Advances in Vibration Engineering

ISSN 0972 - 5768

9 770972 576001

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Advances in Vibration Engineering provides a medium of communication among scientists and engineers engaged in research and development in the field of vibration engineering. It features original papers, in-depth reviews, experimental tests and results, design ideas and application papers of direct relevance to the industry. The journal promotes the objectives of the Vibration Institute of India for creating better awareness about the benefits of vibration analysis in assessing machinery health. Authors are required to transfer copyright of accepted papers to the publisher. The transfer will facilitate widest possible dissemination of information and appropriate reuse of this material by others subject to the following conditions: Personal use of this material is permitted. Subscribers may reproduce the table of contents and abstracts for internal circulation within their organization. However, permission must be sought from the publisher for multiple photocopying, reprinting/republishing this material for advertising or promotional purposes, for creating new collective works for resale or redistribution, or for reuse of any copyrighted component in other works. Permission of the publisher is required to store or use electronically any material contained in the journal. Address all permissions requests to : Krishtel eMaging Solutions Private Limited, B1 Ansary, 39, Madley Road, T. Nagar, Chennai 600 017, India; Tel : +91 44 24345516; Fax : +91 44 24349185; Email : [email protected] Notice: The statements and opinions given in this journal are those of the authors and do not necessarily reflect the views of the publisher or any of its associate companies. © Krishtel eMaging Solutions Private Limited 2011 Registered Office B1 Ansary, 39, Madley Road, T. Nagar, Chennai 600 017, India Published and Distributed by Krishtel eMaging Solutions Private Limited First published 2011 ISSN 0972-5768 Typeset by Krishtel eMaging Solutions Pvt. Ltd., Chennai 600 017 Printed in India at Image Screens # 262, Triplicane High Road, Chennai 600 005

From the Editor’s Desk Prof. J.S. Rao was invited by Prof. R. Rzadkowski from Institute of Fluid Flow Machinery, Polish Academy of Sciences in Gdansk and Gdansk University of Technology as an academic teacher to present a lecture for PhD Students and scientific worker entitled: “Lifing and Optimization of Machine Components with particular reference to Turbomachine Blades” Prof. J.S. Rao gave wonderful and very interesting a week long course of 40 hours lectures. Short description of the lecture content Prior to the evolution of advanced finite element codes in the high speed computation era, the designs are based on Strength of Materials approach coupled with Safety Factors to account for unknowns and then through testing. Today’s designs are based on accurate estimation of the local plastic fields at stress raiser locations using advanced finite element programs and alternating stresses using nonlinear damping from hysteresis and friction. This practically eliminates the need to use factors of safety; this enables a lifing program to be evolved with stress-based, strain-based and fracture mechanics approaches. This simulation has simplified designs considerably by eliminating testing during design process. Once these simulation procedures are developed, the question asked by advanced designers (e.g., aircraft engines, steam and gas turbine power plants . . . ) is concerned with optimum designs for weight, maximum life . . . Topology optimization plays a significant role for aircraft structures and rotors for minimizing peak resonant stresses at critical speeds using shape optimization and weight reduction by removing material sitting in low stress zones. The course will deal with these aspects citing practical examples. Schedule for the lectures: • September 12 AM: The Origins of Machines and What is Lifing? What do we need for lifing? Understanding of rotating machinery • September 12 PM: Linear Vibrations and Damping • September 13 AM: Nonlinear Vibrations and Multidegree Freedom Systems • September 13 PM: Rotor Dynamics + Condition Monitoring • September 14 AM: Stress Based Life Estimation • September 14 PM: Strain Based Life Estimation • September 15 AM: Fracture Mechanics • September 15 PM: Topology Optimization • September 16 AM: Shape Optimization. In 16 Sept 2011 Prof. J.S. Rao and Prof. J. Kiciński Director of Institute signed the Memorandum of Understanding between Koneru Lakshmaiah University, Vaddeswaram, India and The Szewalski Institute of Fluid Flow Machinery, Gdansk, Poland. Two sides agreed in cooperation in advanced research of Turbomachinery in the areas of Structural and Fluid Dynamics, Lifing and Optimization with cooperation between their faculties for over three decades.

R. Rzadkowski Associate Editor Europe Region

Contents

C. L. Zhang and W. Q. Chen

Magnetoelectric Effect of Non-Uniform Multiferroic Composite Nanofibers

287

Jan Kicinski and Grzegorz Zywica

Numerical Analysis of Defects in the Rotor Supporting Structure

297

Y. F. Zheng, L. Q. Deng and F. Wang

Bifurcation and Chaos of Cross-ply Laminated Plates with Piezoelectric Actuators

305

S. Ganesan and C. Padmanabhan

Rotor Dynamic Modeling of High Speed Flexible Coupling

313

Romuald Rzadkowski and Marek Soliński

The Effect of Change in the Number of Stator Blades in the Stage on Unsteady Rotor Blade Forces

325

Vinayak Ranjan and M. K. Ghosh

Free Vibration and Stability Analysis of a Spinning Annular Circular Plate

333

Meng Jigang, Yang Shuhua and Wang Yuefang

Three-Dimensional Seismic Analysis and Safety Evaluation for Nuclear Pump of Nuclear Power Plant Based on the RCC-M Code

343

Y. H. Li, Y. N. Wang and L. Li

Nonlinear Dynamic Behaviors of a ThermoMechanical Coupling Viscoelastic Plate

353

S. M. Li, C. W. Lim and M. J. An

Partial Coherence Method of Transfer Functions on Structural Vibration Under Correlated Excitations

371

Krzysztof Kosowski and Robert Stepie´ n

The Method of Reduction of Aerodynamic Forces Generated in Turbine Blade Seals

389

Magnetoelectric Effect of Non-Uniform Multiferroic Composite Nanofibers C. L. Zhang and W. Q. Chen Department of Engineering Mechanics, Zhejiang University, Yuquan Campus, Hangzhou 310027, China [email protected] Abstract We analyze the magnetoelectric effect of non-uniform multiferroic composite nanofibers composed of both piezoelectric and piezomagnetic phases. The non-uniformity could be induced by the manufacturing error and may have an influence on the magnetoelectric effect. The governing equation is formulated based on the zero-order beam theory, and the state space method is used to compute the magnetoelectric coefficient. Numerical examples are presented for a non-uniform composite nanofiber with different non-uniformity parameters and boundary conditions. The results show that the non-uniformity of the cross-section may increase the magnetoelectric effect, and thus provides a simple and also attracting way for the design of high-performance magnetoelectric structures. Keywords: Multiferroic composite, Magnetoelectric effect, Non-uniform nanofiber

1

Introduction

Magnetoelectric (ME) effect in multiferroic materials has attracted great interest in recent years not only for their fundamental scientific significance, but also for their potential applications in sensing, actuation, data storage and transducers[1–5] . Since magnetoelectric effect in single-phase multiferroics is rather rare and small and often exists only at low temperature[6] , hybrid multiferroic composites consisting of ferroelectric and ferromagnetic phases have been widely explored[4] , where room temperature magnetoelectric effect not existing in either constituent is induced in the composites through the mechanical interactions between piezoelectric and magnetostrictive effects[7–10] . Recently, various composites and laminates of piezoelectric and piezomagnetic materials have been developed[4, 6] . Strong ME couplings in uniform multiferroic fibers and laminates of piezoelectric and piezomagnetic have been demonstrated[11–14] . Both static and frequency dependent ME effects have been studied experimentally and theoretically. To enhance the magnetoelectric effect in multiferroic composites, some research groups have focused on low-dimensional thin films and nanofibers of multiferroic composites[15–17] . The ME effect in nanofibers is higher than in films on a substrate, as revealed in ref. [17]. Multiferroic nanofibers have been proposed as an alternative structure, and have been successfully synthesized by sol-gel based electro-spinning. In all reported works, only nanofibers with uniform cross-section have been considered. It is noted that the non-uniformity may be induced by several factors, such as chemical, physical, or other unpredictable effects, in the manufacturing process. It may change the magnetoelectric effect to a certain degree. To understand better its influence, this paper aims to study the ME effect in non-uniform nanofibers based on the zero-order beam theory derived in ref. [18]. ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011

© KRISHTEL eMAGING SOLUTIONS PRIVATE LIMITED

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C. L. ZHANG AND W. Q. CHEN / ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011

Basic Equations and Solutions

Consider a non-uniform multiferroic composite nanofiber consisting of both ferroelectric and ferromagnetic phases as shown in Fig. 1. We use polarized ceramics PZT-4 for the ferroelectric phase and CoFe2 O4 for the ferromagnetic phase. The PZT-4 and CoFe2 O4 phase are all polarized in the x1 direction. The x1 axis is identical to the centroidal line of the fiber as depicted in Fig. 1. Assume that the cross-section Fig. 1 The sketch of a non-uniform fiber or bar of the fiber is rectangular as shown in Fig. 1 and the crosssectional area varies with x1 , and denoted by A(x1 ). The form of function A(x1 ) can be a linear, power polynomial, exponential or other function of the variable x1 . There is an applied time-harmonic magnetic field in the parallel direction of the x1 such that we have H2 = H3 = 0 and H1 = H exp(iωt). Next, we summarize the zero-order theory equations derived in ref. [18]. For extension there are only u1 = u1 (x1 , t), φ = φ(x1 , t) and ψ = ψ(x1 , t) to be considered. u1 , φ and ψ are the zero order (0,0) displacement, electric potential and magnetic potential, which are the form of u1 , φ (0,0) and ψ (0,0) , (0,0) ” which indicates the component of the zero respectively. And for convenience, the superscript “ order in ref. [18] are all dropped in this paper. The nontrivial strain component is S11 = u1,1 , where the comma followed by the index 1 in the subscript indicates differentiation with respect to x1 . The other zero-order strains due to Poisson’s effect are eliminated in the stress relaxation procedure[19] . The electric and magnetic fields are E1 = −φ,1 and H1 = −ψ,1 . The zero-order constitutive relations in the compact matrix notation are[18] N = A(x1 )(c˜11 S11 − e˜11 E1 − h˜ 11 H1 ) D1 = A(x1 )(e˜11 S11 + ε˜ 11 E1 + α˜ 11 H1 ) B1 = A(x1 )(h˜ 11 S11 + α˜ 11 E1 + μ˜ 11 H1 )

(1)

where N is the extensional resultant force, D1 is the electric displacement and B1 is the magnetic flux, E1 and H1 are the electric field and applied magnetic field, respectively, and c˜11 = 1/s11 ,

e˜11 = d11 /s11 ,

h˜ 11 = q11 /s11 ,

μ˜ 11 = q11 q11 /s11 ,

ε˜ 11 = ε11 − d11 d11 /s11 ,

α˜ 11 = −d11 q11 /s11 .

(2)

In (2) s11 , d11 , ε11 and q11 are the elastic compliance, piezoelectric constant, dielectric permittivity and piezomagnetic constant, respectively. The equation of motion and electrostatics for the nanofibers in extensional vibration are N,1 = ρ u¨ 1 , D,1 = 0,

x1 ∈ (0, l), x1 ∈ (0, l),

(3) (4)

MAGNETOELECTRIC EFFECT OF NON-UNIFORM MULTIFERROIC COMPOSITE NANOFIBERS

289

where ρ is the material density. In this paper, the electrodes are assumed to be open, that is, D = 0, x1 = 0, l. For mechanical boundary conditions, two different cases are considered, N = 0,

x1 = 0, l,

(5)

for the nanofibers with both ends free and u1 = 0,

x1 = 0,

N = 0,

x1 = l,

(6)

for the nanofibers with one end fixed and one end free. To this end, we have established the governing equations, constitutive equations and the boundary conditions for the problem. Next, the State Space Method (SSM) is used to obtain the electric field induced by the applied magnetic field. We choose the axial resultant force N , displacement u1 , the electric potential φ and the electric displacement D as components (or elements) of the state vector. Substitute E1 = −φ,1 into (1)1 and (1)2 , we have u1,1 = φ,1 =

m2 H, f2

(7)

g2 c˜11 e˜11 D + N − H, f 2 A(x1 ) f2 f 2 A(x1 )

(8)

ε˜ 11 2 f A(x

1)

N+

e˜11 2 f A(x

1)

D+

where f, g and m are defined as g 2 = h˜ 11 e˜11 + c˜11 α˜ 11 ,

2 + c˜11 ε˜ 11 , f 2 = e˜11

m2 = h˜ 11 ε˜ 11 − e˜11 α˜ 11 .

(9)

In the following derivation, the dimensionless variables are used and defined as u1 = l u, ¯

x1 = lζ,

¯ N = c˜11 A0 N¯ , D = c˜11 ε˜ 11 A0 D, ¯ H = c˜11 /μ˜ 11 H¯ . φ = l c˜11 /˜ε11 φ,

(10)

Upon substituting (10) into (3), (4), (7) and (8), we obtain the state equation in matrix form as dY = MY + Z, dζ where

⎡

0

(11)

0

⎢ ⎢ ⎢ 0 0 M=⎢ ⎢ ⎢ 2 ⎣− S(ζ ) 0 0 0

ε˜ 11 f 2 S(ζ )

√ e˜11 c˜11 ε˜ 11 f 2 S(ζ )

0 0

⎤ √ e˜11 c˜11 ε˜ 11 2 f S(ζ ) ⎥

⎥ ⎥ 11 − f 2c˜S(ζ ) ⎥, ⎥ ⎥ 0 ⎦ 0

(12)


290

and ¯ T, Y = [u¯ φ¯ N¯ D] Z = m2 c˜11 /μ˜ 11 H¯ /f 2

(13) g 2 c˜11 /μ˜ 11 H¯ /f 2

0

0

T

.

(14)

Y is the dimensionless state vector, the superscript T indicates the transpose of a vector, and is the dimensionless circular frequency. For a non-uniform fiber, S(ζ ) is a function of variable ζ . Thus, (14) is difficult to be solved analytically and directly. Here we employ the approximate laminate model in ref. [20], for which the fiber is equally divided into m segments with each length (l/m) being very small. Therefore, the coefficient matrix M can be assumed constant within each segment. Denote Mj as the constant coefficient matrix in the j th segment. In the following, the matrix Mj is assumed to take its value at the mid-point of each segment. If the fiber has been divided into enough segments, that is, m is enough large, the result will be very close to that of the original non-uniform nanofibers. Now in each segment, the solution of (11) can be written as Y = eMj (ζ −ζj 0 ) Y (ζj 0 ) +

ζ ζj 0

eMj (ζ −t) Zdt,

ζ ∈ [ζj 0 , ζj 1 ],

(15)

where ζj 0 = (j − 1)l/m and ζj 1 = j l/m are the dimensionless coordinates of the left and right ends of the j th segment, respectively. The state vector Y is the function of ζ . The transfer relation between the state vectors at the two ends of each segment can be derived from (15) Y (ζj 1 ) = eMj (ζj 1 −ζj 0 ) Y (ζj 0 ) +

ζj 1 ζj 0

eMj (ζ −t) Zdt,

ζ ∈ [ζj 0 , ζj 1 ].

(16)

By virtue of the continuity conditions of state variables at each fictitious interface, namely, Y (ζj 1 ) = Y (ζ(j +1)0 ), the following relation between the state vectors at two ends of the fiber can be derived from (16) Y l = LY 0 + B.

(17)

where L = 1j =m exp[Mj (ζj 1 − ζj 0 )] is a fourth order square matrix, Y l and Y 0 are values of the state vector at x1 = l and x1 = 0, respectively. According to the matrix form of M, it is easy to demonstrate that among the elements Lij of the matrix L, the components L12 , L32 , L41 , L42 , L43 are zero, and L22 and L44 are unit, that is, ⎡

⎤ ⎡ u(1) ¯ L11 ⎢ φ(1) ⎥ ⎢ ⎢ ¯ ⎥ ⎢L21 ⎢¯ ⎥=⎢ ⎣N (1)⎦ ⎣L31 ¯ 0 D(1)

0

L13

1 0

L23 L33

0

0

⎤ ⎡ ⎤ u(0) ¯ B1 ⎢ φ(0) ⎥ ⎢B ⎥ ¯ L24 ⎥ ⎥⎢ ⎥ ⎢ 2⎥ ⎥⎢ ⎥ + ⎢ ⎥. L34 ⎦ ⎣N¯ (0)⎦ ⎣B3 ⎦ ¯ B4 1 D(0) L14

⎤⎡

(18)

n ζj 1 1 and the vector [B1 , B2 , B3 , B4 ]T = j =1 ζj 0 exp[Mk (ζk1 − t)]Zdt k=j exp[Mk (ζk1 − ζk0 )]. In numerical examples, L and B are calculated numerically by computer. From (18), the electric potential


291

difference between the two ends of the fiber can be directly calculated as U = L21 u¯ 1 (0) + L23 N¯ (0) + ¯ L24 D(0) + B2 when the boundary conditions are known. With the boundary conditions, the electric field induced by the applied magnetic field can be solved from (18). For nanofibers with two ends free, we have N¯ (ζ = 1) = 0,

N¯ (ζ = 0) = 0,

¯ = 1) = 0, D(ζ

¯ = 0) = 0. D(ζ

(19)

Substituting (19) into (18), we obtain the following relations ¯ ¯ φ(1) = L21 u(0) ¯ + φ(0) + B2 , 0 = L31 u(0) ¯ + B3 .

(20)

Solving for u(0) ¯ from (20)2 and substituting it in (20)1 , the electric potential difference between the two ends is given by L21 (21) U = l c˜11 /˜ε11 = l c˜11 /˜ε11 B2 − B3 . L31 Hence, the induced electric field by the applied magnetic field is expressed by U L21 E= B3 . = c˜11 /˜ε11 B2 − l L31

(22)

For the second type of boundary conditions, that is, nanofibers with one end fixed at x1 = 0 and the other end free at x1 = l, we have the following relations ¯ = 1) = 0, N(ζ

u(ζ ¯ = 0) = 0,

¯ = 1) = 0, D(ζ

¯ = 0) = 0. D(ζ

(23)

Similarly, substitution (23) into (18), gives ¯ ¯ φ(1) = L23 N¯ (0) + φ(0) + B2 , 0 = L33 N¯ (0) + B3 . By eliminating N¯ (0) from (24), we can obtain the electric potential difference L23 U = l c˜11 /˜ε11 B2 − B3 . L33 From (25) the induced average electric field is obtained as L23 E¯ = c˜11 /˜ε11 B2 − B3 . L33

(24)

(25)

(26)

292


To study the ME effects through the mechanical coupling in multiferroic nanofibers consisting of piezoelectric and piezomagnetic phase, we define the ME coupling coefficient γ as the ratio of the induced electric field to the applied magnetic field, namely γ =

3

E¯ H

(27)

Examples

In this section, the ME effects of the nanofibers with two boundary conditions are calculated. The multiferroic composite considered in this paper is composed of the PZT-4 phase and the CoFe2 O4 phase. For the two phases in the multiferroic nanofibers, the material properties are given in the following. For PZT-4, we have the density ρ = 7500 kg/m3 , the piezoelectric constant d11 = 289 × 10−12 C/N, the compliant elastic modulus s11 = 15.5 × 10−12 m2 /N, and the permittivity component ε11 = 5.52 × 10−9 C2 N/m2 , and for CoFe2 O4 , we have the density ρ = 5300 kg/m3 , the piezomagnetic constant q11 = 1.88 × 10−9 m/A, the compliant modulus s11 = 7 × 10−12 m2 /N. The effective material constants of the composite are estimated by a simple volume averaging method. In this paper, the volume fraction of PZT-4 in the multiferroic nanofibers is taken as 0.25 and the length of the nanofibers is l = 0.2 m. The structural damping effect is usually introduced by allowing the elastic constant to assume complex values in numerical calculation, and then, in this paper the damping effect is considered by replacing the elastic compliance constant s11 by s¯11 = s11 (1 − 0.01i) in calculation. To guarantee the validity of the SSM, we first make a comparison with the exact solution for a nonfiber with the cross-section varying along the axis in an exponential form of A(x1 ) = A0 exp(a + bx1 / l), where A0 , a and b are constants. The solution is presented in the Appendix A. The comparison of the results of SSM and the exact solution is depicted in Fig. 2, for a freely supported nanofibers with a = 1 and b = 1. It is seen that the two methods give almost identical results. Next, we consider the non-uniform nanofibers with cross-section of the form of A(x1 ) = A0 (a + bx1 / l), and the corresponding dimensionless cross-sectional area is S(ζ ) = a + bζ . For the purpose of comparison, we assume the fiber has the same length l and volume Fig. 2 Comparison of the ME coupling coefficient versus frequency between lA0 . To study the effect on the ME coupling coeffiSSM and the exact solution cient resulted by the non-uniformity of the cross-section, as numerical examples, a is taken as 1, 1.4 and 1.8, respectively. Accordingly, b equals 0, −0.8 and −1.6 with the equivalent volume constraint. When a is unit and b equals zero, the nanofibers becomes uniform. The ME coupling coefficients of the nanofibers with two ends free for different non-uniformity parameters of the cross-sectional area are plotted in Fig. 3 versus the magnetic driving frequency. For the


Fig. 3

The ME coupling coefficient versus frequency for nanofibers with two ends free

Fig. 4

293

The ME coupling coefficient versus frequency for nanofibers with one end fixed and one end free

nanofibers with one end fixed and one end free the ME coupling coefficients are shown in Fig. 4. From Figs. 3 and 4, it is seen that the non-uniform nanofibers has a higher ME coupling coefficient. And for different non-uniformity parameters such as the parameter a, the ME coupling coefficient is different. Figure 4 shows both the ME coupling coefficient and the resonance frequency increase with the parameter a. For the nanofibers with two ends free, the ME coupling coefficient achieves the biggest value when a is taken to be 1.4. The results show that the nanofibers with two ends free have a higher ME coupling coefficient than the nanofibers with one end fixed and one end free, and also show the mechanical boundary conditions at two ends have a significant effect on the resonance frequency.

4

Conclusions

The equations for non-uniform nanofibers are presented based on the zero-order beam theory. Numerical results show that the ME effect in non-uniform nanofibers is higher than that in uniform nanofibers with equivalent length and volume. Different non-uniformity will have different effect on the ME coupling coefficient. The nanofibers with two ends free have a larger ME coupling coefficient than the nanofibers with one end fixed and one end free. And the mechanical boundary conditions at tow ends largely affect the resonance frequency. The effect on the ME coupling coefficient of the non-uniformity of nanofibers may provide an instructive and useful guidance in configurating nanofibers for specific applications.

Acknowledgments The work was supported by the NSFC (Nos. 10725210, 10832009 and 11090333), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20060335107), the National Basic Research Program of China (No. 2009CB623200), and the Zhejiang Provincial Natural Foundation of China (No. Y6090108).

294


Appendix A Assume that the cross-sectional area of the nanofibers varies exponentially along the axis, i.e., A(x1 ) = A0 exp(a + bx1 / l), where A0 , a and b are constants. From (4), the electric potential can be obtained φ,1 =

e˜11 α˜ 11 1 C0 , u,1 + H1 − ε˜ 11 ε˜ 11 ε˜ 11 A(x1 )

(A.1)

where C0 is an undetermined constant. Since the electrodes are open, i.e., D1 (0) = D1 (l) = 0, we can find C0 = 0. Substitution of (A.1) into (3), gives b 1 b ρω2 u,11 + u,1 + 2 u + l l λ2 λ

e˜11 α˜ 11 ˜ − h11 H = 0. ε˜ 11

(A.2)

where λ2 = c˜11 + e˜11 e˜11 /˜ε11 . The solution of (A.2) can be expressed as b 1 u1 = C1 exp(κ1 x1 ) + C2 exp(κ2 x1 ) − l ρω2

e˜11 α˜ 11 ˜ − h11 H, ε˜ 11

(A.3)

where C1 and C2 are undetermined constants and κ1 = κ2 =

−b/ l + −b/ l −

b2 / l 2 − 4ρω2 /λ2 , 2 b2 / l 2 − 4ρω2 /λ2 . 2

(A.4)

From (A.1) and (A.3), and with the boundary conditions at the two ends, we can determine the two arbitrary constants C1 and C2 . For example, when both ends are free, i.e., N (0) = N (l) = 0, we have e˜11 α˜ 11 1 − exp(κ2 l) 1 ˜ C1 = 2 − h11 H, ε˜ 11 λ κ1 exp(κ2 L) − exp(κ1 l) 1 − exp(γ1 l) 1 e˜11 α˜ 11 C2 = 2 − h˜ 11 H, ε˜ 11 λ κ2 exp(κ1 l) − exp(κ2 l)

(A.5)

The average electric field induced by the applied magnetic field is 1 E¯ = L

0

L

Edx1 = −

1 e˜11 α˜ 11 [u(L) − u(0)] − H L ε˜ 11 ε˜ 11

The ME coupling coefficient is then calculated from (27).

(A.6)


295

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[17] Zhang, C. L., Chen, W. Q., Xie, S. H., Yang, J. S. and Li, J. Y., Appl. Phys. Lett., Vol. 94, 12097, 2009. [18] Zhang, C. L., Chen, W. Q., Li, J. Y. and Yang, J. S., One-dimensional equations for piezoelectromagnetic beams and magnetoelectric effects in fibers, Smart Mater. Struct., Vol. 18, 095026, 2009. [19] Mindlin, R. D., Low frequency vibrations of elastic bars, Internat. J. Solids Structures, Vol. 12, pp. 27–49, 1976. [20] Chen, W. Q., Bian, Z. G., Lv, C. F. and Ding, H. J., 3D free vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid, Internat. J. Solids Structures, Vol. 41, pp. 947–964, 2004.

Numerical Analysis of Defects in the Rotor Supporting Structure Jan Kicinski and Grzegorz Zywica Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Fiszera 14 Str., 80-952 Gdansk, Poland [email protected]; [email protected] Abstract In the presented article the authors’ attention was focused on analysis of supporting structure defects. The object of investigation was the large-dimension Rotor Dynamics and Slide Bearing Research Rig. The proposed modeling idea is based on interactions between original and commercial computer codes. Nonlinear bearings properties are described using codes of MESWIR series – originally invented in IFFM PASci to model rotor-bearings systems. Dynamics properties of the supporting structure are analyzed with FEM program ABAQUS. Thanks to effective interactions between both software applications, a full model of the rotating machine was created. A few classes of defects which could appear during operation process were considered. In scope of investigation the time-consuming numerical calculations were performed. It allowed to access the influence of some defects on dynamics characteristics of the rotating machine. It has been proved in the article that some defects of supporting structure might have significant influence on the system vibrations. The article also presents the possibilities of diagnostics of supporting structure defects. Keywords: Rotating machines, Rotor dynamics, Slide bearings, Defects analysis

1

Introduction

In systems of rotor-bearings-supporting structure type certain couplings take place between particular subsystems[1–4] . These couplings result, first of all, from their structure, as none of the rotors supported on slide bearings can be fastened in infinitely stiff bearing bushes. In each real supporting structure, bearing supports affect on the rotor via slide bearings revealing certain dynamic characteristics. Since the rotating machine constitutes a compact whole, its particular subsystems cannot be examined in separation of the remaining parts[5, 6] . This makes it necessary to use methods that produce compact description of the entire machine. The process of developing and advancing methods for modeling of rotating machines has been given much attention for years, but still no satisfying solution has been obtained[8] . It is enough to mention that all large-dimension steam turbines have rotors supported on slide bearings, and their experimental examination is so expensive and dangerous that they are performed extremely rarely in practice[7, 9] . Therefore the numerical models are necessary, especially if we want to improve dynamics properties both existing and new rotating machines. The presented article reports an attempt to assess the effect of the failure having the form of supporting structure defect on the dynamic state of the rotor supported on slide bearings. The investigations were conducted on an experimentally verified numerical model of a selected large-dimension ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011


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rotating machine. The applied method of calculations comprised two basic stages. The first stage consisted in determining complex flexibility characteristics of the supporting structure at the rotor foundation place. In the second stage the obtained flexibility characteristics, after transformation to real matrices of stiffness, damping, and apparent masses, were used for determining parameters of rotor motion. Obtained dynamics characteristics provided opportunities for assessing the effects of the examined defects on rotor vibrations.

2

The Numerical Model

The object of presented investigation was the large-dimension Rotor Dynamics and Slide Bearing Research Rig (Fig. 1), which was placed in the laboratory of IFFM PASci in Gdansk (Poland). The proposed modeling idea is based on interactions between original and commercial computer codes. Nonlinear bearings’ properties and rotor are described using codes of MESWIR series, originally Fig. 1 Schematic of the rotor dynamics and slide created in IFFM PASci[4] . Dynamics probearing research rig (1-rotor, 2-stand with journal perties of the supporting structure are analyzed bearing, 3-disc, 4-coupling, 5-axial ball bearing, with ABAQUS code. Thanks to cooperation 6-frame, 7-bracket, 8-foundation, 9-vibration isolabetween both software applications, a full tion, 10-engine, 11-gear) model of the rotating machine could have been created. The created linear model of the supporting structure consisted of nearly 400 thousands degrees of freedom (Fig. 2a). Its finite element model was made using elements having first (C3D8R, C3D6) and second order (C3D20R, C3D15) shape functions. Model tuning provided opportunities for obtaining a very good agreement between the simulated and experimentally recorded bearing support flexibility characteristics, a situation which is rarely observed in cases of examining supporting structures of largedimension rotating machines. The effect of tuning, having the form of comparison of complex dynamic flexibility amplitude and phase curves for the second bearing support, is shown in Fig. 3.

Fig. 2

Numerical model of the rotating machine (a – model of the supporting structure, b – model of the rotor and bearings)

NUMERICAL ANALYSIS OF DEFECTS IN THE ROTOR SUPPORTING STRUCTURE

Fig. 3

299

Dynamic characteristics of the supporting structure. Simulated and experimental curves of complex dynamic flexibility amplitudes and phases in horizontal and vertical directions for the second bearing support

The rotor-bearings system was studied using Timoshenko beam model and pseudo-diathermal model of slide bearings (Fig. 2b). The model consists of 29 beam elements with four degrees of freedom at each node. The real rotor was 3.4 m long and its diameter was 0.1 m. It was made of two shafts linked together by a coupling. The shafts carried two rigid discs, each of 0.4 m in diameter and mass equal to 185 kg. The rotor rested of three slide bearings, with cylindrical clearance and two lubricating pockets each. The code KINWIR-60, used for kinetostatic calculations, makes use of the diathermal model of slide bearings, which allows such parameters to be determined as: reactions of rotor supports, journals locations within the area of bearing clearances, temperature of oil leaving the oil clearance (and, consequently, the effective dynamic viscosity of the lubricating agent). The data obtained from the kinetostatic calculations are required for performing the dynamic calculation using the nonlinear model of bearing using the code NLDW-80. If we want to analyze the operation of the rotor-bearings-supporting structure system within a wide range of rotational speeds (which is connected with the appearance of large-amplitude journal vibrations, for instance in the rotational speed range close to resonance speeds and to the stability limit of the system), nonlinear modeling becomes necessary. The three-dimensional FEM model of the supporting structure, created in the system ABAQUS, was integrated with the beam model of the rotor and slide bearings via stiffness, damping and mass coefficients[10, 11] . These coefficients replace dynamic properties of the supporting structure (bearing supports and foundation) at rotor foundation points. In order to determine flexibility characteristics of the examined supporting structure, simulation calculations were performed after applying to each support

300


a harmonic force in horizontal and vertical direction. As a result of the calculations, for each excitation frequency ω, complex flexibility matrices L(ω) were obtained. The investigations were performed within the frequency range from 1 to 180 Hz, with the step equal to 1 Hz. As a result, 180 complex dynamic flexibility matrices L(ω) were obtained. These matrices were inverted to obtain the complex dynamic stiffness matrix K(ω). Next, using a two-point method[4] real matrices of coupled masses M(ωi , ωi+1 ), stiffness C(ωi , ωi+1 ) and damping D(ωi , ωi+1 ) were determined. Matrix coefficients were determined within frequency intervals of 1 Hz. This resolution provided opportunities for taking into account even fine changes in flexibility characteristic shapes caused by defects in the supporting structure. The process of determining stiffness, damping, and mass matrix coefficients was automated using the programme MATLAB. Above mentioned calculations were made for each class of failure, but results of those calculations are too vast and are not displayed in this paper. The experimental investigations were earlier performed on the research rig for examining dynamics of rotor and slide bearing. This way experimental verification was possible. Thanks to that, the numerical analysis of supporting structure defects was performed using the verified numerical model of research rig[11] . A few classes of defects which could appear during operation process were considered, such as: cracks of supporting structure, screws crack, screws backlash, decrease of stiffness in some parts of the supporting structure. The adopted crack model, which was used to simulate some cases of defects, had the form of a permanently open crack with the top corner radius equal to zero. The option “crack-assign seam”, available in the ABAQUS system, was used. Such a crack model does not break the linearity of the system, thus allowing the modal superposition method to be used for determining dynamic flexibility of the supports. In scope of investigation the time-consuming numerical calculations were completed. It allowed to access the influence of some defects on dynamics characteristics of the rotating machine. Results of those calculations may be also very useful during diagnosis of machine. We can use obtained results of numerical analysis in data base of diagnostic system. This way, performing numerical analysis, we can receive diagnostic symptoms connected with specified defects without experimental investigations.

