic material; Material is assumed to be homogeneous and isotropic. The material in which stresses are directly proportional to strain and to loads and do not take the material beyond its permanent yield point, is homogeneous and isotropic material. It is assumed that unloaded structure is free of initial or residual stresses.

•

The small displacement assumptions are restricted using formulation of governing equations for linear beam, plate, shell and solid behavior.

•

Gradually applied loads; in linear static analysis our structure is in static equilibrium. Load must be slowly applied which means that they induce no dynamic effects.

However there are methods and software’s to consider geometric and material related nonlinearities.

2.9.4 Basic Steps Followed in FEA The solution of a general continuum problem by the finite element method always follows an orderly step by step process. With reference to static structural problems, the step by step procedure can be stated as follows: Step (i): Discretization of the structure The first step in the finite element method is to divide the structure or solution region into subdivisions or elements. Hence the structure is to be modeled with suitable finite elements. The number, type, size and arrangements of the elements are to be decided. Step (ii): Selection of proper interpolation or displacement model. Since the displacement solution of a complex structure under any specified load conditions cannot be predicted exactly, we assume some suitable solution within an element to approximate the unknown solution. The assumed solution must be simple form a computational point of view, but it should satisfy certain convergence Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

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requirements. In general, the solution or the interpolation model is taken in the form of a polynomial. To satisfy the convergence requirements, the polynomial functions,

Must be continuous within the element

Must contain rigid body displacement or field variables

Must contain constant strain states

Step (iii): Derivation of element stiffness matrices and load vectors. For the assumed displacement model, the stiffness matrix ke and the load vector Fe of element “e” are to be derived by using either equilibrium conditions or a suitable variational principle. Step (iv): Assemblage of elemental equations to obtain the overall equilibrium equations Since the structure is composed of several finite elements, the individual elemental stiffness matrices and load vectors are to be assembled in a suitable manner and the overall equilibrium equations have to be formulated as [k]u=F. where [k] is assembled stiffness matrix, u the vector of nodal displacements and F is called the vector nodal forces for the complex structure. Step (v): Solution for the unknown nodal displacements The overall equilibrium equations have to be modified to account for the boundary conditions of the problem. After the incorporation of the boundary conditions, the equilibrium equations can be expressed as [k]u=F. For linear problems, the vector u is can be solved very easily. But for nonlinear problems, the solution has to be obtained in a sequence of steps, each step involving the modification of the stiffness matrix [k] and/or load vector F. Step (vi): Computational of elemental stress and strains. For the known nodal displacements u, if required the elemental strains and stresses can be computed by using the necessary equations of solid or structural mechanics.


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The terminology used in the above six steps has to be modified if we want to extend the concept to other fields.

2.9.4.1. Process Flow of Finite Element Analysis The general steps followed in a finite element analysis with a commercial FEM package is as shown in figure 2.5:

Problem Definition

Start

Analysis and design decisions

Pre-Processor •

•

•

Reads or generates nodes and elements Reads or generates material property data. Reads and generates boundary conditions (loads and constraints)

Processor • •

•

• • •

Generates element shape functions. Calculates master element equations • Calculates transformation matrices. Maps element equations into global system Assembles element equations. Introduces boundary conditions Performs solution procedure

Stop

Post-processor •

•

•

Prints or plots contours of stress components. Prints or plots contours of displacements. Evaluates and prints error bounds.

Figure2.5. Process Flow of FEA

2.9.5 Engineering Applications of the Finite Element Method The finite element method was developed originally for the analysis of aircraft structures. However, the general nature of its theory makes it applicable to a wide variety of boundary value problems in engineering. A boundary value problem is one in which a solution is sought in the domain or a region of a body subject to the satisfaction of prescribed boundary or edge conditions on the dependent variable or their derivatives. The three major categories of boundary value problems in finite element methods are (1) Equilibrium or steady state or time independent problems, (2) Eigen value problems, and (3) propagation or transient problems.


