Structural Acoustics and Vibration

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Structural Acoustics and Vibration Mechanical Models, Variational Formulations and Discretization

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Structural Acoustics and Vibration Mechanical Models Variational Formulations and Discretization

ROGER OHAYON

CHRISTIAN SOIZE

Conservatoire National des Arts et M6tiers (CNAM) Paris, France

Office National d'Etudes et de Recherches A6rospatiales (ONERA) Chatillon, France

ACADEMIC PRESS San Diego London Boston New York Sydney Tokyo Toronto

ACADEMIC PRESS LIMITED 24-28 Oval Road LONDON NW1 7DX

U.S. Edition Published by ACADEMIC PRESS San Diego, CA 92101

This book is printed on acid free paper

Copyright 9 1998 ACADEMIC PRESS LIMITED

All rights reserved

No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical including photocopying, recording, or any information storage and retrieval system without permission in writing from the publisher.

A catalogue record for this book is available from the British Library

ISBN 0--12-524945-4

Printed and bound in Great Britain by MPG Books Ltd, Bodmin, Cornwall

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I. A Strategy for Structural-Acoustic Problems

I. 2. 3. 4. 5. 6. 7.

5. 6. 7. 8.

..............

1

Introduction, 1 Structural-acoustic master system, 2 Concept of fuzzy structure and structural-acoustic fuzzy system, 2 LE MF and HF frequency ranges, 3 Strategy used for the LF and MF ranges, 4 Excitations and responses, 5 Organization of the book, 6

Chapter II. Bask Notions on Variational Formulations

1. 2. 3. 4.

xi

...............

9

Introduction, 9 Boundary value problem, 9 Strong solution of the boundary value problem, 10 Variational formulation and weak solution of the boundary value problem, 11 Converse, 12 Associated linear operator equation, 13 Ritz-Galerkin approximation and finite element method, 14 Bibliographical comments, 17

Chapter III. Linearized Vibrations of Conservative Structures and Structural Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1. Introduction, 19 2. Conservative elastodynamic boundary value problem in a bounded medium with initial Cauchy conditions, 20 3. Associated spectral problem: eigenfrequencies and structural modes, 23 4. Variational formulation of the spectral problem, 23 5. Associated linear operators and algebraic properties, 25 6. Basic properties of the eigenfrequencies and structural modes for a structure fixed on E0, 28

vi

CONTENTS

7. Basic properties of the eigenfrequencies and structural modes for a free structure, 29 8. Cases of structures with symmetry properties, 33 9. Finite element discretization and generalized symmetric matrix eigenvalue problem, 40 10. Dynamic substructuring procedures for calculation of the structural modes, 41 Chapter IV. Dissipative Constitutive Equation for the Master Structure. 51

1. 2. 3. 4.

Introduction, 51 Notation for the Fourier transform, 52 Damping model with frequency-independent coefficients, 52 Model with frequency-dependent coefficients based on the linear theory of viscoelasticity, 53 5. Summary, 57

Chapter V. Master Structure Frequency Response Function . . . . . . . . . .

1. 2. 3. 4. 5. 6. 7.

61

Introduction, 61 Equations in the frequency domain, 62 Variational formulation, 65 Linear operators and algebraic properties, 66 Frequency response function, 70 Finite element discretization, 75 Boundary impedance operator, 76

Chapter VI. Calculation of the Master Structure Frequency Response Function in the LF Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

1. Introduction, 85 2. FRF model in the LF range, 86 3. Projection of the FRF on the elastic structural modes of the associated conservative system, 87 4. Remark on a nonviscoelastic model diagonalized by the elastic structural modes, 90 5. Introduction of quasi-static correction terms, 94 6. Frequency-by-frequency construction of the FRF, 95 Chapter VII. Calculation of the Master Structure Frequency Response Function in the MF Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 1. Introduction, 97 2. FRF model in the MF range, 98 3. Definition of an MF narrow band, 99

CONTENTS

vii

4. Class of MF narrow band excitation force fields, 100 5. Approximation of the FRF on an MF narrow band due to the frequencydependent damping and stiffness operators, 101 6. Analytical processing of the short time scale in the frequency domain and numerical processing of the long time scale in the time domain, 102 7. Construction of the FRF on an MF broad band, 106 8. Finite element discretization, 107 Chapter VIII. Reduced Model in the MF Range . . . . . . . . . . . . . . . . . . .

109

1. 2. 3. 4. 5.

Introduction, 109 Definition of a narrow MF band, 110 Energy properties of a simple-linear-oscillator response, 110 Variational formultaion of the master structure in the MF range, 117 Construction of an appropriate functional basis for the reduced model in the MF range, 119 6. Construction of a reduced model in the MF range, 123 7. Finite element discretization, 124 8. Construction of the dominant eigensubspace using the subspace iteration method, 127

Chapter IX. Response to Deterministic and Random Excitations

1. 2. 3. 4. 5. 6.

.....

129

Introduction, 129 Operator-valued FRF and associated linear filter in the time domain, 129 LF and MF deterministic cases: time-periodic excitation, 134 LF and MF deterministic cases: time square integrable excitation, 138 LF and MF random cases: time stationary stochastic excitation, 141 Random case: nonstationary stochastic excitation, 153

Chapter X. Linear Acoustic Equations

..........................

169

1. Introduction, 169 2. Inviscid acoustic fluid, 169 3. Dissipative acoustic fluid, 179 Chapter XI. Internal Acoustic Fluid Formulation for the LF and MF Ranges . . . . . . . . . . . .................................... 189

1. 2. 3. 4. 5. 6.

Introduction, 189 Statement of the internal acoustic problem, 190 Boundary value problem, 190 Variational formulation and linear operator equation, 192 Frequency response function, 195 Finite element discretization, 195

CONTENTS

ix

8. Location of the resonant frequencies of the coupled system, 282 9. Structural-acoustic modes of the master structure coupled with an internal acoustic fluid, 283 10. Case of a master structure coupled with an external acoustic fluid, 288 11. Structure coupled with an external and an internal acoustic fluid. Case of a zero pressure condition on part of the internal fluid boundary, 296 12. Case of an axisymmetric structural-acoustic master system, 304 13. Response to deterministic and random excitations, 305 Chapter XIV. Structural-Acoustic Master System in the MF Range . . . 307 1. 2. 3. 4. 5. 6. 7. 8.

Introduction, 307 Statement of the structural-acoustic problem in the frequency domain, 308 Boundary value problem of the structural-acoustic master system, 311 Variational formulation of the structural-acoustic problem, 313 Finite element discretization, 320 FRF calculation in the MF range, 323 Case of a master structure coupled with an external acoustic fluid, 329 Structure coupled with an external and an internal acoustic fluid. Case of a zero pressure condition on part of the internal fluid boundary, 333 9. Case of an axisymmetric structural-acoustic master system, 340 10. Response to deterministic and random excitations, 341 11. Bibliographical comments, 341 Chapter XV. Fuzzy Structure Theory 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

...........................

343

Introduction, 343 Statement of the problem, 344 Fuzzy structure theory stated as an inverse problem, 348 Random equation of the master structure coupled with fuzzy substructures in terms of the displacement field of the master structure, 353 Homogeneous model of a fuzzy substructure, 355 Construction of a homogeneous fuzzy impedance law for a fuzzy substructure and for a local direction, 358 Solving method for the random equation of the master structure coupled with fuzzy substructures, 372 Ritz-Galerkin approximation and finite element discretization, 377 Identification method for the parameters of type I and type II fuzzy impedance laws, 381 Case of a real structure constituted by a master structure coupled with a very large number of simple oscillators. Application of the fuzzy structure theory, 391

viii

CONTENTS

7. Acoustic modes of the acoustic cavity, 197 8. Finite element discretization and generalized symmetric matrix eigenvalue problem, 199 9. FRF calculation in the LF range, 199 10. FRF calculation in the MF range, 205 11. Case of a zero pressure condition on part of the boundary, 209 12. Particular case of an axisymmetric acoustic cavity, 214 13. Response to deterministic and random excitations, 214 Chapter XII. External Acoustic Fluid. Boundary Integral Formulation for the LF and MF Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13.

Introduction, 215 Statement of the external acoustic problem, 217 Exterior Neumann problem related to the Helmholtz equation, 218 Acoustic impedance boundary operator and radiation impedance operator, 221 Free-space Green's function and solution of the inhomogeneous Helmholtz equation, 224 Response to prescribed wall displacement, acoustic source density and incident plane wave, 226 Results of potential theory: single- and double-layer potentials, 231 Helmholtz integral representations and their normal derivatives, 233 Boundary integral equations for the exterior Neumann problem related to the Helmholtz equation, 237 Acoustic impedance boundary operator construction: appropriate symmetric boundary integral method valid for all real values of the frequency, 244 Construction of the radiation impedance operator and asymptotic formula for the radiated pressure field, 246 Symmetric boundary element method valid for all real values of the frequency, 249 Case of a free surface, 253

Chapter XIII. Structural-Acoustic Master System in the LF range . . . . 255 1. 2. 3. 4. 5. 6. 7.

Introduction, 255 Statement of the structural-acoustic problem in the frequency domain, 256 Boundary value problem of the structural-acoustic master system, 259 Variational formulation of the structural-acoustic problem, 261 Finite element discretization, 269 Symmetric reduced matrix model in the LF range, 272 FRF calculation in the LF range, 278

x

CONTENTS

Appendix. Mathematical Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

393

References

407

.................................................

Subject Index

...............................................

Symbol

...............................................

Index

417 423

xi

Preface

This book is devoted to mechanical models, variational formulations and discretization for calculating linear vibrations in the frequency domain of complex structures with arbitrary shape, coupled or not with external and internal acoustic fluids at rest. Such coupled systems are encountered in the area of internal and external noise prediction, reduction and control problems. The excitations can arise from different mechanisms such as mechanical forces applied to the structure, internal acoustic sources, external acoustic sources and external incident acoustic plane waves. These excitations can be deterministic or random. We are interested not only in the low-frequency domain for which modal analysis is suitable, but also in the medium-frequency domain for which additional mechanical modeling and appropriate solving methods are necessary. The main objective of the book is to present appropriate theoretical formulations, constructed so as to be directly applicable for developing computer codes for the numerical simulation of complex systems. Paris, May 1997

Roger Ohayon and Christian Soize


CHAPTER I

A Strategy for Structural-Acoustic Problems

1. Introduction This book is devoted to mechanical models and appropriate numerical formulations for calculating the vibration of complex structures with arbitrary shape coupled with external and internal acoustic fluids. In practice, such coupled systems are encountered in the area of internal and external noise prediction, reduction and control problems. The external noise problem is also called the acoustic radiation of structures. Excitations can arise from different mechanisms such as mechanical forces applied to the structure, internal acoustic sources, external acoustic sources and external incident acoustic plane waves. These excitations can be deterministic or random. Concerning problems solved by analytical methods, such as rectangular thin plates, circular cylindrical thin shells, with particular acoustic and structural boundary conditions, we refer the reader to Junger and Feit, 1993; Bowman et al., 1969; Fahy, 1987; Jones, 1986. In Section 2, we define a structural-acoustic master system. In Section 3, we introduce the concept of fuzzy structure related to the structural complexity and we define a structural-acoustic fuzzy system. Section 4 is devoted to the definition of the LF (low-frequency), MF (medium-frequency) and HF (high-frequency) ranges for structural-acoustic problems. In Section 5, we present the theoretical and numerical strategy used for the LF and MF ranges. In Section 6, we introduce the different types of prescribed deterministic and random excitations for a structural-acoustic system. Finally, in Section 7, we indicate the general organization of the book.

2

STRUCTURAL ACOUSTICS AND VIBRATION

2. Structural-Acoustic Master System In general, a complex structure is composed of a main part called the master structure, defined as the "primary" structure accessible to conventional modeling, and a secondary part called the fuzzy substructure related to the structural complexity and including for example many equipment units attached to the master structure. An internal (or an external) structural-acoustic master system is defined as a master structure coupled with an internal acoustic fluid (or an external acoustic fluid). The acoustic fluid can be a gas or a liquid. A master structure coupled with internal and external acoustic fluids is called a structural-acoustic master system. In this book, mechanical modeling of a structural-acoustic master system is based on the use of the linearized equations of structural and fluid mechanics (acoustic fluids) around a state of equilibrium at rest. For a structuralacoustic problem, the fields of interest are the displacement, velocity and acceleration fields of the master structure, the pressure field in the internal fluid and the pressure field in the external fluid (near field and far field). Since the problem is linear, the structural-acoustic master system can be characterized by a Frequency Response Function (FRF) which allows the response to deterministic and random excitations to be calculated.

3. Concept of Fuzzy Structure and Structural-Acoustic Fuzzy System A fuzzy structure is defined as a master structure coupled with a fuzzy substructure. A fuzzy substructure is defined as a structural complexity consisting of a large number of small secondary dynamical subsystems attached to the master structure. Each secondary subsystem is a discrete or continuous elastodynamic system, whose resonant frequencies lie in the frequency band of analysis considered (for instance, a large number of equipment units together can constitute a fuzzy substructure). More generally, a fuzzy substructure is a part of the structure that is not accessible to conventional modeling because the details of it are unknown, or are known with insufficient accuracy (this explains the choice of the word "fuzzy"). For this reason a statistical approach is proposed for modeling fuzzy substructures, by introducing a random boundary impedance operator related to the interfaces where the fuzzy substructure is connected to the master structure. This approach, called "fuzzy structure theory", was introduced by Soize, 1986. It should be noted that the term "fuzzy" has nothing to

CHAPTER L

STRUCTURAL-ACOUSTICAPPROACH- ORIENTATION OF THE BOOK

3

do with the mathematical theory concerning fuzzy sets and fuzzy logic. In the definition introduced in Section 2, if the master structure is replaced by a fuzzy structure, then we obtain a structural-acoustic fuzzy system.

4. LF, MF and HF Frequency Ranges The different types of dynamical responses of a weakly dissipative complex structure leads to the definition of three frequency ranges of analysis. Fig. 1 is a qualitative diagram showing the frequency response function (FRF) of a component of the displacement in a fixed point of the structure. FRF

~ , ,

LF

,

]

,

.........

Frequency

]

MF

~..

HF

Fig. 1. F r e q u e n c y ranges for a c o m p l e x s t r u c t u r e

The three frequency ranges are defined as follows. - The low-frequency range (LF) is defined as the modal domain for which the associated conservative system has a small number of modes (low modal density). - The high-frequency range (HF) is defined as the frequency band for which there is a uniform high modal density. Among the analysis methods for this band, we can mention the wave approach (see for instance Maidanik and Dickey, 1988 and 1994), the global statistical energy approach and the local energy approach. The most popular global statistical energy approach is the Statistical Energy Analysis. For its basic formulation, we refer the reader to Lyon, 1975; Maidanik, 1981; Lyon and DeJong, 1995; Soize, 1995a; Lesueur, 1988; Cremer et al., 1988; Crighton et al., 1992. For extensions to the case of a structure coupled with an external unbounded liquid, see David and Soize, 1994. Concerning the local energy approach, we can mention the power flow analysis based on continuous energy equations (Nefske and Sung, 1988).

4

STRUCTURAL ACOUSTICS AND

VIBRATION

- For complex systems, an intermediate frequency range called mediumfrequency range (MF) appears. This MF range is defined as the intermediate frequency band for which the modal density exhibits large variations over the band. In addition, if there is a structural complexity related to the presence of a fuzzy substructure, this fuzzy substructure plays an important role in the dynamical behavior of the master structure. The fuzzy substructure induces an "apparent strong dissipation" in the master structure due to the power flow between the master structure and all the dynamical subsystems constituting the fuzzy substructure. Therefore, the frequency response functions of the master structure coupled with the fuzzy substructure have smooth variations which means that this system is not resonant in the MF range. The analysis is similar for internal structuralacoustic systems (master system or fuzzy system). The presence of an external acoustic fluid does not substantively modify the above qualitative analysis. This book is devoted to LF and MF analyses for complex structures and complex structural-acoustic systems.