Fig. 4 Amplitudes of relative journal-bush vibrations in the first bearing for defects compared with base case


301

Fig. 5 Amplitudes of relative journal-bush vibrations in the middle bearing for defects compared with base case

Fig. 6 Amplitudes of relative journal-bush vibrations in the third bearing for defects compared with base case

3

Results of Calculations

The selected results of calculations, having the form of relative journal-bush vibration amplitudes, for all bearings are given in Figs. 4–6. Presented figures show the curves which were obtained for defects in form of screws backlash and crack of supporting structure. Those failures had the biggest influence on dynamic properties of the rotor. In other cases of defects, such as: screws crack, decrease of stiffness in some parts of the supporting structure the clearly changes of the dynamic characteristic were also

Fig. 7 Trajectories of vibrations in the middle bearing at rotational speed 2970 rpm for defects compared with base case

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observed, but those modifications were not as important as presented in Figs. 4, 5 and 6. This is the reason that they are not discussed, and in the following part of the article we will focus our attention on screws backlash and crack of supporting structure. The most remarkable changes were recorded for the crack of supporting structure. In that case considerable increase of amplitude was observed and was accompanied by the shift of the system stability limit. At rotational speed equals to 3000 rpm the large increase of vibration amplitude in each bearing was noticed. After that increase, at highest rotational speed, the amplitude of vibrations decreased. The curves for discussed failure were similar to curves obtained for base model. At high rotational speed we observed the shift of the system stability limit. What is interesting, the system was able to operate at highest speed then structure with no defect. Direct cause of that positive change was increase of dynamic flexibility in the central bearing support at highest frequencies. In case of screws backlash we observed lesser changes of dynamic characteristics. The most important change was connected with the system stability limit. The system with defect in form of screws backlash was able to operate at lower maximal speed than in base case (line with circles in Figs. 4, 5 and 6). Figure 7 compares trajectories of relative journal-bush and absolute bush vibrations in the middle bearing for chosen defects with the base case (without defects). The diagram reveals clear oil whipping, which appeared as a result of crack in the rotor supporting structure. Trajectories of relative journalbush vibrations took here chaotic shapes, characteristic for oil whips. The reason for the appearance of oil whips in this range is the resonance caused by serious defect, which significantly increases dynamic flexibility of the central support, what in case of a real machine would threat with its failure. Figure 7 depict also absolute bearing bush vibrations. In case of screws backlash we can observe that trajectory of vibration took chaotic form, but we can also notice in results of harmonic analysis that the first harmonic is dominant. It means that analyzed system works properly.

4

Conclusions

The combined use of the commercial FEM software with the in-home developed programs for nonlinear dynamics calculations of the rotor and slide bearings has provided opportunities for simulation studies of examined rotating machine within a wide range of rotational speeds. As results of calculations we obtained: courses of complex dynamic flexibility amplitudes and phases, courses of relative journal-bush vibrations, trajectories of relative and absolute vibrations. These results of calculations were presented for selected cases. In presented research a few classes of rotor supporting structure defects were analysed. For examined rotating machine the significant changes were observed only in case of extensive damages, such as crack of supporting structure. Those changes could be used as diagnostic symptoms of the defects. Presented results of investigation clearly show that some supporting structure defects might have significant influence on vibrations and dynamic properties of rotating machine. Supporting structure defects influence the modal model and dynamic state of the rotor and could not be omitted during numerical analysis of any rotating machine. The proposed modeling method makes it possible to both optimise the supporting structure as early as at its design stage, and formulate expectations concerning the behavior of a real structure after the appearance of various defects in it. This can be very helpful when preparing data to be used for decision making by the rotating machine designers, operators, and diagnosticians.

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References [1] Cavalca, K. L., et al., An investigation on the influence of the supporting structure on the dynamic of the rotor system, Mechanical Systems and Signal Processing, Vol. 19, pp. 157–174, 2005. [2] Feng, N. S., et al., Difficulties in predicting vibrations in turbomachinery with hydrodynamic bearings, ISMA International Conference on Noise and Vibration Engineering, Leuven, Belgium, pp. 3621–3630, 2006. [3] Kicinski, J. (ed.), Modeling and diagnostics of mechanics, aerodynamics and magnetic interaction in energetic turbomachinery, Publishers IFFM PAS, Gdansk, (in Polish), 2005. [4] Kicinski, J., Rotor dynamics, Publishers IFFM PAS, Gdansk, 2006. [5] Kicinski, J. and Drozdowski, R., Modeling of dynamics interaction in complicated rotor-support structure, Elaboration of IFFM PAS No. 378/1325/92, Gdansk, (in Polish), 1992. [6] Kicinski, J., et al., The nonlinear analysis of the effect of support construction properties on the dynamic properties of multi-support rotor system, J. Sound Vibration, Vol. 206(4), pp. 523–539, 1997. [7] Pronska, A. and Kabacinski, P., An examination of rotor-bearing system sensitivity to the change of support stiffness on the base of a 13 K215 high power turbine, Machine Dynamics Problems, Vol. 26, No. 1, pp. 19–34, 2002. [8] Sinha, J. K., et al., Estimating the static load on the fluid bearings of a flexible machine from run-down data, Mechanical Systems and Signal Processing, Vol. 18, pp. 1349–368, 2004. [9] Tiwari, R., et al., Identification of speed-dependent bearing parameters, J. Sound Vibration, Vol. 254(5), pp. 967–986, 2002. [10] Zywica, G., Modeling of dynamic reaction in system of rotor-bearing-supporting structure type, Machine Dynamics Problems, Vol. 31, No. 4, pp. 99–109, 2007. [11] Zywica, G., Simulation investigation of the effect of a supporting structure defect on dynamic state of the rotor supported on slide bearings, ASME IDET/CIE 2007 Conference, Las Vegas (USA), 4–7 September, 2007.

Bifurcation and Chaos of Cross-ply Laminated Plates with Piezoelectric Actuators Y. F. Zheng, L. Q. Deng and F. Wang College of Civil Engineering, Fuzhou University, Fuzhou 350108, P. R. China zheng− [email protected] Abstract This paper investigates bifurcation and chaos of cross-ply laminated plates with piezoelectric actuators under a transverse periodic excitation. On the basis of the Von Kármán plate theory and piezoelectric theory, the nonlinear governing equations for cross-ply laminated plates with piezoelectric actuators are established. The Galerkin procedure is applied to truncate the governing equations into a set of ordinary differential equations. By employing the modern methods in nonlinear dynamics, the influences of external loading amplitude, excitation voltage and actuator thickness on the bifurcation and chaos of the plates are discussed qualitatively. We are mainly concerned with tracing the bifurcation diagram and identifying the periodic motions, multiple periodic motions or chaotic motions. Keywords: Bifurcation and chaos, Piezoelectric actuator, Laminated plates, Nonlinearity

1

Introduction

The importance of piezoelectric materials has been considerably intensified by researchers and structural engineers in recent years. Due to the intrinsic direct and converse piezoelectric effects, piezoelectric materials can be effectively used as sensors or actuators for the active shape or vibration control of structures. Tzou[1] studied the dynamic behavior and control of piezoelectric laminated circular plates with an initial nonlinear large deformation. Batra[2] presented the vibration of an elastic rectangular plate forced by piezoelectric actuators under time harmonic electric voltage. Gao[3] investigated the geometrically nonlinear transient vibration response and control of plates with piezoelectric patches. Li[4] discussed the vibration of a simply supported piezoelectric laminated cylindrical shell under a hydrostatic pressure. Zhou[5] quantitatively analyzed the nonlinear electromechanics and active control of a piezoelectric laminated circular spherical shallow shell. Zheng[6] discussed the nonlinear dynamic stability of moderately thick laminated plates with piezoelectric layers. Dash[7] studied the nonlinear free vibration of the laminated composite plate with embedded and/or surface bonded piezoelectric layers in the framework of the higher order shear deformation theory. Xia and Shen[8] presented the nonlinear vibration and dynamic response of a higher order shear deformable Functionally Graded Material (FGM) plate with surface-bonded Piezoelectric Fiber Reinforced Composite actuators (PFRC) in thermal environments. Although extensive studies have been made in vibrations of laminated piezoelectric composite structures, relatively, only a few works have been devoted to study the bifurcation and chaos dynamics of laminated piezoelectric plates. Fu[9] systematically studied the bifurcation and chaos of the damaged piezoelectric plates. Zhang[10] studied periodic and chaotic dynamics of composite laminated piezoelectric rectangular plate with one-to-two internal resonance. ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011


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Y. F. ZHENG, L. Q. DENG AND F. WANG / ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011

In this present work, the bifurcation and chaos dynamics of symmetric cross-ply laminated plates with piezoelectric actuators is studied. Based on piezoelectric theory and Von Kármán plate theory, the nonlinear governing equations for cross-ply laminated plates with piezoelectric actuators are established first, followed by using Galerkin procedure to truncate the governing equations into a set of ordinary differential equations. In numerical simulation, the fourth-order Runge-Kutta algorithm is employed to numerically analyze the periodic and chaotic responses of cross-ply laminated plates with piezoelectric actuators.

2

Basic Equations

Consider an elastic symmetric cross-ply laminated rectangular plate, having length a in the x direction, width b in the y direction and thickness h in the z direction, which consists of N plies, simply supported at four edges, as shown in Fig. 1. The mid-surface of the plate contains the x, y axes and the origin of the coordinate system is taken at the upper left corner of the plate. According to the Von Kármán plate assumption, the nonlinear strain-displacement relations are as follows 1 2 εx = εx0 + zκx = u,x + w,x − zw,xx 2

Fig. 1 Schematic diagram of a piezoelectric laminated plate

1 2 εy = εy0 + zκy = v,y + w,y − zw,yy 2 0 γxy = γxy + zκxy = u,y + v,x + w,x w,y − 2zw,xy

(1)

in which u, v and w are the displacement components on the middle surface at the x, y and z directions, respectively. The comma indicates the partial derivative with respect to the coordinate variable. The constitutive relations for orthotropic elastic materials are of the form ⎤ ⎧ ⎫ ⎧ ⎫ ⎡ e e C C 0 ⎪ σ ⎪ ⎪ε ⎪ 11 12 ⎨ x⎬ ⎢ ⎥ ⎨ x⎬ e e ⎢ ⎥ 0 ⎦ σy = ⎣C12 C22 εy (2) ⎪ ⎪ ⎩σ ⎪ ⎭ ⎩γ ⎪ ⎭ e xy xy 0 0 C66 k where Cije is the reduced stiffness of the elastic materials. Assume that a pair of piezoelectric layers, with thickness hp , is uniformly distributed on the top and bottom surfaces of the elastic plate. Let the piezoelectric layers be polarized along the z direction. The constitutive equation for orthotropic piezoelectric materials can be expressed as ⎤⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎡ p p C11 C12 0 ⎪ σ ⎪ ⎪ ε ⎪ ⎪e ⎪ x ⎨ ⎬ ⎢ ⎥ ⎨ x ⎬ ⎨ 31 ⎬ p p ⎥ εy − e32 Ez σy = ⎢ (3) C C 0 22 ⎣ 12 ⎦⎪ ⎪ ⎪ ⎪ ⎪ ⎩σ ⎪ ⎭ ⎭ ⎭ ⎩ ⎩ p γxy 0 xy 0 0 C 66

BIFURCATION AND CHAOS OF CROSS-PLY LAMINATED PLATES WITH PIEZOELECTRIC ACTUATORS

307

p

where Cij is the reduced stiffness of piezoelectric materials, e31 and e32 are the piezoelectric stress constants, and Ez is the electric field intensity, which can be expressed using the applied excitation voltage V and the thickness hp of the piezoelectric actuators, that is Ez = V / hp . Under action of the transverse load q(x, y, t) and neglecting the effects of the body force, rotatory inertia and in-plane inertia, the governing equations of Von Kármán plate are expressed as follows[11, 12] Nx,x + Nxy,y = 0;

Nxy,x + Ny,y = 0

Mx,xx + 2Mxy,xy + My,yy + Nx w,xx + 2Nxy w,xy + Ny w,yy + q = (hρe + 2hp ρp )w,tt

(4)

where ρe and ρp are the mass of unit volume for the elastic plate and piezoelectric layers, respectively. Moreover, the resultants forces Nx , Ny , Nxy and moments Mx , My , Mxy are defined by ⎫ ⎧ ⎫ ⎧ ⎡ ⎤ ⎧ 0 ⎫ ⎧ p⎫ ε ⎪ ⎪Nx ⎪ ⎪ A 0 A ⎪ ⎪ σ ⎪ ⎪ ⎪ N h 11 12 ⎬ ⎨ ⎬ ⎨ x⎬ ⎨ x⎪ ⎨ x⎬ 2 +hp ⎢ ⎥ A22 0 ⎦ εy0 − Nyp Ny = σy dz = ⎣ 0 ⎪ ⎪ − h2 −hp ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ⎩σ ⎪ ⎭ ⎪ ⎪ ⎩N ⎪ ⎭ 0 0 A66 ⎩γ 0 ⎭ xy xy 0 xy ⎫ ⎧ ⎫ ⎧ ⎡ D11 ⎪ ⎬ h2 +hp ⎪ ⎨ σx ⎪ ⎬ ⎨ Mx ⎪ ⎢ My = σy zdz = ⎣ 0 ⎪ − h2 −hp ⎪ ⎭ ⎩σ ⎪ ⎭ ⎩M ⎪ 0 xy xy p

D12 D22 0

⎤⎧ ⎫ 0 ⎪ ⎬ ⎨ κx ⎪ ⎥ 0 ⎦ κy ⎪ ⎭ ⎩κ ⎪ D66 xy

(5)

p

where Nx and Ny represent the additional actuator forces induced by the electric field, and Aij , Dij respectively are the stiffness elements of the laminated piezoelectric plate, which can be denoted as p

p

Ny = 2e32 V

Nx = 2e31 V ; (Aij , Dij ) =

h 2

− h2

(1, z

2

)Cije dz

+2

h 2 +hp h 2

p

(1, z2 )Cij dz

(i, j = 1, 2, 6)

By substituting (2), (3) and (5) into (4) and introducing the following dimensionless parameters ξ = x/a; V = av/ h2 ;

η = y/b;

λ = a/b;

W = w/ h;

p2 = ρp a 4 /(t12 Eh2 ); G2 = 2e32 a 2 V /(Eh3 );

H = a/ h;

Q = qa 4 /(Eh4 );

τ = t/t1 ;

h¯ p = hp / h;

U = au/ h2

p1 = ρe a 4 /(t12 Eh2 )

G1 = 2e31 a 2 V /(Eh3 )

A¯ ij (τ ) = Aij (τ )/(Eh);

D¯ ij (τ ) = Dij (τ )/(Eh3 )

then, the dimensionless nonlinear governing equations of motions of the laminated plate with piezoelectric actuators in terms of U, V and W can be obtained as follows: A¯ 11 (U,ξ ξ + W,ξ W,ξ ξ ) + A¯ 12 (λV,ξ η + λ2 W,η W,ξ η ) + A¯ 22 (λ2 U,ηη + λV,ξ η + λ2 W,η W,ξ η + λ2 W,ξ W,ηη ) = 0

(6)

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A¯ 12 (λU,ξ η + λW,ξ W,ξ η ) + A¯ 22 (λ2 V,ηη + λ3 W,η W,ηη ) + A¯ 66 (λU,ξ η + V,ξ ξ + λW,η W,ξ ξ + λW,ξ W,ξ η ) = 0 1 1 2 A¯ 11 U,ξ + W,ξ2 + A¯ 12 λV,η + λ2 W,η W,ξ ξ 2 2

(7)

+ 2λA¯ 66 (λU,η + V,ξ + λW,ξ W,η )W,ξ η 1 1 2 W,ηη + λ2 A¯ 12 U,ξ + W,ξ2 + A¯ 22 λV,η + λ2 W,η 2 2 − D¯ 11 W,ξ ξ ξ ξ − 2λ2 (D¯ 12 + 2D¯ 66 )W,ξ ξ ηη − λ4 D¯ 22 W,ηηηη − G1 W,ξ ξ − λ2 G2 W,ηη + Q − (p1 + 2h¯ p p2 )W,τ τ = 0

3

(8)

Solution Methodology

Consider a simply-supported piezoelectric laminated plate, which can be written as ξ = 0, 1 : U = V = W = Mξ = 0 η = 0, 1 : U = V = W = Mη = 0

(9)

A solution in conjunction with upper boundary conditions is sought in the following separable form U=

∞

∞

sin(2πmξ ) sin(π nη)Umn (τ )

m=1 n=1,3,...

V =

∞

∞

sin(π mξ ) sin(2π nη)Vmn (τ )

m=1,3,... n=1

W =

∞

∞

sin(π mξ ) sin(π nη)Wmn (τ )

(10)

m=1,3,... n=1,3,...

The transverse period load is assumed to be of the form Q = F (τ ) sin(π ξ ) sin(π η);

F (τ ) = F0 sin(ωτ )

(11)

in which F0 is the external loading amplitude and ω is the external exciting frequency. Substituting (10–11) into (6–8), and making use of the one-term approximation of the Galerkin method, we can obtain the nonlinear equations in terms of U, V and W . For simplifying calculations, the functions U, V can be expressed in terms of second powers of W . Moreover, introducing the damping ratio μ, then, the resulting equations can be transformed into the function of W .


4

309

Numerical Results

In the following research, the fourth-order Runge-Kutta algorithm is employed to numerically analyze the periodic and chaotic responses of the simply supported symmetric cross-ply laminated rectangular plate with piezoelectric actuators. The elastic material is taken as graphite-epoxy and the piezoelectric material is taken as PZT5. The material properties are taken as[13] Graphite-epoxy: EL = 132.4 GPa; PZT5:

EL = ET = 62 GPa;

ET = 10.8 GPa;

GLT = 23.6 GPa;

GLT = 5.5 GPa;

νLT = 0.31;

νLT = 0.24

e31 = e32 = −19.77 C/m2

and the other parameters are taken as λ = 1;

H = 100;

μ = 0.005;

ω = 0.5;

Figure 2 illustrates the bifurcation diagram of the cross-ply laminated plate with piezoelectric actuators by using the external loading amplitude F0 as the control parameter. We are mainly concerned with tracing the bifurcation diagram and identifying the periodic motions, multiple periodic motions or chaotic motions by Poincare map. From Fig. 2, it can be observed that the motion state of the system is one periodic motion or multiple periodic motion before about F0 < 300. When F0 = 301, the system firstly enters the chaotic motion. In Fig. 3, the Poincare maps for the system are shown for the different loading amplitude. Figure 3(a) indicates the existence of a periodic 3 motion for the cross-ply

p1 = 50;

p2 = 250

Fig. 2 Bifurcation diagram for external loading amplitude F0 (h¯ p = 0.1, V = 100)

Fig. 3 Poincare maps for the different loading amplitude: (a) F0 = 160, (b) F0 = 185, (c) F0 = 301

310


laminated plate with piezoelectric actuators when the external loading amplitude F0 = 160. When F0 = 185 (Fig. 3(b)), the system bifurcates into a period 6 response. The Poincare map has six points. When the external loading amplitude changes to F0 = 301, the chaotic motion of the cross-ply laminated plate with piezoelectric actuators is observed, as shown in Fig. 3(c). Figure 4 shows the bifurcation diagram of the cross-ply laminated plate with piezoelectric actuators by using the excitation voltage V as the control parameter. From Fig. 4, it is observed that the motions of the cross-ply laminated plate with piezoelectric actuators change from the chaotic motion to the multiple periodic motion, and then from the multiple periodic motion to the one

Fig. 4 Bifurcation diagram for control voltage V (h¯ p = 0.1, F0 = 300)

Fig. 5 Poincare maps for the different excitation voltage: (a) V = 30, (b) V = 197, (c) V = 214

periodic motion with the increase of the excitation voltage. In Fig. 5, the Poincare maps for the system are shown for the different excitation voltage. When V = 30 (Fig. 5(a)), the system exhibits the chaotic motion. When V = 197 (Fig. 5(b)), the system exhibits the periodic 5 motion. When the excitation voltage changes to V = 214 (Fig. 5(c)), the system enters the one periodic motion. Hence, increasing the excitation voltage is advantageous to the vibration stability of the structures. Figure 6 gives the bifurcation diagram of the cross-ply laminated plate with piezoelectric actuators by using the actuator thickness h¯ p as the control parameter. From Fig. 6, it is observed that the motions of the cross-ply laminated plate with piezoelectric actuators translates from a chaotic

Fig. 6 Bifurcation diagram for actuator thickness h¯ p (V = 100, F0 = 630)


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Fig. 7 Poincare maps for the different actuator thickness: (a) h¯ p = 0.05, (b) h¯ p = 0.063, (c) h¯ p = 0.066

motion to a double periodic bifurcation, and then from the periodic motion to the chaotic motion with the increase of the actuator thickness. In Fig. 7, the Poincare maps for the system are shown for the different actuator thickness. When h¯ p = 0.05 (Fig. 7(a)), there occurs a crowd of points, which shows the system enters chaotic motion. When the actuator thickness changes to h¯ p = 0.063 (Fig. 7(b)), the crowd of points bifurcates into two points, and then bifurcates into one point at h¯ p = 0.066, as shown in Fig. 7(c).

5

Conclusion

This paper investigates the effects of external loading amplitude, excitation voltage and actuator thickness on the bifurcation and chaos behavior of the cross-ply laminated plates with piezoelectric actuators under a transverse periodic excitation. Numerical simulation is applied to investigate the periodic motion, multiple periodic motion and chaotic motion of the cross-ply laminated plates with piezoelectric actuators. Numerical results show that the cross-ply laminated plates with piezoelectric actuators exhibit complex motions. And we can control the nonlinear dynamic responses of the cross-ply laminated plates with piezoelectric actuators from the chaotic motion to the periodic motion by properly changing the external loading amplitude, excitation voltage and actuator thickness, etc.

Acknowledgements The authors would like to thank for the support from the Young Technological Talents’ Innovation Fund of Fujian Province (2006F3077) and Scientific and Technical Development Fund of Fuzhou University (2007-XQ-20).

References [1] Tzou, H. S. and Zhou, Y. H., Dynamics and control of nonlinear circular plates with piezoelectric actuators, J. Sound Vibration, Vol. 188, pp. 189–207, 1995.

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[2] Batra, R. C., Liang, X. Q. and Yang, J. S., The vibration of a simply supported rectangular elastic plate due to piezoelectric actuators, International Journal of Solids and Structures, Vol. 33, pp. 1597–1618, 1996. [3] Gao, J. X. and Shen, Y. P., Active control of geometrically nonlinear transient vibration of composite plates with piezoelectric actuators, J. Sound Vibration, Vol. 264, pp. 911–928, 2003. [4] Li, H. Y., Lin, Q. L. and Liu, Z. X., Free vibration of piezoelastic laminated cylindrical shells under hydrostatic pressure, International Journal of Solids and Structures, Vol. 38, pp. 7571–7585, 2001. [5] Zhou, Y. H. and Tzou, H. S., Active control of nonlinear piezoelectric circular shallow spherical shells, International Journal of Solids and Structures, Vol. 37, pp. 1663–1677, 2000. [6] Zheng, Y. F., Wang, F. and Fu, Y. M., Nonlinear dynamic stability analysis for moderately thick laminated plates with piezoelectric layers, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 10, pp. 459–468, 2009. [7] Dash, P. and Singh, B. N., Nonlinear free vibration of piezoelectric laminated composite plate, Finite Elem. Anal. Des., Vol. 45, pp. 686–694, 2009. [8] Xia, X. K. and Shen, H. S., Nonlinear vibration and dynamic response of FGM plates with piezoelectric fiber reinforced composite actuators, Composite Structures, Vol. 90, pp. 254–262, 2009. [9] Fu, Y. M. and Wang, X. Q., Analysis of bifurcation and chaos of the piezoelectric plate including damage effects, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 9, pp. 61–74, 2008. [10] Zhang, W., Yao, Z. G. and Yao, M. H., Periodic and chaotic dynamics of composite laminated piezoelectric rectangular plate with one-to-two internal resonance, Science in China Series E-Technological Sciences, Vol. 52, pp. 731–742, 2009. [11] Zheng, Y. F. and Fu, Y. M., Nonlinear dynamic analysis of viscoelastic/damage behavior for symmetric cross-ply laminated plates, Advances in Vibration Engineering, Vol. 3, pp. 185–197, 2004. [12] Zheng, Y. F., Qi, A. and Fu, Y. M., Postbuckling analysis of viscoelastic moderately thick laminated cylindrical panels with damage evolution, Advances in Vibration Engineering, Vol. 8, pp. 27–39, 2009. [13] Varelis, D. and Saravanos, D. A., Mechanics and finite element for nonlinear response of active laminated piezoelectric plates, AIAA Journal, Vol. 42, pp. 1227–1235, 2004.

Rotor Dynamic Modeling of High Speed Flexible Coupling S. Ganesan and C. Padmanabhan Machine Design Section, Department of Mechanical Engineering, Indian Institute of Technology, Madras, Chennai, India [email protected] Abstract In this paper, the influence of rotational speed on the critical speeds of a hollow shaft, with a flexible diaphragm on one end, is investigated using a finite element model. Including a complete threedimensional model of the coupling in the rotor system will be computationally expensive. Hence, an alternate approach, where the coupling is replaced by a speed dependent 6 × 6 stiffness matrix, obtained from a quasi-static analysis of a detailed finite element model of the coupling alone, is proposed. A solid model of the coupling has been generated and a quasi-static analysis including the effect of angular velocity has been done using ABAQUS. To verify the results of the finite element rotor model with centrifugal stiffening, a theoretical estimation of the fundamental critical speed using stiffness and mass parameters with modal mass participation has been carried out. The estimated critical speed matches reasonably well with the finite element centrifugal stiffening model. A limited experimental verification for the axial stiffness value computed from the model has also been performed. Since considerable computational effort is required for generating all the elements of the stiffness matrix of the flexible coupling, the critical speed variation based only on the direct stiffness terms (diagonal terms) of the stiffness matrix is compared to that with the full stiffness matrix (6 × 6) of the coupling. The results indicate that only the second set of critical speeds (only 3 speeds below 50,000 rpm) is affected by the presence of off-diagonal terms. This is due to the bending rotations being significant for the second mode and their significant effect on off-diagonal terms. Analysis performed on the flexible coupling rotor model clearly indicates that the bending critical speeds of the model vary with shaft speed due to centrifugal stiffening and gyroscopic effects. When compared to the gyroscopic effect, centrifugal stiffening is dominant at higher speeds. For speeds up to 20,000 rpm, the increase in the first critical of the flexible rotor is about 15%. Keywords: Flexible coupling rotor, Finite element rotor model, Direct stiffness terms, Cross coupling terms, Centrifugal stiffening

1

Introduction

Misalignment in mechanical systems occurs due to various operating conditions such as thermal expansion of mechanical parts, sinkage and relative movement of mounting beds. Couplings used permit misalignments while transmitting the power from one mechanical system to another. There are many types of couplings used in practice such as universal joint coupling, gear coupling, flector type coupling, diaphragm type disk coupling. Each one of the couplings has merits and demerits for an application. ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011


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S. GANESAN AND C. PADMANABHAN / ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011

For high speed application with large misalignment, diaphragm type disk couplings are preferred since they are constant velocity couplings. A flexible diaphragm coupling rotor designed to permit large misalignment is used to connect the driver and the driven member. The dynamics of the coupling rotor-system needs to be accurately predicted for its performance and reliability. Since the operating speed is very high, the system may have to operate beyond its first critical speed. Then the influence of angular velocity on the modal properties of the rotor has to be understood clearly. The influence of rotational speed on the critical speeds of the flexible rotor is the main focus of the study. To carry out the objective, a time invariant model based on Finite Element (FE) method is developed. Since including a complete three-dimensional model of the coupling, in the rotor system, will be computationally expensive, an alternate approach where the coupling is replaced by a speed dependent 6 × 6 stiffness matrix, obtained from a quasi-static analysis of a detailed finite element model of the coupling alone, is proposed. A solid model of the coupling has been generated and a quasi-static analysis including the effect of angular velocity has been done using ABAQUS. The estimated stiffness in all six Degrees-Of-Freedom (DOF) has been used as a stiffness matrix for the coupling in the time invariant rotor model. To verify the results of the finite element rotor model with centrifugal stiffening, a theoretical estimation of fundamental critical speed using stiffness and mass parameter with modal mass participation has been carried out. The estimated critical speed matches reasonably well with the FEM centrifugal stiffening model. A limited experimental verification of only the axial stiffness value computed from the FEM model has also been performed. A FEM rotor model with 2D beam element has been derived by Nelson and McVaugh[1] to find the influence of gyroscopic couple on a simply supported disk. The kinetic energy, potential energy expressions were derived and equations of motion obtained using Lagrange’s equation. They have clearly shown the effect of gyroscopic couple in splitting the critical speeds. They also showed natural mode variation with speed due to the gyroscopic couple. Modal controllability and observability of a bladed disk and their dependency on the angular velocity was studied by Christensen and Santos[2] . They showed the parametric vibration mode and demonstrated that the blade natural frequencies change depending on the angular velocity due to centrifugal stiffening. Xi[3] has carried out modeling and analysis for active control of circular saw vibrations used in wooden saw mills. In his study, influence of the rotation speed on the natural frequencies was considered. The dynamic response of a spinning Compact Disc (CD) disk studied when the axis of the disk is misaligned with the center of rotation by Heo et al.[4] In their study, effect of rotation speed on the in-plane mode was reported. The dynamic analysis on a rotating rigid-flexible coupled smart structure with a tip mass was done by Huang et al.[5] The geometrical nonlinear effects of axial and transverse displacement of the structure with a tip mass were analysed considering the centrifugal stiffening effect. Frequencies and modes of the rotating flexible bladed disc-shaft assemblies have been studied by Jacquet-Richardet et al.[6] In thier study, the non-rotating mode shapes of flexible bladed disc-shaft assemblies are calculated by a finite element cyclic symmetry approach. Rotational effects, such as centrifugal stiffening and gyroscopic effects are accounted and non-rotating mode shapes are used in a modal analysis method for evaluating the dynamic characteristics (frequencies and mode shapes) of the corresponding rotating systems. The natural frequencies and modes of the longitudinal coupled vibrations in a flexible shaft with multiple flexible disks is investigated by Jial et al.[7] using a substructure synthesis technique. Their rotor model consists of multiple flexible disks attached to a flexible shaft, as used in steam turbines or computer storage devices and taking into account, disk flexibility and centrifugal stiffening effects. Global modeling approaches for dynamic analyses of rotating assemblies of turbo machines are proposed by Eric chatelet et al.[8] The model proposed by

ROTOR DYNAMIC MODELING OF HIGH SPEED FLEXIBLE COUPLING

315

them included centrifugal effect, is applied to a thin-walled composite shaft and to a turbo molecular pump rotating assembly. An effect of centrifugal stiffening on the vibration frequencies of a constrained flexible arm has been studied by Fung and Yau[9] . A clamped-free rotating flexible robotic arm, modeled by the Euler–Bernoulli beam theory, rotates horizontally about the clamped axis and its equation of motion is derived using the Hamilton’s principle and studied under the influence of centrifugal stiffening. Flexible motion of a uniform Euler–Bernoulli beam attached to a rotating rigid hub is investigated by Yang et al.[10] Fully coupled nonlinear differential equations, describing axial, transverse and rotational motions of the beam, are derived by using the extended Hamilton’s principle including the centrifugal stiffening. The dynamic behavior of rotating beams with piezoceramic actuation is studied using bending and shear actuation for application to structures such as helicopter and wind turbine rotor blades by Thakkar and Ganguli[11] . The results show that the centrifugal stiffening effect reduces the tip transverse bending deflection and elastic twist obtained from smart actuation as the rotation speed increases. From the literature review, it is clear that the effect of rotational speed on modal characteristics of many mechanical systems like bladed disk, saw, CD disk and robotic arm have been analysed. These systems have a disk like structure and operate in a single plane that can be modeled without complexity in FEM. Unlike the systems investigated and reported in the literature, the flexible coupling-rotor has four circular plates which operate in multiple planes. This influences the stiffness of the system and estimating it is quite complex. To carry out this exercise, a detailed three-dimensional finite element model of the flexible coupling is developed using ABAQUS. Using this model a quasi-static analysis, including the rotation speed effect, has been carried out. From this analysis, a speed dependent, 6 × 6 stiffness matrix is generated, which is used in a simpler finite element model of the rest of the rotor system. 1.1 Rotor-coupling system A rigid hollow shaft with a flexible diaphragm coupling as a rotor system is shown in Fig. 1. The rigid shaft connected by a flexible diaphragm coupling forms a rotor system, capable of handling large misalignments. Since the coupling is made out of four thin circular plates, it can Fig. 1 Flexible coupling rotor schematic deflect to provide flexibility. For a given deflection, four circular plates share the deflection as series springs, which make the coupling more flexible. Unlike a rigid coupling, the diaphragm coupling provides enough flexibility to take large misalignment with less reaction force imposed on the supporting structure/bearing pedestal. When it is misaligned, it deflects in such a way that the planes of these circular plates are normal to the torque axes. Thereby it achieves constant velocity in output irrespective of misalignment.