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In an equilibrium problem, we need to find the steady state displacement or stress distribution if it is a solid mechanics problem, temperature or heat flux distribution if it is a heat transfer problem and pressure or velocity distribution if it is a fluid mechanics problem. Static analysis of aircrafts wings, fuselages, fins, rockets, missile structure in the areas of aircraft structures and stress concentration problems, stress analysis pressure vessels, pistons, composite material linkages studies in the mechanical design area are some of the equilibrium problem studies. Steady state temperature distribution in solids and fluids in heat conduction; analysis of potential flows, free surface flows, boundary flows and boundary layer flows, viscous flows, transonic aerodynamic problems, analysis of hydraulic structures and dams in hydraulic and water resources; and analysis of excavations, retaining walls, underground openings, rock joints and soil structure interaction problems, stress analysis in soils, dams, layered piles and machine foundations in Geo-mechanics are equilibrium problem studies in their respected fields. In Eigen value problems also, time will not appear explicitly. They may be considered as an extension of equilibrium problems in which critical value of certain parameters are to be determined in addition to the corresponding steady state configurations. In these problems, we need to find the natural frequencies or buckling loads and mode shapes if it is a solid mechanics or structural problem, stability of laminar flows if it is a fluid mechanics problem and resonance characteristics if it is an electric circuit problem. Some Eigen values problems are natural frequencies, flutter, and stability of aircraft, rocket, spacecraft and missile structures in aircraft structures field; natural frequencies and stability of linkages, gears and machine tools in machine design; natural periods and models of shallow basins, lakes and harbors, sloshing of liquids in rigid and flexible containers in hydraulic and water resources engineering; natural frequencies and modes of dam-reservoir systems and soil structure interaction problems in geo mechanics. The propagation or transient problems are time dependent problems. This type of problem arises, for example where we are interested in finding the response of a body under time varying force in the area of solid mechanics and under sudden heating or cooling in the field of heat transfer. Response of aircraft structures to random loads, dynamic response of aircraft and spacecraft to aperiodic loads, crack and fracture problems under dynamic loads, transient heat flow in rocket nozzles, internal Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

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combustion engines, turbine blades, fins and building structures, Analysis of unsteady flow and wave propagation problems, transient seepage in aquifers and porous media, rarefied gas dynamics, magneto hydrodynamic flows, time-dependent soil –structure interaction problems transient seepage in soil and rocks stress wave propagation in soils and rocks are some propagation problems in various fields of study.

2.9.6 Commercial Finite Element Analysis Packages The following are some of the general finite element packages: •

Structural Design Language(Integrated Civil Engineering System, MIT, USA) STRUDEL

•

NASA Structural Analysis (U.S. National Aeronautics and Space Administration) NASTRAN. CSA NASTRAN AND MSC NASTRAN packages are popular in India.

•

Non-linear Incremental Structural Analysis (developed by E Ramm, Institute of Biostatic University of Stuttgart, W Germany) NISA.

•

Engineering Analysis System (Swanson Analysis Inc.) ANSYS.

•

Structural Analysis Programming (Developed by El. Wilson, University of California, USA) SAP.

•

ADAMS

Preprocessor packages: Some of the Preprocessor Packages Used for Finite Element Analysis is listed below: HYPERMESH, Display III/IV, FEMAP, PATRON, ANSA

i) FEM Software-ANSYS The ANSYS computer program is a large-scale multipurpose finite element program, which may be used for solving several classes of engineering analysis. The analysis capabilities of ANSYS include the ability to solve:

Static and dynamic structural analysis


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Steady state and transient heat transfer problems

Mode frequency and buckling Eigenvalue problems.

Static or time varying magnetic analysis.

Various types of field and coupled field application.