5. Strategy Used for the LF and MF Ranges 5.1. Structural-acoustic master systems in the LF and MF ranges

Concerning the internal structural-acoustic master system (master structure coupled with internal fluids), the LF and MF models are based on the symmetric variational formulation of the local equations including appropriate dissipative terms. The master structure is described by the displacement field and the internal fluids are described by scalar fields. An adapted formulation is presented for the LF range and the MF range. The internal structural-acoustic master system is discretized by the finite element method. The external fluid model is based on a symmetric variational formulation of an appropriate boundary integral representation which is valid for all values of the real frequencies and does not exhibit the nonphysical irregular frequencies (spurious frequencies) encountered in standard boundary integral representation. Discretization is carried out using the finite element method which leads to boundary elements. For the structural-acoustic master system, a linear symmetric matrix system is obtained whose dimension is the number of degrees of freedom of the discretized model of the internal structural-acoustic master system due to the use of a boundary integral formulation for the external acoustic fluid.

LF computation strategy. We construct a modal reduction procedure using the structural modes of the master structure in vacuo and the acoustic

CHAPTER I. STRUCTURAL-ACOUSTICAPPROACH- ORIENTATIONOF THE BOOK

5

modes of the internal acoustic cavities. The reduced symmetric matrix equation obtained is solved frequency by frequency. M F c o m p u t a t i o n strategy. In the MF range, the dimension of the matrix equation resulting from finite element discretization is large because the mesh size must be adapted with respect to the wavelengths in the system. Consequently, a frequency-by-frequency solution is not practicable. The method proposed consists in solving the finite element matrix equation MF narrow band by MF narrow band.

5.2. Fuzzy structure in the MF range In this case, the matrix equation obtained is the sum of the finite element matrix systems corresponding to the master structure with a frequencydependent random symmetric matrix corresponding to the fuzzy substructure. An appropriate process is used for solving the linear random matrix equation. 5.3. Structural-acoustic systems in the MF range In this case, the matrix equation obtained is the sum of the finite element matrix systems corresponding to the structural-acoustic master system with a frequency-dependent random symmetric matrix corresponding to the fuzzy substructure and, as above, an appropriate solver is used.

6. Excitations and Responses For deterministic excitations such as mechanical forces applied to the master structure, internal acoustic sources, external acoustic sources and external incident acoustic plane waves, the responses are calculated using the frequency response functions constructed in the LF and MF ranges. Concerning random excitations, two cases are considered. The first case concerns discrete random excitations modeled by vector-valued stochastic processes. This is the case for random mechanical forces or for random aeroacoustic internal sources induced by machines. The second case concerns continuous random excitations corresponding to random wall pressure fields modeled by vector-valued random fields. This is the case for a turbulent boundary layer due to an external flow around a launcher, an aircraft, an automobile, a naval structure, a high-speed train or an external random field due to a spatial distribution of multipole sources such as complex aeroacoustic sources generated by a launcher engine. For time-stationary random excitations, spectral analysis of the linear filtering theory is used and, for nonstationary random excitations, appropriate

6


methods are used. In all these random cases, the frequency response functions are used in the LF and MF ranges.

7. Organization of the Book This book is organized in three main parts. Chapters III to IX are devoted to vibration of the master structures. Chapters X to XIV deal with vibration of structural-acoustic master systems and finally, in Chapter XV, fuzzy structures are introduced. Chapter II is devoted to basic notions on variational formulations, the Ritz-Galerkin method and finite element discretization used throughout the book. In Chapters III, IV and V, we introduce the structural modes, the damping model based on the use of the linear viscoelasticity theory and the frequency response function (FRF) for the master structure. Chapter VI presents the master structure F R F calculation in the LF range using modal reduction and a frequency-by-frequency construction. Chapters VII and VIII give the master structure FRF calculation in the MF range. Chapter VII presents a narrow-frequency-band-by-narrow-frequency-band construction applied to the finite element discretization of the master structure and Chapter VIII introduces a reduced model in the MF range for which a frequency-by-frequency construction can be used. Chapter IX gives the methods for constructing the response of the master structure submitted to deterministic and stationary or nonstationary random excitations in the LF and MF ranges. Chapter X is devoted to the derivation of the linear acoustic equations introducing a damping model, a frequency-dependent wall acoustic impedance and appropriate terms for the internal fluid to ensure that the equations are correctly-stated at zero frequency. Chapter XI introduces the acoustic modes of an internal cavity and presents the construction of the FRF for internal acoustic problems in the LF and MF ranges. Chapter XII presents a symmetric boundary integral formulation for the external acoustic problem which is valid for all frequency values in the LF and MF ranges. Chapter XIII describes the construction of the FRF calculation in the LF range for a structural-acoustic master system using a modal reduced

CHAPTER L

STRUCTURAL-ACOUSTICAPPROACH- ORIENTATION OF THE BOOK

7

symmetric matrix model and a frequency-by-frequency construction. The methods presented in Chapter IX can be applied directly for calculation of the response of a structural-acoustic master system submitted to deterministic or random excitations. Chapter XIV deals with the construction of the FRF calculation in the MF range for a structural-acoustic master system using a narrow-frequencyband-by-narrow-frequency-band construction applied to the finite element symmetric matrix of the system. The methodology presented in Chapter VIII can be extended for constructing a reduced symmetric matrix model of the structural-acoustic master system in the MF range. The methods presented in Chapter IX can be applied directly for calculation of the response of a structural-acoustic master system submitted to deterministic or random excitations. Chapter XV introduces the fuzzy structure theory in structural dynamics for the MF range. The basic tools introduced allow structural-acoustic fuzzy systems to be considered, i.e. structural-acoustic master systems coupled with fuzzy substructures corresponding to additional structural complexities. Mathematical notations are given in the appendix, presenting the main mathematical tools used in the book and particularly elements concerning linear, antilinear, bilinear and sesquilinear forms on Hilbert space and Fourier transforms of functions. A comprehensive list of references is given and is followed by a Subject Index. It should be noted that the footnotes concern purely mathematical aspects which can be skipped in a first reading.


CHAPTER II

Basic Notions on Variational F o r m u l a t i o n s

1. Introduction The objective of this chapter is to review the classical tools for the construction of variational formulations of boundary value problems and the corresponding discretization, which will be used throughout the book. In this chapter, these tools will be illustrated using a classical basic problem. In Sections 2 and 3, we introduce the basic boundary value problem in a bounded medium and its strong solution. Sections 4 and 5 give the method for constructing a variational formulation of the boundary value problem using the test function (weighted function) method. For that, we define the admissible function space of the problem and we use Green's formula. We introduce the weak solution of the boundary value problem which corresponds to the solution of the variational formulation. Finally the converse is established. In Section 6, we introduce the linear operator equation associated with the variational formulation. Section 7 reviews the Ritz-Galerkin and finite element methods which constitute the basic tools used in this book. Section 8 is devoted to bibliographical comments.

2. Boundary Value Problem Let f~ be a three-dimensional open bounded domain of ~3 with a smooth boundary F = 0Ft. The external unit normal to F is denoted as n (see Fig. 1). We denote by x = ( x l , x 2 , x3) any point in f~. Let g be a given


10

real-valued field defined on ~ and G be a given real-valued field defined on F. For k > 0, we consider the following boundary value problem

-V2u+ku-g

in

~

F

,

,

(1)

with the Neumann boundary condition ~u

0--n=G

on

(2)

where u is a real-valued field defined on f~. The Laplacian operator is defined by V 2 = V . V in which V denotes the gradient operator with respect to x = (xl,x2,x3). The normal derivative Ou/On is defined by Vu.n. n

x3

>

x2 x1 Fig. 1. G e o m e t r i c a l c o n f i g u r a t i o n

3. Strong Solution of the Boundary Value Problem For sufficiently regular functions g and G, the boundary value problem defined by Eqs. (1) and (2) has a unique solution u called the strong

solution 1.

1 F r o m a m a t h e m a t i c a l point of view, if g6L2(n) and G6H1/2(F)CL2(F), then strong solution u belongs to H 2 ( R ) . Since u E H 2 ( n ) , then the trace u r of u on r belongs to H3/2(F) and the trace au/an on r belongs to H1/2(F).

l1

CHAPTER II. BASIC NOTIONS ON VARIATIONAL FORMULATIONS

4. Variational Formulation and Weak Solution of the Boundary Value Problem 4.1. Variational formulation

We proceed by the test-function method (also called weighted-function method). In a first step, we introduce the admissible function space C of the problem constituted by the "sufficiently differentiable ''2 real-valued functions Ju defined on f~. In a second step, we consider strong solution u of the boundary value problem defined by Eqs. (1) and (2). Multiplying Eq. (1) by an arbitrary function ~u E C and integrating over the domain fl yields

-/ in which

V2u~udx+k/nu~udx= /ng~udx

dx = dxldx2dx3.

,

(3)

We have Green's formula 3

a -~ 5uds ,

(4)

in which ds is the surface element. Taking into account the Neumann condition defined by Eq. (2) and using Eq. (4), Eq. (3) can be written as

The variational formulation of the boundary value problem defined by Eqs. (1) and (2) is then stated as follows. For given g and G, find u in C such that Eq. (5) is satisfied, i.e.

a(u,Ju)=f(Ju) in which a(u, defined by

Ju)

,

VJueC

,

is the symmetric positive-definite bilinear form on C x C

a(u , ,u) = /a Vu . V , u dx + k / u ,u dx and

f(5u)

(6)

,

(7)

is the linear form on C defined by

f(Ju) - / gSudx-t- frGSUds

.

(8)

2 Space C=H l(fl). 3 Green's formula is used with uEH2(f~) and 5 u E H l ( f l ) . Since H1/2(F)CL2(F), (u,Su)~-+ Ou/On)Suds is the inner product in L2(F) of Ou/anEH1/2(F) with 6uEH1/2(F).

fr(

12


4.2. Weak solution of the boundary value problem

of the boundary value problem defined by Eqs. (1) and (2) is a function u E C satisfying Eq. (6). It should be noted that the strong solution of Eqs. (1) and (2) is a solution of Eq. (6), i.e. also a weak solution of Eqs. (1)and (2). A weak solution

4.3. Existence and uniqueness of the weak solution of the boundary value problem

For sufficiently regular functions 4 g and G, the variational formulation defined by Eq. (6) has a unique solution 5 u E C.

5. Converse Let u be the solution of Eq. (6) (weak solution of Eqs. (1) and (2)) and let us assume that u is sufficiently regular 6. Using Eq. (6) and applying the Green formula defined by Eq. (4) yields (-V2u+ku-g)

hudx+

(-~n - G ) h u d s - O

,

V hu e C

.

(9)

To recover Eqs. (1) and (2) from Eq. (9), we proceed in two steps. First, we choose a test-function 5u in the set of infinitely differentiable functions vanishing in a neighborhood 7 of F. Then Eq. (9) yields J~(-V2u + k u - g)5udx - 0

(10)

From Eq. (10), we deduce that s -V2u+ku-g

in

Ft

.

(11)

For g6L2([2) and G6H1/2(F), Eq. (6) has a unique solution u6C in which C=H1([2). In effect, bilinear form a(u ,6u) is continuous on CxC and coercive ( C - elliptic) on C, and linear form f(6u) is continuous on C. Consequently, the L a x -M i l g ra m t h e o r e m can be used. It should be noted t h a t the weak solution exists for 9 6 C ' (dual space of C) and G6H -1/2 (F) (dual space of H I / 2 ( F ) ) . In this case the strong solution does not exist in the sense defined in f o o t note 1 related to Section 3, but a weak solution exists. 5 We have the regularity p r o p e r t y of the weak solution: for g6L2([2) and G6H~/2(F), t h e n weak solution u belongs to H2 ([2). In this case, the weak solution of Eqs. (1) and (2) coincides with t h e strong solution of Eqs. (1) and (2). 6 It is a s s u m e d t h a t u6H 2 ([2) (see footnote 5). 7 More precisely, test-functions 6u must be taken in the space D(~t) of the infinitely differentiable functions on [2 with compact s u p p o r t . 8 Since Eq. (10) holds for all 6u6D([2), we deduce t h a t - V 2 u + k u - g = O in the space D'([2) of the generalized functions on [2. Since g6L2([2), generalized derivative - V 2 u of u is represented by a function in L2(gt). We can then consider Eq. (11) as an equality of functions belonging to L 2 ([2) and consequently, Eq. (11) is satisfied almost everywhere in ft.

CHAPTER II. BASICNOTIONS ON VARIATIONALFORMULATIONS

13

S u b s t i t u t i n g Eq. ( 1 1 ) i n t o Eq. ( 9 ) y i e l d s

jfr(-~n Ou-

G) 6u ds - 0

,

V~uEC

.

(12)

We then deduce that 9 ~u

On

=G

on

F

.

(13)

6. Associated Linear Operator Equation Bilinear form that ~0

a(u, 5u)

given by Eq. (7) defines a linear operator A such

(14)

< A u, ~u > - a(u, ~u) and the element f such that 11

< f , d u > = f((~u)

.

(15)

Consequently, the linear operator equation corresponding to the variational formulation defined by Eq. (6) is written as 12 Au-f

.

(16)

9 Since ~u~_Hl(i2), the trace of ~u on F belongs to H1/2(F)CL2(fl). Since u E H 2 ( ~ ) (see footnote 6), Ou/On belongs to H 1 / 2 ( F ) and, since GEH1/2(F), function Ou/On-G belongs to H1/2(F)CL2(f~). Then, the left-hand side of Eq. (12) can be considered as the inner product in L2(F) and consequently Eq. (13) is satisfied almost everywhere on F. 10 Since bilinear form a(u,~u) is continuous on CxC (see footnote 4), a(u,~u) defines a bounded linear operator (continuous operator) A from C into its dual space C'. In Eq. (14), the angle brackets < . , .> denote the duality product between C' and C (see M a t h e m a t i c a l Notations in the appendix). 11 Since linear form f(~u) is continuous on C (see footnote 4), f(~u) defines an element f belonging to the dual space C' of C. In Eq. (15), the angle brackets < . , .> denote the duality product between C' and C (see M a t h e m a t i c a l Notations in the appendix). 12 It should be noted t h a t Eq. (16) is an equality in C'.

14


7. Ritz-Galerkin Approximation and Finite Element Method 7.1. Ritz-Galerkin approximation The Ritz-Galerkin method consists in constructing an approximate solution of the variational formulation defined by Eq. (6) in a subspace ~N C C of finite dimension N. To construct CN, we consider a complete countable family {Ua}a=l, 2.... of elements in C such that CN is spanned by { U l , . . . , UN}. Any element u g E CN C C can then be written as N u N --

E

q'~ Ua

in which q = ( q l , . . . , qN) is the RN-vector of components {qa} called the generalized coordinates. A test function 6u in subspace CN c a n be written as

N

6u = ~

6qf~uf~ ,

(18/

f~=l

in which 6q = (6q1,..., 6qN) is in [~N. The restriction of the variational formulation defined by Eq. (6) to subspace CN is obtained by substituting Eqs. (17) and (18) into Eq. (6), N

N

~ a=l ~=I

N

a(ua, u~) qa 6q~ = E

f(u~) 6q~ , V 6q E ~ g .