2

Finite Element Model with Centrifugal Stiffening Effect

2.1 Finite shaft element of rotor The entire rotor-coupling system is modeled using FEM. In this section, a brief description of the model for the rotor based on Nelson and McVaugh[1] is presented. To include axial coupling and torsional

316


modes, the finite element model has been derived including transverse bending, axial and torsion effects, as an extension of the Nelson and McVaugh[1] rotor model. Axial and torsional terms are introduced into the mass and stiffness matrices at the appropriate places. In deriving the bending element, the shaft is assumed to be elastic and flexible and the disk rigid. The nodal variables are u, v which are bending displacements in the x and y directions, while φx and φy are the rotations in bending. The rotor shaft model is shown in Fig. 1. The mass matrices are given by u x 1 [MT ] = μ[ψ]T [ψ]ds; [MR ] = = []{q} = [ψ] {q} ; IP []T []ds; v 2 y (1) where ψ and are standard bending shape functions. The gyroscopic matrices are obtained as [gs ] = 1/2 ω([n] − [n]T )

IP [x ]T [y ]ds;

[n] = [na ] − [nb ];

[na ] =

x = [−dv/ds]x ;

y = [du/ds]y

[nb ] =

(2) IP [y ]T [x ]ds

(3) (4)

The stiffness matrix is given by (with representing the second order spatial derivative) [Kb ] =

l

EI [ψ ]T [ψ ]ds

(5)

0

where μ is mass of the shaft per unit length, IP is polar moment of inertia of shaft element, E is Young’s modulus, I is area moment of inertia and μ is rotational speed. The axial and torsional stiffness terms are obtained by [Ka ] = EA

0

l

T

N N ds;

[KT ] = CIp

l

N T N ds

(6)

0

where, C is the modulus of rigidity, A is the cross-sectional area and [N ] is the standard axial/torsional shape function matrix. The element mass matrices associated with axial and torsion are given below 2 1 2 1 [Ma ] = ρAs ; [MTor ] = ρAs (7) 1 2 1 2 The equations of motion representing all DOF are ⎡ s+d ⎤⎧ ⎫ ⎡ m 0 0 ⎪ 0 g s+d 0 ⎨x¨ ⎪ ⎬ ⎢ ⎥ ⎢ ms+d 0 ⎦ y¨ + ⎣−g s+d 0 ⎣ 0 ⎪ ⎩ z¨ ⎪ ⎭ a 0 0 m 0 0

⎤⎧ ⎫ ⎡ s 0 ⎪ k ⎨x˙ ⎪ ⎬ ⎥ ⎢ 0⎦ y˙ + ⎣ 0 ⎪ ⎩ z˙ ⎪ ⎭ 0 0

0 ks 0

⎤⎧ ⎫ ⎧ ⎫ ⎪ ⎬ ⎨x ⎪ ⎬ ⎪ ⎨fx ⎪ ⎥ 0 ⎦ y = fy ⎪ ⎭ ⎩z⎪ ⎭ ⎪ ⎩f ⎪ ka z (8) 0

where ms+d represents sum of mass matrices of shaft and disk in transverse bending, g s+d represents sum of gyroscopic matrices of shaft and disk in transverse bending, k s represents shaft stiffness in


317

bending direction in x and y, ma represents mass matrices in axial and torsional direction and k a represents stiffness of shaft in axial and torsional direction. In the above equation, the vectors x, y and z represent x i = {ui1 , iy1 , ui2 , iy2 }T ,

y i = {v1i , ix1 , v2i , ix2 }T

and

zi = {w1i , iz1 , w2i , iz2 }T

The equations of motion with all DOFs is given by ¨ + [G]{Q} ˙ + [K]{Q} [M]{Q}

(9)

where {Q} represent the total system DOFs in the model. 2.2 Solution by state space method The equation above is written in state space form where second order differential equations are converted to first order differential equations. ˙ {Q} [0] [M] [−M] [0] {0} ˙ {h} + {h} = {H }; {h} = ; {H } = (10) [M] [G] [0] [K] {F } {Q} For a free vibration problem, a solution of the form {h} = {h0 }eλt is assumed and substituted into the homogenous equation which reduces it to the following eigenvalue problem [0] [I ] 1 (11) {h0 } = {h0 } −1 −1 λ −[K] [M] −[K] [G] Solving the above matrix equation as a eigenvalue problem, the natural frequencies and vibration modes are obtained. Eigenvalues and eigenvector obtained are complex conjugate pairs. The forced response can be obtained using modal superposition.

3

Characterisation of Flexible Coupling

The flexible diaphragm coupling stiffness characteristics are required to understand the behavior of the rotor system. Stiffness of the coupling and its dependence on the angular velocity is to be determined. The flexible diaphragm coupling shown in Fig. 2, can handle large axial misalignment and lateral misalignments. In the axial direction (z), flexible rotor system can expand or compress by 2.5 mm. In the lateral bending directions, it can take angular misalignment of 1.5 or a parallel offset up to 1.5 mm. The diaphragm coupling is made out of four circular plates whose thickness varies from 1 mm at the inner radius to 0.5 at the outer radius. The thin circular plate is supported at the inner and outer radius by the hub and rim which are comparatively thicker. Four circular plates are welded at the rim and hub alternately to make it integral form as a diaphragm coupling. When the rotor is subjected to axial misalignment, either expansion or compression, circular plates in the flexible coupling are subjected to symmetric bending similar to load acting uniformly at the inner radius constraining outer radius or vice versa. But during angular misalignment/parallel offset, circular plates are subjected to asymmetric bending similar to point load acting on the circular plate. Four diaphragms share the deformation when the coupling

318


Fig. 2

Flexible diaphragm coupling

Fig. 3

Discretised coupling model

is subjected to misalignment, allowing the diaphragm coupling to be very flexible. When diaphragm coupling is subjected to rotation, rim at the outer radius of the coupling is subjected to centrifugal force which pulls the circular plates in a radially outward direction. The outward radial force due to rotation stiffens the diaphragm coupling in axial and transverse direction, which affects the critical speed of the rotor. Since centrifugal force varies as the square of the angular velocity of the rotor, it is essential to study its influence for coupling that operates at high speeds. To obtain the stiffness of the coupling, as proposed earlier, a finite element model is generated in the commercial software ABAQUS[12] . The model representing the diaphragm coupling is meshed using quadrilateral shell elements with appropriate aspect ratio, as shown in Fig. 3. Since a shell element is used to model the geometry, a single layer is used to represent the thickness of the thin circular plates. If 3D brick elements are to be used, multiple layers in thickness are required to discretize the geometry to capture bending of the circular plate. A four-noded shell element with six DOF (u, v, w, φx , φy , φz ) at each node can represent the entire thickness as one layer. There are 6266 elements and 6600 of nodes representing the geometry. The total number of DOF in the model is 39564. Mesh refinement has been done to ensure the convergence of the results. The h-type mesh optimization starting from the finer mesh to the present level has been carried out without affecting the convergence to represent the geometry. Nonlinear analysis has been performed to estimate the stiffness of the coupling. At both ends, rigid links are used to connect the nodes at the hub to the center of the diaphragm, as shown in Fig. 3. These two center points of the model have been used to generate the stiffness matrix. A static displacement to one of the DOF, say u, of one center node is given. The other center node is fully constrained and the reaction forces generated in all the constrained DOF are obtained. This set forms one column of the required (6 × 6) stiffness matrix. Similarly by providing the other displacements, (v, w, φx , φy , φz ), one at a time, the six reactions for each displacement are obtained. The reaction forces, thus obtained for all the six DOF displacements, forms the stiffness matrix (6 × 6) for the coupling at the static condition or for zero speed. To include the effect of centrifugal stiffening, a rotational effect is introduced into the ABAQUS model of the coupling. The same exercise of displacing the center node in various directions, (u, v, w, φx , φy , φz ), one at a time, and obtaining the reactions at the other


319

constrained center node, leads to the stiffness matrix of the coupling for non-zero speeds ranging from 2000–100000 rpm. Some of the typical stiffness values obtained such as direct stiffness terms (diagonal element) of the stiffness matrix are plotted in Fig. 4(a to c) and cross coupling stiffness terms (off diagonal element) of stiffness matrix are plotted for various speeds in Fig. 4(d to g). Other stiffness terms exhibiting similar trend or not showing much significant stiffening effect are not plotted. Notations used to represent stiffness term, the first suffix term represents the imposed displacement direction and second suffix term represents reaction force direction. For example, stiffness term kxx represents that first suffix x shows that direction in which displacement is imposed and second suffix x shows the reactions force in that direction.

Fig. 4 Continued

320


Fig. 4 Stiffness matrix elements variation with speed

4

Centrifugal Stiffening Model of Coupling-Rotor

Coupling rotor has been discretized into six two-noded rotor beam elements as shown in Fig. 1. The entire coupling rotor is discretized to have seven nodes with six degrees-of-freedom at each node. The coupling is represented by an element between the second and third nodes. The total number of DOFs in the coupling rotor is 42. The sectional properties such as internal diameter, external diameter, length and material properties like Young’s modulus, density are input to derive the stiffness matrix and mass matrix of each element except for the element that represents the coupling. Based on the speed and sectional properties, the gyroscopic matrix has been derived for each element. The stiffness of the coupling is obtained from the model generated in ABAQUS software for a particular speed. All flanges are represented as a disk at the appropriate nodes including balancing collar provided in the rigid shaft of the coupling rotor. Mass of the coupling is represented as a single disk with equivalent


321

thickness instead of four circular plates and its mass is lumped at the nodes of the coupling element. After obtaining individual elemental mass matrices, gyroscopic matrices and stiffness matrices are assembled to represent the entire coupling rotor. Equations of motion obtained with above matrices are shown below: ¨ + [G]{Q} ˙ + [Kω ]{Q} = {F } [M]{Q}

(12)

In the above equation, stiffness matrix and gyroscopic matrix are dependent on angular velocity of the rotor. The gyroscopic matrix varies with angular velocity of the rotor and the stiffness matrix of the coupling element is influenced by the centrifugal stiffening which varies as the square of angular velocity. Since these matrices are dependent on speed, critical speeds of the rotor will change with speed. To find the critical speeds of the coupling rotor, a state space approach is followed from which the eigenvalues are extracted. The coupling-rotor system’s mass matrix, gyroscopic matrix and stiffness matrix are assembled to have a state space form as mentioned in the section 2.2. After the assembly of the matrix in state space form, the size of the system matrix becomes double (84 × 84). By constraining the DOF at either end of the coupling rotor to simulate simply supported boundary conditions, an eigenvalue analysis is performed to get the eigenvalues and the eigenvectors at that particular shaft speed. For different speeds, the gyroscopic and stiffness matrices are computed and the eigenvalue analysis is repeated to obtain the new eigenvalues and corresponding eigenvectors. Only three bending critical speeds are considered in this study since the higher critical speeds are above 50,000 rpm (the maximum speed of interest). 4.1 Verification of centrifugal stiffening model To verify the critical speeds found by the finite element rotor model, an estimation of the fundamental frequency, based on an analytical method, using stiffness and mass parameters of the coupling rotor, has been carried out. In the coupling-rotor, the flexible coupling has lower stiffness compared to the rigid tube of the shaft and hence it controls the lowest frequency. The rigid tube contributes more to the inertia properties of the rotor. The mass of the coupling is 0.2 kg and accounting for one half of the rigid tube and adaptor to contribute to the disk mass, the total mass of the disk is 1.12 kg. The stiffness of the coupling corresponding to zero rpm is 4.853e6 N/m as seen from Fig. 4(c). To check the modal participation factor, the mode shape of the first mode (obtained from the eigenvalue analysis), corresponding to u and v DOF, is shown in Fig. 5. From the Fig., the modal participation factor can be

Fig. 5 Mode shape of displacement u and v (1st mode)

322


obtained (0.9 in this case). The critical speed of the rotor is, then simply calculated from a SDOF system based on coupling stiffness and rigid tube mass; its value is 2194.2 rad/s. The finite element rotor model for centrifugal stiffening has predicted the critical speed for static condition (zero rpm) as 2194.6 rad/s. These two values agree well, indicating consistency of the ABAQUS model. The stiffness of the coupling obtained from ABAQUS is validated in a limited way. An experiment has been conducted to find the axial stiffness (Z direction). The experimental setup, shown in Fig. 6, has one fixed mounting bracket and one moving bracket. The assembly of the diaphragm coupling is mounted between these brackets and a unit displacement is given to find the resistance. The load sensor mounted in line with displacement measures the load due to the coupling axial displacement. The stiffness of the coupling is obtained Fig. 6 Static thrust test rig and compared with the results obtained from the quasi-static analysis model in ABAQUS, which is specified in Fig. 4c against stiffness in Z direction for static condition (zero rpm). While the finite element model predicts the axial stiffness to be 112.8 N/mm, the experimental value is about 15% higher at 130.2 N/mm. The four thin circular plates form as a diaphragm coupling are very sensitive to small changes in thickness. Rigidity of a plate varies cubic to thickness. Due to machining inaccuracies (tolerance) of the plates in the coupling, experiment results are slightly higher leading to an increase in stiffness of the coupling experimented. 4.2 Results and discussion Initially, a study has been done by including only direct stiffness terms (diagonal terms) of the stiffness matrix of the coupling element to find the influence of angular speed on the critical speed. The results obtained are shown in Fig. 7. The eigen analysis has been repeated by considering direct stiffness element (diagonal terms) and cross coupling terms (off diagonal terms) of the stiffness matrix (6 × 6) of the coupling. The results obtained are shown in Fig. 8. A comparison suggests that out of the three modes of the critical speeds within 50000 rpm, only one mode is affected. The first and third modes are unaffected by the cross coupling terms of the stiffness matrix, while the second mode changes with the presence of the coupling terms. This is probably due to the rotational DOF which influences the off diagonal terms of the matrix whereas translational DOF of bending do not depend on the cross coupling terms. Using only the diagonal terms of the stiffness matrix, eigen analysis can be performed without much loss of computational effort and time for the above coupling-rotor problem at reasonable accuracy for finding the fundamental mode. If sone is interested in getting the complete modal characteristic of the coupling-rotor, it is better to carry out eigen analysis using the full stiffness matrix. Eigenvectors obtained with direct stiffness terms indicates clearly the mode. But with direct and off diagonal terms of the stiffness matrix (6 × 6), clarity is lost. A convergence study has been done by increasing the numbers of nodes of the coupling rotor. The higher the numbers of nodes to represent the coupling rotor, number of critical speeds obtained are higher. By increasing the number of nodes, the rigid tube is discretised with more nodes and the coupling representation is unaffected. By increasing the nodes, the number of frequencies obtained is of higher modes. But the lower modes of our interest, the first three modes which are up to 50000 rpm, are dependent on the coupling stiffness. The results are unaffected because of entire diaphragm coupling is


Fig. 7

Critical speed variation with speed neglecting off diagonal term

Fig. 8

323

Critical speed variation with speed with 6 × 6 stiffness matrix

represented between nodes 2nd and 3rd and discretized as a one element and convergence study does not influence the critical speed of the first few modes. In general, critical speeds in x and y direction at static condition of the rotor (zero speed) is same. But the gyroscopic couple splits the critical speeds and one of the critical speeds keeps increasing and other critical speed keeps decreasing. Compared to the critical speed at zero shaft speed, one critical speed will be always more and other will be always less. But it is observed from the plot that the split critical speeds keep increasing due to gyroscopic couple and centrifugal stiffening. Their values are always greater than the critical speed corresponding to static condition. When compared to the gyroscopic effect, the centrifugal stiffening is predominant at higher speed. Lower modes of bending participate in the centrifugal stiffening effect in a greater way when compared with higher modes of bending. This is because the lower modes of bending are influenced by the coupling element while higher bending modes correspond to the rigid hollow shaft. In the speed range up to 20,000 rpm, the increase in first critical of the flexible rotor is 15%. The gyroscopic effect is not significant since the mass of the flexible coupling is small.

5

Conclusions

In this paper, the influence of rotational speed on the critical speeds of a hollow shaft, with a flexible diaphragm on one end, is investigated using a finite element model. Including a complete threedimensional model of the coupling in the rotor system will be computationally expensive. Hence, an alternate approach, where the coupling is replaced by a speed dependent 6 × 6 stiffness matrix, obtained from a quasi-static analysis of a detailed finite element model of the coupling alone, is carried out A solid model of the coupling has been generated and a quasi-static analysis including the effect of angular velocity has been done using ABAQUS. To verify the results of the finite element rotor model with centrifugal stiffening, a theoretical estimation of the fundamental critical speed using stiffness and mass parameters with modal mass participation has been carried out. The estimated critical speed matches reasonably well with the finite element centrifugal stiffening model. A limited experimental verification for the axial stiffness value computed from the model has also been performed.

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Since considerable computational effort is required for generating all the elements of the stiffness matrix of the flexible coupling, the critical speed variation based only on the direct stiffness terms (diagonal terms) of the stiffness matrix is compared to that with the full stiffness matrix (6 × 6) of the coupling. The results indicate that only the second set of critical speeds (only 3 speeds below 50,000 rpm) is affected by the presence of off-diagonal terms. This is due to the bending rotations being significant for the second mode and their significant effect on off-diagonal terms. Using only the diagonal terms of the stiffness matrix, eigen analysis can be performed without much loss of computational effort and time for the above coupling-rotor problem at reasonable accuracy for finding the fundamental mode. If one is interested in getting the complete modal characteristic of the coupling-rotor, it is better to carry out eigen analysis using the full stiffness matrix. Analysis performed on the flexible coupling rotor model clearly indicates that the bending critical speeds of the model vary with shaft speed due to centrifugal stiffening and gyroscopic effects. When compared to the gyroscopic effect, centrifugal stiffening is dominant at higher speeds. For speeds up to 20,000 rpm, the increase in the first critical of the flexible rotor is about 15%.

References [1] Nelson, H. D. and McVaugh, J. M., The dynamics of rotor-bearing systems using finite elements, American Society of Mechanical Engineers Journal of Engineering for Industry, Vol. 98(2), pp. 593–600, 1976. [2] Rene H. Christensen and Ilmar F. Santos, Modal controllability and observability of bladed disks and their dependency on the angular velocity, J. Vib. Control, Vol. 11, p. 801, 2005. [3] Fengfeng, XI., Aochun george wang, XI. and Zhong qin, Modeling and analysis for active control of circular saw vibrations, J. Vib. Control, Vol. 6, p. 1225, 2000. [4] Jin Wook Heo, Jintai Chiung and Keeyoung Choi, Dynamic time response of a flexible spinning disk misaligned with the axis of rotation, J. Sound Vibration, Vol. 262, pp. 25–44, 2003. [5] Huang Yong-an, Deng Zi-chen and Yao Lin-xiao, Dynamic analysis of a rotating rigid-flexible coupled smart structure with large deformations, Appl. Math. Mech. (English Edition), Vol. 28(10), pp. 1349–1360, 2007. [6] Jacquet-Richardet, G., Ferraris, G. and Rieutord, P., Frequencies and modes of rotating flexible bladed disc-shaft assemblies: A global cyclic symmetry approach, J. Sound Vibration, Vol. 191, Issue 5, 18 April, pp. 901–915, 1996. [7] Jia, H. S., Chun, S. B. and Lee, C. W., Evaluation of the longitudinal coupled vibration in rotating, Flexible disks/spindle systems, J. Sound Vibration, Vol. 208, Issue 2, 27 November, pp. 175–187, 1997. [8] Eric Chatelet, Flavio, D., Ambrosio and Georges Jacquet-Richardet, Toward global modeling approaches for dynamic analyses of rotating assemblies of turbomachines, J. Sound Vibration, Vol. 274, Issues 3–5, 22 July, pp. 863–875, 2004. [9] Fung, E. H. K. and Yau, D. T. W., Effects of centrifugal stiffening on the vibration frequencies of a constrained flexible arm, J. Sound Vibration, Vol. 224, Issue 5, 29 July, pp. 809–841, 1999. [10] Yang, J. B., Jiang, L. J. and Chen, D. CH., Dynamic modeling and control of a rotating Euler–Bernoulli beam, J. Sound Vibration, Vol. 274, Issues 3–5, 22 July, pp. 863–875, 2004. [11] Dipali Thakkar and Ranjan Ganguli, Dynamic response of rotating beams with piezoceramic actuation, J. Sound Vibration, Vol. 270, Issues 4–5, 5 March, pp. 729–753, 2004. [12] ABAQUS version 6.6, user manual, ABAQUS Inc. Dassault systems.

The Effect of Change in the Number of Stator Blades in the Stage on Unsteady Rotor Blade Forces ´ Romuald Rzadkowski and Marek Solinski Institute of Fluid-Flow Machinery, Polish Academy of Sciences [email protected]; [email protected] Abstract Analyzed here was a 100-MW steam turbine in which the unsteady forces acting on the extraction stage rotor blades were found to be large. This was in particular valid for the circumferential components unsteady forces. Results of calculations performed for the same boundary conditions but with different numbers of stator blades in the system are presented. Increasing the number of stator blades considerably affects the level of unsteady forces acting on a single blade. These changes refer to both the stationary components of these forces and their first high-frequency harmonic, which decreases when the number of stator blades is increased. Keywords: Unsteady forces, Rotor blade, Stator blade

1

Introduction

The number of stator blades affects the level of unsteady forces acting on rotor blades[1, 2] . Sokolovskij and Gnesin[1] have shown for the simplified geometry of a stator and rotor blades the minimum value of unsteady forces acting on the rotor blades occurs when the number of stator blades (zs ) is equal to the number of rotor blades (zr ). For each stage with a constant number of real rotor blades[2] found that when the number of stator blades increases, the levels of unsteady forces reach their maximum with zs /zr = 0.25 − 0.3 and then decrease to minimum with zs /zr = 1.2. For the 100-MW turbine extraction stage the unsteady forces acting on the blades were large (zs /zr = 0.33)[3] . This was in particular valid for the circumferential component, the first high-frequency harmonic of which reached even as much as 100% of the stationary component. In this stage the failure of the rotor blade was reported[3] , even though the first five natural frequencies of rotor blade were beyond its resonance regions. This paper discusses the results of calculations performed for the same boundary conditions but with different numbers of stator blades in the system. For each stage, when the number of stator blades increases (zs /zr from 0.33 to 0.76) the levels of unsteady forces are considerably reduced. In order to avoid rotor blade failure we have recommended that turbine manufacturers should increase the numbers of stator blades. The other new results presenting here concern how unsteady components change along the blade length when the number of stator blades is altered.

2

Aerodynamic Model

Considered here is the 3D transonic flow of inviscid non-heat conductive gas through an axial turbine stage, including the Nozzle Cascade (NC) and the Rotor Wheel (RW), rotating with constant ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011


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´ ROMUALD RZADKOWSKI AND MAREK SOLINSKI / ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011

Fig. 1 The tangential and axial section of the turbine stage

angular velocity. Usually both NC and RW have an unequal number of blades of varied configurations. Taking into account the flow unperiodicity from blade to blade (in a pitch-wise direction) it is convenient to choose calculation domain that includes all the blades of the NC and RW assembly, the entry region, the axial clearance and the exit region (see Fig. 1). The main disadvantage of this kind of analysis is its high computational cost. A way to reduce the time of analysis is to use the one-to-one passage model. In simulations presented here a 3DFLOW program package written by V. Gnesin[4] was used. The package (grid generator and CFD solver) was designed for one-to-one passage calculations of three-dimensional, unsteady and inviscid flows through a single axial turbine stage. The governing equations of the 3DFLOW solver, presented in the form of integral conservation laws of mass, impulse and energy, were written in the relative Cartesian coordinate system, rotating with constant angular velocity ω[5–7] . An ideal gas equation of state was also assumed. An H -type stationary structured grid was used for the stator as well as for the moving rotor [7] . In order to discretize the full non-stationary Euler equations, the cell-centered Finite Volume Method was used. The spatial transonic flow through blade passages in turbomachines generally include strong discontinuities in the form of shock waves and wakes behind the exit edges of blades, and for this reason the Godunov-type explicit upwind scheme was used. Such discretization takes into account equations of disturbance propagation and generates numerical viscosity. Thus we were able to obtain stable numerical schemes without the introduction of correcting dissipative terms. In other words, the Godunov-type schemes did not require introducing artificial viscosity to the calculations. A detailed description of the method, complemented by numerous numerical samples, can be found in the monograph[5] . The calculated domain, including all blades on the whole annulus as well as inlet and outlet domains, consists of two sub-domains (NC and RW) having a common part. Let the number of stator and rotor blades be zs and zr , respectively. The difference grid is divided into zs + zr difference segments, each of them includes a blade and expands in circumferential direction which is equal to the pitch of the stator or rotor, respectively (see Fig. 1).

THE EFFECT OF CHANGE IN THE NUMBER OF STATOR BLADES

327

Table 1 Number of stator and rotor blades in each variant

Variant 1 Variant 2 Variant 3

Number of Stator Blades (Ns ) 46 56 112

Number of Rotor Blades (Nr ) 138 147 147

It is assumed that the unsteady flow fluctuations are due to both the rotor wheel rotation and to prescribed blade motions, and the flows far upstream and far downstream from the blade row are at most small perturbations of uniform free streams. The boundary conditions formulation is based on a one t dimensional theory of characteristics, where the number of physical boundary conditions depends on the number of characteristics entering the computational domain. In the general case, when axial velocity is subsonic at the inlet boundary, initial values for total pressure, total temperature and flow angles are used, while at the outlet boundary only static pressure has to be imposed. Non-reflecting boundary conditions can be used, i.e., incoming waves (three at the inlet, one at the outlet) have to be suppressed, which is accomplished by setting their time derivative to zero. In the general case, computations are made using a number of blade passages equal to the number of blades in the cascade. Periodic conditions are applied at the upper and lower boundaries of the calculated domain at each time moment.

3

Numerical Results

We showed[3] that for the 100-MW turbine extraction stage the unsteady forces acting on the rotor blades were large. This was large for the circumferential component, the first high-frequency harmonic of which reached even as much as 100% of the stationary component. This paper discusses the results of calculations performed for the same boundary conditions (inlet To = 440 K, total pressure po = 600000 Pa, static pressure p1 = 550000 Pa, exit static pressure p2 = 370000 Pa) but with different numbers of stator blades in the system (see Table 1). To each variant was applied a numerical grid of a different density, depending on the circumferential periodicity conditions as dictated by the one-to-one passage calculations and other limitations resulting from the assumptions adopted in the calculation package 3DFLOW[4] . Figure 2 presents in the form of two diagrams the changes of stationary components of the total, axial, circumferential and radial forces, and also the torque moment acting in successive rotor blade sections, as calculated for Variants 2 and 3. The results of Variants 1 and 2 are similar. The lowest curve shows the steady part of circumferential force, next radial force, moment, axial and the sum of all forces as they change in each rotor blade cross-section. When analyzing these data, we can conclude that for each abovementioned quantity the qualitative nature of changes remains the same, and the only observed differences are of a quantitative nature. For the axial force, its stationary component increases with increasing distance from the blade root. The stationary components of the unsteady circumferential and radial forces also increase in magnitude, but much less intensively. The stationary component of the unsteady torque moment remains

328


Fig. 2

Stationary components of unsteady axial, radial and circumferential forces as well as moments for different rotor blade cross-sections (Variants 2, 3)

Fig. 3

1st and 2nd harmonics of the unsteady axial force and torsional moment for different blade cross-sections (Variant 1)

at an approximately constant level, close to zero. Changing the number of stator blades reduces the stationary components of particular generalized forces, the effect of which is most visible in the axial force. The next diagrams (Figs. 3–5) present changes of the first and second harmonics of the unsteady axial force and the unsteady torsional moment in successive rotor blade sections. In the first two variants, changes in the dimensional values and percent contributions of the two first harmonics are very similar in both quality and quantity (with slightly larger values in the first variant). The first harmonic achieves its minimum in the fourth section, and its lowest percent contribution is recorded in section five. The highest harmonic level is reached in the rotor blade tip, while the highest percent contribution, in the first blade cross-section. The second harmonic of the unsteady axial force decreases monotonically, both in terms of dimensional value and percent contribution, with increasing distance from section 1 (blade fixing).