The program contains many special features which allow nonlinearity or secondary effects to be included in the solution such as plasticity, large strain, hyper elasticity, creep, swelling, large deflections, stress stiffening, temperature dependency, material anisotropy, and radiation. As ANSYS has been developed other special capabilities such

as

sub-structuring,

sub-modeling,

random

vibration,

kinetostatics,

kinetodynamics, free convection fluid analysis, acoustics, magnetic, piezoelectric, coupled field analysis and design optimization have been added to the program. These capabilities contribute further to making ANSYS a multipurpose tool for the various engineering disciplines. The ANSYS program has been in commercial use since 1970 and has been used extensively in the aerospace, automotive, construction, electronic, energy services, manufacturing, nuclear, plastics, oil and steel industries. In addition, every consulting firm and hundreds of Universities use ANSYS for analysis, research and educational use.

2.9.7 Structural Analysis Structural analysis is probably the most common applications of the finite element method. The term structural (or structure) implies not only civil engineering structures such as bridges and buildings, but also naval, aeronautical and mechanical structure such as ship hulls, aircraft bodies, and machine housings, as well as mechanical components such as pistons, machine parts, and tools.

i) Types of Structural Analysis The seven types of structural analyses available in the ANSYS family of products are explained below. The primary unknowns (nodal degrees of freedom) calculated in a structural analysis are displacements. Other quantities, such as strains, stresses, and reaction forces, are then derived from the nodal displacements. Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

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The various types of structural analyses are briefed here under. Among these analysis types static and buckling analysis are discussed in detail. Static Analysis- Used to determine displacements, stresses, etc. under static loading conditions. Both linear and nonlinear static analyses are possible. Nonlinearities can include plasticity, stress stiffening, large deflection, large strain, hyper elasticity, contact surfaces, and creep. Buckling Analysis- Used to calculate the buckling loads and determine the buckling mode shape. Both linear (eigenvalue) buckling and nonlinear buckling analyses are possible. Modal Analysis– Used to calculate the natural frequencies and mode shapes of a structure. Different mode extraction methods are available. Harmonic Analysis- Used to determine the response of a structure to harmonically time-varying loads. Transient Dynamic Analysis- Used to determine the response of a structure to arbitrarily time-varying loads. All nonlinearities mentioned under Static Analysis above are allowed.

2.9.8 Modal Analysis of Beams using Matlab i) Modal Analysis To model a structure, it is divided into a number of smaller bodies (finite elements). These finite elements are connected to each other at points (nodes). Therefore the whole structure can be analyzed without solving the problem for the entire body, but by formulating equations for every single finite element, which are then combined to receive the solution for the entire structure. If the goal of developing a Finite-Element Model that represents the physical structure entirely is achieved, analysis can be done in order to improve the actual physical structure. To execute the Vibration Analysis, the structure of the rotary machine has to be excited. Therefore a modal hammer is used to excite the structure. The raw data is then measure in the time domain and by applying the Fast- Fourier Transform the data is transformed into the frequency


28


domain. The modal properties are extracted from the Frequency Response Function using the procedure modal analysis. The Modal Properties are Frequency, Mode shapes. It is important to identify where the frequencies occur and in which way they might affect the response of the present structure. Modal Analysis is used in all fields of mechanical engineering, including the automotive, airplane, computer and various other industries. By understanding where the frequencies and mode shapes occur makes it possible for the engineer to find the best solution in terms of noise and vibration applications. The vibration deflection shape of a rotating structure is referred to as the mode shape. For every natural frequency occurring in a structure there is a characteristic deformation shape. It describes the axial distribution of the vibration amplitude and phase along the structure. They are defined by the material properties, mass, stiffness and damping. Experimental modal parameters are obtained from measured Operating Deflection Shapes (ODS). Operating Deflection Shapes are the visualization of motion of two or more points on a structure. The motion at a point in a direction is called a Degree of Freedom (DOF). Since both, location and direction are correlated with motion it is a vector quantity. In the experimental modal analysis the actual physical structure is excited and the motion of each chosen point (DOF) is measured. At a resonance peak the Operation Deflection Shape is dominated by the natural frequency and therefore can be seen as a good approximation to the mode shape.[18]

ii) MATLAB Program (Algorithms) 1) The coordinates of all nodes are defined. 2) The number of members, free nodes and fixed nodes are assigned. 3) The connection of members in between is described so that the DOFS of each member can be sequentially assigned to the nodes.