(19)

~=i

From Eq. (191, we deduce the linear matrix equation of dimension N [A]q=~

(20)

,

in which [,4] is an (N x N) matrix such that

[ A ]~. = a(u., ~ )

(21)

and .%" = (~'1,...,-~'N) E ~ N such that

~/3-- f(u~)

(22)

9

Since bilinear form a(u, 5u) is symmetric positive definite, ~(~, ~ ) = ~ ( ~ , ~)

,

(23)

CHAPTER II.

BASIC NOTIONS ON VARIATIONAL FORMULATIONS

,

a(u,u)>O

rue0

,

15

(24)

we deduce that [,4] is a symmetric positive-definite matrix and consequently, that matrix [At] is invertible. 7.2. Finite element method

Let us briefly recall that the finite element method is a powerful tool for constructing the basis used in the Ritz-Galerkin method described in Section 7.1. In the context of the finite element method, we denote the dimension of the finite approximation as n and the basis of Cn as { e l , . . . , en}. In this method, the generalized coordinates are denoted as Uj and coincide with the values u n (xj) of u n at the n nodes xj of a finite element mesh of domain Ft = ftOF (see Fig. 2). Approximation u n is defined by a low-order F

Fig. 2. Finite element mesh of domain

polynomial interpolation from its nodal values. rewritten as

Eq.

(17) can then be

n =

.

(25)

j=l

The values Uj = u n (xj), j = { 1 , . . . , n}, are called the degrees o f f r e e d o m (DOFs). It should be noted that ej(xk) = 5jk (Sjk denotes the Kronecker symbol) and that basis function ej (x) is equal to zero for all x belonging to the finite elements which are not connected to node xj (this basis, defined on ft, is called a local basis). For this particular basis related to the finite element method, Eq. (20) is rewritten as [A]U=F

,

(26)

16


in which U = (U1,..., U~) e ~n, [A] is an (n • n) symmetric positivedefinite sparse matrix such that

(27)

[ A ]ej = a(ej, ee) and F = ( F 1 , . . . , Fn) C ~n

is s u c h

that

Fe = f (ee)

(28)

.

Remark on the use of the linear operator equation. We emphasize that linear operator equation A u -- f introduced in Section 6 has a direct counterpart for its corresponding finite dimension approximation, i.e., [,4] q = ~ for the Ritz-Galerkin method and [A]U = F for the particular case of the finite element method. In addition, the algebraic properties of operator A hold for the matrices [,4] and [A] which are symmetric positive definite. This presentation will be used throughout the book in order to have similar expressions for the continuous cases and the corresponding finite dimension matrix equations. 7.3. Comments concerning convergence of the Ritz-Galerkin method

The Ritz-Galerkin method allows a finite dimension approximation u N of solution u of Eq. (6) (weak solution of Eqs. (1) and (2)) to be constructed. The convergence of sequence { u N } N to u is obtained in space C independently 13 of the regularity of solution u, lim

-

N-++c~

Niic -

0

,

(29)

in which [I. [[c denotes the norm in space C. Let us assume that g belongs to L2(ft) (space of square integrable functions) and G belongs to a space Cr of regular 14 functions on F, such that trace Ou/On on F of the normal derivative of weak solution u belongs 15 to Cr (in this case the weak solution coincides with the strong solution, see Sections 4.2 and 4.3). In addition, we assume that basis functions {u~}~ are such that trace Ou g / O n on F of the normal derivative of u N belongs to Cr. In general, we have 16 lim

N--++oo

uGH2(Ft)CHI(~t), Cr----H1/2(F).

13 From footnote 5, 14 See footnote 1,

I[

OUN 0n

-

G I]c7 r

but we have only

0

lim[[u--uN[[gl(a)--O

(30) as N--++c~.

15 See footnote 5. 16 For all

vEH1/2(F)C_+L2(F)C_+H-1/2(F),we have

[[V[]H_I/2(F)~C[]V[[H1/2(F), C>0.

CHAPTER II. BASIC NOTIONS ON VARIATIONAL FORMULATIONS

17

in which C~. denotes the dual space 17 of Cr. This means that the convergence of u g in C (see Eq. (29)) does not imply the convergence of Ou N / O n to G in space C~. Consequently, the constructed approximation defined by Eq. (17) does not allow G to be retrieved on boundary F for any value of N. Nevertheless, since Eq. (29) implies the convergence of V u N to V u in ~pace L2(f~), then the first derivatives are convergent inside f~, but not on boundary F (see Fig. 3).

no

0

0no/\

Ou (Xo sn~ 0% -

G(x0)

OuN(xo-sn o)

O%

>s

Fig. 3. Convergence of the normal derivative

8. Bibliographical Comments Concerning the mathematical aspects of functional analysis using generalized functions, generalized derivatives, Sobolev spaces and trace theorems in the context of the variational formulations of boundary value problems, we refer the reader, for instance, to Brezis, 1987; Dautray and Lions, 1992; Reed and Simon, 1980. For a general presentation of variational methods for boundary value problems, see for instance Oden and Reddy, 1983; Dautray and Lions, 1992; Washizu, 1975. For more details on the RitzGalerkin method and variational principles, see Dautray and Lions, 1992; Finlayson, 1972; Mikhlin, 1964. Presentation of the finite element method and its applications in science and technology can be found in Zienkiewicz and Taylor, 1989; Hughes 1987; Bathe and Wilson, 1976; Ciarlet, 1979; Strang and Fix, 1973; Dautray and Lions, 1992; Raviart and Thomas, 1983; Ofiate et al., 1991.

17 Space Cr--H1/2(F) and space C~,--H-1/2(F).


19

C H A P T E R III

Linearized Vibrations of Conservative Structures and Structural Modes

1. Introduction The objectives of this chapter are to give a synthesis of linear vibrations of conservative elastic structures and to introduce the fundamental notion of structural modes, the formulations appropriate to their calculation for complex structures and the corresponding numerical procedures. Among the structural vibration textbooks, we can mention Argyris and Mlejnek, 1991; Clough and Penzien, 1975; Landau and Lifchitz, 1992a; Meirovitch, 1980 and 1990; Roseau, 1980. In Section 2, we introduce the equations and the boundary conditions in the time domain for linear vibrations of an undamped structure occupying a bounded domain. In Section 3, we present the associated spectral problem which allows the introduction of structural modes. Sections 4 and 5 deal with the variational formulation of the spectral problem and the corresponding linear operator equation. In Sections 6 and 7, we introduce the basic properties of the spectral problem for a fixed structure (zero displacement on a part of the boundary) and for a free structure: countable number of positive eigenfrequencies, discussion of rigid body modes, orthogonality and completeness of the structural modes. In Section 8, we discuss the cases of structures having symmetry properties: structures with one plane of symmetry, axisymmetric structures and structures with cyclic symmetries. We present the appropriate formulation allowing us to simplify the calculation of the structural modes.

20


VIBRATION

Section 9 is concerned with the finite element discretization of the spectral problem and the corresponding generalized symmetric matrix eigenvalue problem. The finite element method is the most efficient method for computing the structural modes in the case of an arbitrary domain, boundary conditions and materials. Finally, in Section 10 we briefly discuss substructuring techniques allowing a large eigenvalue problem to be replaced by several smaller eigenvalue problems posed on substructures.

2. Conservative Elastodynamic Boundary Value Problem in a Bounded Medium with Initial Cauchy Conditions 2.1. General equations, boundary conditions and initial conditions The physical space ~3 is referred to a Cartesian reference system (i, j, k) and we denote the generic point o f ~ 3 as x - (xl,x2,x3). Let f~ be a threedimensional bounded connected domain of ~3 occupied by the structure at equilibrium. It is assumed that boundary 0f~ is smooth. The external unit normal to OFt is denoted as n (see Fig. 1).

G

F X

x3

1 /

ro

u= uF0

x2

x1

Fig. 1. G e o m e t r i c a l configuration

Let u(x, t) - (ul (x, t), u2(x, t), u3(x, t)) be the displacement at time t of a particle located at point x in f~. For any function a(x, t), we use the

21

CHAPTER III. LINEARIZED VIBRATIONS AND STRUCTURAL MODES

notation a,j(x, t) - 0a(x, t)/Oxj, Ota(x, t) - 0a(x, t)/Ot and 02a(x, t) = 02a(x,t)/Ot 2. We also use the classical convention for summations over repeated Latin indices, but not over Greek indices. In this chapter, we consider linear vibrations of a structure around a position of static equilibrium taken as reference configuration Ft (for the sake of brevity, we do not consider a reference configuration with prestresses: we use the natural state at rest). We operate within the framework of three-dimensional linearized elasticity theory and refer the reader to basic books (Truesdell, 1960; Fung, 1968; Malvern, 1969; Washizu, 1975; Abraham and Marsden, 1978; Germain, 1973 and 1986; Marsden and Hughes, 1983; Salen~on, 1988; Ciarlet, 1988; Landau and Lifchitz, 1992a). The symmetric stress tensor aij is related to the linearized symmetric strain tensor eij by the constitutive equation aij = aijkh ~kh , (1) in which aijkh is the tensor of the elastic coefficients of the material having the usual properties of positive definiteness (for all second-order real symmetric tensors Xij, we have aijkh X k h X i j ~ C X i j X i j with c > 0) and symmetry ( a i j k h -- ajikh -- aijhk -- a k h i j ) . Strain tensor ~ij is related to displacement field u by 1

=

(uk,h + Uh,k)

(2)

Denoting as g(x, t) = (gl (x, t), g2(x, t), g3(x, t)) a given body force field applied in f~, the elastodynamic equation is written as

pOYui(x,t)-aij,j(x,t)=gi(x,t)

in

fl

,

t>0

,

(3)

in which p(x) is the mass density field defined on ft. For prescribed displacements Uro(X, t) on the part F0 of the boundary Oft of Ft, we have the boundary condition u(x,t)=Uro(X ,t)

on

Fo

9

(4)

For a given surface force field G(x, t) = (G1 (x, t), Gz(x, t), G3(x, t)) applied to the part F = 0f~\Fo, we have the boundary condition (x, t) nj (x) = a (x, t)

oil

r

(5)

.

Finally, the above equations must be completed by the Cauchy initial conditions

u(x, 0) =u0(x)

,

0tu(x, 0) =v0(x)

in

f~ ,

(6)

22


VIBRATION

in which u0 and v0 are given displacement and velocity fields respectively. 2.2. Boundary value problem in terms of displacement field u

In order to introduce the spectral problem in the next section, we consider the following time boundary value problem for which Uro - 0 on F0, g - 0 in [2 and G - 0 on F (see Fig. 2-a). n

F

n

F=0~

x

f~

X3

~

~

X3

r0 _x2

/

Xl

x2

x1 b- Free structure

a- Structure fixed on PO

Fig. 2. Geometrical configuration for the spectral problem

Below, aid(u) denotes the stress tensor aid as a function of u obtained by substituting "Ckh defined by Eq. (2) into the constitutive equation defined by Eq. (1), i.e. (u) = kh(U) . (7) Substituting Eq. (7) into Eq. (3) and from Eqs. (4) to (6), we deduce the boundary value problem in terms of u:

pO2 u i - a i j , j ( u ) - O u(x,t)=O

o u(x, 0) - Uo(X)

j(u)

,

j=0

in

f~

,

on

Fo

r

t>0

,

(8)

,

(9)

,

0tu(x, 0) - v0(x)

(10)

in

f~

,

(11)

in which u0 and v0 are given initial conditions. Eqs. (8) to (11) correspond to a structure fixed on Fo. For a free structure (Po = 0), Eq. (9) does not exist and Eq. (10) holds on the global boundary P = 0f~ of domain f~ (see Fig. 2-5).


23

3. Associated Spectral Problem: Eigenfrequencies and Structural Modes We consider all the solutions u(x, t) = u(x) exp(iwt) of the boundary value problem defined by Eqs. (8) to (10), where i is the pure imaginary complex number. The spectral problem consists in searching for {w, u}, with u ~= O, satisfying the boundary value problem

-w 2pui-aij,j(u)=O u=O

on

aij(u)nj=O

in

F0

,

on

F

f~

,

(12) (13)

.

(14)

We will see in Section 6 that for a structure fixed on F0, this spectral problem has a countable number of solutions {w~,u~}~=l,2,..., in which w~ > 0 and u~ are real vector fields. For a free structure (F0 = 0), Eq. (13) does not exist and Eq. (14) holds on the global boundary F = 0f~ of domain f~. In this case, we will see in Section 7 that w~ >_ 0 and u~ are real vector fields. For the two cases, the angular frequencies w~ and the corresponding ua are called the eigenfrequencies (or natural frequencies) and the structural modes (or eigenmodes or mode shapes of vibrations) respectively.

4. Variational Formulation of the Spectral Problem The variational formulation is constructed using the test-function method presented in Chapter II. 4.1. Construction of the variational formulation

First, we introduce the vector space C of "sufficiently differentiable" 1 functions 5u - (SUl, 5u2, 5u3) defined on f~ with values in R3. In a second step, multiplying Eq. (12) by an arbitrary function 5u E C and integrating over domain f~ yields

--032 /f~ pU i ~Ui d x - /a 1 Space C is the Sobolev space (Hl(f~)) 3.

CTij'j(U) ~ui

dx - 0 .

(15)

24

STRUCTURAL

ACOUSTICS A N D

VIBRATION

The second term on the left-hand side of Eq. (15) is transformed using the following identity

(aij 5ui),j dx .

(16)

The second integral on the right-hand side of Eq. (16) is transformed to an integral on 0f~ using Stokes's formula

/ (aij 5ui)5 dx - fo aij nj 5ui ds

,

(17)

f~

where ds is the surface element. Green's formula

From Eqs.

(16) and (17), we deduce

/f aij,j Sui dx - /f aij Sui,j dX - fof~ Crijnj Su~ ds .

(18)

Substituting the left-hand side of Eq. (18) into Eq. (15) yields

-w 2/fpui 5u~ d x + / a ~ j ( u ) 5u~,j dx-fof~

crij ( u ) n j 5 u i ds = 0

. (19)

In a third step, we transform the second term of the left-hand side of Eq. (19) in order to get a symmetric formulation in terms of u and 5u. From the symmetry of the stress tensor and the definition of the strain tensor given by Eq. (2), we deduce the equalities aij ~ui,j = aij((~ui,j -Jr-~uj,i)/2 - a~j ~j (Su). Substitution into Eq. (19) yields

-w 2 ff puiSuidx+/f aij(u)~ij(Su)dx- fof~aij(u)njSui ds=O In a fourth step, we introduce the must be distinguished.

admissible function space.

. (20)

Two cases

Structure fixed on F0. In this case, F0 r 0 and the admissible function space must satisfy the constraint on u defined by Eq. (13) and consequently is the subspace Co of C defined by 2 C0-{uEC

;

u=0

on

ro}

2 Space Co is the Sobolev space HoX,ro( ~ ) - - { u E ( H I ( ~ ) ) 3 ,U--0 on t o ) .

9

(21)

CHAPTER III.

25

LINEARIZED VIBRATIONS AND STRUCTURAL MODES

Let 6u be any test function in Co C C. Since 6u = 0 on F0 and using Eq. (14), Eq. (20) yields ,

V~uECo

.

(22)

Free Structure. In this case, Fo = ~) and Eq. (13) does not exist. Therefore,

the admissible function space of the problem is C (there is no constraint on u). Let ~u be any test function in C. Due to Eq. (14), the integral over 0f~ in Eq. (20) is equal to zero and Eq. (20) yields

-w 2/~ pui ~ui dx +/f~ crij (u)eij(6u)dx

,

= 0

VSuEC

.

(23)

4.2. Variational formulation Structure fixed on Fo. The variational formulation of the spectral problem

defined by Eqs. (12), (13) and (14) is stated as follows. Find real w2 > 0 and u in Co such that

faaij(u)

eij(Su) dx = w 2 / ~ p u. 6u dx

,

v~ueCo

.