329

Fig. 4


Fig. 5


A different situation is observed in variant 3 in which the number of stator blades is equal to 112. In this case the first harmonic of the unsteady axial force achieves the highest levels in the sections situated close to the blade fixing, and then decreases rapidly both in terms of dimensional value and percent contribution (from c. 16% to c. 1%). It is also worth noting that compared to the previous two variants, in which the number of stator blades was equal to 46 and 56, respectively, the maximum of this harmonic decreased much more markedly. The second harmonic is at approximately the same level in all sections, increasing slightly in successive sections, while its percent contribution slightly decreases with increasing distance from the blade root. For the unsteady torque moment (Variant 1), the minimum of the first harmonic, expressed in Nm, is found in section five, while in the first and last sections it reaches its maxima. The percentage contribution of the first harmonic is the highest and equal to slightly over 30% in the first section, and it then decreases to reach its minimum at 18% in the fifth section. The curves of the unsteady torque moment (right plots in Figs. 3, 4, 5) have similar shapes to those describing the unsteady axial force (left plots in Figs. 3, 4, 5). For stator blades 46 and 56 in the first two variants, the first harmonic of the unsteady torque


330

Table 2 High frequency components of unsteady axial, radial and circumferential forces as well as the moment acting on rotor blades of a 100-MW steam turbine for different numbers of stator blades Variant

1

2

3

Har. 0 1 2 3 0 1 2 3 0 1 2 3

f [Hz] 0 2300 4600 6900 0 2800 5600 8400 0 5600 11200 16800

Axial [N] 351 44.2 8.7 4.0 299 31.6 3.75 1.75 148 8.63 2.3 0.14

Axial [%] – 12.5 2.5 1.1 – 10.5 1.25 0.58 – 5.82 1.55 0.095

Circum. [N] −192 126.3 4.5 2.5 −209 88.45 0.75 1.87 −177 3.27 1.01 0.21

Circum. [%] – 65 2.3 1.3 – 42.12 0.35 0.89 – 1.84 0.57 0.11

Radial [N] −87 7 0.27 0.14 −88 5.04 0.11 0.03 −88 0.43 0.02 0.013

Radial [%] – 8 0.32 0.2 – 5.68 0.12 0.03 – 0.49 0.02 0.0156

Moment Moment [Nm] [%] −124 – 80.6 64 2.82 2.27 1.59 1.28 −135 – 56.39 41.66 0.33 0.45 1.16 0.86 −114 – 2.13 1.87 0.64 0.56 0.13 0.11

moment reached its minimum in the fifth section and two maxima in the first and last sections. The curve had a virtually parabolic shape (in both peripheral cross-sections the values were approximately the same). In the third variant (Fig. 5) the first harmonic was the largest in the first section and then decreased monotonically to its minimum in the blade tip section. The second harmonic of the unsteady torque moment in variant 1 (Fig. 3) decreased monotonically in successive sections along the blade span. In variant 2 (Fig. 4) and variant 3 (Fig. 6) it had a local minimum in section 5. Table 2 collects the stationary components and high-frequency harmonics, expressed in dimensional values and percent of the stationary component, of particular generalised forces acting on a single blade, for Variants 1, 2 and 3. These data reveal that changing the number of stator blades remarkably affects the first harmonics of all quantities. After comparing the corresponding data for variants 1 and 2, we can notice that they are close to each other. A slight increase in the number of stator and rotor blades in the stage system results in the reduction of particular harmonics and a change in the stationary components which decreases for the axial forces and increases for the remaining generalized forces. What is most interesting from in the case of system dynamics is the comparison of the results obtained in Variants 2 and 3. Doubling the number of stator blades in the system, while the number of rotor blades and boundary conditions remains the same, leads to a twofold decrease of the axial force stationary component. The first harmonic decreases in magnitude to nearly one fourth (from 31,6 N to 8,6 N), while its percentage contribution decreases by nearly a half (from 10,5% to 5,8%). As for the remaining generalized forces, the decrease of their stationary components is not so marked, and is almost negligible in the case of the radial force. Particular harmonics decrease markedly (practically by an order in magnitude). Therefore we can conclude that changing the number of stator blades considerably affects the dynamic characteristics of the system. These relations are even more noticeable when we analyze the amplitude-frequency characteristics of particular generalized forces, the diagrams of which are shown in Figs. 6–8.


Fig. 6 Harmonics of unsteady axial force and moment (Variant 1)



331

332

4


Conclusions

This paper discussed the results of calculations of unsteady high-frequency forces acting on the rotor blades in the extraction stage of a 100-MW steam turbine for different numbers of stator blades. Increasing the number of stator blades affects considerably the level of unsteady forces acting on a single blade. These changes refer to both the stationary components of these forces and their first highfrequency harmonic, which decreases when the number of the stator blades is increased. Unfortunately, the small number of analyzed variants is still insufficient for us to formulate a unique relation between these two quantities, but it suggests the need for further studies of this subject. Increasing the number of stator blades improves the harmonic distribution of unsteady forces along the length of the rotor blades. On the other hand, increasing the number of stator blades reduces the efficiency of the stage. Therefore this subject also needs to be analyzed.

Acknowledgments All numerical calculations were made at the Academic Computer Center TASK (Gdansk, Poland).

References [1] Rao, J. S., Turbomachine unsteady aerodynamics, Wiley Eastern (New Delhi, London), 1994. [2] Sokolovskij, G. and Gnesin, V., Nonstationary transonic and viscous flow in turbomachinery, Naukova Dumka, Kijev (in Russian), 1986. [3] Rzadkowski, R., Gnesin, V. and Radulski, W., Aerodynamic unsteady forces of the rotor blades in the control stage and stage with steam extraction, Advances in Vibration Engineering, Vol. 5(1), pp. 1–10, 2006. [4] Gnesin, V., Program package, 3D Flow, Theory Manual, Rep. IMP PAN, 1652/01, Gdańsk, 2001. [5] Godunov, S. K., et al., Numerical solution of multidimensional problems in gasdynamics, Nauka, Moscow (in Russian), 1976. [6] Gnesin, V. and Rzadkowski, R., A Coupled fluid-structure analysis for 3D inviscid flutter of IV standard configuration, J. Sound Vibration, Vol. 251(2), pp. 315–327, 2002. [7] Gnesin, V., Kolodyazhnaya, L. and Rzadkowski R., A numerical model of stator-rotor interaction in a turbine stage with oscillating blades, Journal of Fluids and Structures, Vol. 19(8), pp. 1141–1153, 2004.

Free Vibration and Stability Analysis of a Spinning Annular Circular Plate Vinayak Ranjan1 and M. K. Ghosh2 1 Department of Mechanical Engineering and Mining Machinery Engineering, Indian School of Mines University, Dhanbad, Jharkhand, India [email protected] 2 Department of Mechanical Engineering, Institute of Technology, Banaras Hindu University, Varanasi 221 005, India Abstract In this paper, Rayleigh Ritz method has been applied to investigate free vibration behavior and stability of a rotating annular plate with an attached point mass. The classical thin plate theory assumptions have been used to calculate the strain energy and kinetic energy. The coordinate functions are combinations of polynomials which satisfy boundary conditions at the outer boundary. The frequency speed diagram is obtained for (0, 0) vibration mode. The plate is assumed to be spinning and the initial stress fields in the plate are derived assuming the plane stress condition. The problem is of practical importance in many engineering applications, e.g., silicon wafers used for LSI chips are produced by slicing the cylindrical crystal ingot with a disk like tool which is free along the inside hole and clamped at the outer radius. It has been shown that mode (0, 0) correspond to two modes at frequencies f 1 and f 2. Mode f 2 experiences a flutter type of instability after the critical speeds. The critical speeds are independent of attached point masses but depend upon the inner radius and bending rigidity of the plate. With increase in bending rigidity, critical speeds increase. For a plate i.e., for D = 0.18e4, f 2 mode meets f 1 mode and makes the f 1 mode unstable and the corresponding rotational speed of the plate is dependent on inner radius. With increase in inner radius, this rotational speed decreases. Keywords: Free vibration, Stability, Spinning plate

1

Introduction

Circular plates have many engineering applications and are widely used in ocean engineering, mining engineering, spacecrafts and missiles applications, land based vehicles, underwater vessels and structures. In variety of cases, plates carry loads in eccentric fashion as well as support engines or motors on elastic foundations. Besides, dynamics of spinning disks have importance in computer disks memory units and circular saw blades, turbines and compressor disks also. Therefore, one must know the fundamental frequencies and the critical speeds of the plates with different boundary conditions with point masses placed at any arbitrary position. Leissa[1] and Laura et al.[2] have reviewed analytically most of the early works in the area of free vibration of a circular plate with concentrated mass attached at the center of the plate. Natural frequencies of spinning annular plates were determined by Ramaiah[3] using ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011


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VINAYAK RANJAN AND M. K. GHOSH / ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011

Rayleigh-Ritz method. Lamb and Southwell et al.[4] , Barasch and Chen[5] and Ewan and Moeller [6] have reported earlier works on the free vibration of flexible spinning disks. Chen and Bogy[7] investigated on the natural frequencies and stability of thin spinning disks with a stationary load system. The load system included influence of spring, mass system in contact with plate and effect of centrifugal force. Chonan and Sato[8] presented a theoretical analysis of vibrations and stability of rotating annular plate using Galerkin method. They compared the instability due to rotation for two different boundary conditions of the plate. Shaw et al.[9] studied the free vibration of rotating rectangular plates with an exponentially varying thickness. In his experiment, the plate was clamped at the rim of hub and is free along the other three edges. Finite difference method was used to solve the differential equation governing the in plane displacements. Chen[10] investigated on the active control scheme to suppress vibration in spinning disks in both sub-critical and super-critical speed ranges. Bambill et al.[11] determined the fundamental frequency of vibration of circular plate carrying a concentrated mass at an arbitrary position with different boundary conditions using variational approach with Rayleigh Ritz formulation. Lim and Liew[12] investigated free vibration characteristics of perforated plates with rounded corners using Ritz minimum energy method. A set of orthogonally generated polynomial functions which satisfied appropriate boundary conditions was employed to describe the deflection of the plate. Eigenvalue equations were developed by minimizing the energy functional. Results of eigenvalues of rectangular and super elliptical clamped plates were presented. It was shown that method resulted in very accurate results of frequency parameters and is computationally more efficient compared to finite element analysis. Liew, Kitipornchai and Lim[13] extended this method to analyze free vibration behavior of thick super elliptical plates. Vinayak and Ghosh[14] presented forced vibration response of a rectangular thin plate with single discrete mass and patch placed at the center of plate using finite element method. The effects of varying thickness of area of patch on the dynamic vibration absorption have been investigated. They showed that there exists optimal thickness of patch for a given area as well as an optimal area of patch for a given thickness, for which patches behave optimally as dynamic vibration absorber. Duan et al.[15] showed that an axisymmetric vibration mode of circular plate with free edges can be realized by increasing the bending rigidity of the outer rim of the circular plate by using a larger thickness or by using a material with a larger young’s modulus or both. They adopted Midline plate theory. Sinha[16] investigated on free vibration of thin spinning disk stiffened with an outer reinforcing ring. Dasgupta and Hagedorn[17] presented theoretical analysis to calculate critical speeds of thin spinning disks with an external ring using Von Karman theory of plates. Vinyak and Ghosh[18] using finite element method studied the transverse vibration behavior of spinning disk with attached distributed patch and discrete point masses. Both free and forced vibration behavior was investigated. It was shown that these attached masses can very significantly influence the natural frequencies of the disk. Further, it was also demonstrated that discrete patches of piezoelectric material can be used to control the vibration of the disk. Kyo-Nam Koo and George A. Lesieutre[19] studied the stress distributions for a rotating disk composed of two annular disks, of which the inside is made of isotropic material and the outside is made of orthotropic material. The dynamic equation for a composite-ring disk in rotation is formulated to calculate its natural frequencies and critical speeds. For the solution of transverse vibration, a rotational symmetry condition is applied in the circumferential direction and a finite element interpolation with Hermite polynomials is performed in the radial direction. The results showed that reinforcing a disk at the rim increases critical speeds drastically, and can cause buckling in mode (0, 0) which occurs above the lowest critical speed.

FREE VIBRATION AND STABILITY ANALYSIS OF A SPINNING ANNULAR CIRCULAR PLATE

2

335

Theory and Formulations

The plate geometry and its physical properties are shown in Fig. 1. The annular plate rotates at a speed in the inertial reference frame (r, θ ). The plate is free at inner radius r = b and clamped at outer radius r = a. The point mass is attached to the plate at (r1, θ 1). The energy functional J of the annular plate is given as 2 2 1 ∂W 1 ∂ 2W ∂ W ∂ 2W + + 2ν J = D/2 r ∂r r 2 ∂θ 2 ∂r 2 ∂r 2 A +

1 ∂W 1 ∂ 2W + 2 r ∂r r ∂θ 2

×

∂ 2W

2

1 1 ∂W − 2 r ∂r∂θ r ∂θ 2π

+ 2(1 − ν) 2 r∂r∂θ −

a 2 w 2 ρh 2

Fig. 1 Schematic diagram of circular plate with attached discrete mass

1

1 W 2 (r, θ )rdrdθ − Mω2 (W (r1, θ 1))2 2 0 b/a ∂W 2 1 ∂W 2 1 r∂r∂θ + σθ h σr + ∂r r ∂θ 2 A 2 1 ∂W 2 ∂W ∂W 2 ∂W − ρ r∂r∂θ h + 2 + 2 ∂t ∂t ∂θ ∂θ A ×

(1)

where W (r, θ ) is the transverse vibration amplitude, is the uniform rotational speed of the plate about its axis, w is the natural frequency, ρ is plate density, h is the plate thickness, ν is the poison’s ratio, D, the bending rigidity of the plate is defined as Eh3 /12(1 − ν 2 ), σr and σθ are initial radial and hoop stress respectively resulting due to rotation. For the plate clamped at outer radius and free at inner radius σr and σθ are given as: ⎫ σr = 18 ρa 2 2 (F 1 + F 2 − F 3) ⎪ ⎬ (2) ⎪ σθ = 1 ρa 2 2 (F 1 − F 2 + F 4)⎭ 8

where F1 = F2 = F3 =

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ((b/a)2 (r/a)2 )((1−ν)(3+ν)(b/a)2 −(1−ν 2 )) ⎬

(1+ν)(3+ν)(b/a)2 +(1−ν 2 ) (1+ν)(b/a)2 +(1−ν)

((1+ν)(b/a)2 +(1−ν))

(3 + ν)(r/a)2

F 4 = (1 + 3ν)(r/a)2

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(3)

336


The candidate mode shape taken is taken as W (R, θ ) = cos nθ

p

j =0

Cj (1 + αj R t + βj2 )R (2j +n)

(4)

The boundary conditions, at outer radius are W (R, θ ) = 0

and

∂W (R, θ ) =0 ∂R

(5)

Rayleigh Ritz method requires minimizing the energy functional with respect to the Cj ∂J (W ) =0 ∂Cj

(6)

This gets an equation in w(t) using the non-triviality condition. The parameter t allows for improved natural frequencies. The plate configuration and its properties are taken as inner radius b = 0.03 m, outer radius a = 0.3 m, plate thickness h = 0.01 m, Young’s modulus of elasticity, E = 206.8 GPa, density ρ = 7820 kg/m3 , Poison’s ratio ν = 0.3.

3

Results and Discussion

The modes of vibration of circular plate are represented by (m, n) where m represents the number of nodal circles and n represents the number of nodal diameter. For the present study, the numerical simulations are carried out for the annular plate clamped at outer radius and rotating with uniform speed about its axis for mode (0, 0). Since the plate is circular and its frequency diagram is symmetric about its axes, hence only the region defined by positive λ and is depicted. Figures 2, 3 and 4 present the effect of the rotational speed on the frequencies of plate for (0, 0) mode without any attached point mass with varying bending rigidity for configuration b = 0.03 m, a = 0.3 m. It is obvious from the Figs. 2, 3 and 4 that for (0, 0) mode, we get two eigenvalues (denoted by f 1 and f 2). The f 2 mode in Figs. 2, 3, 4 meets the axis and then afterwards eigenvalues become complex numbers. The critical speed is defined as the rotational speed at which frequency f 2 of mode (0, 0) becomes zero. For f 1 mode we do not get any critical speed. The critical speeds for f 2 in Figs. 2, 3, 4 are 815.95, 2580.3 and 8159.5 radians/sec respectively which increase with increase in D. Afterward the eigenvalues become complex numbers. When the real part of the eigenvalue becomes positive, the corresponding mode is unstable. When the imaginary part of the complex mode is non zero, the corresponding mode experiences a flutter type of instability. In the Figs. 2, 3, 4 after their critical speeds f 2 mode become complex with non-zero imaginary part, and therefore, flutter type instability is induced in the corresponding mode. With increasing bending rigidity, this flutter type instability shifts towards higher rotational speeds and we get more stable regions of operation. Figures 5, 6 and 7 present the effect of the rotational speed on the frequencies of plate for (0, 0) mode without any attached point mass for same variation of bending rigidity, D for another configuration b = 0.15 m, a = 0.3 m. The critical speeds for f 2 modes are 1332.2, 4212.8, 13322 radians/sec respectively which are higher than those obtained in Figs. 2, 3, 4 respectively. Clearly, when the inner radius b is decreased to 0.15 m, then for same variation of bending rigidity,


337

Fig. 2

The comparison of the effect of the rotational speed on the frequency coefficients for plate having zero nodal diameter without any attached point mass with D = 0.18938e4 (b = 0.03 m, a = 0.3 m)

Fig. 3


Fig. 4


Fig. 5


respective critical speeds enhances, and therefore instability regions further shifts toward higher rotational speeds. It is apparent that the critical speed depends upon the inner radius as well as bending rigidity. Figure 8 presents the variation of critical speeds for different inner radius for different values of bending rigidity. It is clear from the Fig. 8 that increases in critical speed of plate with increasing bending rigidity is significant for some range of inner radius. Say, for of a given value of D = 18.938e4, critical speed increases significantly for the values of b ranging from 0.2 m to 0.25 m but is not significant for inner radius less than 0.1 m. To avoid the flutter type of instability, we need to enhance the critical speed. And for a given critical speed we may have a choice of bending rigidity and inner radius. As for example, in Fig. 8, for a critical speed of 20000 rad/sec, we have choice of b and D corresponding to points A and B. Figures 9, 10, 11 present the effect of the rotational speeds on the frequencies of plate for (0, 0) mode with a attached point mass equal to 5% of plate mass for same variation of bending rigidity, D

338


Fig. 6


Fig. 7


Fig. 8

The variation of critical speed and the inner radius for plate having zero nodal diameter without any attached point mass with varying D(a = 0.3 m)

Fig. 9

Comparison of the effect of the rotational speed on the frequency coefficients for plate having zero nodal diameters with a point mass for D = 0.18938e4 (b = 0.03 m, a = 0.3 m)

for the configuration b = 0.03 m, a = 0.3 m. The critical speeds for f 2 modes are 815.95, 2580.3, 8159.5 radians/sec respectively which are same in comparison of the plate without any point masses in Figs. 2, 3, 4 respectively. It is apparent that critical speeds are not affected by the attachment of point masses to the plate. But it is interesting to note that the f 2 mode crosses the f 1 mode at point A in the Fig. 9. As we get the flutter type of instability for f 2 mode after the critical speed, and because f 2 mode crosses the f 1 mode at A, it is expected to make the f 1 mode instable. Therefore, the rotational speed corresponding to point A in Fig. 9 becomes very critical to stability of mode f 1. In Figs. 10 and 11, when bending rigidity is increased, the f 1 mode remains stable throughout the rotational speeds considered here as f 2 mode does not cross it. It is obvious that to avoid the instability of f 1 mode, there is an optimum value of bending rigidity of plate where the f 1 mode remains stable. Figures 12, 13 and 14 present the effect of the rotational speed on the frequencies of plate for (0, 0) mode with a attached point mass equal to 5% of plate mass for same variation of bending rigidity, D for another configuration b = 0.15 m, a = 0.3 m. The critical speeds for f 2 modes are 1332.2, 4212.8,


339

Fig. 10

Comparison of the effect of the rotational speed on the frequency coefficients for plate having zero nodal diameters with an attached point mass with D = 1.8938e4 (b = 0.03 m, a = 0.3 m)

Fig. 11

Comparison of the effect of the rotational speed on the frequency coefficients for plate having zero nodal diameters with attached point mass with D = 18.938e4 (b = 0.03 m, a = 0.3 m)

Fig. 12

Comparison of the effect of the rotational speed on the frequency coefficients for plate having zero nodal

Fig. 13

Comparison of the effect of the rotational speed on the frequency coefficients for plate having zero nodal diameter with attached point mass with D = 1.8938e4 (b = 0.15 m, a = 0.3 m)

13322 radians/sec respectively which are same as obtained for the plate without any point mass in Figs. 5, 6 and 7. This indicates that the critical speeds are not affected by the attached point masses for the given configuration. Again, it is noteworthy that f 2 mode crosses f 1 mode at point B in Fig. 12 induces the instability in f 1 mode. However, the rotational speed corresponding to the point (point B in Fig. 12) where the f 2 mode cuts the f 1 mode is lower than that of point A in Fig. 9. This indicates that with increasing inner radius the crossing point of f 2 and f 1 mode shifts towards lower rotational speeds and therefore reducing the range of stable rotational speed for f 1 mode. The dependency of the crossing point of f 1 and f 2 mode on inner radius, is further indicated in Fig. 15 for another configuration (b = 0.003 m, a = 0.03). By increasing the bending rigidity of the plate, as evident in the Figs. 13 and 14, mode f 1 and f 2 do not cross each other and therefore f 2 mode only experiences the flutter type of instability after the critical speed.


340

Fig. 14

4


Fig. 15


Conclusions

1. There are two natural frequencies for (0, 0) mode viz. f 1 and f 2. 2. Mode f 2 experiences a flutter type of instability beyond the critical speed. 3. The critical speeds are independent of attached point masses but depend upon the inner radius and bending rigidity of the plate. Increase in bending rigidity increases critical speed. 4. For a low bending rigidity of plate i.e., D = 0.18938e4, f 2 mode meets f 1 mode and makes the f 1 mode unstable and the corresponding rotational speed of the plate is dependent on inner radius. With increase in inner radius, the rotational speed where f 2 meets f 1 curves intersects each other decreases.

References [1] Leissa, A. W., Vibration of Plates, NASA SP, 160, 1969. [2] Laura, P. A. A., Laura, A. P., Diez, G. and Cortinez, V. H., A note on vibrating circular plates carrying concentrated masses, Mech. Res. Comm., Vol. 11(6), pp. 397–400, 1984. [3] Ramaiah, G. K., Natural frequencies in spinning annular plates, J. Sound Vibration, Vol. 74(2), pp. 303–310, 1981. [4] Lamb, H. and Southwell, R. V., The vibrations of a spinning disk, Proceedings of the Royal Society, Vol. 99, pp. 272–280, 1982. [5] Barasch, S. and Chen, Y., On the vibration of rotating disks, American society of mechanical engineers, Journal of Applied Mechanics, Vol. 39, pp. 1143–1144, 1972. [6] Ewan, W. D. and Moeller, T. I., The stability of a spinning elastic disc with transverse load system, American Society of Mechanical Engineers, Journal of Applied Mechanics, Vol. 43, pp. 485–490, 1976.


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[7] Chen, J. S. and Bogy, Effects of load parameters on the natural frequencies and stability of a flexible spinning disk with a stationary load system, Transactions of American Society of Mechanical Engineers, Journal of Applied Mechanics, Vol. 59, S230–S235, 1992. [8] Chonan, S. and Sato, S., Vibration and stability of rotating free-clamped slicing blades, J. Sound Vibration, Vol. 127(2), pp. 245–252, 1988. [9] Shaw, D., Shen, K. Y. and Wang, J. T. S., Flexural vibration of rotating rectangular plates of variable thickness, J. Sound Vibration, Vol. 126, pp. 373–385, 1989. [10] Chen, J. S., Vibration control of a spinning disk, International Journal of Mechanical Sciences, Vol. 45, pp. 1269–1282, 2003. [11] Bambill, D. V., Malfa, S. La., Rossit, C. A. and Laura, A., Analytical and experimental investigation on transverse vibration of solid, Circular and annular plates carrying a concentrated mass at an arbitrary position with marine applications, Ocean Engineering, Vol. 31(2), pp. 127–138, 2004. [12] Lim, C. W. and Liew, K. M., Vibrations of perforated plates with rounded corners, ASCE, Journal of Engineering Mechanics, Vol. 121, No. 2, pp. 203–213, 1995. [13] Liew, K. M., Kitipornchai, S. and Lim, C. W., Free vibration analysis of thick superellitical plates, ASCE, Journal of Engineering Mechanics, Vol. 124(2), pp. 137–145, 1998. [14] Vinayak Ranjan and Ghosh, M. K., Forced vibration response of thin plate with attached discrete dynamic absorbers, Thin-Walled Structures, Vol. 43, pp. 1513–1533, 2005. [15] Duan, W. H., Wang, C. M. and Wang, C. Y., Modification of fundamental vibration modes of circular plates with free edges, J. Sound Vibration, Vol. 317, pp. 709–715, 2008. [16] Sinha, S. K., On the free vibrations of a thin spinning disk stiffened with an outer reinforcing ring, American society of mechanical engineers, Journal of Vibration and Acoustics, Stress and Reliability in Design, Vol. 110, pp. 507–514, 1988. [17] Dasgupta, A. and Hagedorn, P., J. Sound Vibration, Vol. 283, pp. 765–779, 2005. [18] Vinayak Ranjan and Ghosh, M. K., Transverse vibration of spinning disk with attached distributed patch and discrete point masses using finite element analysis, International Journal of Science, Engineering and Technology, Vol. 1(1), pp. 74–89, 2009. [19] Kyo-Nam Koo and George A. Lesieutre, Vibration and critical speeds of composite-ring disks for data storage, J. Sound Vibration, Vol. 329, pp. 833–847, 2010.

Three-Dimensional Seismic Analysis and Safety Evaluation for Nuclear Pump of Nuclear Power Plant Based on the RCC-M Code Meng Jigang1,2 , Yang Shuhua2 and Wang Yuefang1,3 1 Department of Engineering Mechanics, Dalian University of Technology, Dalian 116023, China 2 Technical Center, Shenyang Blower Works (Group) Ltd, Shenyang 110142, China 3 State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian 116024, China jigang− [email protected] Abstract Pumps are important equipments of nuclear power plants. The most significant difference between nuclear power pumps and ordinary pumps is strict requirements on reliability and safety by design codes. The nuclear pumps must keep structural intactness and be able to function what they are designed for under strong earthquakes. In this paper, a three-dimensional model of a nuclear power pump based on real project designs is established by SolidWorks software. Hexahedron-dominated elements are used to mesh the finite element model of the pump by ANSYS to obtain solution of seismic analysis. A static analysis in normal conditions defined by Technical Specification of the pump is performed to determine the distribution of basic initial stress. Then, a modal analysis considering the initial stress is carried out to solve the free vibration of the structure. Afterwards, a spectrum analysis considering two horizontal loads and one vertical load is accomplished to obtain seismic responses of deformation and stress distribution of the pump. A post-procession is performed to get the superposition and linearization of dynamical and static results. Finally, the safety of the pump under seismic circumstance is evaluated through comparison of linearized stress on specific paths with the stress intensity limit urged by the RCC-M Code at the level of the given operating condition. The present analysis ensures that the pump will be safe and be able to fulfill the requirement by the RCC-M Code. Keywords: Nuclear pump, Seismic analysis, Safety evaluation, RCC-M Code

1

Introduction

Nuclear power is one of the high-efficiency and environmentally-friendly resources. It is one of main orientations for the new energy development strategy in China related to national energy security. To develop nuclear power plants is very important for the promotion of energy diversification and the optimization of power industrial structure. With the development of nuclear power, issues such as energy developing and supplying will be resolved partly. Besides, developing nuclear power is necessary for the strategy of sustainable development of the country[1, 2] . For general mechanical devices it is required that the stress is below its allowable value. Considering the potential disasters caused by nuclear radiations to both humans and natural environment when the power plant fails, the requirement on strength for nuclear power-related devices must be higher by fully admitting the damage to the device by earthquakes. It has been recognized that additional stress and ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011


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MENG JIGANG, YANG SHUHUA AND WANG YUEFANG / ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011

deformation can be developed in the device by transient seismic forces which are transmitted inside through structural bases. Failures of the device due to loss of structural integrity and/or performability will jeopardize the whole device and may even lead to disastrous consequences of e.g., severe leakage. Hence, it is crucial that seismic analyses should be performed for nuclear devices considering all kinds of loading cases. Nuclear pumps are coral devices of the nuclear power stations just like the heart of a human body. The most significant difference between nuclear power pumps and ordinary pumps is strict requirements of reliability and safety by design codes. The nuclear pumps must keep structural intactness and be able to function under normal operation condition and survive strong earthquakes. The security of nuclear pumps in situations of emergency guarantees a safe operation of the nuclear power station and maintains its effectiveness and service continuity. Hence, the safety evaluation for nuclear pumps under seismic circumstance shall be performed indispensably. In this paper the analysis object is an important nuclear pump of a domestic nuclear power station under construction. A three-dimensional model of a nuclear power pump based on real project designs is established by SolidWorks software. Hexahedron-dominated elements are used to mesh the finite element model of the pump by ANSYS software to perform finite element analysis. Firstly, a static analysis in normal conditions defined by the Technical Specification of the pump is performed to determine the distribution of basic initial stress. Secondly, a modal analysis considering the initial stress is carried out to solve the free vibration of the structure. Afterwards, a spectrum analysis considering two horizontal loads and one vertical load is accomplished to obtain seismic response of deformation and stress distribution of the pump. A post-procession is then performed to get the superposition and linearization of dynamical and static results. Finally, the safety of the pump under seismic circumstance is evaluated through comparison of linearized stress on specific paths with the stress intensity limit requested by the RCC-M Code at the level of the given operating condition.

2

Model Overview

The nuclear pump structure investigated in this paper consists of a pump casing, an end cover, inlet & outlet nozzles, an impeller, a shaft and bearing housing etc. With SolidWorks software, the entire threedimensional model of the nuclear pump is assembled in accordance with reasonable assembly sequence. Figure 1(a) shows the two-dimensional division drawing of the pump, and the three-dimensional geometric model of the whole pump assembly is plotted in Fig. 1(b).

3

Basic Theory of Seismic Analysis

The theory of structural seismic analysis is one of most important focuses in civil engineering research. The earthquake response of structures depends on characteristics of seismic forces and the intrinsic dynamics of the structure. It has been noticed that seismic forces are highly stochastic and very difficult to predict for their magnitudes and types are uncertain at the specific location of the structure on decade basis. Further, the dynamic response varies with external excitation on it. Nevertheless, the accuracy and efficiency of seismic analysis have been greatly improved due to the rapid growth of computational science and technology. In general, three seismic analysis theories have been developed, namely, the Static Analysis method, the Response Spectrum method and the Dynamic Analysis method based which various computational approaches have been proposed for solving structural seismic responses.