29


4) The length is calculated of all the members are computed so as to determine the global coordinates of the nodes. 5) The characteristic constants of the structure calculated above such as are given as inputs the Area (A), Modulus of elasticity (E), Moment of inertia (I), and m mass per unit length. 6) For each element the Stiffness matrix is assigned into a 4X4 matrix in local coordinates. 7) Then each of them is transposed to the Global coordinates according to their position. 8) The individual Stiffness matrixes are combined to form the Global stiffness matrix of the total structure, according to their degree of freedoms. 9) The degree of freedoms of the fixed nodes is assigned as 0, whereas the free nodes have each 3 DOFS. 10) The components of common DOFS are added in the Global stiffness matrix. 11) Accordingly the mass matrix is also calculated for each member and combined as global matrix. 12) The Eigen value and Eigen vectors of K and M are calculated. The Eigen values depict the frequency of the natural vibration of the nodes and the Eigen vectors will give the mode shape of the displacements. The natural frequencies of each node of the beam is computed though the MATLAB Programmed. The Eigen values of the Stiffness matrix and the mass matrix gives the Natural frequencies and the Eigen vector gives the modal nodes. The mode shape of different cross section of beams such as flat and taper for varying the element size with same length. Plots of one second third mode shapes of flat and taper beams with element size 5 10 15 20 and 25 are shown in Appendix –B and The natural frequencies of each node of the beam is computed though the MATLAB Programmed, the different cross section beam written program are described in Appendix –C. Ozgur Turhan, Gokhan Bulut[14]Accurate prediction of vibration characteristics is crucial in the design stage of turbomachinery because prototyping and testing costs are exceptionally high and failure is generally disastrous in the practical applications


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of these systems. As the vibratory failures generally occur in the blades, the researchers are mostly focused on the blade vibrations. Mohammed .W. Alhazmi[15] Turbine blades are subjected to very strenuous environments inside a gas turbine. They are subjected to high temperatures, high stresses, and high speed. All these factors can lead to blade failure like Cracking, bulging, twisting, bending and breaking of blades. The damage or failure of turbine blades will affect the performance of the gas turbine engine hence early detection of blade problems is necessary to ensure the availability, reliability and good performance of gas turbine engines or other machines with blades. Due to the harsh environment the gas turbine blades are operating in, it is very difficult to directly monitor the blades working conditions. Some studies have shown that rotating blades produce mechanical vibration and there is a coupling between blades and rotor vibration. Therefore, blade vibration and problem can be indirectly detected from the vibration of the rotor-disk-blade system. Rishi Kumar Shukla[16] It is well known that blades are very common types for engines, different type turbines and other components and can be classified according to their geometric configuration as uniform or tapered, and slender or thick. It has been used in many engineering applications and a large number of studies can be found in literature about transverse vibration of uniform isotropic beams. But if practically analyzed, the non-uniform beams may provide a better or more suitable distribution of mass and strength than uniform beams and therefore can meet special functional requirements in architecture, aeronautics, robotics, and other innovative engineering applications and they has been the subject of numerous studies. Nonprismatic members are increasingly being used in diversities as for their economic, aesthetic, and other considerations. Dr. Negahban[17] In this paper the Measurements of thin film properties are difficult when compared to bulk materials. One method for finding the modulus of elasticity of a thin film is from frequency analysis of a cantilever beam. A straight, horizontal cantilever beam under a vertical load will deform into a curve. When this force is removed, the beam will return to its original shape; however, its inertia will keep the beam in motion. Thus, the beam will vibrate at its characteristic frequencies. If a thin film is sputtered onto the beam, the flexural rigidity will be altered. This change Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

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causes the frequency of vibrations to shift. If the frequency shift is measured, the film’s elastic modulus can be calculated. Jerry H. Griffin

[18]

.One of the main causes of turbine blade failure is high cycle

fatigue or HCF. Fatigue failure is caused by repeated cyclic loads on a structural member. The fatigue life of a part is defined by the number of load cycles it can survive. The fatigue life depends on the stress cycles magnitude and the part’s material properties. In most cases the higher the stress the shorter the fatigue life. William 0. Hughes