(24)

Free structure. The variational formulation of the spectral problem defined

by Eqs. (12) and (14) is stated as follows. Find real w2 > 0 and u in C such that

/ a~j(u) c~j(~u) dx = w2/a p u. ~u dx

,

V6uEC

.

(25)

5. Associated Linear Operators and Algebraic Properties We proceed as in Chapter II in order to introduce a linear operator equation corresponding to the variational formulation defined by Eq. (24) or (25). 5.1. Mass operator

We introduce the bilinear form m(u. ~u), called the structural mass bilinear form, defined by

,n(u, ~u) = / p Ja

u. ~u dx ,

(26)

26


VIBRATION

which is symmetric because

= m( u, u)

,

(27)

and which is positive definite because p(x) > 0 (for all u r 0, we have re(u, u) > 0). This result holds for the fixed and free cases. We then introduce the linear operator M, called the mass operator, such that 3 <Mu,6u>=m(u,6u)

(28)

.

5.2. Stiffness operator

Using Eq. (7), we introduce the bilinear form k(u, ~u), called the structural stiffness bilinear form, defined by k (u, 5u) - ]~

aij (u) eij (Su) dx

- / ~ aijkh Ckh(U) Cij(6U) dx

(29)

From the symmetry property a i j k h - - a k h i j of the tensor of the elastic coefficients of the material, we deduce that bilinear form k is symmetric:

k(u,

= k( u, u)

(30)

From the positive definiteness of tensor a i j k h , w e deduce that bilinear form k is positive semidefinite. The fixed and free cases must be distinguished. Structure fixed on F0. Bilinear form k(u, 5u) defined on Co • Co is positive

definite: k(u,u)>0

,

for all

u#0

We then introduce the linear operator K, called the that 4 < K u , 6u > = k(u, 6u) .

.

(31)

stiffness operator, such (32)

Bilinear form m ( u , 6 u ) is continuous on H • with H - - ( L 2 ( F t )) 3 and is also continuous on C• T h e n o p e r a t o r M is continuous from C into its dual space C' and the angle brackets in Eq. (28) denote the duality p r o d u c t between C ' and C. Bilinear form k( u, 6u) is continuous on Co• T h e n o p e r a t o r K is continuous from Co into its dual space C~ and the angle brackets in Eq. (32) denote the duality p r o d u c t between C~ and Co.

CHAPTER III.

27


Free structure. Bilinear form k(u, 5u) defined on C • C is positive semidef-

inite k(u,u)>__0

,

for all

ur

.

(33)

In effect, from Eq. (29), we deduce that k ( u , u ) = 0 implies tensor eij(u) = 0 due to the positive definiteness of tensor aijkh. Since f~ is a connected domain, the general solution of the system of equations Eij (u) = 0 is written as Urig(X) - - t + 0 x x , Vx E [2 , (34) in which t and 0 are two arbitrary constant vectors in ~3. We denote as ~rig the subset of C spanned by t and 0. This space is of dimension 6 and Urig belongs to Crig. From a mechanical point of view, Urig represents the rigid body displacement field (in the linearized theory). We then introduce the linear operator K, called as above the stiffness operator, such that 5 =k(u,6u)

(35)

,

which is symmetric positive semidefinite. 5.3. Rewriting the variational formulation Structure fixed on F0. Using the above notations, the variational formula-

tion defined by Eq. (24) can be rewritten as follows. Find real w > 0 and u in Co such that k(u,Su)-w 2m(u,6u)

,

V6ueC0

9

(36)

Free structure. Similarly, the variational formulation defined by Eq. (25)

can be rewritten as follows. Find real w > 0 and u in C such that k(u,Su)-w

2m(u,Su)

,

VSueC

.

(37)

5.4. C o m m e n t s about slender structures

In the case of slender structures such as beams, plates and shells, the structural mass and stiffness bilinear forms defined by Eqs. (26) and (29) for the three-dimensional case are simply replaced by appropriate expressions, see Leissa, 1993a and 1993b; Novozhilov, 1964; Soedel, 1993. For beam, plate and shell finite elements see Bathe and Wilson, 1976; Dautray and Lions, 1992; Hughes, 1987; Ohayon and Nicolas-Vullierme, 1981; Zienkiewicz and Taylor, 1989.

5 Bilinear form k(u,Su) is continuous on C XC. Then operator K is continuous from C into its dual space C' and the angle brackets in Eq. (35) denote the duality product between C' and C.

28


VIBRATION

6. B a s i c P r o p e r t i e s o f t h e E i g e n f r e q u e n c i e s a n d S t r u c t u r a l M o d e s for a S t r u c t u r e F i x e d o n ro

The eigenvalue problem. Setting ~

the spectral problem defined by Eq. (36) is rewritten as the following eigenvalue problem. Find ~ and u r 0 in Co such that -

k(u,~u)=~m(u,~u)

0) 2 ,

,

V~ueC0

9

(38)

Countable number of positive eigenvalues. Since fl is a bounded domain and from the properties indicated in Section 5 for a fixed structure (symmetry and positive definiteness of the mass and stiffness bilinear forms), it can be shown that there exists an increasing sequence of positive eigenvalues 0 < A ~ _ < A 2 _ < . . ~ 1 form a complete set in Co which means that an arbitrary function u belonging to Co can be expanded as -~-C~

u -

qo uo

,

(40)

in which {qa }a is a sequence of real numbers.

Orthogonality of the eigenfunctions. It can be shown that the sequence of eigenfunctions {u~ }~ satisfies the orthogonality conditions with respect to mass and stiffness, m ( u ~ , u ~ ) - #~ 5 ~ , (41) 2 (laf~ ,

(42)

in which ,

(43)

CHAPTER III.

29


and where #a is a positive real number depending on the normalization of eigenfunction ua, because the eigenfunctions are defined up to a multiplicative constant. Linear operator equation. The linear operator equation corresponding to the variational formulation defined by Eq. (38) is written as Ku=AMu

,

ueC0

9

(44)

This problem is called a generalized eigenvalue problem because operator M is not the identity operator. The algebraic properties defined by Eqs. (41) and (42) can be rewritten as < M u~, u/~ > = #~ ~/~

,

(45)

< K u ~ ,u/~ > = ~aw~2 6aZ

(46)

Bibliographical comments. Concerning the mathematical aspects, we refer the reader to Dautray and Lions, 1992; Kato, 1966; Dieudonn~, 1969; HSrmander, 1985; Raviart and Thomas, 1983; Sanchez-Hubert and SanchezPalencia, 1989; Yosida, 1966. Terminology. In structural vibrations, wa defined by Eq. (43) is called the eigenfrequency (or the natural frequency) of elastic structural mode ua (or the eigenmode or mode shape of vibration) whose normalization is defined by generalized mass #~. On the right-hand side of Eq. (40), real numbers qa are called the generalized coordinates. An elastic structural mode ~ is defined by the three quantities {wa, ua, #a }.

7. B a s i c P r o p e r t i e s o f the E i g e n f r e q u e n c i e s a n d S t r u c t u r a l M o d e s for a F r e e S t r u c t u r e 7.1. The eigenvalue problem Setting A - w2, the spectral problem defined by Eq. (37) is rewritten as the following eigenvalue problem. Find A and u ~ 0 in C such that

k(u,,~u) = ~, m(u,,~u)

,

V ,~u e C

.

(47)

7.2. Rigid body modes (solutions for A = 0) Let us show that there exist solutions of the type {A -- 0, u ~ 0}. Taking ), = 0 in Eq. (47) yields k(u, ~u) = 0 for all ~u E C. The general solution

30


VIBRATION

of this problem corresponds to the rigid body displacement field Brig E Crag (see Section 5.2 and Eq. (34)). Since the dimension of Crag is 6, then ), - 0 can be considered as a "zero eigenvalue" of multiplicity 6, denoted as A - 5 , . . . , A0. Let u - 5 , . . . , u0 be the corresponding eigenfunctions which are constructed such that the following orthogonality conditions with respect to the mass and stiffness bilinear forms are satisfied, for a and ~ in

{-5,...,0}, m(u~,u~)

= #~ 5 ~

k(u~,uf~) -- 0

,

(48)

.

(49)

These eigenfunctions, called the rigid body modes, form a basis of Crag C C and any rigid body displacement Urig in Crag can be expanded as 0 Urig

--

qc~ Ua

'~

(50)

9

c~---5

7.3. Elastic structural modes (solutions for ,k ~ 0) Definition of the vector space Celas. We introduce the subset Celas of C corresponding to all the displacement fields Uelas belonging to C and which do not belong to ~rig (~elas -- ~ \ ~rig)- Consequently, k(Uelas, 5Uelas) defined on Celas X Celas is positive definite and we then have ]g(Uelas , Uelas ) > 0

,

V Uelas # 0 E Celas

(51)

9

Eigenvalue problem restricted to Celas- The eigenvalue problem defined by Eq. (47) and restricted to space Celas is written as follows. Find A r 0 and Uelas r 0 in Celas such that k(Uelas, (~Uelas)- ~ m(Uelas, (~Uelas)

,

V (~Uelas E Celas

9

(52)

Countable number of positive eigenvalues. We consider the solutions of Eq. (52). Since gt is a bounded domain and due to the symmetry and positive definiteness of bilinear forms m and k on Celas • Celas, it can be proved that there exists an increasing sequence of positive eigenvalues

0 < A~ __5 is a sequence of real numbers. From a mathematical point of view, Eq. (65) can be written as the direct sum C - Crig (~ Celas

(67)

,

w i t h t h e u s u a l properties Crig C C, Celas C C a n d Crig N Celas :

{0}.

CHAPTER III.


33

7.6. Linear operator equation The linear operator equation corresponding to the variational formulation defined by Eq. (47) is written as Ku-AMu

,

ueC

,

(68)

and corresponds to a generalized eigenvalue problem. The orthogonality conditions of the rigid body modes and elastic structural modes (see Eqs. (48), (49), (55), (56), (61) and (62)) can be rewritten as

< Mu~, u~ > - #~ 6a~

,

< K u,~, u~ > = = #a w~2 ~c~

(69)

(70)

in which c~ _> - 5 and/~ _ - 5 .

7.7. Terminology In structural vibrations, the terminology is the same as the terminology defined in case. Nevertheless, a structural mode ~, {w~,u~,#~}, is either a rigid body mode quency (wa = 0) or an elastic structural eigenfrequency (w~ > 0).

used for the free structure case Section 6 for the fixed structure defined by the three quantities associated with a zero eigenfremode associated with a positive

8. Cases of Structures with S y m m e t r y Properties In this section, we briefly discuss the results concerning the structural modes of structures with a plane of symmetry, of axisymmetric structures and of structures with cyclic symmetries.

8.1. Case of a structure with a plane of symmetry The structure has a plane of symmetry (lr) if its domain f~ is symmetric with respect to (~r) and if for any two symmetric points x and x ~with respect to (lr), the mass density and the tensor of the elastic coefficients of the material are such that p(x) = p(x') and aijkh(X) = aijkh(X') respectively. In addition, if the structure is fixed on F0, F0 must be symmetric with respect to (lr). The set of structural modes is decomposed into two subsets,

34


VIBRATION

the subset of symmetric modes C + and the subset of antisymmetric modes g - (see Fig. 3). (n)

(n)

a- Symmetric mode

b- Antisymmetric mode

Fig. 3. Case of a structure with a plane of symmetry

The initial spectral problem can then be split into two independent spectral problems posed in a half domain. Their variational formulations are obtained by restriction to g+ and C-. 8.2. Case of an axisymmetric structure

Let (i,j,k) be the Cartesian frame of reference and ( x l , x 2 , x 3 ) the coordinates of x in (i,j,k). We introduce the cylindrical coordinates (r, 0, z) such t h a t xl = r cos 0, x2 = r sin 0, x3 = z and the local cylindrical frame (er, eo, ez) attached at each point x such that for 0 = O, we have er = i, eo - - j and ez - k (see Fig. 4).

""-....... "'"-............. k

g--. x1

i

J

Fig. 4. Coordinate systems for an axisymmetric structure

CHAPTER IlL

35


The structure is axisymmetric with respect to the z axis if its domain f~ has the z axis as a rotational symmetry, if the mass density and the tensor of the elastic coefficients of the material are independent of 0. In addition, if the structure is fixed on P0, boundary Po must have rotational symmetry with respect to the z axis. Let u(r, 0, z) be the displacement field expressed in the local cylindrical frame (er, ee, ez). The Fourier series expansion of u with respect to 0 yields --}-o o

o, z) + . ;

o, z) -

0, z)}

n--0

in which for each circumferential wave number n >_ 0, u + (r, 0, z) - (u + (r, z) cos nO, v+ (r, z) sin nO, w + (r, z) cos nO) ,

(72)

u~ (r, O, z) = (~n (r, z) sin n O , - v n (r, z) cos nO, wn (r, z) sin nO) ,

(73)

where for n >_ 0, (~n~, Vn ~, W~) are the harmonic components or order n. The reference plane (i, k), denoted as (0r), is the generating plane of threedimensional domain f~ and is a plane of symmetry for the structure. Consequently, the set of structural modes is decomposed into two subsets, the subset of symmetric modes u+, n > 0, with respect to (lr) and the subset of antisymmetric modes u~, n > 0, with respect to (~r) (see Fig. 5). x~ (~)

x2

a- Symmetric mode

b- Antisymmetric mode

Fig. 5. Deformation of a cross section in the plane xl,x2 for n - 3

The subset of symmetric modes is decomposed into subsets C+, n >_ 0. The subset of antisymmetric modes is decomposed into subsets C~, n >__0. The initial three-dimensional spectral problem can then be split into a countable number of independent two-dimensional spectral problems 7)f, n _> 0 posed in plane (lr). Their variational formulations, expressed in terms of the

36


VIBRATION

harmonic components, are obtained by restriction to C~. For each n >_ 1, problems P + and P n are identical (same eigenvalues). Problem P0+ related to harmonic components (~+, 0, w+) is different from problem P o related to harmonic components (0,-Vo, 0). In conclusion, the eigenvalue problems to be solved are related to P0+, P o and P+, n _ 1, and the structural modes of the three-dimensional structure are reconstituted using Eqs. (72) and (73). 8.3. Case of a structure with cyclic symmetries

Let (A) be the rotational axis of symmetry and N _> I be a positive integer. Domain f~ has cyclic symmetry with axis (A) and angle 21r/N if N-1

,

= Um=o

(74)

in which f~0 is the generating sector of angle 21r/N and if, for all m _> 1, sector tim is deduced from sector f~m-1 by the rotation ((A), 2r/N) (see Fig. 6). Let Rot2~m/g be the rotation ((A),2~rm/N) which maps any point x0 in f~0 into its corresponding point Xm in f~m"

Xm - Rot2~m/N(xO)

,

VX0 E f~0

9

(75)

Each sector f~m has its own Cartesian reference system (i,j,k)m which rotates with the sector.

tl5= 0

Fig. 6. Structure with cyclic symmetry


37

The structure has cyclic symmetry with axis (A) and angle 21r/N firstly, if its domain D has cyclic symmetry with axis (A) and angle 21r/N and secondly, if for all m > 1, the components of the tensor of the elastic coefficients of the material in (i, j, k)m at any point Xm in ~~m a r e equal to the components of this tensor in (i, j, k)0 at the corresponding point Xo in flo and thirdly, if for all m >_ 1, mass density p(Xm) at any point Xm in f~m is equal to p(x0) at the corresponding point x0 in flo. In addition, if the structure is fixed on Fo, boundary F0 must have cyclic symmetry with axis (A) and angle 2~r/N. Let u be the displacement field defined on ft. Let u m be the restriction of u to sector ~-~m whose components are expressed in (i,j,k)m. For two points Xo E f~0 and Xm E ~'~m satisfying Eq. (75), it can be shown, see Arnold (Appendix 10), 1978; Bluman and Kumei, 1989; Ludwig and Falter, 1988; Ohayon, 1985, that N-1

um(xm) -- ~e

2i~rmn }

~Un(Xo) exp(-~ N

)

,

(76)

n----O

where u~ represents the complex harmonic component of order n defined on generating sector f~0. Eq. (76) is a Discrete Fourier Transform (DFT). Two cases must be distinguished. N even. The structure has an even number N of sectors. Instead of the complex harmonic components, it is convenient to introduce N + l new real variables {u~ } defined in D0 and called the cyclic components, such that - o+ -

and for n -- 1 , . . . , N 2

{-0}

,

=

,

(77)

1,

Un+ -- ~ e {U n -~- UN_ n }

,

U n -" ~]~e { - - i ( u n -- U N _ n ) }

.