THREE-DIMENSIONAL SEISMIC ANALYSIS AND SAFETY EVALUATION

Fig. 1a

The two-dimensional design drawing of the nuclear pump

Fig. 1b

345

The three-dimensional geometric model of the overall nuclear pump

In the present paper the Response Spectrum Method is used to analyze the structural reactions to designate seismic loadings. It is assumed with this method that no rotation of the ground is considered. The ground motion is decomposed into one vertical and two horizontal components, each of which is computed for actions to the structure. Then, all reactions to the motion component are superposed to give the structural response to the seismic loading in the sense of statistics. Denote that the vector of absolute general displacement of the structure by {δ} which is composed of three translational degrees-of-freedoms i.e., xi , yi , zi , i = 1, 2, . . . , n, where n is number of total degree-of-freedoms of the structure. The vectors of connected displacement due to seismic ground motion and relative displacement are denoted as {δg } and {δr }, respectively, satisfying {δ} = {δg } + {δr }

(1)

The equation of motion of the structure can be expressed as [M]{δ¨r } + [C]{δ˙r } + [K]{δr } = −[M]{δ¨g }

(2)

For convenience we assume the base motion of the structure is a x-directional translation, and the acceleration of the motion is −ax . Hence, the connected acceleration is {δ¨g } = −{δax } · ax , where {δax } = [100000100000100000 . . . ]T is the directional identification vector. The n natural frequencies, Pi and the modes, {A(i) }, i = 1, 2, 3, . . . , n, can be solved through the undamped free vibration problem: [M]{δ¨r } + [K]{δr } = 0

(3)

The corresponding normalized modal vectors are: 1 (i) {AN } = √ {A(i) }, Mi

(4)

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Considering the orthogonality conditions of the modes and assuming the damping matrix [C] is proportional damping, the governing equation of motion can be obtained: [I ]{x¨N } + [CN ]{x˙N } + [KN ]{xN } = [AN ]T [M]{δax }ax

(5)

x¨Ni + 2ξi pi x˙Ni + pi2 xNi = ηi,x ax ,

(6)

i = 1, 2, 3, . . . , n

(i)

where ηi,x = {AN }[M]{δax } is the participating coefficient of the ith mode. Notice that this (6) are n decoupled single degree-of-freedom oscillators with excitation of seismic acceleration on the right hand side compared with ordinary vibrating systems. The structural response of x-displacement can be obtained through coordinate transformation {δr } = [AN ]{xN }. The y- and z-displacements can be obtained by the same procedure with seismic inputs in y- and z-directions, respectively[3] . The overall, three dimensional displacements can be obtained by superposing peak values of x-, y- and z-displacements using the method of SRSS (Square of Sum Square). Based on this method the maximum displacement is calculated through: L 2 δr(max) = (7) δri (max) i=1

where L is the number of superposed modes. Similarly, the element stress can be computed for each modal response and then superposed to give the maximum stress through the SRSS method.

4

Finite Element Analysis

4.1 Meshing After geometric model of the pump is generated, the meshing process is carried out by ANSYS. In order to ensure the accuracy of numerical simulation, two kinds of high order 3-D solid elements that exhibit quadratic displacement behavior are selected to create the mesh used in the computational process: One is type Solid 186 which is hexahedron structural solid with 20 nodes, and the other is type Solid 187, which is tetrahedral structural solid defined with 10 nodes. The method of mixing free meshing and mapped meshing is used to generate the finite element model which is hexahedron-dominated[4] . As shown in Fig. 2, there is the hybrid finite element model of the whole pump, which is composed of 144623 elements and 402287 nodes. 4.2 Load conditions 4.2.1 Definitions of operating conditions According to the Technical Specification for the pump, different operating conditions defined by distinct loadings require different level of criteria whose details are shown in Table 1. The normal loads include design initial pressure, the thermal expansion under design temperature and the dead weight. NN , NP and N represent nozzle loads for all design conditions. The analysis results of the pump in normal condition are presented.


347

Fig. 2 The hybrid finite element model of the overall nuclear pump Table 1 Loads combination and level criteria Type of Operating Conditions Normal Upset Faulted

Design Loads Normal loads + NN Normal loads + NP + 1/2 SSE loads Normal loads + N + SSE loads

Level Criteria A B B

Table 2 Horizontal floor acceleration spectrum on 1/2 SSE (Frequency: Hz, Acceleration: g) Frequency Acceleration

0.20 0.050

1.00 0.220

2.00 0.520

3.38 0.820

4.56 1.220

6.22 1.220

7.50 0.570

14.70 0.570

18.00 0.300

35.00 0.215

100.00 0.215

Table 3 Vertical floor acceleration spectrum on 1/2 SSE (Frequency: Hz, Acceleration: g) Frequency Acceleration

0.20 0.035

2.40 0.380

5.00 0.380

9.80 0.690

14.20 0.690

20.00 0.200

35.00 0.140

100.00 0.140

4.2.2 Seismic loads The nuclear pump is classified as a “Seismic IA” equipment as required by the technical specification. This kind of pumps should be able to resist the Safe-Shutdown Earthquake (SSE). Specifically, the structural integrity and operation of “Seismic IA” equipment during and after SSE should be completely guaranteed. It is assigned in Technical Specifications that the pump unit is installed in elevation of ±0.00 m. The acceleration response spectrum of the floor at the height of 4 m, with the damping ratio of 2% and one half of the SSE, is shown in Tables 2 and 3. The SSE spectrum can be obtained by multiplying the data of Tables 2 and 3 by 2. In the present analysis, floor response spectrum at the height of 4 m about the reactor plant is regarded as the input seismic spectrum, conservatively. 4.3 Material properties The nuclear pump structure is consisted of various kind of material, such as pump body, import & export flange, pump cover and etc. The materials are from RCC-M Code, appendix Z, whose properties and allowable stress are listed in Table 4[5, 6] .

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Table 4 Material properties about main components

Material Z2CN18-10 ZG230-450 Z5CND16.04

Young’s Modulus E(105 MPa) 2.0 2.0 2.0

Density (103 Kg/m3 ) 7.8 7.8 7.8

Poisson’s Ratio 0.3 0.3 0.3

Tensile Strength σb (MPa) 450 450 785

Yield Strength σs (MPa) 175 230 635

Allowable Stress S (MPa) 112 183

Table 5 Stress limits about materials on deferent level criteria (MPa)

Material Z2CN18-10 ZG230-450 Z5CND16.04

Allowable Stress S 101 112 183

Level Criteria A σm + σb ≤ 1.5S 151.5 168.0 274.5

Level Criteria B σm + σb ≤ 1.65S 166.6 184.8 302.0

Material stress limits corresponding to level criteria A & B of the main component are listed in Table 5. In Table 5, σm and σb are corresponding to a membrane stress and a bending stress respectively.

5

Analytic Processes

5.1 Response spectrum method In this paper, the natural frequency analysis for the whole structure of the nuclear pump is carried out. Based on the modal solution, the earthquake response of the pump in two horizontal spectrums and a vertical spectrum provided by the Technical Specification are obtained separately. The results of the three separate spectrums are combined in accordance with the SRSS combination, which gives the forced response of the pump under seismic loads. 5.2 Combination of operation conditions Since there exist various factors that contribute to the security of the pump, the safety analysis under various load conditions needs to be taken into consideration. In this paper, the method of solution combination with operation conditions is used[7] . Firstly, a series of calculations considering single load factor, such as the dead weight, design pressure, nozzle loads and seismic loads, are completed separately. Secondly, solution combination of these conditions defined by special loads shown in Table 1.

6

Seismic Safety Evaluation

6.1 Stress evaluation According to the combined solution in accordance with the operation conditions, the safety evaluation of the pump’s integrity on the pressurized boundary and the machine’s performability under the seismic


Fig. 3a

The overall displacement cloud for the pump under faulted conditions

Fig. 3b

349

The overall Stress Intensity cloud for the pump under faulted conditions

circumstance is completed individually. The overall displacement cloud and the overall Stress Intensity cloud for the nuclear pump under faulted conditions is shown in Fig. 3(a) and 3(b). The RCC-M code provides a specific way to assess structural integrity of the pressurized boundary of the pump. The stress in high stress areas on the is categorized into general primary membrane stress, local primary membrane stress, primary bending stress, secondary stress and peak stress. Then, different criteria are used to evaluate the stress according to the level of safety requirement. Therefore, the key issue is categorization of stress through precise decomposition of computed stress into the abovementioned stresses. What commonly used for the decomposition is the Equivalent Linearization method which has been adopted as a standard postprocess procedure in many commercial Finite Element Softwares. By using the Equivalent Linearization method the analyzer assigns one or more path that virtually penetrates (usually in the normal direction) at several possible “dangerous” locations in the wall on the pressurized boundary, and decompose the calculated elastic stresses along the path into membrane and linear bending stresses based on equivalences of resultant force and resultant moments. In ANSYS Postprocessor the Equivalent Linearization can be easily realized through Path Mapping and Stress Linearization. The categorization of stress can be achieved based on the Code consequently[8–10] . The path operation used in the above linearization involves three major steps: path definition, mapping and processing the path, including parameterization, mathematical operation, save/resume data and save/resume path to file. A typical path operation can be described with the following stream command lines[7] : ! Step 1. Defining a Path CSYS, KCN PATH, Name, nPts, nSets, nDiv PPATH, POINT, NODE, X, Y, Z, CS ! Step 2. Mapping Path Data PATH, Name RSYS, KCN PDEF, Lab, Item, Comp, Avglab

! Assign the global coordinate ! Assign path parameters ! Define the path ! Assign path name ! Choose result coordinate ! Map results

350


! Step 3. Postprocessing among Path Items PLPATH, Lab PLSECT, S, INT, PRSECT, RHO, KBR PASAVE, ALL, P− TOTAL, PATH PADEL, ALL PARESU, ALL, P− TOTAL, PATH

! Display the path graphically ! Draw Stress Intensity ! Print results ! Save all paths to the file named p− total.path ! Delete all paths from current data ! Resume path data from the file p− total.path

Table 6 Level criteria and stress limits Type of Operating Conditions Level Criteria Membrane Stress Membrane

Normal A σm ≤ S σm + σb ≤ 1.5 S

Upset B σm ≤ S σm + σb ≤ 1.5 S

The safety criteria and corresponding stress limits of three operating conditions are presented in Table 6. In Table 6, σm and σb are corresponding to a membrane stress and a bending stress respectively. In this paper, three special paths in the high stress region on the pressure boundary along the thickness direction are set, linearized stresses distributed on the paths are generated, and values of evaluation indexes are properly formed. For the nuclear pump in this paper, we defined 3 paths on the pressure boundary, shown in Fig. 4(a). And the linearizing result on path no. 1 suffering normal loads is shown in Fig. 4(b). The results of stress evaluations are shown in Table 7.

Fig. 4b Linearizing result on path no. 1

Faulted B σm ≤ 1.1 S σm + σb ≤ 1.65 S

Fig. 4a Definition of paths on the pump body


351

Table 7 The results of stress evaluations about main components Component Normal Pump body

Upset Faulted Normal

Pump cover

Upset Faulted

Maximum Stress (MPa) Membrane Membrane + Bending Membrane Membrane + Bending Membrane Membrane + Bending Membrane Membrane + Bending Membrane Membrane + Bending Membrane Membrane + Bending

28.02 42.88 30.82 49.08 45.94 61.09 8.957 13.69 11.04 17.08 16.60 25.66

Stress Limits (MPa) 101.0 151.5 111.1 166.6 111.1 166.6 101.0 151.5 111.1 166.6 111.1 166.6

Table 8 Max deformation difference on locations with different seismic types Locations Front end of the impeller Back end of the impeller Others

7

Seismic Type 1/2 SSE SSE 1/2 SSE SSE 1/2 SSE SSE

Limits (mm) 0.330 0.495 0.240 0.360 Min 0.150 Min 0.225

Max Deformation Difference (mm) 0.32 0.35 0.22 0.27 0.10

Performability Evaluation

As for the performability under seismic loads required by the Technical Specification, it is ensured that the equipment is able to keep functioning during and after the event of SSE. The deformation caused by the event results in nothing but slight friction and interference between the stationary parts and rotating parts, and the operation of equipments should not be affected. These slight friction or interference can only be observed during the earthquake and the necessary analysis should prove that the pump structure can immediately return normal after the earthquake. In general, the performability is assessed through evaluation of relative deformation between the rotor and the stator using the following formula: Y = k(δ − ε)

k = 0.6 for 1/2 SSE event k = 0.9 for SSE event

Y is the calculated clearance; δ is the original clearance and ε is tolerance due to manufacture.

352


In the paper, the author selects the nodes at some key locations on the rotor and stator. Radial displacements of the selected nodes are picked up to calculate the clearance and to compare with the design clearance. The evaluation methods and process are shown in Table 8. It is obvious that the performability under seismic loads is fairly acceptable.

8

Conclusions

Through the above-mentioned modeling and analysis, the safety of a nuclear pump of a nuclear power plant is evaluated strictly in accordance with the RCC-M codes. The conclusions are as follows: (1) The present analysis ensures that the pump will be safe and be able to fulfill the requirement by the RCC-M Code. The pump has ability of continuous operation during and after the event of SSE, with a fairly large design margin. (2) The response spectrum analysis can be used to simulate the structure’s performance in response to the earthquake loads. (3) It is a convenient and accurate way of using the path function in ANSYS software to obtain linearization stress categories. That makes the stress evaluation more feasible.

Acknowledgement The authors are grateful for supports from the Exploration Projects on Nuclear Pumps of Dalian University of Technology, the National Science Foundation of China (10721062) and the Chinese National Programs for High Technology Research and Development (2007AA04Z405) and the State Key Development Program for Basic Research of China (Project 2009CB724300).

References [1] Sheng Xuan-yu, Luo Xiao-wei and Fu Ji-yang, Aseismatic strength analysis of main nuclear reactor pump based on real three dimension model, Nuclear power engineering, No. 5, pp. 25–30, 2005. [2] Yang Xiao-feng, Zhang Yong, Sun Bai-tao and Hu Shao-qing, Seismic analysis on canned motor pump used in nuclear power generating station, World Earthquake Engineering, Vol. 23, No. 3, pp. 47–53, 2007. [3] Rao, J. S. and Gupta, K., Theory and practice of mechanical vibration, John Wiley and Sons, 1984. [4] ANSYS Modeling and Meshing Guide, ANSYS, 2006. [5] RCC-M SECTION I, Institute of standardization for nuclear industry, 2002. [6] RCC-M SECTION II, Institute of standardization for nuclear industry, 2002. [7] ANSYS Structural Analysis Guide, ANSYS, 2006. [8] Lu Mingwan and Chen Yong, Primary structure method of stress classification for design by analysis, Nuclear power engineering, Vol. 19, No. 4, pp. 330–337, 1998. [9] Lu Mingwan and XU Hong, Discussion on some important problems of design by analysis (1), Pressure Vessel Techology, Vol. 23, No. 1, pp. 15–19, 2006. [10] Lu Mingwan and XU Hong, Discussion on some important problems of design by analysis (2), Pressure Vessel Techology, Vol. 23, No. 2, pp. 28–32, 2006.

Nonlinear Dynamic Behaviors of a Thermo-Mechanical Coupling Viscoelastic Plate Y. H. Li1 , Y. N. Wang2 and L. Li1 1 School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China [email protected] 2 Mechatronic Engineering College, Southwest Petroleum University, Chengdu 610500, China Abstract In this paper, the nonlinear dynamic model of a thermo-mechanical coupling viscoelastic rectangular plate is established. This plate is with varied temperature field and subjected to both actions of an alternating periodic transverse external excitation and in-plane uniform distributed force. The influence of heat conduction, thermal expansion and viscosity is considered in this model. This model is established through constitutive description of thermo-viscoelastic material which obeys the Boltzman’s superposition principle, and through the dynamic equilibrium equation of the rectangular plate on the basis of the Karman theory for thin plates with large deflection and the thermo-viscoelastic energy theory. It can be converted to a nonlinear differential-integral dynamical system by using Galerkin’s method. Based on the nonlinear integral-ordinary differential dynamical system of the thermo-mechanical coupling viscoelastic plate, the general nonlinear numerical method for viscoelastic plate is obtained by introducing difference method. And then, a kind of special dynamic problem with thermo-mechanical coupling is solved. Finally, synthetically using several methods in dynamic systems, the dynamic properties of the thermo-mechanical coupling viscoelastic plate are sufficiently revealed. It shows that the dynamic properties of the thermo-mechanical coupling viscoelastic plate, which is subjected to both actions of an alternating periodic transverse external excitation and in-plane uniform distributed force, are abundant, and the chaos is greater than the viscoelastic plate’s without the influence of temperature. Especially, the motion state of hyperchaos appears. Keywords: Viscoelastic plate, Thermo-mechanical coupling, Nonlinear dynamic model, Hyperchaos, Bifurcation

1

Introduction

Recently, thin-wall structure made by composite viscoelastic material, which has complicated nonlinear dynamic behaviors, is used widely. Abundant dynamical phenomena appear due to the influence of the load, temperature and environment. James etc.[1] derived a thermodynamically consistent nonlinear viscoelastic constitutive theory including such phenomena as yield, enthalpy relaxation, nonlinear stress– strain behavior in complex loading histories, and physical aging. Zhang Yitong etc.[2] derived the creep constitutive law for all kinds of thermo-viscoelasticity materials under nonconstant temperature states. Drozdov[3] derived constitutive equations for the viscoelastic response with temperature of a semicrystalline polymer and to determine adjustable parameters in the stress–strain relations by fitting the ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011


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Y. H. LI, Y. N. WANG AND L. LI / ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011

observations. Yang Tingqing etc.[4] derived the correlation between material viscosity or free volume and temperature and stress level. Akhtar etc.[5] developed a phenomenological one-dimensional constitutive model characterizing the complex and highly nonlinear finite thermo-mechanical behavior of viscoelastic polymers. Fu Yiming etc.[6] studied the nonlinear dynamic models and behaviors with the effect of shift factors induced by the change of temperature and humidity. Anastasia[7] derived a viscoelastic multiscale model and analyzed time–stress–temperature behaviors of graphite/epoxy laminated composite materials and structures. Fu Yiming etc.[8] studied the influence of the nonlinear dynamic response of viscoelastic rectangular plate with damage under the varied temperature field, and analyzed the effect of thermal expansion for the response of the damaged structure. Ilyasov[9] studied the dynamic stability of viscoelastic plates with a general isotropic viscoelastic constitutive relations. Qin Yuming[10] studied the existence of a global attractor for a semiflow, which is governed by the weak solutions to a nonlinear one-dimensional thermoviscoelasticity with a non-convex free energy density. However, there are few results about nonlinear dynamic models and behaviors of viscoelastic plate, which with the influence of heat conduction, thermal expansion and viscosity. Generally, the govern equations for dynamic behaviors of viscoelastic structures are infinite dimensional dynamical systems, and can be simplified into a nonlinear integral-differential equation by using Galerkin’s method. Numerical methods in nonlinear dynamics are synthetically applied to investigate the dynamical properties of viscoelastic structures. Rao[?] studied vibrations of laminated plates using mixed theory. Argyris[11] examined chaos motion of a nonlinear viscoelastic beam using the methods of Poincare mapping and Lyapunov exponent. Cheng Changjun etc.[12] investigated the stability analysis and chaotic motion of a homogeneous, simple supported column by terms of Poincare mapping and power spectrum. Yen-Liang Yeh[13] studied the nonlinear dynamic behaviors and chaos motion about rectangular plate with thermo-mechanical coupling the time-history, phase plane plots, Poincare maps, power spectra, maximum Lyapunov exponent, bifurcation diagrams and fractal dimension. Huang and Li[15] studied the Nonlinear Vibration of a Viscoelastic Beam Subjected to Both Axial Forces and Transverse Magnetic Field. The objective of this paper is to study the nonlinear dynamic response for thermo-mechanical coupling viscoelastic rectangular plate, which is affected by heat conduction, thermal expansion and viscosity. The governing equations for thermo-mechanical coupling viscoelastic plate are established firstly, and the general nonlinear numerical method for viscoelastic plate is obtained by introducing difference method. Using this method, a kind of special dynamic problem with thermo-mechanical coupling is solved. Finally, synthetically using several methods in dynamic systems, the dynamic properties of the thermomechanical coupling viscoelastic plate are sufficiently revealed.

2

Nonlinear Mathematical Model of a Viscoelastic Plate

Consider a viscoelastic plate with the thickness h and edge lengths a and b respectively, as shown in Fig. 1. The plate is subjected to uniform distributed forces px and py on the edges respectively, in x and y directions, transverse load Q = f cos(ωt) and thermal loading. 2.1 The nonlinear constitutive equations of material Based on Von Karman theory of the plate with large deflection and the Boltzmann superposition principle, the nonlinear constitutive equations for the thin plate can be obtained as following:

NONLINEAR DYNAMIC BEHAVIORS OF A THERMO-MECHANICAL COUPLING VISCOELASTIC PLATE

σx =

355

E [(εxx + νεyy ) − (1 + ν)αT ] 1 − ν2 t ˙ E(t − τ ) [(εxx (τ ) + νεyy (τ )) − (1 + ν)αT (τ )]dτ + 1 − ν2 0

E [(εyy + νεxx ) − (1 + ν)αT ] 1 − ν2 t ˙ E(t − τ ) [(εyy (τ ) + νεxx (τ )) − (1 + ν)αT (τ )]dτ + 1 − ν2 0 Fig. 1 The geometry model of visco t ˙ E(t − τ ) elastic rectangular plate (1) εxy (τ )dτ = Gεxy + G E 0

σy =

σxy

where E(t) is the stress relaxation function. G is shear modulus, α is the thermal expansion coefficient, T is the temperature field acting on the plate. 2.2 Geometry equations The nonlinear strains of plate are 0 εxx = εxx + zKxx , 0 εxx

∂u 1 = + ∂x 2

∂w ∂x

Kxx = −(∂ 2 w/∂x 2 ),

0 εyy = εyy + zKyy ,

2 ,

0 εxx

∂v 1 = + ∂y 2

0 εxy = εxy + zKxy

Kyy = −(∂ 2 w/∂y 2 ),

∂w ∂y

2 ,

0 εxy =

∂u ∂v ∂w ∂w + + ∂y ∂x ∂x ∂y

Kxy = −2(∂ 2 w/∂x∂y) (2)

0 , ε 0 and ε 0 are the neutral plan strains; K , K where (·),x = ∂(·)/∂x, (·),xx = ∂ 2 (·)/∂x 2 , . . . , εxx xx yy yy xy and Kxy are the plate curvatures; u, v and w are the displacements neutral plane of the plate in x, y and z directions, respectively.

2.3 Equilibrium-motion equations The equilibrium-motion equations of the rectangular plate are Nx,x + Nxy,y = 0 Nxy,x + Ny,y = 0 Mx,xx + 2Mxy,xy + My,yy + Nx w,xx + 2Nxy w,xy + Ny w,yy − px w,xx − py wyy + Q = ρhw,tt (3) where ρ is the material density of the plate. Nx , Ny and Nxy are the membrane forces, defined by h/2 h/2 h/2 Nx = −h/2 σx dz = F,yy , Ny = −h/2 σy dz = −F,xy , Nxy = −h/2 σxy dz = F,xx , where F is

356


h/2 the stress function. Mx , My and Mxy are the membrane moments defined by Mx = −h/2 zσx dz, h/2 h/2 My = −h/2 zσy dz, Mxy = −h/2 zσxy dz. By (1) and definitions of Mx , My , Mxy , following equations are obtained: Eh3 [(w,xx + νw,yy ) + (1 + ν)αM T ] 12μ t ˙ E(t − τ )h3 [(w,xx (τ ) + νw,yy (τ )) + (1 + ν)αM T (τ )]dτ − 12μ 0

Mx = −

Eh3 [(w,yy + νw,xx ) + (1 + ν)αM T ] 12μ t ˙ E(t − τ )h3 [(w,yy (τ ) + νw,xx (τ )) + (1 + ν)αM T (τ )]dτ − 12μ 0 ˙ Gh3 Gh3 t E(t − τ) =− w,xy − w,xy (τ )dτ 6 6 0 E

My = −

Mxy

(4)

where μ = 1 − ν 2 · M T and N T are the bending moment and membranous force due to temperature h/2 h/2 respectively, defined by M T = h123 −h/2 T (x, y, z, t)zdz, N T = h1 −h/2 T (x, y, z, t)dz. By (3), (4) and the definitions of Nx, Ny, and Nxy , following equation is obtained: E 0 h3 T T + M,yy )] [w,xxxx + w,yyyy 2w,xxyy + (1 + ν)α(M,xx − 12μ t T T ˙ − E(t − τ )[w,xxxx (τ ) + w,yyyy (τ ) + 2w,xxyy (τ ) + (1 + ν)α(M,xx (τ ) + M,yy (τ ))]dτ 0

+ w,xx F,yy − 2w,xy F,xy + w,yy F,xx + Q − px w,xx − py w,yy − ρhw,tt = 0

(5)

2.4 Thermo-mechanical coupling response with process of heat transfer Assuming that the stain field and temperature field are interdependent, the energy equation based on the linear thermo-elastic theory is k(T,xx + T,yy + T,zz ) − Cε T,t − T0 β(εxx,t + εyy,t ) = 0

(6)

where k is the thermal conductivity. Cε = ρc, where c is the specify heat. T0 is the reference temperature. β = αE(0)/1−ν is the thermo-mechanically coupling factor. Taking into account the plate is a thin one, the assumption that the temperature field varies linearly along the z-direction is valid. So the temperature field is expressed as T (x, y, x, t) = [(T )h/2 + (T )−h/2 ]/2 + [(T )h/2 − (T )−h/2 ](z/ h)

(7)

where (T )h/2 and (T )−h/2 are the temperatures on the upper and lower surfaces of the plate, respectively.

357


Assuming (T )h/2 = B1 (t) sin(π x/a) sin(πy/b),

(T )−h/2 = B2 (t) sin(π x/a) sin(πy/b)

(8)

By (7), (8) and the definitions of M T and N T field, following equations hold. M T = [(T )h/2 − (T )−h/2 ]/ h = (B1 (t) − B2 (t))/ h · sin(π x/a) sin(πy/b) N T = [(T )h/2 + (T )−h/2 ]/ h = (B1 (t) + B2 (t))/ h · sin(π x/a) sin(πy/b)

(9)

Let B(t) = (B1 (t) − B2 (t))/ h, C(t) = (B1 (t) + B2 (t))/ h, (9) can be written as M T = B(t) sin(π x/a) sin(πy/b)

(10a)

T

N = C(t) sin(π x/a) sin(πy/b)

(10b)

The convective boundary conditions on the top and bottom surfaces of the plate are kT,z |z=h/2 = H [T∞ − (T )h/2 ],

kT,z |z=−h/2 = −H [T∞ − (T )−h/2 ]

(11)

where H is the boundary conductance and T∞ is the temperature of surrounding medium. Substitute (2) and (7)–(9) to (6), multiplying the resulting equation by z0 and z1 , and then integrating along the thickness h, the governing equation can be obtained as T T kN,xx + kN,yy +k

2H T T0 β (N − T∞ ) − N,tT (Cε + 2αT0 β) − (1 − ν)(F,xxt + F,yyt ) = 0 kh hE (12)

T T kM,xx −k + kM,yy

where M =

3

12 h2

hH + 1 M T − Cε M,t + T0 β[w,xxt + w,yyt ] = 0 2M

(13)

Hh 2k .

Stress Function

The boundary conditions of the plate are w = w,xx = 0 at x = 0, a;

w = w,yy = 0 at y = 0, b

(14)

Based on Galerkin’s method, the displacement function which satisfied the boundary condition can be assumed as follows: w(t, x, y) = A(t) sin(π x/a) sin(πy/b)

(15)

The compatibility equation of the plate is 2 εx,yy + εy,xx − εxy,yy = w,xy − w,xx w,yy

(16)


358

Substitute (2) and (15) to (16), the stress function can be obtained F (x, y, t) = fa cos(2π x/a) + fb cos(2πy/b) + fc sin(π x/a) sin(πy/b) t ˙ − τ )/E0 · [fa cos(2π x/a) + fb cos(2πy/b) + E(t 0

+ fc sin(π x/a) sin(πy/b)]dτ

(17)

where fa = a 2 E0 A2 (t)h/(32b2 ), fb = b2 E0 A2 (t)h/(32a 2 ), fc = 2a 2 b2 (1 + U )C(t)h/[π 2 (a 4 + b4 + 2a 2 b2 )]. Substitute (17) to (5), (12), (13), three governing equations which just have three unknown functions w, M T and N T can be obtained.

4

Dynamic Governing System with Thermo-Mechanical Coupling

Substitute (10), (15), (17) to (5), (12)–(13), the nonlinear dynamic system about A(t), B(t) and C(t) can be obtained (a1 + a2 + 2a3 )A(t) + (a4 + a5 )B(t) t ˙ E(t − τ ) + [(a1 + a2 + 2a3 )A(τ ) + (a4 + a5 )B(τ )dτ E 0 ˙ A(t) 48π 2 fc − τ ) 48π 2 fc 9π 4 (fa + fb ) A(t) t E(t 9π 4 (fa + fb ) − − − − dτ Dx 18ab 18ab Dx 0 E 18ab 18ab ρha8 ∂ 2 A(t) a7 A(t)

− Q=0 px a4 + p y a5 + Dx Dx ∂t 2 Dx 12 hH + 1 a8 − B (t)(Cε a8 ) + A (t)T0 β(a4 + a5 ) = 0 B(t) ka4 + ka5 − k 2 2k h 2H 2H ∂C(t) C(t) a4 + a5 + k a8 − k a7 T ∞ − (Cε + 2αT0 β)a8 kh kh ∂t ∂C(t) 1 a b ∂A(t) + T0 β(1 − ν) αa8 − + A(t) ∂t 3 b a ∂t t ˙ E(t − τ ) b ∂A(τ ) ∂C(τ ) 1 a + T0 β(1 − ν) αa8 − + A(t) dτ = 0 E ∂τ 3 b a ∂t 0 +

where a1 =

π 4b , 4a 3

a8 =

ab , 4

a2 =

π 4a , 4b3

= α(1 + υ),

a3 =

π4 , 4ab

Dx =

Eh3 12μ

a4 = −

π 2b , 4a

a5 = −

π 2a , 4b

a7 =

4ab π2

(18) (19)

(20)


359

In order to non-dimensionalize the above equations, the following variables and some non-dimensional parameters are introduced ¯ A(t) = A(t)h, θ = ωt, τ0 =

B(t) =

h ¯ B(t), a2α

dθ = ωdt,

a 2 Cε , k

β¯ =

a , b

λ=

αT0 β , Cε

C(t) = λ1 =

h2 ¯ C(t) a2α

h , a

ω2 =

a4 a2 Dx 4 1 + , + 2 π ρha 4 b4 b2

e(t) = E(t)/E(0)

Equations (18)–(20) are transformed as ¨ = −A1 A(t) − A2 B(t) − A3 A3 (t) + A4 A(t)C(t) + A6 cos(t) − A5 A(t)

t

− A2

t 0

t

e(t ˙ − τ )B(τ )dτ − A3 A(t)

0

e(t ˙ − τ )A(τ )dτ

t

e(t ˙ − τ )A (τ )dτ + A4 A(t) 2

0

e(t ˙ − τ )C(τ )dτ

0

(21a) ˙ ˙ B(t) = B1 B(t) + B2 A(t) ˙ ˙ C(t) = −C1 C(t) + C2 αT∞ + C3 A(t)A(t) + C3

t

− C4

(21b)

t

˙ )A(τ )dτ e(t ˙ − τ )A(τ

0

˙ )dτ e(t ˙ − τ )C(τ

(21c)

0

where Ak1 = 1 − p1, Ak4 =

Ak2 = −

128μλ2 , π 4 (1 + λ2 )3

Bk1 = −

1−υ π 2 (1 + λ2 )

Ak5 = 1,

,

Ak3 =

3μ 1 + λ4 , 4 (1 + λ2 )2

Ak6 = f 1

π 2 (1 + λ2 ) + 12(M + 1)/λ21 , ωτ0

¯ + λ2 ), Bk2 = −π 2 β(1

Ck1 =

−π 2 (1 + λ2 ) + 4M/λ21 , ¯ − υ) − (1 + 2β)] ¯ ωτ0 [β(1

Ck3 =

¯ − υ)(1 + λ2 ) 4β(1 , ¯ − υ) − (1 + 2β)] ¯ 3[β(1

Ck3 =

(px + py λ2 )a 4 , Dx b2 π 2 (1 + λ2 )2

16f a 4 π 6 Dx h(1 + λ2 )2

p1 =

f1 =

Ck2 =

64M ¯ − υ) − (1 + 2β)] ¯ ωτ0 λ41 [β(1

¯ − υ) β(1 ¯ ¯ β(1 − υ) − (1 + 2β)

,

360


Expressions (21) are the integral-differential dynamic system, which reflect coupling relationship with transverse A(t), temperature difference B(t) between top and bottom surface and the temperature C(t) of the middle surface.