[19]

A study of the matrix displacement method for modeling the

vibrations of structures is presented and the model can analyze both the free and forced vibrations of a structure. Static loading on a structure is treated as a special case of the forced vibration analysis. F.T.K. Au[20] This paper presents a unified method for the analysis of vibration and stability of axially loaded non-uniform beams with abrupt changes of cross-section under various support conditions. J.T. Katsikadelis, G.C. Tsiatas[21] In this paper a direct solution to dynamic problem of beams with variable stiffness undergoing large deflections has been presented. B. Rama Sanjeeva Sresta,[22] The FEM method is applied for dynamic analysis of cantilever beam and T-structure using the MAT lab. Natural frequencies are obtained for cantilever and T-structure beams using FEM. the polynomial regression method is used for obtaining natural frequencies of cantilever beam and T-structure by varying width and depth for dynamic reanalysis. G. Jeyaraj Wilson[23] The finite element method is applied to the vibration analysis of axial flow turbine rotors. Using the axi-symmetric properties of the configuration of such rotors, several new finite elements are developed to describe the bending and stretching of thin or moderately thick circular plates, and which are characterized by only four or eight degrees of freedom. These elements incorporate the 'desired number .of diametric nodes in their dynamic deflection functions, and allow for any specified thickness variation in the radial direction. Hemanta Kumar Rana[24]The natural frequency is an important parameter in the dynamic analysis of structures. If a system is excited by an external force and both the Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

32


exciting and natural frequencies are very close to each other or equal, the resonant condition will occurs, thereby resulting violent vibration of the structure. This condition often leads to the catastrophic failure of the system. Therefore, it becomes necessary to design the dynamic system for its safe operation. Marcelo Areias Trindade[25],It is well-known that flexible beams become stiffer when subjected to high speed rotations. This is due to the membrane-bending coupling resulting from the large displacements of the beam cross-section. This effect, often called geometric stiffening, has been largely discussed in the last two decades. Several methodologies have been proposed in the literature to account for the stiffening effect in the dynamics equations. Junping Pu, Peng Liu[26] The finite element method program is worked out and an entire dynamic response process of the beam with non-uniform cross sections subjected to a moving mass is simulated numerically, the results are compared to those previously published for some simple examples. Miss Meera[27] The present study deals with experimental investigation on free vibration of laminated composite beam and compared with the numerical predictions using finite element method (FEM) in ANSYS environment. A program is also developed in MATLAB environment to study effects of different parameters. Nathalie Gotin[28] MATLAB is a high-performance programming language based environment that integrates computation, visualization, and programming. Problems and solutions can be expressed in common mathematical notation. And FiniteElement Modeling is the mathematical and numerical process that translates a physical structure into a mathematical model. The dynamic characteristics can be estimated from this mathematical model. Those include the natural frequencies and mode shapes. Brandon C. Rush

[29]

Finite element analysis of a periodic beam was used to

demonstrate and investigate aspects of periodic structure theory. A direct solution static finite element code was developed in MATLAB capable of solving static and dynamic beam flexure problems. Peng He[30] Many beams in engineering have variable axial parameters. Tapered beam,

which has variable axial geometric parameters, is a typical case. Moreover, the beam Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

33


under non-uniform axial temperature distribution has variable material mechanical parameters. The variable axial parameters bring obstacles in vibration analysis of these beams. In order to take the variable axial parameters into account conveniently, this work is motivated to improve the traditional beam element. Hemanshu[31]A dynamic analysis is carried out which involves finding of natural frequencies and mode shapes for different L/H ratios and different stacking sequences. Finally the non-dimensional natural frequencies of the beam are calculated by using MATLAB and ANSYS model of corresponding composite beam. Henri P. Gavin[32] The formulation of each element involves the determination of gradients of potential and kinetic energy functions with respect to a set of coordinates defining the displacements at the ends, or nodes, of the elements. The potential and kinetic energy of the functions are therefore written in terms of these nodal displacements. Keny Ordaz-Hernandez[33] This paper introduces a new way to obtain the dynamical response of Euler–Bernoulli beams under large deflections. They are simulated with parsimonious models based on a two integrated neural networks. Our article proves that neural network based-reduced modeling allows mechanical simulations to benefit shorter time processing without a great loss of accuracy.