(78)

Using Eqs. (77) and (78), we deduce that Eq. (76) can be rewritten, for m - 0,...,N-l, as N_ 1 2

u

(x

l=u0

m,+

. m,+

m,-

, (79)

n--1

in which 27rmn m,+ Un (X m ) -- U ~ (X 0) COS( g

um'- (Xm) - u~ (Xo) sin(

)

(80)

2rmn g )

(81)

38


VIBRATION

It can be shown that the initial three-dimensional spectral problem can then be split into 2~-1 independent spectral problems {P~ ,n - 0 , . . . , N/2} in terms of the cyclic components, posed in the generating sector fl0, with appropriate boundary conditions on the two sides of f~0. These sides are denoted as Sr and St and are such that St - R o t 2 ~ / N ( S r ) . Let K and M be the operators defined by Eq. (32) or Eq. (35) and Eq. (28) respectively for the three-dimensional domain f~ (for a fixed or free structure). Let Ko and M0 be the corresponding operators related to domain f~o. P r o b l e m Po 9 The eigenvalue problem P0 is defined by

K0u + - X M o u 0

+

,

u +E/g0

,

(82)

in which/go is the appropriate admissible function space (including the boundary condition if the structure is fixed) containing the following constraint equation on sides Sr and St

(Uo+)~,- (Uo+)~

.

(83)

Problem 79o has a countable number of eigenvalues and eigenvectors denoted as {)~,u0+ }a> 1-

~)N/2" The

Problem

~N/2 is defined by

eigenvalue problem

Ko.+i~- ~ Mo.+i~

'

" + i ~ ~ u ,< i ~

(s4)

,

in which l~N/2 is the appropriate admissible function space (including the boundary condition if the structure is fixed) containing the following constraint equation on sides Sr and Sl +

Problem ~N/2 has a countable number of eigenvalues and eigenvectors denoted as {A, u+/2 }~>1. P r o b l e m Pn

9

For n fixed in {1 ~ ' ' ' ~

N 2

1} the eigenvalue problem 7~n is

defined by

[Oo

[Oo

[an+]u ,an+

in which/An is the appropriate admissible function space (including the boundary condition if the structure is fixed) containing the following constraint equation on sides Sr and S~

(ux)s,

- sin(2~/N)

cos(2~n/N)

("X)s~

"

39


Problem :P~ has a countable number of eigenvalues and eigenvectors denoted as { A, (u +, u n ) } ~>1. Structural modes of the three-dimensional structure. The structural modes of the three-dimensional structure occupying domain f~ are constituted

(1)- by {A,u om'+}a>l in which the structural modes are reconstituted by m,+ Eq. (80), that is to say by u o (Xm) - u+(xo), where {A,u+}~___l are the solutions of problem Po. (2)- by {),, UN/2 }~___1 in which the structural modes are reconstituted by Eq. (80), that is to say by UN/2 (xm) -- u+/2(Xo), where {),, u+/2}~>l are the solutions of problem PN/2. (3)- for all n fixed in { 1 , . . . , N2 1}, by {A, u nm'+}a>l_ and {A,u m n ' - - } a > l_ in which the structural modes are reconstituted by Eqs. (80) and (81), where { A, (u +, u n ) } ~> 1 are the solutions of problem :Pn. Consequently, for each ),, there is a pair {unm'+, u m n , - } of structural modes associated with the same eigenvalue ),. N odd. The structure has an odd number N of sectors. As was done for the case with N even, instead of the complex harmonic components, we introduce ( N + l ) / 2 cyclic components {u~} defined in f~0 and such that Uo+ -

{uo }

,

(88)

and for n = 1 , . . . , ( N - l ) / 2 , a n+ - - ~ e {U n -+- U N - n }

,

U n - - ~:~e { - - / ( U n -- U N - n ) }

9

(89)

Using Eqs. (88) and (89), we deduce that Eq. (76) can be rewritten, for m = 0,...,N-l, as (N-1)/2

U

m,+

(Xm) -- Uo

+

m~-~-

E

m~-

(90)

cos( 2~rmn N )

,

(91)

2rmn N )

"

(92)

in which m,+

an

(Xm)--

Un'-(Xm)

u+

(X0)

- - U n (X0)

sin(

It can be shown that the initial three-dimensional spectral problem can then be split into (N + 1)/2 independent spectral problems { P n , n -

40


VIBRATION

0 , . . . , ( N - 1 ) / 2 } in terms of the cyclic components, posed in the generating sector f~0, with appropriate boundary conditions on the two sides S~ and Sl of f~0. Let K and M be the operators defined by Eq. (32) or (35) and Eq. (28) respectively, for the three-dimensional domain f~ (for a fixed or free structure). Let Ko and M0 be the corresponding operators related to domain f~0.

Problem 7)0. The eigenvalue problem 790 is defined by Eqs. (82) and (83). Problem Pn. For n fixed in {1,..., ( N - I ) / 2 } , the eigenvalue problem Pn is defined by Eqs. (86) and (87). Structural modes of the three-dimensional structure. The structural modes of the three-dimensional structure occupying domain f~ are constituted (1)- by {A, u~ '+ }a> ~ in which the structural modes are reconstituted by m,+ Eq. (80), that is to say by u 0 (Xm) - u+(x0), where {A, u0+}a>l are the solutions of problem P0. (2)- for all n fixed in {1,..., (N-1)/2}, by {A, u,r n ' - b } a >_l and {A, u m n ' - }c~>l in which the structural modes are reconstituted by Eqs. (80) and (81), where { A, (u +, u~) }a___1 are the solutions of problem Pn. Consequently, for each A, there is a pair {unm'+,u ~ ' - } of structural modes associated with the same eigenvalue A.

9. Finite Element Discretization and Generalized Symmetric Matrix Eigenvalue Problem The matrix equation of the generalized symmetric eigenvalue problem corresponding to the finite element discretization of the three-dimensional continuous problem is directly deduced from the eigenvalue problem defined by Eqs. (38) or (47) (see Chapter II) [K]U=A[M]U

,

(93)

in which U = (U1,..., Un) is the vector of the DOFs which are the values of the displacement field at the nodes of the finite element mesh of domain f~. The (n x n) mass matrix [M] is symmetric positive definite. For a fixed structure on F0, the (n x n) stiffness matrix [K] is symmetric positive definite and then invertible. For a free structure, stiffness matrix [K] is symmetric positive semidefinite and singular of rank n - 6. The finite element procedures in structural vibrations can be found in Argyris and Mlejnek, 1991; Bathe and Wilson, 1976; G~radin and Rixen, 1994; Petyt,

CHAPTER III.


41

1990; Zienkiewicz and Taylor, 1989. The numerical solution of the generalized symmetric eigenvalue problem defined by Eq. (93) requires specific eigenvalue solvers, see Parlett, 1980; Bathe and Wilson, 1976; Golub and Van Loan, 1989; Chatelin, 1993.

10. Dynamic Substructuring Procedures for Calculation of the Structural Modes 10.1. Objective of the substructuring procedures

The strategy of dynamic substructuring consists in replacing the global model of the structure by an assemblage of substructures. Dynamic substructuring techniques based on the use of the fixed-interface modes or free-interface modes of each substructure have been widely developed in the literature: see for example Hurty, 1965; Craig and Bampton, 1968; MacNeal, 1971; Rubin, 1975; Craig, 1985; Min, Igusa and Achenbach, 1992; Leung, 1993; Farhat and Geradin, 1994; Morand and Ohayon, 1992 and 1995; Ohayon, Sampaio and Soize, 1997. Below we present the main method based on the use of the fixed-interface modes and completed by static boundary functions (Hurty, 1965; Craig and Bampton, 1968; Morand and Ohayon, 1992 and 1995). To clarify the main ideas, we consider a structure composed of only two substructures that interact through a common boundary (the extension to the case of more than two substructures is straightforward). In the first subsection, we present the variational formulation for the case of a free structure decomposed into two substructures. The second subsection is devoted to a dynamic substructuring method using the fixed-interface modes and boundary static functions of each substructure, presented in a general framework allowing various other decomposition procedures to be obtained. After constructing the reduced matrix model of each substructure, we construct the eigenvalue problem for the structure as the assemblage of the substructures. 10.2. Variational formulation for a structure made of two substructures Interface conditions. Domain f~ of the free structure is decomposed into two subdomains, ftl with boundary 0gtl - F1 U ~ and ft2 with boundary 0f~2 = F2 U ~, that interact through the common boundary ~ (see Fig. 7). For r - 1 and r = 2, let u ~ be the restriction of the displacement field u to ftr. Let a r - {ai~. } i j be the stress tensor related to f~ such that

aij (u

-

a jkh 6kh (u

(94)

42

STRUCTURAL

ACOUSTICS AND

VIBRATION

in which ai~kh is the restriction of aijkh to f~r- Finally, we introduce pr aS the restriction of mass density p to f~r. The coupling conditions on ~ are written as u 1-u

1(ul O'ij

) n j1 _

2

on

)2

2(u2)n2 -- ff ij

,

(95)

on

E

(96)

where n r - (nrl,n~,n~) is the unit normal to )2, external to f~. Equation (95) expresses the continuity of the displacement on 5] and Eq. (96) expresses the surface-forces interaction through ~.

r X3

_X2 X1

F i g . 7. S t r u c t u r e m a d e of two s u b s t r u c t u r e s

V a r i a t i o n a l f o r m u l a t i o n in t e r m s of (u 1, u2). The variational formulation of

the eigenvalue problem is defined by Eq. (47) for the free structure case. Let Ca,. be the admissible function space 6 defined on f~r (which does not contain any constraint on 0f~r - Fr t_J )2). For r - 1 and r -- 2, let

=

pr Ur.

5U~ dx

(97)

be the structural mass bilinear form and k r (u r , 6u r) - / ~

airj (u r) eij (Su r) dx

(98)

be the structural stiffness bilinear form, defined on C ~ x C ~ . The variational formulation defined by Eq. (47) can be rewritten as follows. Find 6 Space C a r = ( H l(f~r)) 3.

CHAPTER III.

43


A, u i E Ca~ and U 2 E Cf~2 satisfying u i = u 2 (Eq. (95))7 such t h a t for all 6u i E Ca~ and 6u 2 E Ca~ satisfying 6u i = 6u 2 on ~, kl(ul,~u

1) + k 2 ( u 2 , Su2) -- )~ { m l ( u l , ~ u

1) -+- m 2 ( u 2 , ~ u 2 ) }

.

(99)

Using Green's formula, it can easily be seen that the above variational formulation yields all the local equations of the problem. 10.3. Dynamic substructuring using the fixed-interface modes of each substructure

The substructuring technique consists in using a reduced matrix model for each substructure f~ (r = 1 or r = 2) based on the decomposition of displacement field u ~ first, on elastic structural modes u~ of substructure f~r fixed on ~, and secondly on static functions Ustat. Fixed-interface modes of substructure f~r. A fixed-interface mode of substructure f~r (for r = 1 or r = 2) is defined as an elastic structural mode of substructure ~ r fixed on ~ (see Fig. 8). This type of problem was studied in Section 6.

x3

_x2

Fr

n

r

xI

Fig. 8. Eigenvalue problem for each s u b s t r u c t u r e with zero displacement on the interface

Introducing admissible function space C o defined by eL

-

{ u ~ e c~

;

u~ -

o

o.

r~ }

,

(i00)

If cgf~r is the boundary of f~r, then the trace operator from (Hi(f~r)) 3 onto (H1/2(0~r))3 is a continuous linear surjection. Equality u i : u 2 must be read in the sense of the traces on ~ = 0 f ~ l n 0f~2.

44


VIBRATION

each eigenvalue A~ and its corresponding eigenvector u r E C ~ satisfy the following variational formulation kr(ur,6u ~)-A~mr(u~,6u r)-0

,

V6u r e C ~

,

(101)

in which m r and k r are defined by Eqs. (97) and (98) respectively. In Eq. (101), if test function ~u r belongs to Ca~ and not to subspace C ~ , then the left-hand side of Eq. (101) is not equal to zero and we have kr(u~,6u ~ ) - A ~ m ~ ( u ~ , ~ u ~ ) - e ~ ( 6 u ~)

,

V~u ~ e C a .

,

(102-1)

where s r) is a linear form on C ~ which represents the work of the force field induced by constraint uar _ 0 on ~. From a mechanical point of view, if F~ - a r ( u r) n r denotes this reaction force field applied to ~, we have P ~ (~u r) - Jp~ Fra 9~u r ds (102 2) The right-hand side of Eq. (102-2) is a notation which is used below.

Static functions of substructure ~ , introduction of the linear operator S r. We consider the solution Ustatr of the elastostatic problem of substructure f~r subjected to a prescribed displacement field Upresc on Z (see Fig. 9).

r

r

= Upresc x3

~x 2

ar

l'~r

x1

Fig. 9. E l a s t o s t a t i c p r o b l e m for each s u b s t r u c t u r e with prescribed displacement on the interface

The mapping Upres cr have

~

is a linear operator 8 denoted as S r. We then

Usta tr r

Ustat

__

S r

r

(Upresc)

(103)

Linear operator S r is called a lifting operator and is continuous from the trace space Vr, C ( H 1 / 2 ( a ~ r ) ) 3 on E into ( H l ( ~ r ) 3.

45


By construction, the value o f Sr(upresc) on the boundary E is equal to presc of all the Upresc .r To construct operator S r, we introduce the set Cn~ r functions Ustatr defined on f~r whose values on E are equal to Upres c ~presc nr

__

{

r Ustat E CNr

r r Ustat -- Upres c

;

on

~] }

.

(i04)

Linear operator S ~ defined by Eq. (103) is constructed by solving the r presc such that following problem. Find Ustat in Cfl~ k r ( u r t a t , (~ur) = 0

,

V~U r E C~r

,

(105)

where C ~ is the space defined by Eq. (100). It should be noted that (1) the space of rigid body displacement fields (introduced in Section 7.2) is a subset of the range space of operator St; (2) the discretization of S r by the finite element method is obtained by a classical static condensation procedure of the stiffness matrix of substructure f~r with respect to degrees of freedom on E (see Section 10.4). ~ Taking 5u ~ - Ua ~ in Eq. (105) Conjugate relationships between ua and Ustat.

yields r

r

k r (Ustat , u a ) -- 0

.