5

Discussions About Models

5.1 The elastic plate with thermo-mechanical coupling When the non-dimensional stress relaxation function e(t) = E(t)/E0 is constant, i.e., the integral items of the system (21) are all ignored, the dynamic system of the elastic plate with thermo-mechanical coupling can be simplified from the one of the viscoelastic plate with thermo-mechanical coupling to following: ¨ = −A1 A(t) − A2 B(t) − A3 A3 (t) + A4 A(t)C(t) + A6 cos(t) A(t)

(22a)

˙ ˙ B(t) = B1 B(t) + B2 A(t)

(22b)

˙ ˙ C(t) = −C1 C(t) + C2 αT∞ + C3 A(t)A(t)

(22c)

The differential dynamic system is the same with (39)–(41) from article[13] , so it is a special case. 5.2 The model of viscoelastic plate Let B(t) = C(t) ≡ 0, the dynamic system of viscoelastic plate without heat effect can be obtained as following: t ¨ = −A1 A(t) − A3 A3 (t) + A6 cos(t) − A5 e(t ˙ − τ )A(τ )dτ A(t) 0

t

− A3 A(t)

e(t ˙ − τ )A2 (τ )dτ

(23)

0

The integral-differential dynamic system corresponds to (6) from article[14] . 5.3 The model of thermal viscoelastic plate without heat transfer 5.3.1 The model of viscoelastic plate with varied temperature field When the effect of heat transfer is ignored, the temperature of middle plane is equal to the one of top and bottom surface. So the temperature difference between top and bottom surface B(t) = 0, and the temperature of middle plane C(t) = Tm (t). Then, the dynamic system of viscoelastic plate with varied temperature field is simplified as following: ¨ = −A1 A(t) − A3 A3 (t) + A4 A(t)Tm (t) + A6 cos(t) A(t) t t − A5 e(t ˙ − τ )A(τ )dτ − A3 A(t) e(t ˙ − τ )A2 (τ )dτ 0

0 t

+ A4 0

e(t ˙ − τ )A(τ )Tm (τ )dτ

(24a)


˙ T˙m (t) = C2 αT∞ − C3 A(t)A(t) − C3

t

˙ )A(τ )dτ − C4 e(t ˙ − τ )A(τ

0

t 0

361

e(t ˙ − τ )T˙m (τ )dτ (24b)

5.3.2 The model of viscoelastic plate with constant temperature field When it is a constant temperature, i.e., C(t) = Tm (t), T0 , (24b) is an identity and the dynamic system of viscoelastic plate with constant temperature field is obtained as following: ¨ = −A1 A(t) − A3 A3 (t) + A4 T0 A(t) + A6 cos(t) A(t) t t t 2 − A5 e(t ˙ − τ )A(τ )dτ − A3 A(t) e(t ˙ − τ )A (τ )dτ + A4 T0 e(t ˙ − τ )A(τ )dτ 0

0

0

(25) Furthermore, let T0 = 0, the dynamic system (23) of viscoelastic plate without temperature field can be simplified. 5.3.3 The model of elastic plate with varied temperature field When the integral items of system (24) are all ignored, the dynamic system of elastic plate with varied temperature field is simplified as: ¨ = −A1 A(t) − A3 A3 (t) + A4 A(t)Tm (t) + A6 cos(t) A(t) ˙ T˙m (t) = C2 αT∞ − C3 A(t)A(t)

(26a) (26b)

5.3.4 The model of elastic plate with constant temperature field When the non-dimensional stress relaxation function e(t) = E(t)/E0 is constant and the integral items of system (26a) are all removed, the dynamic system of geometrical nonlinear thin plate with constant temperature field and subjected to both actions of an alternating periodic transverse external excitation and in-plane uniform distributed force is simplified from (26a) to following: ¨ = −A1 A(t) − A3 A3 (t) + A4 T0 A(t) + A6 cos(t) A(t)

(27)

Let T0 = 0, the dynamic system of elastic plate without temperature field is established. ¨ = −A1 A(t) − A3 A3 (t) + A6 cos(t) A(t)

6

(28)

Nonlinear Numerical Calculation Method

It is difficult to find the exact solution of an integral-differential dynamical system. But by introducing difference method, an ordinary differential one, which is the general form for nonlinear dynamical system of the viscoelastic plate with thermo-mechanical coupling, can be obtained. Arbitrary functions h(t) and g(t) satisfy the following formula: t R ˙ − τ )g(τ )dτ = 1/2 h(t [g(τr+1 ) + g(τr )][h(τR+1 − τr+1 ) − h(τR+1 − τr )] (29) 0

r=1

where the time scale is divided into intervals by the time values, τr = 1, 2, . . . , R + 1, with τ1 = 0 and τR+1 = τ .

362


According to (29), replacing integral items with difference forms, the system (21) is transformed as: ¨ = −A1 A(t) − A2 B(t) − A3 A3 (t) + A4 A(t)C(t) + A6 cos(t) A(t) − A5 /2

R

[A(τr+1 ) + A(τr )][e(τR+1 − τr ) − e(τR+1 − τr )]

r=1

− A2 /2

R

[B(τr+1 ) + B(τr )][e(τR+1 − τr ) − e(τR+1 − τr )]

r=1

− A3 A(t)/2

R

[A2 (τr+1 ) + A2 (τr )][e(τR+1 − τr ) − e(τR+1 − τr )]

r=1

− A4 A(t)/2

R

[C(τr+1 ) + C(τr )][e(τR+1 − τr ) − e(τR+1 − τr )]

r=1

˙ ˙ B(t) = B1 B(t) + B2 A(t) ˙ ˙ C(t) = −C1 C(t) + C2 αT∞ + C3 A(t)A(t) − C4 /2

R

˙ r+1 ) + C(τ ˙ r )][e(τR+1 − τr ) − e(τR+1 − τr )] [C(τ

r=1

+ C3 /2

R

˙ r+1 )A(τr+1 ) + A(τ ˙ r )A(τr )][e(τR+1 − τr ) − e(τR+1 − τr )] [A(τ

(30)

r=1

where the time scale is divided into intervals by the time values, τr = 1, 2, . . . , R + 1, with τ1 = 0 and τR+1 = τ .

7

A Kind of Special Coupling Model of Viscoelastic Plate

Assume that the material of the plate is a standard linear solid. In this case, e(t) = e0 + e1 e−α1 t ,

e(t ˙ − τ ) = −e1 e−α1 t · αeατ = −ϕ(t) · ψ(τ )

(31)

where e1 is a parameter about the material, α1 is reciprocal of the relaxation time of the material. Introducing the new variables as following: t t ˙ y1 = A(t), y2 = A(t), y3 = ϕ ψ(τ )A(τ )dτ, y4 = ϕ ψ(τ )A2 (τ )dτ, y5 = ϕ

0 t

˙ )A(τ )dτ, ψ(τ )A(τ

y6 = B(t),

y7 = ϕ

0

y9 = ϕ 0

0 t

ψ(τ )C(τ )dτ,

y10 = ϕ 0

t

˙ )dτ ψ(τ )C(τ

0 t

ψ(τ )B(τ )dτ,

y8 = C(t),


363

Non-autonomous ten dimensional dynamical system is obtained as following, y˙1 = y2 , y˙2 = −Ak1 y1 − Ak2 y6 − Ak3 y13 + Ak4 y1 y8 + Ak5 y3 + Ak2 y7 + Ak3 y1 y4 − Ak4 y1 y9 + Ak6 cos(t), y˙3 = α1 (e1 y1 − y3 ), y˙4 = α1 (e1 y12 − y4 ), y˙5 = α1 (e1 y1 y2 − y5 ), y˙6 = Bk1 y6 + Bk2 y2 , y˙7 = α1 (e1 y6 − y7 ), y˙8 = −Ck1 y8 + Ck2 αT∞ + Ck3 y1 y2 + Ck4 y10 − Ck3 y5 , y˙9 = α1 (e1 y8 − y9 ), y˙10 = α1 (e1 y˙8 − y10 ) = α1 [e1 (−Ck1 y8 + Ck2 αT∞ + Ck3 y1 y2 + Ck4 y10 − Ck3 y5 ) − y10 ]) (32) Its initial values are: y1 = 0, y2 = 1, y3 = 0, y4 = 0, y5 = 0, y6 = 0, y7 = 0, y8 = 0, y9 = 0, y10 = 0.

8

Numerical Results and Discussions

The Runge-Kutta fourth order method is adopted for the numerical analysis for dynamic system with various parameters. The time-history, phase plane plots, Poincare maps, power spectra, maximum Lyapunov exponent, bifurcation diagrams for the viscoelastic plate with thermo-mechanical coupling versus the external in-plane force p1 and external lateral force f1 are obtained. (1) Figure 2 reveals the time-history, phase plane plots, Poincare maps, power spectra for the plate with the effect of heat conduction, thermal expansion and viscosity, the parameters are λ = 1; λ1 = 0.01; μ = 0.3; M = 0.02; β1 = 0.3; r = 3.5e6; e1 = 0.9; α1 = 0.006; p1 = 1.6; f1 = 6.0. It shows that: The phase trajectory is disorderly, the power spectrum is substantially broadened and Poincare map is constituted with infinite points, and two Lyapunov exponents are positive according to Table 1. Above results show that hyperchaos motion appears in this case. Figure 3(a) presents the global bifurcation diagrams of the plate amplitude “A” versus the external in-plane force p1 for f1 = 7.3. It can be found in Fig. 3(a) that the period of 2T motion of the plate amplitude A appears in the range of 0 < p1 < 0.1, a period of 4T motion when 0.1 < p1 < 0.32, and the alterative between period and chaos motion begins to occur as the in-plane load is greater than p1 = 0.32. Figure 3(b) presents the global bifurcation diagrams of the plate amplitude “A” versus external lateral force f1 for p1 = 1.3. It can be observed from Fig. 3(b) that period window appears, and then complete chaos as transverse load increasing.

364


Fig. 2

Dynamic response of thermo-mechanical coupling viscoelastic rectangular plate (λ = 1; λ1 = 0.01; μ = 0.3; M = 0.02; β1 = 0.3; r = 3.5e6; e1 = 0.9; α1 = 0.006; p1 = 1.6; f1 = 6.0)

Table 1 Lyapunov exponent of different models The Models The viscoelastic plate with thermo-mechanically coupled The viscoelastic plate without temperature field The elastic rectangular plate with thermo-mechanically coupled

Positive Lyapunov Exponent λ1 = 0.17001 λ2 = 0.00217 λ1 = 0.18649 λ1 = 0.05623

State Hyperchaos Chaos Chaos


365

Fig. 3 Global bifurcation diagram of thermo-mechanical coupling viscoelastic rectangular plate

Fig. 4

Dynamic response of viscoelastic rectangular plate (λ = 1; λ1 = 0.01; μ = 0.3; M = 0.0; β1 = 0.0; r = 0.0; e1 = 0.9; α1 = 0.006; p1 = 1.6; f1 = 6.0)

(2) Prescind from the effect of the temperature field (M = 0.0; β1 = 0.0; r = 0.0), dynamic behavior for viscoelastic plate subjected to both actions of an alternating periodic transverse external excitation and in-plane uniform distributed force can be obtained. Figure 4 reveals the time-history, phase plane plots, Poincare maps, power spectra for the plate with the effect of viscosity, the parameters are

366


Fig. 5 Global bifurcation diagram of viscoelastic rectangular plate

λ = 1; λ1 = 0.01; μ = 0.3; e1 = 0.9; α1 = 0.006; p1 = 1.6; f1 = 6.0. The phase trajectories and Poincare map of the plate amplitude motion are irregular under these particular system conditions, meanwhile, the power spectrum is very broad. It can be seen that there are a large number of excited frequencies and periodic motion is no longer apparent. The maximum Lyapunov exponent is positive in Table 1. From above results, it can be found that the chaos motion appears in this case. Figure 5(a) presents the global bifurcation diagrams of the plate amplitude “A” versus the external in-plane force p1 for f1 = 7.3. Figure 5(a) shows that it is a period of 2T motion when 0 < p1 < 0.53, intermittent chaos appear in the range of 0.53 < p1 < 2.08, and then replaced by a period of 4T motion when p1 > 2.08. Figure 5(b) presents the global bifurcation diagrams of the plate amplitude “A” versus external lateral force f1 for p1 = 1.3. It can be found in Fig. 5(b) that the period motion appears in the range of 0 < f1 < 1.74 and period and chaos motion alternate as the transverse load increasing, complete chaos occur finally. (3) Let α1 = 0.0, e = 0.0, the dynamic behaviors of the thermo-mechanically coupled elastic rectangular plate with the effect of heat conduction are obtained. Figure 6 shows the corresponding time-history, phase plane plots, Poincare maps and power spectra of the elastic plate with the effect of heat conduction and thermal expansion, the parameters are λ = 1; λ1 = 0.01; μ = 0.3; M = 0.02; β1 = 0.3; r = 3.5e6; e1 = 0.0; α1 = 0.0; p1 = 1.6; f1 = 6.0. The phase trajectory is disorderly, Poincare map shows that chaos contains some holes, and the power spectrum is broadened, meanwhile, the maximum Lyapunov exponent is positive in Table 1 chaos motion appears. Figure 7(a) presents the global bifurcation diagram of the plate amplitude “A” versus the external in-plane force p1 for f1 = 7.3. Figure 7(a) shows that period of 2T motion of the plate amplitude A appears in the range of 0 < p1 < 0.59, and then period doubling bifurcation obviously as p1 increasing, and finally, complete chaos appear when p1 > 1.80. Figure 7(b) presents the global bifurcation diagram of A versus external lateral force f1 for p1 = 1.3. Figure 7(b) reveals that period window and chaos appear alternately as f1 increasing, and the interval are smaller and smaller, and then complete chaos occur. (4) Table 1 shows the positive Lyapunov exponent about three kind of dynamic models. It shows that the chaos is greater than those of viscoelastic plate, which without the influence of temperature, and thermo-mechanically coupled elastic rectangular plate. Especially, the motion state of hyperchaos appears.


Fig. 6

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Dynamic response of thermo-mechanical coupling elastic rectangular plate (λ = 1; λ1 = 0.01; μ = 0.3; M = 0.02; β1 = 0.3; r = 3.5e6; e1 = 0.0; α1 = 0.0; p1 = 1.6; f1 = 6.0)

Fig. 7 Global bifurcation diagram of thermo-mechanical coupling elastic rectangular plate

368

9


Conclusions

The study indicates that: (1) If the effect of heat expand and thermal conduction are ignored, the model can be reduced to the one of the viscoelastic plate[14] . (2) If the thermal conductive coefficient is zero and the heat expand coefficient is non-zero, the model is the same with the dynamic model of a viscoelastic plate with the influent of heat expand and the damage coefficient is zero[8] . (3) If the effect of viscosity is ignored, i.e., the model does not include the integral items, the model can be simplified to the thermo-mechanical coupling dynamic model of elastic plate[13] . (4) The dynamic properties of the thermo-mechanical coupling viscoelastic plate subjected to both actions of an alternating periodic transverse external excitation and in-plane uniform distributed force are abundant, and the chaos is greater than the viscoelastic plate’s without the influence of temperature, especially, the motion state of hyperchaos appears.

Acknowledgments This research is supported by the Foundation of Key Lab. of Bridge Structure Eng. of Ministry of Communications, Chongqing Jiaotong University (2006–1).

References [1] James, M. and Caruthersa, D. B., A thermodynamically consistent, Nonlinear Viscoelastic Approach for Modeling Glassy Polymers, Polymer, Vol. 45, pp. 4577–4597, 2004. [2] Qu, J. L. and Zhang, Y. T., Themoviscoelasticity finife element method, Journal of Hebei Mining and Civil Engineering Institute, Vol. 1, pp. 11–15, 1997. [3] Drozdov, A. D., Agarwal, S. and Gupta, R. K., Linear thermo-viscoelasticity of isotactic polypropylene, Computational Materials Science, Vol. 29, pp. 195–213, 2004. [4] Luo, W. B., Yang, T. Q. and Wang, X. Y., Effects of temperature and stress level on the free volume in high polymers, Polymer Materials Science and Engineering, Vol. 21, pp. 11–15, 2005. [5] Akhtar, S. K., Oscar, L. P. and Rehan, K., Thermo-mechanical large deformation response and constitutive modeling of viscoelastic polymers over a wide range of strain rates and temperatures, International Journal of Plasticity, Vol. 22, pp. 581–601, 2006. [6] Peng, F. and Fu, Y. M., The Analysis of nonlinear dynamic stability for viscoelastic plates in hygrothermal environment, Natur. Sci. J. Xiangtan Univ., Vol. 27, pp. 51–66, 2005. [7] Anastasia, M. and Aravind, N., Characterization of thermo-mechanical and long-term behaviors of multi-layered composite materials, Composites Science and Technology, Vol. 66, pp. 2907–2924, 2006. [8] Fu, Y. M., Tang, K. K. and Wang, Y., Analysis of nonlinear dynamic response for viscoelastic rectangular plates with damage under varied temperature field, Act Mechanic Solid Sinic, Vol. 27(3), pp. 243–248, 2006.


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[9] Ilyasov, M. H., Dynamic stability of viscoelastic plates, Internat. J. Engrg. Sci., Vol. 45, pp. 111–122, 2007. [10] Qin, Y. M. and Fang, J. A., Global attractor for a nonlinear thermoviscoelastic model with a nonconvex free energy density, Nonlinear Analysis, Vol. 65, pp. 892–917, 2006. [11] Argyris, Chaotic vibrations of a nonlinear viscoelastic beam, Chaos, Solitons and Fractals, Vol. 7(2), pp. 151–163, 1996. [12] Chen, L. Q. and Cheng, C. J., Stability and chaotic motion in columns of nonlinear viscoelastic material, Appl. Math. Mech., Vol. 21(9), pp. 890–896, 2000. [13] Yeh, Y. L., Chaotic and bifurcation dynamic behavior of a simply supported rectangular orthotropic plate with thermo-mechanical coupling, Chaos, Solitons and Fractals, Vol. 24, pp. 1243–1255, 2005. [14] Cheng, C. J. and Zhang, N. H., Chaotic and hyperchaotic behaviors of viscoelastic rectangular plates under transverse periodic load, Act Mechanic Sinic, Vol. 30(6), pp. 690–699, 1998. [15] Huang, Z. H., Du, C. C. and Li, Y. H., Nonlinear vibration of a viscoelastic beam subjected to both axial forces and transverse magnetic field, Advances in Vibration Engineering, Vol. 10(2), pp.

Partial Coherence Method of Transfer Functions on Structural Vibration Under Correlated Excitations S. M. Li1 , C. W. Lim2 and M. J. An1 1 College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China [email protected] 2 Department of Building and Construction, City University of Hong Kong, Kowloon, Hong Kong, China [email protected] Abstract Aim at the causality and priority of inputs in the vibration system of multi-inputs and multi-outputs, a new method has been proposed based on the comparison between calculated value and true value of transfer function. By comparing the value of transfer function between inputs and outputs, the causality of inputs and priority of causal correlated inputs can be distinguished directly. The method has been verified by simulation and experiment on an automobile chassis. Simulation analysis shows effectiveness of the method. The partial coherence approach has been able to analyze the relations successfully between the correlated inputs and outs of automobile classis based on the new method. Experiment and simulation results are consistent. Calculation of the actual contribution of each input to the outputs in the correlated frequency are also presented. Keywords: Correlated inputs, Source identification, Partial coherence, Transfer function

1

Introduction

For systems containing a number of excitation sources, it is necessary to accurately identify the role of each source to the key points in order to solve the noise and diagnosis problems of vibration. Because excitations of components come from the same mechanical system, it results in correlation between excitation sources. Hence the role of excitation source will be enlarged during calculation and incorrect conclusions could be drawn[1, 2] . Therefore, the impact of correlation between excitation with response has an important and practical engineering significance. At present, the partial coherence approach is the most common method for calculating the impact of correlation between excitation and response[3–5] . The prior conditions for successful application of the partial coherence method are whether or not there exists causes-and-effects and the correct priority of excitations due to the causes-and-effects. Three methods to judge the priority of causal correlation of source signals are the coherence function method[1] , the impulse response function method[6, 7] and Hilbert transform[8, 9] . The coherence function method determines the priority by comparing the coherence function between sources and output response signals, the impulse response function method determines the priority by calculating the difference of two related inputs in time domain, and Hilbert transform determines the priority according to the Cauchy Riemann condition for calculating the frequency response function between correlated inputs, as well as whether or not the real and imaginary ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011


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S. M. LI, C. W. LIM AND M. J. AN / ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011

parts of the frequency response function meet the Hilbert transform. Each of above the methods has insufficiency. Before to application, they need other methods to identify whether the correlations between of inputs belong to causes-and-effects relations. Furthermore the ordering between of input sequences under the causes-and-effects relations is limited in some fields[10] . This paper compares the transfer function between inputs and responses to the transfer function of two actual points in the system. It determines the cause-and-effect correlations between the inputs in order to avoid unnecessary calculation for unknown causality. At the same time, we can directly determine the priority of cause-and-effect correlative between the inputs by comparing the transfer functions.

2

Partial Coherence Theory of Transfer Function

2.1 Response expression of multi correlated inputs Multi-input/multi-output means that multiple outputs are produced when there are Multiple numbers of inputs. Figure 1 shows there are n inputs xi (t), (i = 1, 2, . . . , n). Responds to each output yk (t), there are m impulse response functions, that are hik (t), so there are m frequency response functions Hik (t). There are m × n impulse response functions as there are m outputs. The matrix is ⎡ ⎤ h11 (t) h12 (t) · · · h1n (t) ⎢ h (t) h (t) · · · h (t) ⎥ 22 2n ⎢ 21 ⎥ h(t) = ⎢ (1) .. .. ⎥ ⎢ .. ⎥ ⎣ . . ··· . ⎦ hm1 (t) hm2 (t) · · · hmn (t)

Fig. 1 Linear system model

The relationship between frequency response function and impulse response functions is +∞ h(t)e−2πf tj dt H (f ) = −∞

Transform (1) according to (2), m × n frequency response function matrix H (f ) is got, ⎡ ⎤ H11 (f ) H12 (f ) · · · H1n (f ) ⎢ H (f ) H (f ) · · · H (f ) ⎥ 22 2n ⎢ 21 ⎥ ⎥ H (f ) = ⎢ .. .. ⎢ .. ⎥ ⎣ . ⎦ . ··· . Hm1 (f ) Hm2 (f ) · · · Hmn (f ) Fourier transforms of its input, and be expressed as n × 1 column vector, ⎡ ⎤ X1 (f ) ⎢X (f )⎥ ⎢ 2 ⎥ ⎥ X(f ) = ⎢ ⎢ .. ⎥ ⎣ . ⎦ Xn (f )

(2)

(3)

(4)

PARTIAL COHERENCE METHOD OF TRANSFER FUNCTIONS

373

Expressed its outputs as m × 1 column vector, ⎡

⎤ Y1 (f ) ⎢ Y (f ) ⎥ ⎢ 2 ⎥ ⎥ Y (f ) = ⎢ ⎢ .. ⎥ ⎣ . ⎦ Ym (f )

(5)

According to (3), (4) and (5), following equation is available, Y (f ) = H (f )X(f )

(6)

And the multi correlated inputs equation is, Xi (f ) = Hijf (f )Xii (f )

(7)

Put (7) into (6), Y (f ) = H (f )X(f ) = H (f )Hijf (f )Xii (f )

(8)

Or it can be expressed as, ⎡

⎤ ⎡ Y1 (f ) H11 (f ) ⎢ Y (f ) ⎥ ⎢ H (f ) ⎢ 2 ⎥ ⎢ 21 ⎢ . ⎥=⎢ . ⎢ . ⎥ ⎢ . ⎣ . ⎦ ⎣ . Ym (f ) Hm1 (f ) ⎡

1

H12 (f ) H22 (f ) .. . Hm2 (f )

⎤ H1n (f ) H2n (f ) ⎥ ⎥ ⎥ .. ⎥ ⎦ ··· . · · · Hmn (f ) ··· ···

0 1

⎢H12f (f ) ⎢ ⎢ H13f (f ) H23f (f ) ×⎢ ⎢ ⎢ .. ⎣ . H1nf (f ) H2nf (f )

0

0

1 .. H3nf (f )

⎤ X11 (f ) ⎥ ⎢ X22 (f ) ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ × ⎢ X33 (f ) ⎥ ⎥ ⎢ ⎥ .. ⎥ ⎢ .. ⎥ .⎦ ⎣ . ⎦

0

.

...

1

⎤

⎡

(9)

Xnn (f )

Equations (8) or (9) is the response expression of multi correlated inputs. 2.2 Principle of partial coherence function One of the coherence functions is the partial coherence function which shows the linear relationship of condition data. Partial coherence function method determines the contributions of every vibration source to response by calculating the value of partial coherence function[1] . Figure 2 shows q random inputs xi (t), (i = 1, 2, . . . , q), they produce a measuring output y(t) according a stationary parameters linear system of q frequency response function Hi (f ), (i = 1, 2, . . . , q). The output y(t) is the sum of expected ideal linear output and possible noise in measuring. n(t) is noise which is not related to xi (t) and y(t). According to the partial coherence function method,

374


we assume the priority of inputs in frequency is x1 (t), x2 (t), . . . , xq (t). According to the condition spectrum, we have the self-spectrum of output y(t) Syy = GY :X + GNN = GY :X1 + GY :X2·1 + · · · + GY :Xq:(q−1)! + GNN

(10)

where GY :X is linear output spectrum, GNN is noise spectrum. Xq:(q−1) is the Finite Fourier transforms of the q input xq (t), it’s the input without the first (q − 1) inputs, and it is the real part of xq (t) to y(t). Equation (10) shows that the output spectrum can be expressed as each input-independent part after removing linear correlation in proportion of outFig. 2 MISO system puts. Each part in (10) can be expressed as ⎧ 2 G 2 G ⎪ GY :X2·1 = γ2y·1 GY :X1 = γ1y yy yy·1 ⎪ ⎪ ⎨ .. . ⎪ ⎪ ⎪ ⎩G 2 Y :Xq:(q−1)! = γqy·(q−1)! Gyy·(q−1)! GNN = Gyy·q!

(11)

2 is the partial coherence function between the input xq (t) after removing the first (q − 1) where γqy·(q−1)! 2 inputs. The output γqy·(q−1)! Gyy·(q−1)! is the part of the q inputs in y(t) after removing the linear impact of the first (q − 1) inputs. We observe in (11) that the partial coherence function reflects the real part of each input in output.

2.3 Identification of the causality correlation using the transfer function method Correlation relationship is divided into two parts: causality correlation and not-causality correlation. Because of interaction of signals, there is correlation in causality correlation, for example, the front wheel connected by a rigid shaft of an automobile, when it’s moving, there are shaft ends inputs come from road roughness, and there is correlation because of transfer effect produced by rigid shaft. There is no interaction among signals in not-causality correlation, but the calculated value of correlation function is not zero, correlation exists, for example, there is no rigid shaft between the front wheel and the rear wheel, but the inputs come from road roughness are same, there is correlation among inputs of the front wheel shaft ends and the rear wheel shaft ends. In the field of signal correlation analysis, the causation or non-causation relationships between signals are not directly identified. The impact of incentives on the response by causation or non-causation relationship is not the same, on the condition of the linear systems, the classic theory of linear systems analysis can be used to resolve the incentive role by non-causal related input signals. However, the causal relevance of input response is relatively complex, in order to get an accurate analysis, the analysis of the interaction between input signals is necessary. Thus, in the process of actual data processing, it is essential that clearing correlation between signals is causal relevance or non-causal relationships.


375

The Double-Input and Single-Output (DISO) system as shown in Fig. 3 is used as an example. The x1 (t) and x2 (t) can be assumed as the measured values of excitation signals. x3 (t) is the part of x2 (t) which is not relevant to x1 (t). H12 (f ) is the transfer function of x1 (t) and x2 (t), H1 (f ) and H2 (f ) are the transfer functions between x1 (t), x2 (t) and y(t), respectively. n(t) is the outFig. 3 DISO system put noise which is not relevant to x1 (t) , x2 (t) and y(t). We further assume that x1 (t) and x2 (t) are 2 ≤ 1 in the frequency domain, and correlated, that is the value of coherence function is nonzero, 0 < γ12 2 is the coherence function between x (t) and x (t). γ12 2 1 If the correlation between x1 (t) and x2 (t) is a causality, we have H12 (f ) = 0, that is, x1 (t) and x2 (t) affect each other, and x1 (t) and x3 (t) are the real inputs. If the correlation is not a causality, that they have no effect on each other, then H12 (f ) = 0 and x1 (t) and x2 (t) are the real inputs to every excitation point. For convenience, we assume H1y (f ) and H2y (f ) are the real of transfer function which is x1 (t) and x2 (t) transfer to y(t), but H1y (f ) and H2y (f ) are the compute values of transfer function of x1 (t) and x2 (t) to y(t). From Fig. 3, we observe no effect between relevant inputs x1 (t) and x2 (t) of non-causality, that is, H12 (f ) = 0. The real value of transfer function between two inputs and output is

H1y (f ) = H1 (f ) (12) H2y (f ) = H2 (f ) For causality correlated inputs, H12 (f ) = 0. The real value of transfer function between inputs and output is

H1y (f ) = H1 (f ) + H12 (f )H2 (f ) (13) H2y (f ) = H2 (f ) Whether the relationship of inputs is causality or not, the transfer function of correlative inputs and output can be expressed in the general form

H1y (f ) = H1 (f ) + H12 (f )H2 (f ) (14) H2y (f ) = H2 (f ) + H21 (f )H1 (f ) where H1y (f ) and H2y (f ) are the values of transfer function from the excitation points to output point. H21 (f ) is the transfer function from x2 (t) to x1 (t). Comparison (12) and (14), the calculated values of transfer function are higher than the actual values of transfer function because the values of H12 (f ) and H21 (f ) are zero with respect to the non-causality inputs. Because the transmission direction of signal is known in causality, the calculated values of transfer function between the main inputs and output are equal to the actual value of transfer function. Therefore, according to (14) we can know whether the correlation between inputs is causality or not. From experiment data of correlative input-output, and calculated input-output values of transfer function, the input correlation is non-causality for frequency when the calculated values of transfer function

376


are higher than the actual value. On the other hand, it is causality for frequency when the calculated values of transfer function are equal to the actual value. Thus we can identify the causality by comparing the amplitude of transfer function in every frequency. 2.4 The priority of correlation inputs of transfer function Input priority is very important according to the principle of partial coherence when it is causality between the correlative inputs. In order to better describe the problem, we assume that x1 (t) and x2 (t) are the correlative inputs, and source 2 is partially caused by source 1, which is caused by the transmission of H12 (f ). Comparing (13) and (14), because H21 (f ) = 0, we have

H1y (f ) = H1y (f ) (15) H2y (f ) > H2y (f ) From (15), source 1 is actual while source 2 is enlarged and we have x1 (t) prior to x2 (t). For causality correlation, the priority can be determined. When the calculated transfer functions are equal to the actual transfer functions, the corresponding inputs are prior to other inputs. When the calculated transfer functions are more than the actual transfer functions, the inputs has the next priority. Comparing the transfer function value with the real one of each input and output calculation under the conditions of multi correlated inputs. If all the calculations are equal to the real ones, the multi correlated inputs are not-causality correlation; otherwise, the multi correlated inputs are causality correlation. The priority input is the one which is consistent with them, then to determine the remaining priorities without the linear influence of priority input by continue comparing the transfer function value with real. From the discussion on transfer function method, it only needs to compare the calculated value and the actual value of transfer function. It then identifies causality between correlative inputs and the priority with reference to causality.