34


CHAPTER-III

CANTILEVER RECTANGULAR BEAM



3. CANTILEVER RECTANGULAR BEAM

3.1. Mathematical Analysis Cantilever Rectangular Beam For a cantilever beam subjected to free vibration, and the system is considered as continuous system in which the beam mass is considered as distributed along with the stiffness of the shaft, the equation of motion can be written as, % !

"#$ !

& = ' ( ! % !

! !

3.1

Where, E is the modulus of rigidity of beam material, I is the moment of inertia of the beam cross-section, Y(x) is displacement in y direction at distance x from fixed end, ω is the circular natural frequency, m is the mass per unit length, m = ρA(x), ρ is the material density, x is the distance measured from the fixed end.

Figure.3.1.A Cantilever Beam

Figure.3.2.The Beam under Free Vibration


35


Fig. 3.1 shows of a cantilever beam with rectangular cross section, which can be subjected to bending vibration by giving a small initial displacement at the free end; and Fig. 3.2 depicts of cantilever beam under the free vibration. % !

=0 !

We have following boundary conditions for a cantilever beam (Fig. 3.1)

,- ! = 0, % ! = 0,

,- ! = 1,

2 % !

% !

= 0, =0 ! 2 !

3.2

3.3

For a uniform beam under free vibration from equation (3.1), we get 3 % !

− β3 5 ! = 0 ! 3 with

β3 =

ω m EI

:; ! = sin ?; @ − sinh ?; @ sin ?; ! − sinh ?; ! A

3.4

The mode shapes for a continuous cantilever beam is given as. + > cos ?; @ − cosh ?; @ cos ?; ! − cosh ?; ! AC

3.5

Where

E = 1,2,3 … . ∞,E?; @ = EG

A closed form of the circular natural frequency ωnf, from above equation of motion and boundary conditions can be written as #$ ';H = I; J 3 (@

3.6

where

I; = 1.875,4.694,7.885

The natural frequency is related with the circular natural frequency as ';H :;H = OP 2G

3.7

Where


36


I, the moment of inertia of the beam cross-section, for a circular cross-section it is given as $=

G 3 64

Where, d is the diameter of cross section and for a rectangular cross section

Q2 $= 12

3.8

3.9

Where

b and d are the breadth and width of the beam cross-section.

Figure 3.3: The First Three Undamped Natural Frequencies and Mode Shape of Cantilever Beam

3.2 Finite Element Formulation of Cantilever Rectangular Beam Theorem In this method a simplified uniform element is considered. The beam element has the length L, the mass density (ρ), the modulus of elasticity E, the cross –sectional area (A), and the moment of inertia (I). The beam element experiences a transverse motion. The displacement can be simply interpolated over 0≤ x ≤ L:


37

Effect of Elastic Modulus and Density Ratio on Vibration Characteristics of Rotating Turbine Blade 3

v x. t S TU ! XU - 3.10

UVW

The shape functions have to satisfy following boundary conditions: TW 0 1, TWY 0 TW @ TWY @ 0 T 0 1, TY 0 T @ T Y @ 0 T2 0 1, T2Y 0 T2 @ T2Y @ 0 T3 0 1, T3Y 0 T3 @ T3Y @ 0

Figure 3.4 Uniform Elements Undergoing Transverse Deflection

The shown beam in Figure 3.4 shows an element of a regarded beam statically loaded by end shears and bending moments. The result produces numerous deflection shapes. Therefore the equilibrium equation for a uniform end loaded beam is: EIX YY YY =0, the general solution for a uniform beam therefore is

! ! 2 ! X ! ZW B Z B Z2 [ \ B Z3 [ \ 3.11 @ @ @

By substituting the boundary condition in this cubic polynomial solution of the equilibrium equation, following shape functions are obtained: ! ! 2 TW 1 1 4 3 [ \ B 2 [ \ @ @


38


! ! 2 T 1 = ! − 2@ [ \ + @ [ \ @ @ ! ! 2 T2 1 = 3 [ \ − 2 [ \ @ @

! ! 2 T3 1 = −@ [ \ + @ [ \ @ @

By inserting the shape functions in the expressions for mij and kij for Bernoulli-Euler Beams, `

]U^ = _ #$TUYY T^YY !