(106)

r Since C p~esc is included in Ca~ we can choose Ustat as (~ur in Eq. (102) where Ustat is the solution of Eq. (105) for an arbitrary prescribed displacement field Upresc o n )-]. F r o m Eqs. (102) and (106) and the symmetry property of bilinear forms k ~ and m r, we deduce that

r , UX)-m r (Ustat

(107)

1 /r~ F a" 9Upres " c ds

because the trace of Ustat r r c. Finally, the properties defined o n E is Upres by Eqs. (106) and (107) can be rewritten as follows. For all prescribed displacement fields Upres c o n ~] and eigenvector ua~ solution of Eq. (101) we have r

r

k r ( s r ( U p r e s c ) , u a ) -- 0 m r ( S r ( u p resc)' u a ) --

,

~r J~ F~ 9Upres c ds

(108)

.

(109)

Decomposition of Cn~. Consider an arbitrary displacement field u r in Cn~

whose value on 2 is u~. Taking Upresc = u~ and from Eq. (103), we deduce that the displacement field u r _ Usta t r is equal to zero on ~. Then

46


VIBRATION

this difference belongs to C ~ and can then be spanned by fixed-interface ~ ua. ~ Therefore, introducing modes ua according to u ~ - U ~s t a t - E c ~N~ = l qa the space c s t a t of all the solutions of Eq. (105) for all displacement fields u~ on E, we have the following decomposition into a direct sum Ca~ = ~past, at @ C ~

,

(110)

G u~

(111)

O0

u" - S ' ( u ~ ) +

E

Construction of the reduced matrix model for substructure ftr. We use the Ritz-Galerkin method. Let us introduce the finite dimension subspace of C ~ spanned by eigenvectors {u[ , ... , u ~N ~ } w i t h N r >_ 1. F r o m E q . (111) and using the same notation u r to simplify the writing, we deduce that N~

q~r u~r

u ~ - S~(u~) + ~

,

(112)

and the associated test function is then denoted as NT5u r

-

S~(bu~)+ ~

5q~ u r

.

(113)

Let qr _ ( q [ , . . . , q~v~) be the vector of the generalized coordinates. Substituting Eqs. (112) and (113) into Eqs. (97) and (98), we obtain the reduced matrix model of substructure f~ with respect to {u ~2, q~ }

Mr~ed--A ~

[A4 ~]

'

K~ed--

0

o]

[KT]

'

in which [A4 r] and [/Cr] are diagonal matrices defined by Eq. (41) and (42) (for the fixed-interface modes of substructure gtr) and the linear operators M~ and K~ are defined by 9

< M~ u~, ~u~ > - .~(S ~(u~), S~(~u~))

,

(115)

9 The trace space related to E is denoted as VEC(H1/2(cg~r)) 3. Linear operators M~ and K~ are continuous from VE into V~ in which V~ is the dual space of V~. In Eqs. (115) and (116), t h e angle brackets < . , . > denote the duality product between V~ and V~.

CHAPTER III.

47


-

k

S

(116)

,

where m r and k r are defined by Eqs. (97) and (98). It should be noted that these operators are related to surface F and correspond to the static condensation on E of the mass and stiffness operators (Guyan, 1965). Finally, j t r is the linear operator defined by N~

( A r u~). 6q r - E

mr (St (u~), u r) 6q;

(117)

a--1

From Eq. (109), we deduce that

N" ( A r u~:). aq" - - E

{1s aqr

a=l

"~r

} F ; - u ~ ds

(118)

AC~

Operator tar is the adjoint 10 of operator j F . In conclusion, M~red and K~ed defined by Eq. (114) are called the "reduced matrix model" of substructure f~r relative to the displacement field u~ on E and to the generalized coordinate vector q~ (which can be viewed as "internal generalized degrees of freedom").

Construction of the reduced matrix model for structure f~. Let u~ - u~p. = u~p~ be thedisplacement field on E. Substituting Eqs. (112) and (113) into Eq. (99) and using Eq. (114) for r = 1 and 2, we obtain the reduced matrix model of the generalized eigenvalue problem for structure f~:

0

o o I iu l

[K~1] 0 0 [K:2]

ql-A q2

[

,42

[.M 1] 0 0 [A42]

ql q2

.

(119)

10.4. Finite element discretization The numerical implementation of the substructuring technique presented above needs the discretization of two types of variational formulation which are related to the calculation of fixed-interface modes of each substructure (Eq. (101)) and to linear operator S r (Eq. (105)), in order to calculate the finite element discretization of the reduced matrix model of each substructure (see Eq. (114)). Below we use a compatible finite element mesh on 10 Linear o p e r a t o r .,4.r is continuous from V~ into ~ N r , in which V~ is defined in footnote 9. The adjoint o p e r a t o r ~ t r is continuous from ~Nr into the dual space V~ of V~.

48


VIBRATION

interface ~ (for the case of an incompatible mesh, see Farhat and Geradin, 1994 and Ohayon, Sampaio and Soize, 1997). Calculation of fixed-interface modes. The generalized eigenvalue problem defined by Eq. (101) has to be solved for each substructure. As presented in Section 9, the corresponding finite element discretization can be written as

[Kg]U

=

[M ]U

,

(120)

in which U r = ( U ~ , . . . , U r ) is the vector of the DOFs which are the values of the displacement field at the nodes of the finite element mesh of domain ~r- Matrices [M~] and [K~] of dimension ( n r • n r ) are symmetric positive definite. In a first step, the numerical solution of this generalized symmetric matrix eigenvalue problem (see Section 9) gives {A~, u r, #r } for - 1 , . . . , Nr, in which coefficients #~ are the generalized masses defined by nq. (41). We then deduce diagonal matrices [M r] and [K:r]. In a second step, for each c~, we calculate the finite element discretization of the linear form associated with reaction force field F r on ~E using Eq. (102) which allows us to calculate matrix [.,4r] from Eq. (118). Calculation of linear operator S r. For each substructure, we construct the

finite element discretisation [S r ] of operator Sr defined by Eq. (103). Let r Ustat be the vector of the DOFs which are the values of the elastostatic displacement field at the nodes of the finite element mesh of domain ~rWe then introduce vectors U[ of the DOFs related to boundary E and U~ of the other DOFs. Therefore, we have Usrtat -- (U~, U~). We introduce the following block splitting of the stiffness matrix [K r ] corresponding to Eq. (98) [Kr]=

[ [K[I] [gr2] T

[Kr2]] [g~2]

(121) "

From Eqs. (103) to (105), we deduce that U~ - - [ g ~ 2 ] - i consequently, [Sr] -

[

I,l

_[K~2I_I[Kr2] T

]

[ ~(r~.12j]T|Ir~,land (122)

.

From Eqs. (116) and (121), we deduce that [ g ~ ] - [ g ; 1 ] - [g[2] [g~2] -1 [ g ~ 2 ] T

,

(123)

which corresponds to the classical static condensation. Matrix IMP] is deduced from Eq. (115) and (121) and corresponds to the Guyan condensation.

CHAPTER III.


49

Calculation of the structural modes of the global structure. By assemblage of the reduced matrix of each substructure, we obtain the finite element discretization of the generalized eigenvalue problem defined by Eq. (119). This generalized symmetric matrix eigenvalue problem is solved as indicated in Section 9 for a free structure (the case now considered).


51

CHAPTER IV

Dissipative Constitutive Equation for the Master Structure

1. Introduction In dynamics, the master structure must always be modeled as a dissipative continuum. In Chapter III, for the conservative part of the master structure, we used the linear elasticity theory which allowed us to introduce the structural modes. This was justified by the fact that, in the low-frequency range (see Chapter I), the conservative part of the master structure can be modeled as an elastic continuum. In this chapter, we introduce damping models for the master structure. In Section 2, we introduce some general notation for the Fourier transform with respect to time and in particular for the displacement field, and the strain and stress tensors. In Section 3, we introduce the damping model with frequency-independent coefficients based on the linear theory of viscoelasticity without memory. In Section 4, we present a constitutive equation with frequency-dependent coefficients based on general linear viscoelasticity theory. Finally, in Section 5, we summarize the viscoelastic constitutive equation which will be used for low- and medium-frequency ranges. This constitutive equation is the sum of an "elastic" part and a "damping" part which are generally frequency-dependent. In the low-frequency range, the "elastic" part is frequency-independent and the "damping part" is frequencydependent. The model presented in Section 3 for which the damping part is also frequency-independent (viscoelasticity without memory) is a particular case. On the other hand, in the medium-frequency range we use the general model based on linear viscoelasticity theory (with memory) presented in Section 4 for which both the "elastic" and the "damping" parts are frequency-dependent.

52


2. Notation for the Fourier Transform Let u(x, t) be the displacement field at time t and eij(u(x, t)) be the strain tensor (see Eq. (III.2)). At each time t, we denote the stress tensor as aij(t) and we introduce the following notation for the strain tensor ~j(t) = ~j(u(x,t))

,

~j(t) = ~j(O~u(x,t))

(1)

.

As explained in Chapter I, we are interested in the formulations of structural-acoustics and vibration problems in the frequency domain. Therefore, we have to introduce the Fourier transform for various quantities (see Mathematical Notations in the appendix). For the displacement field, and the stress and strain tensors, we use the following simplified notation consisting in using the same symbol for a quantity and its Fourier transform u(x, w) = j f e -i''t u(x, t) dt

a~j(w) = ~ e -~t

aij(t) dt

,

eij

(2)

,

(w) = ~ e -~t

eij(t) dt

.

(3)

From Eqs. (1) and (2), we deduce that ~ j ( ~ ) = i~ ~ j ( ~ )

,

(4)

in which ~ij(w) denotes the Fourier transform of ~ij(t).

3. Damping Model with Frequency-Independent Coefficients The damping model with frequency-independent coefficients is based on the linear theory of viscoelasticity without memory. Constitutive equation in the time domain. At each time t, symmetric stress tensor aij(t) is decomposed into an elastic part ai~as(t) and a dissipative damp (damped) part oij (t) such that (Tij(t)

--

_elas (t) + %damp (t) o~

,

(5)

in which, using Eq. (1), ~j ( t ) - - ~ j k h ffelas

~kh(t)

,

(6)

CHAPTER IV. DISSIPATIVE CONSTITUTIVE EQUATION damp

aij

(t)

-

bijkh ~kh(t)

53

(7)

9

aijkh of the elastic coefficients and tensor bijkh of the damping coefficients of the material depend on x, are independent of t and have the usual properties of symmetry and positive definiteness. Denoting as ?~ijkh either aijkh or bijkh, these properties can be written as Tensor

(8)

?~ijkh "- ?~jikh -- ~Tijhk -- ?~khij

and for all second-order real symmetric tensors Xij ?~ijkh X k h X i j ~ c X i j X i j

(9)

,

with c > 0. The model defined by Eqs. (5) to (7) is said to be without memory because the stress tensor depends on the strain tensor locally in time (the past history of the strain tensor does not affect the present state of the stress tensor). Constitutive equation in the frequency domain. Taking the Fourier transform with respect to time t of the two sides of Eqs. (5) to (7) and using Eqs. ( 2 ) t o (4)yields r

)

-

-

oij-elas(W) -[-iw aijdamp(W)

,

(10)

in which 7elas(03) -- aijkh Ckh (03) ij damp

sij

(w) -- bijkh ~kh (w)

,

(11)

.

(12)

This damping model is said to be frequency-independent because coefficients aijkh and bijkh depend only on x and are independent of w.

4. Model with Frequency-Dependent Coefficients Based on the Linear Theory of Viscoelasticity The damping model with frequency-dependent coefficients is based on the linear theory of viscoelasticity with memory in the general case of a material whose constitutive equation is invariant for shifts in the time domain (see Truesdell, 1984; Mandel, 1966; Fung, 1968; Bland, 1960). Constitutive equation in the time domain. At each x fixed in gt and for all time t in R, symmetric stress tensor aij(t) is written as

(t) - V jkh(O) 6kh(t) + fO

Gijkh(T) 6kh(t--T) dT

54


VIBRATION

in which ~ij(t) is the symmetric strain tensor defined by Eq. (1) and realvalued functions t ~ Gijkh(X, t), denoted simply as Gijkh(t), are called the relaxation functions at x in domain ~. At each x fixed in ~ and each t fixed in R, t e n s o r Gijkh (and t h u s Gijkh) has the properties of symmetry

Oijkh -- Ojikh -- Oijhk -- Gkhij

9

(14)

In addition, at x in ~, initial value Gijkh(O) of the elasticity tensor must have the property of positive definiteness, i.e., for all second-order real symmetric tensors Xij, we have

Oijkh (O) Xkh X i j >_ c X i j X i j with c > 0. Below (in the in the frequency domain), tions which are necessary replacing t - T by r in the

,

(15)

subsection relative to the constitutive equation we add two assumptions on the relaxation functo obtain a stable mechanical system. Finally, integrand of Eq. (13), we can rewrite Eq. (13)

as

aij (t) - Gijkh (0) Ckh (t) +

Gijkh (t-- T) 6kh (r) dT

.

(16)

oo

The model defined by Eq. (13) or (16) is said to be with memory because the stress tensor at time t depends on the past history r < t of the strain tensor.

Integrability properties of the relaxation functions and Fourier transform of their derivative. The relaxation functions are defined on [0, +c~[ and differentiable with respect to t on ]0, +c~[. Their derivatives are denoted as (~ijkh (t). In addition, we assume that functions t ~-~ Gijkh (t) are integrable on [0, +c~[. Consequently, for all x fixed in ~, G~jkh (t) can be written as

G jkh(t)

--

G jkh(0) + f0 t Gijkh(r)

dr

9

(17)

The limit of G~jkh(t) is finite as t tends to +c~ and is denoted as G~jkh(C~),

G jkh(cr -- G jkh (0) + f0

Gijkh(r) dT

For all x fixed in ~, we introduce the functions t and simply denoted as gijkh(t), such that

gijkh(t)--O

if

t c(w)

,

(33)

in which, for all real w, positive constant c(w) is such that c(w) >_ Co > 0 where co is a positive real constant independent of w. This damping model is said to be frequency-dependent because coefficients aijkh (w) and bijkh (w) depend on x and w.

Simple example of a constitutive equation in the frequency domain. Let us consider a linear differential equation o n aij and eij in the time domain. We limit the presentation to the following first-order linear differential operator in time Cijkh akh (t) + &ij(t) = Aijkh Ckh (t) + Bijkh ~kh (t)

(34)

,

which can be denoted using global tensor notation as C a(t) + I &(t) - A c(t) + B ~(t)

(35)

,

in which I is the fourth-order identity tensor and where A, B and C are time-independent fourth-order tensors. We have to find the appropriate initial condition at time t = 0 such that the solution of Eq. (35) for t > 0 yields a constitutive equation of the type defined by Eq. (13) (in the time domain), or equivalently, by nqs. (25) to (27) (in the frequency domain). For real numbers ~ and w, introducing the complex number p = ~ + iw and taking the Laplace transform of the two sides of Eq. (35) yields, C Y(p) + I (p Y(p) - a(0)) - A ~(p) + B (p ~(p) - ~(0))

,

(36)

57

CHAPTER IV. DISSIPATIVE CONSTITUTIVE EQUATION

in which ~5(p) -

jr0+c~ e -pt

a ( t ) dt

,

~0+c~ e -pt

~(p) -

c(t) dt

.

(37)

In this Laplace transform calculus, we assume that p belongs to the righthalf complex plane (~ > ~0) in which ~(p) and ~(p) are defined. To obtain a linear relation between Y(p) and ~(p) (transfer function), Eq. (36) shows that the initial condition must satisfy the following equation ~(o) - B ~(o)

,

,.

(3s)

and therefore Eq. (36) yields

(C + p I)Y(p)

( A + p B)~(p)

.