3

Simulation Analyses

A three correlated excitations/one output system which is a simple multi correlated excitations system is applied to simulation analysis. In order to express simply, the time parameter t and frequency parameter f are omitted in later expression. By multi correlated excitations (7) and response (9), we can get three correlated inputs equation ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ X1 (f ) 1 0 0 X11 (f ) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 0⎦ × ⎣X22 (f )⎦ (16) ⎣X2 (f )⎦ = ⎣H12f (f ) X3 (f ) H13f (f ) H23f (f ) 1 X33 (f ) Three correlated inputs three outputs system equation ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Y1 (f ) H11 (f ) H12 (f ) H13 (f ) X1 (f ) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣Y2 (f )⎦ = ⎣H21 (f ) H22 (f ) H23 (f )⎦ × ⎣X2 (f )⎦ H31 (f ) H32 (f ) H33 (f ) Y3 (f ) X3 (f )

(17)


377

Fig. 4 A three correlated inputs/one output system

Multi correlated inputs/multi outputs vibration system can be simplified many multi inputs/single output system, so by (16) , (17) a three correlated excitations/one output system model is built, ⎤T ⎡ ⎤ ⎡ ⎤ ⎡ X11 (f ) H1y (f ) 1 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 1 0 ⎦ × ⎣X22 (f )⎦ Y1 (f ) = ⎣H2y (f )⎦ × ⎣H12f (f ) (18) H3y (f ) H13f (f ) H23f (f ) 1 X33 (f ) In the equation, H1y (f ), H2y (f ), H3y (f ) are respectively the transfer functions of x1 , x2 , x3 transfer to y, they are the first-line elements of (17). Equation (18) can be expressed as Fig. 4. It is shows in Fig. 4 that, x11 , x22 , x33 are independent parts of three inputs. x1 , x2 , x3 are three inputs of the system, y is output. Assume x11 , x22 , x33 are respectively the force random signals in frequencies bands 1.0 kHz–1.4 kHz, 1.6 kHz–2.0 kHz, 2.2 kHz–2.6 kHz, and acceleration transfer function are respectively H12f (f ) = 0.57 × exp(−j × 0.0035 × f )

H13f (f ) = 0

H23f (f ) = 0.44 × exp(−j × 0.0015 × f )

H1y (f ) = 0.3 × exp(−j × 0.005 × f )

H2y (f ) = 0.7 × exp(−j × 0.0018 × f )

H3y (f ) = 0.45 × exp(−j × 0.003 × f )

(19)

In (19), assuming H13 (f ) = 0 is to avoid complexity of later frequency correlated analysis, then the three correlated excitation signals of correlation between x1 and x2 , correlation between x2 and x3 is produced. By MATLAB, set sampling frequency as fs = 1000 Hz, sampling Points are 20000; to produce random signals as x11 , x22 , x33 request, set each transfer function value in Fig. 4 according to (19), execute numerical calculation, get the multi inputs x1 , x2 , x3 and output y. In order to be closer to the actual situation, add Gaussian white noise which amplitude are 0.6, 1.0, 1.2, 3.0 respectively, then to produce correlated inputs and outputs signals with noise. Figure 5 is the input/output time domain signal through MATLAB numerical simulation, frequency spectrum of it, as shows in Fig. 6, each input and output signal which satisfies (18) are got. According to the analysis of transfer function partial coherence, we first divide the inputs into several correlative frequency bands. Then we judge the correlation causality and identify the priority by

378


Fig. 5 Inputs signals and outputs signals got through simulation

Fig. 6 The frequency spectrum of Simulation of input signal and output signal


379

comparing the transfer function in each frequency band. Finally, we calculate the real contribution of correlative inputs to output with the partial coherence method. 3.1 Partition of correlation frequency band Choose the right frequency resolution, the coherence function between inputs and outputs are shown in Fig. 7 while the coherence function of inputs is shown in Fig. 8. The dashed line in each of the figures shows that the value of coherence function is 0.25. It is used to determine whether there exists correlation between signals and to divide the frequency bands of correlation. In the frequency bands of 1.0 kHz–1.4 kHz, 1.6 kHz–2.0 kHz, 2.2 kHz–2.6 kHz in Fig. 7, the correlation between input signals and output signals is near to 1, in the frequency bands of 1.0 kHz–1.4 kHz, 1.6 kHz–2.0 kHz, coherence function between the input signals is near to 1, so there exist correlation between input signals and output signals. From Fig. 8, X1 and X2 are relative in the frequency bands of 1.0 kHz–1.4 kHz, X2 and X3 are related in the frequency bands of 1.6 kHz–2.0 kHz, while X1 and X3 are not related in all the frequency bands. Combing the spectrum analysis of signals in Fig. 6, there are three correlative frequency bands: 1.0 kHz–1.4 kHz, 1.6 kHz–2.0 kHz, 2.2 kHz–2.6 kHz. In Fig. 7, the response signal (Y ) is almost entirely contributed by input X3 in the frequency band of 2.2 kHz–2.6 kHz. The contribution to response signals (Y ) needs to be calculated in the frequency bands of 1.0 kHz–1.4 kHz and 1.6 kHz–2.0 kHz. 3.2 Correlation analysis based on transfer function As shown in Fig. 7, coherence function between x1, x2 and y is near to 1, x1 and x2 are related to y in the frequency band of 1.0 kHz–1.4 kHz. However x3 is not related to y because most of the coherence function values are less than 0.25. Hence the output signal in the frequency band of 1.0 kHz to 1.4 kHz

Fig. 7 Simulation of coherence function between inputs and outputs

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Fig. 8 Simulation of coherence function of inputs

is mainly contributed by inputs x1 or x2 , while the contribution of x3 is very small. It is shows in Fig. 8 that, coherence function between x1 and x2 is near to 1 in this frequency bands, so we need to judge the correlated causality between x1 and x2 first, if the correlation between x1 and x2 is not-causality correlation, then the 1.0 kHz–1.4 kHz parts is caused by x1 and x2 ; if the correlation between x1 and x2 is causality correlation, then judge the priority input between x1 and x2 . So we need to decide the correlated relationship between inputs x1 and x2 by comparing transfer function, calculate each transfer function between input and output in frequency bands 1.0 kHz–1.4 kHz, then compare with its actual value, as shown in Fig. 9. The solid line is the calculated value while the dashed line is the actual value. From Fig. 9(a), we observe that the calculated value of transfer function fluctuates with the actual value. The transfer function has input signal x1 and output signal y. The fluctuation is due to noise of input x1 and output y, hence we can determine that the calculated transfer function of x1 and y is consistent with the actual value. Comparison of the transfer function of x2 and the transfer function of y, as shown in Fig. 9(b), the calculated value is higher than the actual value. Form Fig. 9(c), the calculated value of transfer function between x3 and y is much larger than the actual value. Furthermore, x3 is contributes the smallest to the response in this frequency band. The correlation of x1 and x2 in the frequency of 1.0 kHz to 1.4 kHz is causality. Hence x1 is prior to x2 . Similarly, we can analyze the relationship of inputs in the frequency of 1.6 kHz to 2.0 kHz. The comparison of calculated value and actual value of the respective transfer functions is shown in Fig. 10. There is causality correlation of x2 and x3 and x2 is prior to x3 . The contribution of x1 to output y in this frequency band is very small. In the frequency range of 2.2 kHz to 2.6 kHz as shown in Fig. 7 and Fig. 8, we know that output signal y is only related to input signal x3 . Because it is not related to other inputs, the contribution to response comes from x3 . From Figs. 9 and 10, we observe that the noise has a great influence on the transfer function of inputs and output. Therefore, noise should be filtered and the appropriate frequency for effectively reducing the impact of noise on calculated value before determining the transfer function should be chosen.


Fig. 9 Calculated values and true value in the frequency of 1.0 kHz to 1.4 kHz

Fig. 10 Calculated value and true value in the frequency of 1.6 kHz to 2.0 kHz

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Table 1 The priority of inputs in each correlated frequency bands Frequency Band Priority of Inputs 1

1.0 kHz–1.4 kHz x1 → x2 → x3

1.6 kHz–2.0 kHz x2 → x3 → x1

2.2 kHz–2.6 kHz x3 → x1 → x2 or x3 → x2 → x1

Through the above simulation analysis, we have analyzed the causality and priority in each frequency band using the transfer function method. As shown in Table 1.

4

Experiment Analysis of Multi-Correlative Excitation on Auto-Mobile Frame

4.1 Experimental conditions description Experiment instruments and equipments used are as follows: one chassis, 2 piezoelectric accelerometers, 3 impedance heads, power amplifier, signal generator, 1 set of DH5935 N dynamic signal measurement and analysis system (including signal conditioning, A/D transfer, data storage, computer communication and so on), one computer and some wires. DH5935 N dynamic signal measurement and analysis system is introduced as data acquisition module. When collecting data, first, set the sensitivity of each channel according to the sensitivity of the sensor, make the measured data is the actual vibration rather than relative value; second, initialize the hardware; last, clear and balance the channels, make sure the accuracy of test data. During the experiment, after each data collection, repeat clearing and balancing the channels, to ensure the validity of test data. We use a rear-wheel driven automobile frame as a specimen of experiment. Three Fig. 11 The position of exciting point and response point excitation points and a response point with of automobile frame reference to the actual vibration of frame are selected as shown in Fig. 11. We can measure the true value of transfer function and spectrum between excitation points and response point in advance. 4.2 Experimental process Open the test bed; make the chassis vibrate for a while under the experimental conditions. After the chassis vibration reaches a steady-state, collect force signal and response signal of each point using DH5935 N. The collected signal should meet the Signal Sampling Theorem. Set sampling frequency as 5.12 kHz, the limit of the analyzed frequency is 2 kHz; sampling time is over 5 seconds to satisfy the frequency resolution.


Fig. 12

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The time-domain signal of each excitation and response point under multi correlated excitation in frequency bands

Each signal generator produces random signals of different frequency bands. The frequency bands of signal generator 1, 2, 3 are 300–400 Hz, 150–250 Hz, 30–100 Hz. 4.3 Results analysis Under the multi correlated excitations of frequency bands, first analyze the response signals and excite signals relationship of response point 1. In the same time bands, intercept 20000 points in the original data to analyze. The time domain waveform and spectrum analysis shown in Figs. 12 and 13. Considering the correlation between excitation signals, the correlation between excitation signals and response signals, calculate the coherence function of excitation signals, also the coherence function of excitation signals and response signals, as shown in Figs. 14 and 15. In Figs. 14–15, the coherence function value of the transverse line is 0.25, to ensure contribution of each input is greater than a fixed one, and to avoid the influence of noise come from inputs and outputs. It can be seeing from Fig. 13, there are three correlative frequencies bands 20 Hz–98 Hz, 150 Hz–250 Hz and 300 Hz–400 Hz. At first, take a pre-judgment on the effect and priority of coherence function in each frequency band, for uncertain frequency bands, determine the priority input by comparing the real value of transfer function and it’s calculated one. From Fig. 14, in frequency band 20 Hz–98 Hz, only the coherence function of x3 and y1 is near to 1, 2 of x , y and the coherence function the correlation is strong, however, both the coherence function γ1y 1 1 2 of x , y are less than 0.25, the correlation is weak. We can make a pre-judgment that the major γ2y 2 1 part of the output y in frequency band 20 Hz–98 Hz is caused by x3 , and then from Fig. 12 we know the coherence function value among x1 , x2 , x3 is far less than 1 in this frequency band, it means that correlation among x1 , x2 , x3 is so weak that it can be ignored. From Fig. 13 the spectrum we can see that, only x3 is obvious, both x1 and x2 are very small. Above all, we can know that y1 is mainly caused by x3 in frequency band 20 Hz–98 Hz; the contributions of other inputs are small.

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Fig. 13 Spectrum of exciting points and response point under multi-correlative excitation

Fig. 14 Coherence function between exciting signal and response signal

2 , γ2 , γ2 In frequency band 150 Hz–250 Hz, the coherence functions γ1y 2y 3y of x1 , x2 , x3 and y are all greater than 0.25, and near to 1. From Fig. 15 we can see that the coherence functions 2 of x and x is almost 1, also the correlation of x , x in this frequency band is strong, combining γ23 2 3 2 3 Fig. 13 frequency spectrum analysis we know that, for response signals in frequency band 150 Hz– 250 Hz, the greatest contribution come from x2 and x3 , but we can not decide the causality relationship of inputs correlation of x2 and x3 in this frequency band. Therefore, we determine the causality of x2 and x3 in this frequency band and the priority by comparing the transfer function value, as shown in Fig. 16.


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Fig. 15 Coherence function of exciting points

Fig. 16

Calculated value and true value of exciting points and response points in the frequency of 150 Hz to 250 Hz

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Fig. 17 Calculated value and true value of exciting points and response points in the freq. of 300 Hz–400 Hz Table 2 The priority of inputs in different frequency bands Frequency Band Priority of Inputs 1

20 Hz–98 Hz x3 → x1 or x3 → x2

50 Hz–250 Hz x2 → x3 → x1

300 Hz–400 Hz x1 → x2 → x3

From Fig. 16 we know the calculated curve of transfer function of x2 and y1 is consistent with the curve of actual value but with some errors observed, the calculated transfer function of x3 and y1 is much higher than the actual value and the maximum value of x1 and y1 is 6 × 104 . Therefore, there is causality correlation between x2 and x3 . In addition, x2 is prior to x3 in the frequency of 150–250 Hz. 2 , γ 2 , γ 2 of x , x , x and y are all In frequency band 300 Hz–400 Hz, the coherence function γ1y 1 2 3 2y 3y greater than 0.25, and near to 1, also the coherence function value among x1 , x2 , x3 are all near to 1. However from Fig. 13 spectrum we know that, comparing with x1 , x2 , value of x3 is small and shown little in this frequency band, it means that there is nearly no excitation come from x3 in the frequency, thus, y is mainly caused by x1 , x2 in this frequency band, but we can not decide the causality of x1 , x2 in the frequency band, as shown in Fig. 17. Similar to Fig. 16, the calculated transfer function of x1 and y1 is consistent with the curve of actual value but with some errors observed. The calculated transfer function of x2 and y1 is much higher than the true value and the maximum value of x3 and y1 is 1.5 × 105 , which far away from the true value. Therefore, there is causality correlation between x1 and x2 . In addition, x1 is prior to x2 in the frequency of 300–400 Hz. The priority of inputs in different frequency bands is shown in Table 2.


Fig. 18

The contribution of inputs to output in the frequency of 20–98 Hz

Fig. 19

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The contribution of inputs to output in the frequency of 150–250 Hz

According to the priority of inputs in different frequency bands, Figs. 18–20 shows the calculated contribution of inputs to output with partial coherence method in correlative frequency bands. As observed from the figures, contributions in the frequency ranges of 20–98 Hz, 150–250 Hz and 300–400 Hz are produced by x3 , x2 and x1 , respectively. 4.4 Experimental summary Through the analysis above, it is feasible to determine the causality correlation of inputs using the transfer function method. The errors between calculated and actual values are induced by noise in the experiment. In addition, there are other factors Fig. 20 The contribution of inputs to output in the frequency of 300–400 Hz which induce the difference between calculation and actual values which include the difference in experiment condition, length of data and frequency in analysis. Therefore, it is necessary to minimize the effect of these factors in the experiment.

5

Conclusion

This paper investigated the transfer function of vibration system through correlative inputs. Some conclusions are drawn as follows: (1) For inputs with causality correlation, the role of sub-priority source is enlarged because the transfer function is enlarged in calculation.

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(2) The inputs are causality correlated when the true value of transfer function is equal to the actual value; otherwise, they are non-causality correlated (3) From simulation analysis; there is calculation error using the transfer function method. However, the error does not affect the correlative causality and priority. (4) In the experiment of multi-correlative excitations on an automobile frame, we have successfully identified causality and priority which have illustrated validity of the proposed method.

Acknowledgments The research is supported by the National Natural Science Foundation of China (50675099).

References [1] Bendat, J. and Piersol, A., Engineering applications of correlation and spectral analysis, New York: John Wiley, pp. 25–30, 1993. [2] Huang Dunpu and Wang Youzhi, Identifying main path of vibration transmission, Journal of Vibration Engineering, Vol. 2, No. 1, pp. 32–38, 1989. [3] Chung, J. Y., Crocker, M. J. and Hamilton, J. F., Measurement of frequency response and multiple coherence function of the noise generation system of a diesel engine, Journal of the Acoustical Society of America, Vol. 58, pp. 635–542, 1975. [4] Trethewey, M. W. and Evensen, H. A., Identification of noise source forge hammers during production and application of residual spectrum technique to transients, J. Sound Vibration, Vol. 77, pp. 357–374, 1981. [5] Xu Qihong, The computation of partial coherence function used for sound source identification, Acta Acustica, Vol. 14, No. 5, pp. 377–382, 1989. [6] Park, J. S. and Kim, K. J., Determination of priority among correlated inputs in source identification problems, Mechanical Systems and Signal Processing, Vol. 6, pp. 491–502, 1992. [7] Park, J. S. and Kim, K. J., Source identification using multi-input/single-output modeling and causality checking of correlated inputs, Journal of Vibration and Acoustics, Vol. 116, pp. 232–236, 1994. [8] Bae, B. K. and Kim, K. J., A Hilbert transform approach in source identification via multiple-input single-output modeling for correlated inputs, Mechanical Systems and Signal Processing, Vol. 12, No. 4, pp. 501–513, 1998. [9] Konmpella, M. S., et al., A technique to determine the number of incoherent contributing to the response of a system, Mechanical System and Signal Processing, Vol. 8, pp. 363–380 1994. [10] Chen Moli and Li Shunming, Coherence functions method for signal source identification, Chinese Mechanical Engineering, Vol. 18, No. 1, pp. 95–100, 2007.

The Method of Reduction of Aerodynamic Forces Generated in Turbine Blade Seals n´ Krzysztof Kosowski and Robert Stepie Gdańsk University of Technology, ul. Narutowicza, Gdańsk, Poland [email protected] Abstract The distribution of pressure in the seal gaps does not only affect the so called “leakage losses” and the turbine stage overall efficiency but also plays an important role in the generation of aerodynamic forces which may cause self-excited rotor vibrations. The paper describes a chamber seal applied for the reduction of the aerodynamic forces created in shroud seals. This kind of turbine seal was patented and tested. Special attention was paid to the pressure field in the rotor blade shroud clearance in the situation when rotor-stator eccentricity leads to asymmetrical conditions of flows in blade passages and shroud gaps. The investigations into the pressure field in the shroud gap were performed by means of CFD Fluent Code and compared to the results measured on a single-stage air model turbine of the impulse type. The performed experimental investigations and numerical calculations prove that the new turbine seal can reduce pressure forces acting on the stage shroud more than by 50%, depending on the type of the shroud. Keywords: Aerodynamic forces, Turbine seals, Self-excited vibrations, Turbine rotors

1

Introduction

An enormous theoretical and experimental effort has been made to achieve better understanding of the phenomenon of self-excited vibrations due to aerodynamic forces generated in turbine seals. Rotorstator eccentricity or rotor-stator misalignment change the distribution of the stage clearances, which influences the fluid parameters in the flow channels and affects the pressure distribution in radial gaps. The asymmetrical clearance distribution above the blade shroud (and in glands) creates aerodynamic forces (i.e., “pressure forces”) acting on the turbine rotor. They depend on the rotor displacement (rotor-stator radial and axial displacement as well as rotor-stator misalignment), on the shroud type, shroud dimensions, radial and axial clearances, the swirl velocity of the leakage flow in the tip gap and stage output. Various theoretical models have been elaborated to describe fluid motion in labyrinth seals. These methods and their experimental verification are presented in a number of published works. Diettzen and Nordman[5] described turbulent flows in seals by means of Navier-Stokes equations and k-ε turbulence model. Tam et al.[20] directly applied Computational Fluid Dynamic to the numerical study of forces generated in seals and bearings. Averaged Navier-Stokes equations for determining rotordynamic forces were used by Rhode et al.[17] , Guo et al.[6] , Arghir and Frêne[1, 2] and others. Tallman and Lakshminarayana[19] performed numerical simulation of tip leakage flows in axial turbines and came to the conclusion that the distribution of the pressure field inside the gap was non-smooth. Kostyuk and ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011


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´ KRZYSZTOF KOSOWSKI AND ROBERT STEPIE N / ADVANCES IN VIBRATION ENGINEERING, 10(4) 2011

Shan[13] analysed non-stationary flows in impulse turbine stages with different types of shroud seals. The physical aspect of the flow including vorticity and turbulence fields, was described and experimentally researched by Xiao et al.[26] and McCarter et al.[14] . Pfau et al.[16] , investigating an annular cascade with a labyrinth seal, emphasised vorticity and periodic static pressure field in the inlet and exit cavities. CFD calculations were applied for the optimization of a turbine seal geometry[9, 15, 21, 24] , the numerical calculations were followed by the experimental research[4, 7, 22, 28] , the nonlinear aspects of the flows in turbine shrouds were also taken into account [8, 23] and the relations between the aerodynamic forces and rotor dynamics were investigated[25, 27] . The paper describes a chamber seal applied for the reduction of the aerodynamic forces created in typical, labyrinth turbine shroud seals. Our investigations into the pressure field in the shroud gap were performed by means of CFD Fluent Code and compared to the results measured on a single-stage air model turbine of the impulse type. An asymmetrical pressure distribution (due to rotor-stator eccentricity) in the blade tip clearance leads to the creation of pressure forces acting on the blade shroud. By increasing the volume above the shroud, the pressure distribution becomes more uniform on the whole shroud circumference and the effect of rotor eccentricity is reduced. It can be easily performed by introducing an extra chamber (A) to the typical turbine blade seals, Fig. 1. The circumferential flows in the chamber result in the decrease in the pressure differences and finally in the decrease in aerodynamic forces acting on the whole turbine rotor. Figure 1 presents the examples of three typical turbine blade shroud seals (Figs. 1a, 1b, 1c) and the same seals modified by introducing an additional chamber (Figs. 1d, 1e, 1f). The chambers can have different shapes, different volumes and they can be located in different places, and connected with a shroud clearance in different ways. The effectiveness coefficient ηA of the chamber seal was calculated for each variant of the shroud. It is defined as the ratio of the value of pressure force FCA , generated in the seal with the chamber, to the value of pressure force FC in the seal without the chamber: ηA = FCA /FA (the lower value of ηA the better effectiveness of the chamber seal).

Fig. 1

Examples of typical turbine blade shrouds with and without the additional chamber A (EA [mm]–chamber width, HA [mm]–chamber height)

THE METHOD OF REDUCTION OF AERODYNAMIC FORCES GENERATED IN TURBINE BLADE SEALS

391

The pressure distribution is shown for the characteristic cross-sections of the shroud: at the top (T), at the bottom (B), on the right (R) and on the left (L), Fig. 2. The values of the pressure given in all the pictures ought to be understood as the pressure difference relative to ambient pressure.

2

Research Methods

Our investigations were performed into the model turbine stage shown in Fig. 3. A vast number of tests was performed and numerical calculations were compared with the experimental results. 2.1 3D Calculations The geometry of the flow part of the turbine stage was modeled by a calculating mesh. It was divided into four blocks: an inlet to the stage with nozzle channels, rotor blade channels, a shroud clearance and a stage exit. An example of the blocks and the mesh of the flow parts is presented in Figs. 4 and 5

Fig. 2 Schema of rotor-stator eccentricity (B, L, R, T–positions of the shroud cross-sections)

Fig. 3 Turbine flow part and the nozzle and rotor profiles

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Fig. 4


Blocks of meshes representing the turbine flow part (the example)

Fig. 5

Fragment of the mesh of the stage flow

respectively. In all the calculated variants the rotor passages were modeled by the moving mesh and the rotor blades were treated as a “moving wall”. All the other meshes were assumed to be stationary (details are given in[3, 18] ). In the case of our turbine stage (relatively short blades) the stage flow calculations carried out using the Sliding Mesh method and the Multiple Reference Frame with the structural shroud mesh led to the results of the averaged pressure distribution which correspond well with the timeaveraged values of the pressure recorded during the experiment. The Sliding Mesh method showed range of nonstationary pressure fluctuations which well matched to the experimental data[12] . 2.2 Experimental investigations The experimental rig for the investigation into aerodynamic forces causing self-excited vibrations consists of an air model turbine stage of the impulse type with moderately loaded prismatic nozzle and blade profiles, equipped with a special measuring system. The construction of the turbine makes it possible to change the blade shroud (i.e., its type or dimensions) and the nominal value of the radial and axial clearances. The support of the bearings can also be easily replaced and, in this way, the support stiffness can be changed. The elastic support enables the displacement of the stator disk in radial and

Fig. 6 The longitudinal section of the model turbine and location of measurement points


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longitudinal directions. This stand and the measuring system were described in detail by Kosowski and Piwowarski[10, 11] . The longitudinal section of the turbine is shown in Fig. 6. The pressure distribution in the blade tip clearance was measured at 48 points which were placed on the shroud circumference and in 4 cross-sectional planes located downstream along the turbine axis, Fig. 6. In addition, some extra traverse probes were used for the measurement of the pressure at the inlet and exit of the turbine stage. The pressure value was transmitted to a piezoresistive converter (tubes in-between the tap and the converter were used), where it was transformed into an electrical signal. This signal was amplified and transmitted to a computer extension card. The experimental stand enables the measurement of the pressure distribution in the blade tip clearance and the registration of the rotor trajectory for different rotor eccentricities, different rotor speeds and different turbine load. Knowing the pressure distribution, the “pressure forces” can be determined and the influence of the above mentioned parameters on those forces can be examined. 2.3 Calculations versus experiment The calculations of flows in the model turbine were carried out for the variants of the rotor speed and the corresponding inlet and outlet air parameters which had been measured experimentally. According to the results of our calculations, the Sliding Mesh and Multiple Reference methods give the very similar results of average values of the pressure distribution or the velocity field in the shroud Fig. 7 Pressure distribution along the shroud width (ω = 645 rad/s) red line–maximum presclearance[12] . The results obtained by these methods sure calculated by means of Sliding Mesh correspond well to the values of the pressure at the method, blue line–minimum pressure calmeasuring points recorded during the experiment. culated means of Sliding Mesh method, The calculated results obtained by applying the both bars–range of pressure changes according to of the methods fairly refers to the experimental data. experimental data The best coincidence between the numerical and experimental results were obtained for the nominal conditions of the turbine operation. Only the Sliding Mesh technique appeared to describe non-stationary effects and the pressure pulsations in the turbine flow channels and clearances. In Fig. 7 the pressure pulsations determined by the Sliding Mesh method are compared to the experimental results. In this example the range of calculated pressure pulsations refers quite well to the range of the pressure changes recorded during the experiment (the turbine speed rotor equal to ω = 645 rad/s).

3

Results of Investigations

The effectiveness of the chamber seal is shown using as an example a turbine blade seal with four teeth (2 teeth placed in the shroud and 2 teeth placed in the stator), Fig. 1a. All the presented results refer to the nominal value of the radial gap equal to 0.5 mm, the value of rotor-stator eccentricity (in vertical direction) amounting to 0.3 mm and the turbine rotor speed equal to 566 rad/sec. The pressure distributions along the blade shroud in horizontal and vertical sections are presented in Fig. 8. It is visible that by introducing the additional chamber we reduce the pressure difference on the whole

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Fig. 8

Pressure distribution along the shroud width in different circumferential positions vertical axis–pressure ([Pa], above atmospheric pressure), horizontal axis–shroud width ([mm]) dotted line–pressure in the seal without the chamber, solid line–pressure in the seal with the chamber

shroud circumference and the most noticeable effect is observed in the horizontal direction (direction perpendicular to the rotor eccentricity). As a result a decrease in the resultant pressure force is achieved. In the presented example, the chamber seal effectiveness coefficient ηA was equal to about 43%. It means that the value of pressure force was reduced by nearly 60%. The effect of the chamber shape and its dimensions on the effectiveness coefficient ηA was also examined. The investigations were carried out for 3 variants: • the seal with the base chamber, • the seal with the chamber with the volume of 50% of the base chamber, • the seal with the chamber with the volume of 150% of the base chamber. The results are presented in Fig. 9. For the considered examples the effectiveness coefficient ηA varies from 55% to 35%. The effectiveness coefficient ηA as a function of the chamber volume is shown in Fig. 10. The volume of the base chamber was approximately equal to the volume of the seal (the volume of the whole clearance between the shroud and the stator) and we may conclude that the increase in the chamber volume above that value shows only slight effect on the effectiveness coefficient ηA , while the decrease of the chamber volume affects the effectiveness coefficient remarkably. The effect of introducing one extra chamber, placed at the seal inlet part, is presented in Fig. 11 and explained in Fig. 12 where pressure distribution along the shroud width is shown. The reduction of the pressure difference acting on the shroud is especially visible in the horizontal direction, but this relatively small chamber shows weaker influence on the seal effectiveness (ηA = 76%). The base chamber reduces the pressure forces much more remarkably (ηA = 43%). If the small additional chamber is applied


395

Fig. 9

Schema of the seal with the additional chamber and the vectors of the resultant pressure forces a) 50% of the base chamber volume b) the base chamber, c) 150%. of the base chamber volume FC –pressure force for the seal without the chamber, FCA –pressure force for the seal with the chamber EA [mm]–chamber width, HA [mm]–chamber height

Fig. 10

Effectiveness coefficient ηA as a function of the relative chamber volume (ratio of the base chamber volume)

together with the base chamber we observe the reduction in the total effectiveness coefficient by only additional 5% (ηA = 37%). Similar investigations were carried out for different types of shrouds, different shapes and dimensions of the chambers, and different turbine working conditions. The effectiveness of the chamber seal was investigated for three different turbine blade seals:

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Fig. 11


Schema of the seal with the extra chamber at the inlet and the vectors of the resultant pressure forces a) single extra small chamber b) the base chamber, c) base chamber and extra chamber FC –pressure force for the seal without the chamber, FCA –pressure force for the seal with the chamber EA [mm]–chamber width, HA [mm]–chamber height

• with four teeth (2 teeth placed on the shroud and 2 teeth placed in the stator), Fig. 1a, • with two teeth placed on the shroud, Fig. 1b, • without teeth, Fig. 1c. In the case of the seal with two teeth and in the case of the shroud without teeth the effectiveness of the chamber seal was even more favorable. The example of the pressure distribution in the shroud clearance for the case of the seal with two strips (Fig. 1b) is shown in Fig. 13. The effect of the reduce in pressure difference acting on the shroud due to application of the extra chamber is clearly visible. The comparison of the appropriate effectiveness coefficients obtained by calculations and experiments is presented in Fig. 14. The experiments shows that the chamber seal is characterised by even more favoruable values of the effectiveness coefficient than those obtained by calculations. It must be emphasized that, according to the calculations and experiments, applying the chamber to the typical labyrinth turbine shroud seals does not effect the flows in the turbine blade channels. This conclusion is justified by the results of the numerical calculations and the experimental research. The observed changes of mass flow rate in the blade channels and in the seals as well as the changes of the turbine stage output and efficiency were less than the accuracy of the investigations. Special designs of the chamber seal allow to change the aerodynamic forces generated in shroud clearance in an active control method.


Fig. 12

4

397

Pressure distribution along the shroud width in different circumferential positions dotted line–pressure in the seal without the chamber, solid line–pressure in the seal with the chamber

Conclusions

The results of the numerical and experimental calculations led to the following conclusions: 1. The proposed chamber seal can remarkably reduce aerodynamic forces generated in the labyrinth type shroud seals, sometimes even more than by 60%. 2. The choice of the shape of the chamber, its dimensions and the design details depend on the particular type of the labyrinth seal.