3.12

`

(U^ = _ a1 + I − 1 yA>1 + ? − 1 yA 1] P + = } I − 1

I y } >1 + − 1 yA + >1 + I − 1 yA |

4.10

The Eq.4.10 is the final equation of motion for a double-tapered beam with rectangular cross-section. It was solved by numerical integration to give values of (lk) for various taper ratios for clamped-free beam with boundary conditions i.e. at x=0 or u=0, d2z/du2= 0 and z= 0, at x= l or u= 1, dz/du= 0 and z= 0. This after solving leads

toω = k hW W . The relation can be used as a comparison while solving with FEA

to show the effect of taper ratio on the fundamental frequencies and mode shapes.

4.2. Finite Element Formulation Most numerical techniques lead to solutions that yield approximate values of unknown quantities i.e. displacements and stiffness, only at selected points in a body. A body can be discretized into an equivalent system of smaller bodies. The assemblage of such bodies represents the whole body. Each subsystem is solved individually and the results so obtained are then contained are then combined to obtain solution for the whole body. Of the numerical techniques, the finite element technique is the most suitable for digital computers. It is applicable to wide range of problems involving non-homogeneous materials, nonlinear stress-strain relations, and Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

44


complicated boundary conditions. Such problems are usually tackled by one of the three approaches, namely (i)

displacement method or stiffness method

(ii)

the equilibrium or force method and

(iii)

Mixed method. The displacement method, to which we shall confine our discussion, is widely used because of the simplicity with which it can be handled on the computer.

In the displacement approach, a structure is divided into a number of finite elements and the elements are interconnected at joints called as nodes. The displacements in each element are then represented by simple functions. The unknown magnitudes of these functions are the displacements or the derivatives of the displacements at the nodes. A displacement function is generally expressed in terms of polynomial. From the convergence point of view such a function is so chosen that it i) Is continuous within the elements and compatible between the adjacent elements. ii) It includes the rigid body displacements and rotations of an element. iii) Has a consistent strain state. Further while choosing the polynomial for the displacement function, the order of the polynomial has to be chosen very carefully.

4.2.1 Calculation of shape function The analysis of two dimensional beams using finite element formulation is identical to matrix analysis of structures. The Euler-Bernoulli beam equation is based on the assumption that the plane normal to the neutral axis before deformation remains normal to the neutral axis after deformation. Since there are four nodal variables for the beam element, a cubic polynomial function for y(x), is assumed as v ! = , + ,W ! + , ! + ,2 ! 2

4.11

From the assumption for the Euler-Bernoulli beam, slope is computed from Eq. (4.1) is

! = , + 2, ! + 3Z2 !

4.12

Where α0, α1, α2, α3 are the constants The Eq. (4.11) can be written as Dept. of Mechanical Engg., The Oxford College of Engg., Bangalore-068

45


I IW % ! = >1 ! ! ! 2 A cI e I2

4.13

I IW >A = >1 ! ! ! 2 A ,E >IA = cI e I2

4.14

∴ % ! = >A>IA Where

For convenience local coordinate system is taken x1=0, x2=l that leads to vW = , ;

W = ,W ;

v = , + ,W 1 + , 1 + ,2 1 2 ; = ,W 1 + , 1 + ,2 1 2 ;

4.15

This can be written as vW 1 0 0 W 0 1 1 cv e = c 1 1 1 0 1 21

0 I 0 IW ec e 12 I 312 I2

4.16

Eq. (4.13) can be written as

=,C = >A>OA=,C

4.17

4.18

Where

>OA = >A>OA = >OW ! , O ! , O2 ! , O3 ! A Where Hi(x) are called as Hermitian shape function whose values are given below O ! = 1 −