(39)

We assume that fourth-order tensor C is such that, for all fixed real w, tensor (C + iw I) is invertible. Consequently, we obtain in the frequency domain a ( w ) - (C + iw I) -~ (A + iw B) E(w) . (40) Identifying nq. (40)with nqs. (25)to (27)yields

a(w) - ~e { (C + iw I) -1 (A + iw B)} w b(w) - ~ m { (C + iw I) -1 (A + iw B)}

,

(41) ,

(42)

in which a(w) and b(w) are the fourth-order tensors aijkh (w) and bijkh (w) introduced in Eqs. (26) and (27). It should be noted that tensors A, B and C must be such that aijkh(W) and bijkh(W) satisfy the symmetry and positivity properties defined by nqs. (32) and (33).

5. Summary Sections 3 and 4 can be summarized as follows. In the frequency domain, the general constitutive equation is written as O.ij(O)) __

_elms(w) + iw ~j _damp(03) o~j

(43)

where a i j ( w ) has complex values and in which

:7~jelas ( ~ ) damp

~j

a~jkh (03) Ckh

(~)

(44)

(~1 - b~jkh(~) ~kh(~)

(45)

58


Equations (43) to (45) can be rewritten as

o~j (~) = (~jkh(~) + i~ b,jkh (~)) ~ h (~)

9

(46)

For each real w and at each x in domain fi, tensors aijkh(W) and bijkh(W) must satisfy the symmetry properties

~kh(~)

= ~j~kh(~) = a~jhk(~) = ~kh~j(~)

,

(47)

bijkh(W) -- bjikh(W) = bijhk(W) = bkhij(W)

,

(48)

and the positive-definiteness properties, i.e., for all fixed real w, for all second-order real symmetric tensors Xij,

aijkh (W) Xkh Xij >__ca (w) Xij Xij

,

(49)

bijkh (w) Xkh Xij > Cb(O))Xij Xij

:

(50)

in which, for all real w, positive constants ca(w) and Cb(W) are such that ca(w) >_ co > 0 and Cb(W) >__ Co > 0 where Co is a positive real constant independent of w.

5.1. LF range: constitutive equation with frequency-independent elastic coefficients and frequency-dependent damping coefficients This model, described in Section 3, is given by Eqs. (46) to (50) in which tensor a i j k h , related to the elastic stress tensor ai~as, is independent of w (but depends on x) and tensor bijkh (W), related to the damping stress tensor damp ij (w), depends on w (and on x). In addition, at each x in domain f~, function w ~ bijkh (w) must be continuous on ~ and, for Iwl --+ +cr must satisfy the asymptotic properties defined by Eq. (31). It should be noted that the damping model with frequency-independent coefficients presented in Section 3 is a particular case for which Eq. (31) does not need to be satisfied (as proved in Section 3).

5.2. MF range: constitutive equation with frequency-dependent coefficients This model, described in Section 4, is given by Eqs. (46) to (50) in which tensors aijkh(W) and bijkh(W) depend on w (and on x). In addition, at

CHAPTER IV. DISSIPATIVECONSTITUTIVE EQUATION

59

each x in domain ~, functions w ~ aijkh(W) and w ~ bijkh(W) must be continuous on ~ and, for Iwl --+ +c~, must satisfy the asymptotic properties defined by Eqs. (30) and (31).


61

CHAPTER V

Master Structure Frequency Response Function

1. Introduction In Chapter III, we introduced the equations of the vibrations of a conservative structure in terms of the structural displacement field. Using the test function method, we presented the variational formulation and introduced the structural mass and stiffness operators and studied their properties. In this chapter we present the variational formulation of the vibrations of the master structure submitted to given forces, for which the damping effects are modeled using the dissipative constitutive equation introduced in Chapter IV for the low- and medium-frequency ranges. Finally, we introduce the operator-valued Frequency Response Function (FRF) which is intrinsic to the structure. This operator allows calculation of the dynamical response of the structure for arbitrary deterministic or random excitations. In Section 2, we introduce the equations of the master structure in the frequency domain. In Sections 3 and 4, we present the variational formulation of these equations and the corresponding linear operator equation. We introduce the structural mass, damping and stiffness operators. In Section 5, we introduce the frequency response function which is explicitly constructed in Chapters VI and VII for the LF and MF ranges respectively. In Section 6, we introduce the corresponding finite element approximation. In Section 7, we consider the coupling between two substructures on an interface. The problem consists in eliminating the displacement field in one substructure as a function of the displacement field at the interface. For this purpose, we construct the boundary impedance operator which

62


VIBRATION

relates the value of the velocity field on the interface with the force field on this interface for this substructure.

2. Equations in the Frequency Domain 2.1. Master structure configuration We consider linear vibrations of the master structure around a position of static equilibrium taken as reference configuration ~ (for the sake of brevity, we do not consider a reference configuration with prestresses: we use the natural state at rest). The physical space ~3 is referred to a Cartesian reference system (i,j,k) and we denote the generic point of ~3 as x = (xl, x2, x3). Let ~ be a three-dimensional bounded connected domain of ~3 occupied by the master structure at equilibrium with a smooth boundary 0~. The external unit normal to 0~2 is denoted as n (see Fig. 1).

G

z/

x2

=X2

x1

X1

a- Master structure fixed on r 0

b- Free master structure

Fig. 1. Master structure configuration

Let u(x, t) - (ul (x, t), u2(x, t), u3(x, t)) be the displacement of a particle located at point x in ~ and at a time t. As in Chapter III, two cases are considered. - The master structure is fixed on a part Fo of boundary 0 ~ (u - 0 on Fo), a given surface force field G(x, t) = (G1 (x, t), G2(x, t), G3(x, t)) is applied to the part F = 0 ~ \ F o and a given body force field g(x, t) = (gl (X, t), g2 (X, t), g3 (X, t)) is applied in ~ (see Fig. l-a).

63

CHAPTER V MASTER STRUCTURE FREQUENCY RESPONSE FUNCTION

- The master structure is free (F0 = 0), a given surface force field G(x, t) is applied to the total boundary F = 0f~ and a given body force field g(x, t) is applied in [2 (see Fig. l-b). 2.2. Equations in the time domain

For any function a(x,t), We use the notation a s ( x , t ) =

Ota(x, t) = Oa(x, t)/Ot

and

02t a(x, t) = 02a(x, t)/Ot 2.

cOa(x,t)/Oxj,

We also use the clas-

sical convention for summations over repeated Latin indices. The elasto-

dynamic equation is written as

pO2tui(x,t)-aij,j(x,t)=gi(x,t)

in

f~ ,

t>0

,

(1)

in which p(x) is the mass density field on ~ at equilibrium and a~j(x, t) is the stress tensor. The constitutive equation in the frequency domain is defined in the next subsection. Concerning the boundary conditions, since there is a given surface force field G(x, t) applied to F, (x, t) n j(x ) = G (x, t)

on

r

(2)

.

In addition, if the master structure is fixed on F0, we have

u(x,t)=o

r0

.

(3)

For a free master structure (F0 = 0), the boundary condition defined by Eq. (3) does not exist and Eq. (2) holds on the total boundary F = On. 2.3. Fourier transform notation

As explained in Chapter IV, in order to obtain the equations in the frequency domain, we introduce the Fourier transform for various quantities using the same symbol for a quantity and its Fourier transform (see Mathematical Notations in the appendix). For all fixed x, body force g(x, t), surface force G(x, t), displacement u(x, t) and stress tensor aij(x, t) have the following Fourier transforms with respect to t g(x, w)

u(x,w)

=

= f~ e -i~ g(x, t) dt , G(x, w) = f~ e -iWt G(x, t) dt , f e-i~

aij(x, w)

=

f e -i~t

a~j(x,t)dt

(4)

.

(5)

It should be noted that these Fourier transforms are complex-valued vectors a n d tensors.

64

STRUCTURAL

ACOUSTICS A N D

VIBRATION

2.4. Equations in the frequency domain Taking the Fourier transform of Eqs. (1) to (3) and using Eqs. (4) and (5), we obtain, for all real w,

-~

p~,~(x.,,) - o~j.j(x.,,) - g~(x.,,) aij(x, w) nj(x) = Gi(x, w)

on

in f t ,

(6)

,

(7)

r

and, if the master structure is fixed on F0, u(x,w)=O

on

r0

(8)

9

For a free master structure (F0 = 0 and F = Oft), the boundary condition defined by Eq. (8) does not exist. In the frequency domain, the constitutive equation of the master structure (which is dissipative) are given by Eqs. (IV.43) to (IV.50). 2.5. Boundary value problem in the frequency domain The Fourier transform of the strain tensor defined by Eq. (III.2) is written as 1 ~k~(u) - ~(~k,h(x,~) + ~h,k(X,~)) 9 (9) From Eqs. (IV.43) to (IV.45) defining the constitutive equation of the master structure and using Eq. (9), we deduce that Telas ~j

(u)-

~ j k h ( ~ ) ~kh(U)

,

sijd a m p ( u ) - bijkh(W) ~kh(U) O'ij(U) - - o~j _elas

(10)

,

(u) + i~ ~jd a m p (.)

(II)

,

(12)

in which aijkh (w) and bijkh (W) depend on x and w and satisfy the properties introduced in Section IV.5 (the dependence on x has been omitted for brevity). Substituting Eq. (12) into Eqs. (6) and (7), we obtain --0)2 p U i -- O'ij,j(U) -- gi

aij(u)nj--Gi

on

in

ft

r

,

,

(13)

(14)

and if the master structure is fixed on F0, u=0

on

F0

9

(15)

For a free master structure (F0 = q) and F = Oft), the boundary condition defined by Eq. (15) does not exist. The problem can be summarized as follows. For each real w and at each x in ft, find complex-valued vector u(x,w) as a solution of Eqs. (13) and (14) with the additional boundary condition defined by Eq. (15) if the master structure is fixed on F0.

65

CHAPTER V. MASTER STRUCTURE FREQUENCY RESPONSE FUNCTION

3. Variational Formulation The variational formulation of the damped master structure equations in the frequency domain is constructed using the test-function method presented in Sections 4 and 5 of Chapter III for a conservative elastic structure whose principles are explained in Chapter II. As the solution u is a complex-valued vector, it is necessary to introduce the admissible function c space of complex functions C~ and C0 instead of the admissible function space of real functions C and C0. Consequently, all the linear and bilinear forms related to real vector space C or Co are replaced by the corresponding antilinear and sesquilinear forms related to complex vector space C~ or C0 (see Mathematical Notations in the appendix). 3.1. Construction of the variational formulation

We introduce the complex vector space C~ of sufficiently differentiable 1 functions defined on f~ with values in C 3 9 Multiplying Eq. (13) by ~-u E C e and integrating over domain f~ yields

-w2 / pui ~--uidx - jfa aij,j(u) ~u--idx - jf gi ~--uidx .

(16)

The second integral on the left-hand side of Eq. (16) is transformed using the same calculations performed in Section III.4.1. We then obtain -

-w

ui r dx

ij(u) cij(~u) dx

Finally, we introduce the distinguished.

ij(u)

nj ~ui ds =

admissible function space.

i ~ui dx.

(17)

Two cases must be

Structure fixed on Fo. In this case, Fo ~- 0 and the admissible function

space must satisfy the constraint on u defined by Eq. (15). Consequently, 12 this admissible function space is the subspace Co of C~ defined by 2 C

CC

C0 - { u e

~

;

u-0

on

F0}

(lS)

C Let ~iu be any test function in CO C C C o Taking into account Eq. (14) and c since J u - 0 on F0, for all 5u in Co, Eq. (17) yields

- w 2 / pui ~---uidx + / aij(u) cij(~u) dx - / gi ~'ui dx + fr Gi ~-ui ds . Space C c is t h e Sobolev space ( H l ( f ~ ) ) 3. c

Space CO is t h e Sobolev space { u E ( H l ( f 2 ) ) a , u - - 0 on Fo}.

(19)

66


VIBRATION

Free structure. In this case, F0 = q}and Eq. (15) does not exist. Therefore, the admissible function space of the problem is C~ (there is no constraint on u). Let 5u be any test function in (7~. Taking into account Eq. (14) and for all ~u in C~, Eq. (17) yields

3.2. Variational formulation Structure fixed on F0. Substituting Eq. (12) into Eq. (19), the variational formulation of the boundary value problem defined by Eqs. (13), (14) and (15) is stated as follows. For all fixed real w, find u in CO such that, for all c

c

5u in Co,

--032 /f p

dx- -i d/flsdi3mP(U)gij( -u)dx' "/[.l(Tieljas(u) ij( 'U)dx

in which ~_elas(U) and ~-ij damp (u) are defined by nqs. (10) and (11) respectively. Substituting Eq. (12) into Eq. (20), the variational formulation of the boundary value problem defined by Eqs. (13) and (14) is stated as follows. For all fixed real w, find u in C~ such that, for all 5u in

Free structure.

- w 2 /f~ p u. -5u dx + iw

/f ~ijdamp(u) r -

fog

(~'~) dx _[_/f oij _elas(u)eij (5-u) dx

dx + fr G.

6---uds

,

(22)

in which ~_elas(u) and ~ij .damp (u) are defined by Eqs. (10) and (11) respectively.

4. Linear Operators and Algebraic Properties We introduce the linear operator equation corresponding to the variational formulation defined by Eq. (21) or Eq. (22).

CHAPTER V. MASTER STRUCTURE FREQUENCY RESPONSE FUNCTION

67

4.1. M a s s o p e r a t o r

We introduce the sesquilinear form re(u, 6u) on C~x C~, called the structural mass sesquilinear form, defined by m

r e ( u , 6u) =

(23)

p u . 6 u dx

which is Hermitian, (24) and which is positive definite because p(x) > 0 (for all u # O, we have re(u, u) > 0). This result holds for the fixed and free cases. The linear operator M, called the mass operator, is defined by 3 <Mu,6u>=m(u,6u)

(25)

.

4.2. Stiffness o p e r a t o r

Using Eq. (10), for all fixed real w, we introduce the sesquilinear form k(w ;u , 6u) defined on C ~x C ~ and called the structural stiffness sesquilinear form, defined by /k(w ;u,

~u) -

. / o _elas

(u) % (~U) dx

s

-

(26)

From the properties defined by Eqs. (IV.47) and (IV.49), we deduce that sesquilinear form k(w ;u, 6u) is Hermitian:

k@;u,

= k(w ; 5u, u)

(27)

The fixed and free cases must be distinguished. S t r u c t u r e fixed on F0. For all fixed real w, the restriction of sequilinear form k(w ;u, 6u) to C0 x Co is positive definite, c

c

k(w;u,u)>O

,

Vu~O

.

(28)

Sesquilinear form m ( u , 6 u ) is c o n t i n u o u s on H c x H c w i t h H c = ( L 2 (~I))3 a n d is also c o n t i n u o u s on Cc• c. T h i s sesquilinear f o r m defines a c o n t i n u o u s o p e r a t o r M f r o m C c into its a n t i d u a l space C c' a n d in Eq. (25), t h e angle b r a c k e t s d e n o t e t h e a n t i d u a l i t y p r o d u c t b e t w e e n C c' a n d Cc .

68


Stiffness operator K(w)

is defined by 4

k(w ;u,

< K(w) u, ~u > -

5u)

.

(29)

F r e e structure.

For all fixed real w, sesquilinear form on C ~ • C~ is positive semidefinite k(w;u,u)>0

Stiffness operator K(w)

,

Vu#O

k(w ; u,

~u) defined

.

(30)

is then defined by 5

< K(w) u , 6u > - k @ ; u ,

6u)

(31)

.