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Fig. 13

Pressure distribution in seal clearance along the shroud width in different circumferential positions. Shroud with 2 strips: standard version without the chamber (upper) and version with the chamber (bottom)

Fig. 14

Comparison of the effectiveness coefficients determined by calculations and the experimental tests (red line–experiment, black line–calculations), shroud seal with 2 strips

3. The proposed method may be applied in active control of aerodynamic forces generated in turbine shroud seals. 4. The proposed seal modification can be easily introduced to turbines in operation and to the newlydesigned ones.

References [1] Arghir, A. and Frene, J., Forces and moments due to misalignment vibrations in annular liquid seals using the averaged navier-stokes equations, ASME Journal of Tribology, Vol. 119, pp. 279–290, 1997.


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[2] Arghir, A. and Frene, J., Analysis of a test case for annular seal flows, ASME Journal of Tribology, Vol. 119, pp. 408–414, 1997. n, R. and Piwowarski, M., Research into flows in turbine blade [3] Badur, J., Kosowski, K., Stepie´ seals, Part I: Research Methods, Task Quarterly, Vol. 7(3), 2003. [4] Bohn, D., Krewinkel, R., Tummers, C. and Sell, M., Influence of the radial and axial gap of the shroud cavities on the flowfield in a 2-stage turbine, Proceedings of ASME Turbo Expo, GT-2006– 90857, June 2006. [5] Dietzen, F. J. and Nordman, R., Calculating rotordynamic coefficients of seals by finite-difference techniques, ASME Journal of Tribology, Vol. 109, pp. 388–394, 1987. [6] Guo, Z., Rhode, D. L. and Davis, F. M., Computed eccentricity effects on turbine rim seals at engine conditions with a mainstream, ASME Journal of Turbomachinery, Vol. 118, pp. 143–152, 1996. [7] Gupta, M. and Childs, D., Rotordynamic stability predictions for centrifugal compressors rotordynamic stability predictions for centrifugal compressors using a bulk-flow model to predict impeller shroud force and moment coefficients, Proceedings of ASME Turbo Expo, GT-2006– 90374, June 2006. [8] Huang, D. and Li, X., Rotordynamic characteristics of a rotor with labyrinth gas seals, Part 2: A nonlinear model, Journal of Power and Energy, Vol. 218, pp. 179–185, 2004. [9] Jun, L., Xin, Y. and Zhenping, F., Effects of pressure ratio and fin pitch on leakage flow characteristics in high rotating labyrinth seals, Proceeding of ASME Turbo Expo, GT-2006–91145, June 2006. [10] Kosowski, K. and Piwowarski, M., Experimental research into aerodynamic forces leading to selfexcited vibrations of turbine rotor, ASME Paper GT-2002–30638, Proceedings of ASME Turbo Expo, Amsterdam, 2002. [11] Kosowski, K. and Piwowarski, M., Experimental investigation into pressure field in tip clearance of shrouded rotor blades, ASME Paper GT-2002–30397, Proceedings of ASME Turbo Expo, Amsterdam, 2002. [12] Kosowski, K., Stepi e´ n, R., Piwowarski, M. and Badur, J., Research into flows in turbine blade seals, Part III: Numerical calculations versus experiment, Task Quarterly, Vol. 7(3), 2003. [13] Kostyuk, A. G. and Shan, T., Analysis of non-stationary flows in turbine stage due to rotor whirl, (Russ.), Teploenergetika, Vol. 7, pp. 26–33, 1997. [14] McCarter, A. A., Xiao, X. and Lakshiminarayana, B., Tip clearance effects in a turbine rotor: Part II-Velocity field and flow physics, ASME Journal of Turbomachinery, Vol. 123, pp. 305–313, 2001. [15] Paolillo, R., Wang, C., Cloud, D., Vashist, T., Fons, M. and Kool, G., Rotating seal rig experiments: Test results and analysis modeling, Proceeding of ASME Turbo Expo, GT-2006–90957, June 2006. [16] Pfau, A., Treibe, M., Sell, M. and Gyarmathy, G., Flow interaction from cavity of an axial turbine blade row labyrinth seal, ASME Journal of Turbomachinery, Vol. 123, pp. 342–351, 2001. [17] Rhode, D. L., Hensel, S. J. and Guidry, M. J., Labyrinth seal rotordynamic forces using a three-dimensional navier-stokes code, ASME Journal of Tribology, Vol. 114, pp. 683–689, 1992. [18] Stepie´ n, R., Kosowski, K., Piwowarski, M. and Badur, J., Research into flows in turbine blade seals, Part II: Numerical analysis, Task Quarterly, Vol. 7(3), 2003.

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[19] Tallman, J. and Lakshminarayana, B., Numerical simulation of tip leakage flows in axial flow turbines with emphasis on flow physics: Part I-Effect of tip clearance height, Part II-Effect of outer casing relative motion, ASME Journal of Turbomachinery, Vol. 123, pp. 314–333, 2001. [20] Tam, L. T., Przekwas, A. J., Muszynska, A., Hendricks, R. C., Braun, M. J. and Mullen, R., Numerical and analytical study of fluid dynamic forces in seals and bearings, ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 110, pp. 315–325, 1987. [21] Vakili, A. D., Meganathan, A. J. and Sricharan, A., Advanced labyrinth seals for steam turbine generators, Proceeding of ASME Turbo Expo, GT-2006–91263, June 2006. [22] Volker, L., Influence of close proximity of a blade row on probe flow measurements, XX-th Turbomachinery Workshop, September 2006. [23] Wang, W. Z., Liu, Y. Z., Meng, G. and Jiang, P. N., Nonlinear analysis of orbital motion of a rotor subject to leakage air flow through an interlocking seal, Journal of Fluids and Structures, Vol. 25(5), pp. 751–765, 2009. [24] Wang, Y., Young, C., Snowsill, G. and Scanlom, T., Study of airflow features through step seals in the presence of disengagement study of airflow features through step seals in the presence of disengagement due to axial movement, Proceeding of ASME Turbo Expo, GT-2004–53056, June 2004. [25] Xi, J. and Rhode, D. L., Rotordynamics of turbine labyrinth seals with rotor axial shifting, International Journal of Rotating Machinery, 2006. [26] Xiao, X., McCarter, A. A. and Lakshminarayana, B., Tip clearance effects in a turbine rotor: Part I-Pressure field and loss, ASME Journal of Turbomachinery, Vol. 123, pp. 296–304, 2001. [27] Yuan, Z., Chu, F., Hao, R. and Wang, S., Clearance-excitation force of shrouded turbine rotor accounting for pitching motion, Journal of Mechanical Engineering Science, Vol. 221(2), pp. 187–194, 2007. [28] Yun, Y., Porreca, L., Kalfas, A., Song, S. and Abhari, R., Investigation of 3D unsteady flows in a two stage shrouded axial turbine using stereoscopic PIV and FRAP, Part II: Kinematics of shroud cavity flow, Proceeding of ASME Turbo Expo, GT-2006–91020, June 2006.

Message from IFToMM President, Prof. Marco Ceccarelli Dear IFToMM officers and IFToMMists: The end of the term is near and this is my last message to say farewell as IFToMM President for a term that has been exciting since a dense activity and fruitful interaction with the community has brought to relevant results for the growth of IFToMM. The main results of this term can be summarized within the planned program for Visibility-Activity in – we have increased the number of MOs (Member Organizations) – we have helped MOs and TCs (Technical Committees) in difficult conditions to achieve proper functioning – we have outlined procedures in EC regulations and Constitution has been properly updated – we have increased the number of IFToMM sponsored conferences – we have increased publications under the umbrella of IFToMM – we have increased the number of IFToMM affiliated journals and started book series linked to IFToMM for better dissemination of MMS – we have reinvigorated the activities of TCs with more participation of individuals – we have started or restarted TCs in hot topics in MMS – we have determined clear benefits for MOs and individuals in being involved in IFToMM activity – we have celebrated the 40-th year anniversary of IFToMM also with a ceremony at the foundation city Zakopane, Poland – we have started new initiatives such as student Olympiad, summer schools, and tutorials – we have enhanced the finance of IFToMM with increased income sources and budget returns During the term 2008–2011 the presidency activity has been focused on guiding IFToMM activity with the help of members of the IFToMM Executive Council (EC) by also improving the functioning of the IFToMM bodies. The Visibility-Activity plan has been intended to increase the visibility of IFToMM and its activity by promoting new and existing activities with an explicit mention of IFToMM and to facilitate new and existing initiatives under the umbrella of IFToMM. During the term 2008–2011, results have been achieved in increasing meetings, exchanges, publications, teaching, and international collaborations, as prescribed for the mission of IFToMM in the IFToMM constitution. Efforts have been also spent to have clear increased benefits for IFToMM Member Organizations and their affiliated individuals in being involved in IFToMM. In particular, a book series on MMS has been started in 2010 by the international publisher Springer and with more IFToMM affiliations, 6 journals are now available and linked for paper publications. The journals are Mechanism and Machine Theory (http://www.elsevier.com), Problems of Mechanics (http://pam.edu.ge), Open-access Mechanical Sciences (http://www.mech-sci.net), Chinese Journal of Mechanical Engineering (http://www.cjmenet.com), Advances in Vibration Engineering (http://www.tvi-in.com/index.asp), and Mechanics Based Design of Structures and Machines (http://www.tandf.co.uk/journals/titles/15397734.asp). The book series are Book series on MMS (http://www.springer.com/series/8779) and Book series on History of MMS (http://www.springer.com/series/7481). Sources for IFToMM finance have been increased through returns from IFToMM supported initiatives, donations, and royalties from publications mainly within the above mentioned book series.

Functioning of the IFToMM bodies has been supervised with an efficient help of the IFToMM Executive Council also through specific activity of EC Working Groups in attaching specific matters. Chairs of MOs in GA have been continuously informed on IFToMM activity that has been coordinated by EC with supervision of the President. In general they have been successfully reactive in providing more representatives in TCs and PCs. Some critical situations related to due arrears and lack/insufficient participation to IFToMM activity have been solved, but others have required unpleasant strong actions as decided at the last GA (General Assembly). In this term new MOs have been accepted from Turkey, Portugal, Egypt, Denmark, after preliminary negotiations for proper candidature submissions. Now, IFToMM Federation has 48 MOs with presences of active IFToMM communities in all continents. Permanent Commissions (PCs) have shown different situations reflecting different interests from IFToMM community. PC for Standard and Terminology has worked out relevant activity with a new multilingual publication of MMS terminology that is now available also on line. This PC has also started a link with ISO for collaboration on standards. The PC for History of MMS has carried out impressive achievements both in scientific works and organization of the field, with the largest commission in IFToMM. The PC has also improved the storage of documents in a rich and rich IFToMM archive. The PC for Communications has served with difficulties in the task of improving IFToMM webpage and IFToMM newsletter mainly because of a very weak response from the IFToMM community, also with a very limited number of active PC members. Similarly, PC publications has encountered difficulties in fulfilling the By-Laws duties because of a poor participation from IFToMM community, although the frames of IFToMM publications have been increased considerably both in journals and books (textbooks and conference proceedings). Thus, at the 2010 EC meeting the EC has proposed to merge the two PCs into a new PC for Communications, Publications and Archiving with a reshaped plan of duties under the direct responsibility of the IFToMM Presidency desk; at the 2011 GA the new PC was approved and it has just started the activity. Success of Technical Committees (TCs) strongly depends of individuals, who are representatives of MOs in their respective fields with the aim to fulfil the By-Laws prescriptions in order to develop activities in the main aspects of meetings, collaborations, conference events, transfer of expertise, and publications. During the past two terms, also thanks of repeated requests, TCs have been enriched with more representatives from MOs, but not yet all MOs have TC members in all TCs. Most of the TCs have worked very successfully in the above mentioned aspects and in general with trends towards good results in a near future. In particular, several new conferences have been started within TC frames and existing conferences have received a clear sponsorship by IFToMM through the TCs. The new book series on MMS is also available for publication of books and proceedings of IFToMM sponsored conferences with royalty to IFToMM. Two TCs have been started, namely on Sustainable Energy Systems and on Biomedical Engineering. A TC on Gearing and Transmissions has been re-established with a reinvigorated group of colleagues. Unfortunately, there are still few TCs with a weak activity that has required help and strong actions by the EC with the hope of a quick revitalization referring to the hot topics which they are related to. Now IFToMM has 14 TCs in the hottest topics in MMS. The Executive Council (EC) has been very efficient in dealing with IFToMM activity during the term by attaching problems and situations with a well thought attention towards suitable solutions, and in general by achieving considerable improvements of the functioning of IFToMM both in visibility and activity. Most of EC members have been more active and reactive than expected as in the constitution prescriptions. Particular mention for a continuous dedication to IFToMM is deserved to the Secretary General prof. Carlos Lopez-Cajùn and the Treasurer Joseph Rooney. EC has worked out not only at the time of EC meeting, but EC members have been active both for telemeetings that have been started

in 2010 and for email postal ballots, and for a continuous monitoring and discussion of the IFToMM activity mainly by email. Those continuous activities have permitted a continuous feedback with the community and its operations. EC regulations have been outlined to clarify and simplify procedures for IFToMM functioning and EC leadership. Another important new means of EC work has been successfully experienced with Working Groups (WGs) that have been established to solve specific problems with short time targets. The President has continuously stimulated EC members in a friendly collaborative ways and great satisfaction seem to be achieved from both sides. As Chair of IFToMM Committee for Honours and Awards, the President has stimulated successfully the community for annual awards of significant personalities, who can be also ambassadors of the IFToMM mission. Finally, I will like to outline few visionary thoughts as from experience gained in serving as Secretary General in 2004–2007 and President in 2008–2011. IFToMM has a relevant significant role in the development both of Technology and Society. We need to be well aware of this and increase the visibility of it. In today aggressive world, nobody advertises or even recognizes merits of others and therefore we IFToMMists are forced to stress continuously the role of IFToMM by ourselves. We are a worldwide community, who develop mechanical engineering in terms of formation and innovation. We can form properly new generations of engineers with modern vision for the future when we are well aware (i.e. visibility) and well skilled (i.e. activity) in the developments of mechanical/mechatronic systems for the benefit of society. We can work and transfer innovation when we are well aware (i.e. collaboration) and well expert (i.e. research) in attaching challenging novel problems for technological developments. As President I have experienced that IFToMM is a unique worldwide body in the fields of Mechanical Engineering with specific focus on MMS, and it has a worldwide significance because of the community worldwide presence and its activity. But this is not an abstract entity, and each of us can contribute to IFToMM success with large or small participation depending of her/his own possibilities. Without individuals with common views and activities a community cannot exist and cannot be a reference also to help for other individuals. Thus, the future of IFToMM with a great success as in the past, and even with more influence in the Society for the benefit of Society, much depends on us, on how we participate, disseminate, and transfer the feeling and activity of a community. This is fundamental and instrumental for innovation and enhancement. The IFToMM community works successfully when its members helps each other and all together contribute to the intellectual and scientific growth of the individuals within specific technical fields. The vitality of a community as the IFToMM community is also appreciated by giving turn of MOs representatives in the leadership of the community. Thus, beside being careful to properly distribute TC Chairmanships among MOs depending of the recognized reputation of candidates with high levels of expertise, it is also very important to have a turn in EC members as from MOs, even with the aim to recognize the most active MOs or to stimulate more MO activities. This turn plan is also fundamental not only to recognize IFToMM membership without any preference or prominence of one MO with respect to other MOs but mainly to show an open-mind community in agreement with the mission of IFToMM. Today IFToMM has a great potentiality more than in the past and it is shown as a reality. Let us look at the past, let us work with the enthusiasm in the present to shape the future, our future, our IFToMM. The considerable activity and significant achieved results in this term have been possible thanks to the support, participation, and help of all IFToMM officers (Chairs of MOs, PCs, and EC members) and also many individual IFToMMists. I will like to thank all the IFToMM officers since we had indeed direct

contacts and I received from each of them valuable feedbacks, suggestions, and inputs. But I cannot forget individual IFToMMists, whom I have met at conferences and meetings, and whom I have exchanged emails with: thanks to all of them! I have had always the feeling of doing well for the interest and benefit of IFToMM, because of the smiles and nice words I received when meeting persons of our IFToMM community. Those appreciations together with claims and arguments have been for me a continuous stimulus to spend more and more efforts and time for the wonderful experience as IFToMM President. The expressed gratitude that I have received also with few simple words has been and is for me of great value and memory. I owe deep gratitude to my family, my daughters Elisa and Sofia, my son Raffaele and my granddaughter Greta and mainly to my wife Brunella. Their patience has been immense in permitting me to dedicate lot of time in replying emails and preparing material even when home, but mainly for the many travels I spent in name and in occasion of IFToMM and also in receiving warmly the many IFToMM hosts home. They are the great base of my success, if you will consider successful my presidency activity. I also take the opportunity to wish Prof Y. Nakamura, President-elect, and new Executive Council a fruitful activity in the term to come. I wish you all happy end-year holidays and all the best in 2012. Thanks to all and let us be proud to be IFToMMist for ever!

Cassino: November 2011 Prof Marco Ceccarelli (IFToMM President 2008–2011) LARM: Laboratory of Robotics and Mechatronics DIMSAT, University of Cassino Via Di Biasio 43, 03043 Cassino, Italy Email: [email protected] LARM webpage: http://webuser.unicas.it/weblarm/larmindex.htm

First Announcement and Invitation for Participation

The Eight International Conference on Vibration Engineering and Technology of Machinery (VETOMAC-VIII) ´ Gdansk, Poland 3–6 (Monday – Thursday) September, 2012 Co-Chairs Prof. Romuald Rzadkowski Prof. J. S. Rao Prof. Jan Kiciński

Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdansk, Poland Protem KL University/Altair Engineering, India Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdansk, Poland

Scientific Committee Prof. J.S Rao Prof. K. Gupta Prof. N. S. Vyas Prof. Yuantai Hu Prof. S. Narayanan Prof. A.D. Raj Kumar Prof. K. Narayana Rao Prof. D.N. Reddy Prof. Tielin Shi Prof. Qingguo Meng Prof. Cheng Wang Prof. Guang Meng Prof. R. Kielb Prof. J. Kiciński Prof. R. Rzadkowski Prof. C. W. Lim M.Sc. W. Radulski Prof. R. Szczepanik Prof. J. Sawicki

Protem KL University/Altair Engineering, India Indian Institute of Technology, Delhi, India Indian Institute of Technology–Kanpur, India Huazhong University of Science and Technology, P.R. China Indian Institute of Science, Bangalore, India Osmania University, Hyderabad, India Member Secretary, AICTE, New Delhi, India Jawaharlal Nehru Technological University, India Huazhong University of Science and Technology, P.R. China Meng NSFC, China Hohai University, P.R. China Shanghai Jiao Tong University, China Duke University, Durham, North Carolina, U.S.A Polish Academy of Sciences, Poland Polish Academy of Sciences, Poland City University of Hong Kong, Hong Kong, China ALSTOM, Elblag, Poland Air Force Institute of Technology, Warsaw, Poland Cleveland University, U.S.A

Organizing Committee Dr. Marcin Drewczyński M.Sc. Artur Maurin Dr. Sławomir Banaszek

Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdansk, Poland ([email protected]) Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdansk, Poland ([email protected]) Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdansk, Poland ([email protected])

Conference Secretariats Dr. M. Drewczyński M.Sc. Artur Maurin Dr. M. Drewczyński

Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdansk, Poland Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdansk, Poland Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdansk, Poland

Venue Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdansk, Fiszera 14, Hotel Mercure, 80-890 Gdańsk ul. Jana Heweliusza 22 Aims VETOMAC-VIII (Vibration Engineering and Technology of Machinery) is the eight in a series of conferences being held as an effort to bring together researchers from diverse areas in Vibration Engineering and Technology of Machinery. The conference is organized by Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdansk, Poland, in collaboration Koneru Lakshmaiah University, Vaddeswaram, India and the Vibration Institute of India. Major Topics • • • • • • • • • • • •

non-linear vibrations fluid structure interaction vibration in energy and power systems machinery and structural dynamics rotor dynamics condition monitoring, tip-timing, experimental techniques composites and nano-structures microturbines vehicle dynamics vibration and waves MEMS, smart structures and systems wave propagation

The purpose of the meeting in Gdańsk (2012) is to promote the vital exchange of knowledge, ideas and information on the newest developments and applied technologies, theoretical, experimental and computational methodologies.

On behalf of Institute of Fluid Flow Machinery, Polish Academy of Sciences the organizing committee cordially invites representatives from universities, research institutes and industry all over the world to attend this conference. History VETOMAC I VETOMAC II VETOMAC III VETOMAC IV

VETOMAC V VETOMAC VI VETOMAC VII

organized by the Indian Institute of Science, Bangalore in 2000. organized by the Bhabha Atomic Research Centre, Mumbai in 2002. organized by the Indian Institute of Technology, Kanpur in 2004. organized by the University College of Engineering (Autonomous), Osmania University, Hyderabad, India, R&D Division Bharat Heavy Electricals Limited, Hyderabad, India, in collaboration with The Vibration Institute of India. organized by Huazhong University of Science and Technology, Wuhan in 2009. organized by by Indian Institute of Technology Delhi, New Delhi in 2010. organized by Shanghai Jiao Tong University and City University of Hong Kong, in collaboration with the Vibration Institute of India in 2010.

Call for Papers Key researchers and engineers from universities, private and public research centers and industry are invited to submit papers. All accepted papers, after a review, will be published in the Proceedings of Symposium. Some papers, after a view by the Scientific committee, will be published in Advances in Vibration Engineering, Key Dates Abstract submission deadline Abstract acceptance notification deadline Full length paper submission deadline Final paper acceptance notification deadline

February 1, 2012 March 1, 2012 April 1, 2012 May 1, 2012

Conference Registration Fee Early-bird registration: Regular registration: Late/On site registration: ∗

1 May – 26 June, 2012 EUR 27 June – 30 July, 2012 EUR after 30 July 2012 EUR

450.00 520.00 550.00

This fee will cover all meals during the days of the conference.

Submission of Abstract and Papers Abstract and Papers are to be sent by email to the following address: [email protected]

´ Gdansk City A thousand-year history, a location at the crossroads of important commercial and communication routes, an extensive port and mercantile traditions – all this makes Gdańsk a meeting place of many cultures, nationalities and denominations. The first written mention of Gdańsk comes from 997. The defensive and urban complex as well as a port started to really form in the second half of the 10th century. The dynamic development of trade, fishery and craft guilds soon pushed the city to the leading position in Pomerania. It maintained this even despite being taken over by the Teutonic Knights in 1308. The city continued to develop dynamically. Joining the League of Hanseatic Cities (in 1361) and the fast development of the port are just some of the factors contributing to the strong position of Gdańsk in Europe. The following years are traditionally called the “golden age.” During this time Gdańsk was one of the wealthiest and most significant cities in Europe. The religious freedom gained in the 16th century turned the city into a true melting pot of nationalities and denominations, giving it yet another stimulus for development, thanks to the specific “community of differences.” It was one of the few such places in the world at the time. In 1919, the Free City of Gdańsk was established under the Treaty of Versailles, which brought the city back to the elite of European ports. The tragic December 1970, and then August 1980 and the martial law period are the successive dates symbolizing the fight of the citizens of Gdańsk against the prevailing communist regime. It was Gdańsk that became the cradle of “Solidarno´sc´ ” which was to transform the then map of Europe. History has come full circle. Contemporary Gdańsk – a halfa-million, dynamically developing agglomeration – is vibrant with life as before and again deserves to be called “the Pearl of the North.” Post-Conference Tour Several exciting post-conference tours will be arranged upon participants’ request. Possible tourism sites include: Gdańsk Old Town sight seeing, Gdansk, Sopot, Gdynia excursion, Oliwa Cathedral sight seeing. Hotel Fee Hotel lodging will be provided after your arrival. • Hotel Mercure, 80-341 Gdańsk ul. Jana Heweliusza 22, The hotel fee approximately 80 EUR per room. • Hotel “Dom Nauczyciela”,ul. Uphagena 28, Gdańsk Wrzeszcz. The hotel fee approximately 35 EUR one single room, 40 EUR two bed room. Web Site http://www.imp.gda.pl/vetomac-viii

Guidelines for Contributors Authors are requested to submit their papers by e-mail followed by three copies sent by post of to the Editor-in-Chief whose address is listed in the inside first cover of the journal. Submission of papers to the journal implies that it has not been previously published, and is not under consideration elsewhere. Papers submitted to the journal are peer-reviewed, and in this process, the author may be asked to revise and resubmit the manuscript according to the comments made by the reviewer. It is the responsibility of the authors to obtain written permission from the publishers for use of any previously printed material in their paper. The journal considers for publication: (i) invited papers from experts discussing the latest trends in a specialized area, (ii) original research papers and (iii) application-oriented papers. Manuscript Specifications Manuscripts must be typed in 12 pt font with double line spacing on A4 size paper (on one side only) with wide margins. Paper length should not exceed 25 pages in the typed format, inclusive of figures and tables, and must have the following format: Title: Title page must contain (i) the title (which must be short and contain words useful for indexing), (ii) the initial(s) and name(s) of the author(s) and the name and address of the institution where the work was done, (iii) e-mail ID, and (iv) a brief running title of not more than 60 letter spaces. Abstract must be informative and not just indicative, and must contain the significant results reported in the paper. Keywords, not more than about ten in number, must be provided for indexing and information retrieval. The text must be divided into sections, generally starting with ‘Introduction’ and ending with ‘Conclusions’. The main sections should be numbered 1, 2 etc., subsections 1.1, 1.2 etc., and further subsections (if necessary) 1.1.1, l.2.1 etc. Tables and Figures: Tables and figures are to be numbered in consecutive numerical order in the paper. They should be selfcontained and have a descriptive title. Do not place figure title within the figure, type it below the figure; type table caption above the table. For figure labels, use Times Roman font, except in case of Greek symbols where use of SymbolProp BT font is recommended. Line drawings, graphs, charts, photos and gray scale diagrams will be scanned electronically for final production, and should be original proofs and not photocopies. All line drawings and photos should be in black and white (for colour graphics, authors will be charged at printings costs). Line drawings must be in India ink on good quality tracing paper or Bristol board, preferably of the same size as the text paper. Lines should be sufficiently thick (axes about 0.3 mm, curves about 0.6 mm). All letterings on the illustrations must be done in lower case letters except for symbols or acronyms. The printout should be 600 dpi for line art; 220 dpi for figures with grayscale shading and black and white photographs; and 300 dpi for colour graphics. Graphics may be submitted in TIFF/PS/EPS formats; CorelDraw/Adobe illustrator files are also acceptable.

A list of symbols must be provided, with each symbol identified typographically (e.g. Gk, alpha, script oh, Latin ell etc.) for the printer; this list will not appear in print. Authors may if necessary provide another list of symbols for the reader (to be printed). Mathematical material: Equations must be written clearly, each on its own line, well away from the text but punctuated to read with it. Complicated expressions should be avoided in the text; when absolutely necessary they should be displayed separately like equations. All equations must be numbered consecutively in arabic numerals with the number in parentheses near the right hand margin. Superscripts and subscripts (which must be indicated with vee and wedge respectively) must be kept as simple as possible; those of greater than second order (e.g. subscript on superscript on superscript) must be avoided. Similarly, symbols or letters with accents or unusual marks placed above or below them must be avoided. ‘Oh’ and ‘zero’, K, k and kappa, ‘ell’ and ‘one’ etc. must be clearly distinguished. Authors must indicate in pencil in the margin wherever special characters (Greek, German, script, scalar, vector, tensor, matrix etc.) are required. All other mathematical letter-symbols will be set in italic type. Vectors must be underlined by a wavy line and tensors by two wavy lines. Units and associated symbols must invariably follow SI practice. References: A numbered list of references sorted alphabetically by last name of the first author must be provided at the end of the paper. In the text, citations of references should be given as numbers (corresponding with the reference list at the end) in square brackets placed in superscript position at the appropriate context. Author names need not be mentioned in the text unless it is relevant to the text. Do not give dates of references in the text. A detailed style for references is available at www.krishtelemaging.com/vibeng.htm, and authors are required to strictly adhere to the format specified. Electronic submission: The final manuscript after acceptance is to be submitted in both electronic and hard copy formats. Please do not submit PostScript and PDF files. Manuscripts may be submitted as word document. Manuscripts prepared in TeX, LaTeX are also welcome. In this case, please include all the macros or definitions that are required to produce the document in one file. Otherwise, include an ASCII version on the disk with the word-processor version. Graphic files are to be submitted on a separate disk. Copyright: Authors must sign the Copyright Transfer Form sent along with the acceptance letter before a paper can be published. This transfer enables Krishtel eMaging Solutions Pvt. Ltd., to protect the copyright material, but does not relinquish the author’s right to use it for his/her personal use. Permission must be obtained from the publisher to reprint/republish this material for advertising or promotional purposes or for reusing any copyrighted component in other works. Proofs: Proofs will be sent to the corresponding author and must be returned without delay. Corrections should be limited to typesetting errors.

Edited and published by M. R. Dhayanithy for Krishtel eMaging Solutions Private Limited, 39, Madley Road, T. Nagar, Chennai 600 017 and printed by K. S. Ajay Kumar at Image Screens, # 262, Triplicane High Road, Chennai 600 005.

Advances in Vibration Engineering The Scientific Journal of the Vibration Institute of India www.tvi-in.com EDITOR-IN-CHIEF J S Rao The Vibration Institute of India #1039, II Cross Road, II Block, BEL Layout Vidyaranyapura, Bangalore 560 097, India email: [email protected]

PRINCIPAL ADVISER Ronald L Eshleman The Vibration Institute Suite 212, 6262 South Kingery Highway Willowbrook IL 60514 - 2986, USA

ASSOCIATE EDITORS ASIA PACIFIC REGION

EUROPE REGION

AMERICA REGION

C W Lim

R Rzadkowski

C Nataraj

UNITED KINGDOM

Jyoti K. Sinha

Department of Building and Construction City University of Hong Kong Tai Chee Avenue, Kowloon Hong Kong SAR email: [email protected]

Head of the Aeroelasticity Department Institute of Fluid Flow Machinery Polish Academy of Sciences ul. Fiszera 14, 80-231 Gdansk, Poland email: [email protected]

Head of Mechanical Engineering University of Villanova Villanova, PA 19085 USA email: [email protected]

School of Mechanical Aerospace and Civil Engineering (MACE) The University of Manchester P.O. Box 88, Manchester M60 1QD email: [email protected]

HONORARY EDITORIAL BOARD Dr V Arunkumar, Bangalore, India Prof N Bachschmid, Milan, Italy Prof A K Bajaj, West Lafayyette, IN, USA Dr K V Bhaskara Sarma, Hyderabad, India Prof Chong-Won Lee, Taejon, Korea Prof Daizhong Su, Nottingham, UK Prof D J Ewins, London, UK Prof M K Ghosh, Varanasi, India Prof Y Hori, Tokyo, Japan Prof H Irretier, Kassel, Germany Prof S C Jain, Roorkee, India Dr P B Jhala, Ahmedabad, India Prof G Kirk, Blacksburg, VA, USA

Prof A V Krishnamurty, Bangalore, India Prof K M Liew, Hong Kong Prof B C Majumdar, Kharagpur, India Prof J Mathew, Brisbane, Australia Prof V Ovarsky, Bratislava, Republic of Slovakia Dr K G Pandey, Ranchi, India Prof A M Sharan, St. Johns, New Foundland, Canada Prof A Sinha, University Park, PA, USA Prof N S Vyas, Kanpur, India Prof H S Yan, Tainan, Taiwan Prof D Qin, Chongqing, China Prof Rama Bhat, Concordia University, Montreal, Canada

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Advances in Vibration Engineering

Advances in Vibration Engineering

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