O ! =

3! 2! 2 2! !2 + , OW ! = ! − + 1 1 1 1

2! 2 ! !2 3! − , O2 ! = − + 1 1 1 1

4.2.2 Stiffness calculation of tapered beam The Euler-Bernoulli equation for bending of beam is t t v t v u#$ w + a< u w − !, -

t! t! t-

4.20

Where y(x, t) is the transverse displacement of the beam ρ is the mass density,

EI is the beam rigidity, q(x, t) is the external pressure loading, t and x represents the time and spatial axis along the beam axis. We apply one of the methods of the weighted residual, Galerkin’s method, to the above beam equation to develop the finite element formulation and the corresponding matrices equations. The average weighted residual of Eq. (4.20) is $ = _ ua

t v t v t v + u#$ !

w − w ! = 0 tt! t!

4.21

Where l is the length of the beam and p is the test function. The weak formulation of the Eq. (4.21) is obtained from integration by parts twice for the second term of the equation. Allowing discretization of the beam into number of finite elements gives


47


∂ y ∂ y∂ p dp W I = S _ ρ p dx + _ EI x

dx − _ qp dxA + −Vp − M = 0 4.22

∂x ∂x dx ∂x >A

Where V = −EI x

∂ y is the shear force ∂x 2

= −EI x

∂ y is the bending moment ∂x

Ψe is an element domain and n is the number of elements for the beam.

Applying the Hermitian shape function and the Galerkin’s method to the second term of the Eq. (5.12), results in stiffness matrix of the tapered beam element with rectangular cross section i.e. > A

n

= _ >BA EI x >¡A !

Where

>¡A = =O,, OW,, O ,, O2,, C

4.23

4.24

And the corresponding element nodal degree of freedom vector = C = =vW W v C¢

4.25

In Eq. (4.23) double prime denotes the second derivative of the function. Since the beam is assumed to be homogeneous and isotropic, so, E that is the elasticity modulus can be taken out of the integration and then the Eq. (4.23) becomes ]WW ] = # c W ]2W ]3W

]W ] ]2 ]3

]W2 ] 2 ]22 ]32

]W3 ] 3 e ]23 ]33

4.26

Where Kmn (m, n = 1, 4) are the coefficients of the element stiffness matrix. n

]m; = ];m = # _ $ !

∂ H ∂ H dx ∂x ∂x


4.27

48


Solving the above equation, we get the respective values of coefficients of the element stiffness matrix for rectangular cross-sectioned beam. 6 $ + $W

~ 12 } } 2 $ + 2$W

} 1 > A = # } 6 $ + $W

}− 12 } } 2 2$ + $W

| 1

2 $ + 2$W

1 $ + 3$W 1 2 $ + 2$W

− 1 $ + $W 1

Here $ = Q 2 /12 and $W = QW W2 /12

6 $ + $W 2 2$ + $W

12 1 2 $ + 2$W

$ + $W − 1 1 6 $ + $W

2 2$ + $W 12 1 2 2$ + $W

3$ + $W − 1 1 −

4.28

Eq. (4.28) is called as the element stiffness matrix for tapered beam with rectangular cross-sectioned area.

4.2.3 Mass matrix of tapered beam Since, for dynamic analysis of beams, inertia force needs to be included. In this case, transverse deflection is a function of x and t. the deflection is expressed with in a beam element is given below

v !, - = O ! v - + +OW ! W - + O ! v - +O2 ! 2 - 4.29

mWW m W = a cm 2W m3W

mW m m2 m3

mW2 m 2 m22 m32

mW3 m 3 m23 e m33

4.30

The coefficients of the element stiffness matrix are n

(m; = (;m = _ < ! >HA >HA dx


4.31

49


> A

1 10

Effect of Elastic Modulus and Density Ratio on

Effect of Elastic Modulus and Density Ratio on

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