4.3. D a m p i n g o p e r a t o r

For the "damping" part of the model, we proceed as for the "elastic" part (stiffness operator). Using Eq. (11), for all fixed real w, we introduce the sesquilinear form d(w;u , 5u) defined on C ~ x C ~ and called the structural damping sesquilinear form, defined by d a m p (u) cij (~-U) dx sij

d(w ; u, 5u) -

/o

(32)

b jkh(w) Skh(u) e j( u) ax -

From the properties defined by Eqs. (IV.48) and (IV.50), we deduce that sesquilinear form d(w ; u, 5u) is Hermitian, d(w;u,Su) - d(w;Su,u)

.

(33)

The fixed and free cases must be distinguished. S t r u c t u r e fixed on F0. For all fixed real w, the restriction of sesquilinear

form

d(w ; u,

c

c

5u) to CO • CO is positive definite, d(w;u,u)>0

, c

Vu#0

.

c

(34) c

4 Sesquilinear f o r m k(w ; u,6u) is continuous on CO xC O , coercive on CO due to Eq. (IV.49). T h i s c

cg

sesquilinear form defines a continuous o p e r a t o r K(w) from Co into its a n t i d u a l space Co and CI

C

in Eq. (29), the angle brackets denote the a n t i d u a l i t y p r o d u c t between Co and Ca. e Ce 5 See f o o t n o t e 4, replacing Co by C c. It should be noted t h a t k(w ;u,u) is coercive on \Crig in which CrigCC is the set of the rigid b o d y displacement fields (see Section III.5.2).

69

CHAPTER V. MASTER S T R U C T U R E F R E Q U E N C Y R E S P O N S E FUNCTION

Damping operator D(w) is defined by 6 < D(w) u, 5u > = d(w ; u, 5u)

.

(35)

Free structure. For all fixed real w, sesquilinear bilinear form d ( w ; u , 5u) defined on C~ • C~ is positive semidefinite

d(w;u,u)>__O

,

Vu#0

.

(36)

Damping operator D(w) is then defined by 7 < D(w) u, 5u > - d(w ;u, 5u)

.

(37)

4.4. Given force vector For all fixed real w, we introduce the antilinear form f(w ;Su) defined on C~ by f(w ;Su) = / a g(x, w). 5u(x) dx

(38)

+fr G(x, w). 5u(x) ds(x)

which defines the element f(w), called the given force vector, such that s < f(w), 5u > = f(w ;Su)

.

(39)

4.5. Rewriting of the variational formulation

Structure fixed on F0. With the above notation, the variational formulation defined by Eq. (21) can be rewritten as follows. For all fixed real w, find u in Co such that, for all 6u E Co, c

c

- w 2 re(u,

+ iw d(w ; u,

+ k(w ;u, c

=/(w

c

;

(40)

c

6 Sesquilinear form d(w;u,Su) is continuous on CO x Co , coercive on CO due to Eq. (IV.50). This c

c!

sesquilinear form defines a continuous o p e r a t o r D(w) from Co into its a n t i d u a l space Co and c? c in Eq. (35), t h e angle brackets denote the a n t i d u a l i t y p r o d u c t between Co and Co . c cc 7 See footnote 6, replacing Co by C c. It should be noted t h a t d(w;u,u) is coercive on \Crig in which CrigCC is the set of the rigid b o d y displacement fields (see Section III.5.2). 8 For all fixed real w, m a p p i n g x~-~g(x,w) defined in f2 and m a p p i n g x~-+G(x,w) defined in I" are Cc c . C~, a s s u m e d to be such t h a t 6u~-~f(w;6u) is continuous on (or Co) E l e m e n t f(w) E (or ct t Co ). In Eq. (39), the angle brackets denote the a n t i d u a l i t y p r o d u c t between C c and C c (or c!

c

between Co and Co).

70

STRUCTURAL

ACOUSTICS AND

VIBRATION

The associated linear operator equation is written as 9 (-w 2M +

iw D(w) + K(w)) u - f(w)

.

(41)

Free structure. The variational formulation defined by Eq. (22) can be rewritten as follows. For all fixed real w, find u in C~ such that, for all

6u EC ~, - w 2 re(u, 6u) +

iw d(w ;u, ~u) + k(w ;u, ~u) - f(w ;6u)

.

(42)

The associated linear operator equation is then written as 10 (-w2M +

iw D(w) + K(w)) u - f(w)

.

(43)

i

5. Frequency Response Function In this section, we introduce the operator-valued Frequency Response Function (FRF). This operator allows the calculation of the dynamical response of the structure for arbitrary deterministic or random excitations.

5.1. Dynamic stiffness and impedance operators For the two cases defined in Section 4.5, we introduce the so-called dynamic stiffness sesquilinear form a(w ; u, 5u) and the impedance sesquilinear form z(w ; u , 5u) defined by a ( w ; u , 6 u ) -- - w 2 m ( u , 6 u )

+'iwd(w;u,(fu) + k ( w ; u , 6 u )

iw z(w ;u, 6u) -- a(w ;u, 6u)

,

.

(44) (45)

We then define the associated dynamic stiffness operator A(w) and impedance operator Z(w) such that 11 < A(w) u, 6u > = a(w ; u, ~u) A(w)--w

2M+iwD(w)+K(w) c!

,

(46)

,

iwZ(w)-A(w)

.

(47)

c

9 Eq. (41) is an e q u a l i t y in t h e a n t i d u a l space CO of Co. 10 Eq. (43) is an e q u a l i t y in t h e a n t i d u a l space C c' of C c . C

C

11 S e s q u i l i n e a r form a ( w ; u , 6 u ) is c o n t i n u o u s on Co XC o or

~') E(Co,C o

or

') A(w)E/:(Cc,C c

cC

xC c

9

L i n e a r o p e r a t o r A(w)E

(see M a t h e m a t i c a l N o t a t i o n s in t h e a p p e n d i x )

71

CHAPTER V. MASTERSTRUCTURE FREQUENCYRESPONSE FUNCTION

5.2. Operator-valued FRF for a structure fixed on F0

The operator-valued frequency response function is defined by the linear operator which associates each given force vector with the displacement field as the solution of the variational formulation defined by Eq. (40). From Section 4, it can be proved 12 that for all fixed real w and given force vector f(w), the variational formulation defined by Eq. (40) has a unique solution u(w) in C~ which depends linearly on f. From EQ. (39), we deduce that there exists a linear operator f(w) ~ u(w) such that 13 u(w) - T(w)f(w)

,

(48)

in which w ~ T(w) - A(w) -1 is called the operator-valued frequency response function (FRF). 5.3. Operator-valued FRF for a free structure

In this case, we use the variational formulation defined by Eq. (42). For all real w, we assume that given body force field g(w) and given surface force field G(w) are in equilibrium in the following sense. Let Crig C C c be the set of the rigid body displacement fields Urig (see Section III.5.2). Taking 6u = Urig in Eq. (38), for all fixed real w, the equilibrium hypothesis is written as f ( w ; U r i g ) = ./Q g(X,W)" Urig(X)dx-~- ./p G ( x , w ) . Urig(X ) d s ( x ) ~

=

0

.

(49)

We deduce that the condition defined by Eq. (49) is equivalent to the set of linear equations obtained for c~ - { - 5 , . . . , 0}

/ng(

x , w). ua(x) dx + Jfr G(x, w). ua(x) ds(x) = 0

(50)

in which vectors u - 5 , . . . , u0 s p a n Crig (see Section III.7.2). Two cases must be considered. c

12 For co>0, the proof of the existence and uniqueness of the solution in C o is obtained by

applying the Lax-Milgram theorem to the variational formulation ac(w;u,6u)-'J'c(w;6u) in which ac(w;u,6u)=-ia(w;u,6u) and fc(W;6u)=-i f(w;6u), because ~eac(w;u,u)--wd(w;u,u) is coercive due to Eq. (IV.50). For w . (55) N

From Eq. (54), we deduce that f(W;Urig) -- 0, which proves that force vector f(w) is associated with a system of external forces in equilibrium. ~v

5.4. Introduction of a unified notation for the fixed and free structure cases c

We introduce the admissible ~function space CS such that c

for a structure fixed on

Co

c

F0 (56)

CS-

Celas c

for a free structure

,

c

in which CO and ~elas are defined by Eqs. (18) and (52) respectively. From Sections 5.2 and 5.3, response u(w) in CS defined by c

u(w) = T(w)f(w)

,

(57)

is the unique solution of the following variational formulation written in the unified notation. For all fixed real w, find u E CS such that, c

a(w;u,~u)-y(w;~u)

c

,

V~UeCs

,

(ss)

in which a(w ;u, 5u) is defined by Eq. (44) and it is assumed that f(w; 5u) satisfies Eq. (49). 5.5. Associated linear filter and operator, valued impulse response function

In this section, we define the linear filter whose operator-valued frequency response function is T(w). Let f(t) be such that its Fourier transform is f(w), jtD

f(w) - / e -iWt f(t)dt Je

(59)

. c

For all fixed real w, let u(w) be the unique solution in CS of Eq. (58). Let u(t) be such that its Fourier transform is u(w), u(w) - j f

e -iWt

u(t)

dt

.

(60)

74


VIBRATION

It should be noted that Eq. (60) corresponds to Eq. (5) in which all the values of x are considered. Let us consider the linear filter whose input is f(t), whose output is u(t) and whose operator-valued impulse response function is h(t). Consequently, its operator-valued frequency response function is T(w) such that, T(w) = [ e - i W t h ( t ) dt J~

(61)

.

Since h(t) is a function with values in a real vector space, we then have T ( - w ) = T(w)

.

(62)

This linear filter denoted as h *t is such that

u(t) = (h *t f)(t)

,

(63)

which means that u(t) = 9~~ h ( r ) f ( t - r ) d r - j f h(t - r ) f ( r ) d r

(64)

.

Eq. (63) or (64) appears as a convolution product with respect to the time variable. Due to the properties of the viscoelastic constitutive equation introduced in Chapter IV and used in this chapter, it can be verified that linear filter h *t is causal (or physically realizable), which means that h(t)=O

,

Vt between C s and Cs of g with 6u is equal to the usual inner product (g,6u)Hc = ~ yo g(x).6u(x) dx of H c. In this case, we have f(w)=g(w). 2 Integral operator TN(W) considered as an operator in HC--(L2(fl)) 3 can be written as

~o~

TN(W)-"

N [T(td)]c~3 Ua~)HcU~3. 1 ~'~D__I

3 From a m a t h e m a t i c a l point of view, TN(W)* is the adjoint operator of bounded operator T N(W) (see Mathematical Notations in the appendix).


90

VIBRATION

3.3. Finite element discretization of the projection of the FRF on elastic structural modes Consider a finite element discretization of domain ~. Let n be the number of degrees of freedom. Let {U~ },~=l,...,g be the set of N eigenvectors defined by Eq. (III.93) which corresponds to the finite element discretization of the continuous eigenvalue problem. Each U~ is an ~n vector representing the finite element discretization of elastic structural mode u~(x). The matrix [TN(w)] of operator T N(w) obtained from the finite element discretization, can then be written as N

N

-

(30)

u .

~=1/3=1

in which [T(w)] is the (N • N) complex symmetric matrix defined by Eq. (21). Matrix [TN(w)] is an (n • n) complex symmetric matrix, _

.

(31)

This property corresponds to Eq. (29).

4. R e m a r k on a N o n v i s c o e l a s t i c M o d e l D i a g o n a l i z e d by the Elastic Structural M o d e s 4.1. Statement of the problem We saw above that with the theory of linear viscoelasticity, damping operator D(w) was not diagonalized by the elastic structural modes (matrix [7:)(w)] is dense with [T)(w)]~ = d(w ;u~,u~) and there is no reason for matrix [7:)(w)] to be diagonal). However, this does not raise any problems because the inversion of (N • N) dense matrix [A(w)] to construct [ T ( w ) ] - [A(w)] -~ is straightforward because g - d(w ;u~, u~)

(32)

=

(33)

= 2 #~ w~ ~ (w) 5 ~

(34)

in which #a and wa are defined by Eqs. (15) and (16) respectively. The critical damping rate ~a (w) of elastic structural mode ua is a positive real number depending on f~, ~r

,

Vwe[R

,

Va6{1,2,...}

.

(35)

A small damped structure is a structure such that 00 for all gr in H) and finally is a trace operator, i.e. the series of its eigenvalues is convergent. c

11 The range space of operator E~v is such that R{E~v } : { E ~ e , V e E H } : ( H 2 ( g t ) ) 3 N C o . 12 The adjoint of a bounded operator in a Hilbert space is defined in Section 6.2 of Mathematical Notations in the appendix.

122


VIBRATION

in which ~e denotes the real part. For the proofs of these results, we refer the reader to Soize, 1997a. 5.4. Spectral properties of the energy operator

From the mathematical properties of energy operator EB established in Section 5.3, we deduce that operator Es~ has a countable number of decreasing positive eigenvalues with finite multiplicity, possibly excepting zero, AI_>),2>_...--40 , (77)

in which the Aa terms are the repeated eigenvalues of EB. sponding eigenfunctions {ea }a>_I, such that Ee, e . - A. ea

The corre-

,

(78)

are functions ea (x) from f~ into ~3 and form a complete orthonormal family in H, (ea, eta), - 5,f~ . (79) Since EB~ is a positive-definite symmetric trace operator, we have +oo

+oo

a--I

a--I

(80) The trace norm of Es~ denoted as tr EB~ is such that --~--oo

tr E~u

E Aa < +c~

(81)

c~-'l

and Ee~ can be written as -~-OG

EB u

(.,e.).e.

(82)

which means that, for all g in H,

E~v g --

(83)

CHAPTER VIII.

123

REDUCED MODEL IN THE MF RANGE

5.5. Fundamental property of the eigenfunctions of the energy operator The set {ea}a=l,...,+oo of eigenfunctions of operator EB~ is a complete fame ily in admissible space CO c H, orthonormal for the inner product of H. In addition, each eigenfunction ea is a continuous function from f~ into ~3. For the proof of these properties, we refer the reader to Soize, 1997a. This result allows a Ritz-Galerkin projection of the variational formulation defined in Section 4 to be constructed using a truncation strategy based on the use of the property defined by Eq. (81).

6. Construction of a Reduced Model in the MF Range Taking into account the result of Section 5.5, the reduced model adapted to medium-frequency band B~, is obtained using the Ritz-Galerkin projection of the variational formulation defined by Eq. (60) on the subspace Co,N of CO spanned by the eigenfunctions { e l , . . . , eN} which correspond to the N highest eigenvalues {A~,..., AN } of energy operator E B . Let u(w) E C0 be the unique solution of Eq. (60) and let u N (w) be the projection of u(w) on Co,g such that c

c

c

c

N

u~(x,~)-

~

0~(~o)~(x)

,

(a4)

~'-'1

in which O~(w) E C. From Eq. (60), we deduce that for all w in ~, U ~,, O(w) = (0~ (w),..., ON(w)) E C N is the solution of the linear equation

[AN (~)] 0(~) = ~(~) J=(~)

,

(85)

in which [AN(W)] is the symmetric (N x N) complex matrix defined by [ A g ( w ) ] ~ = a(w ;e~,e/~)

,

(86)

and where .T(w) - (3c1(w),..., 3ON(W)) E C N is such that ~7(w) ~c~ (co) -- f (ca ;e~)

(87)

.

For all w in By U 13v, matrix [,AN(w)] is invertible and the solution of Eq. (85) is written as

o(~)

-

~(~)[TN(~)] a:(~)

,

(88)

in which [Tg(w)] is the symmetric (N x N) complex matrix such that

[TN(~)] = [.4N(~)] -~

9

(89)

124

STRUCTURAL ACOUSTICS AND N

VIBRATION

C

For all w in By U By, UN (W) ~ U(~g) in CO as N --+ +c~. The reduced model adapted to medium-frequency band B~ of the master structure described by Eq. (60) is constituted by the set

{

JAN(.,)].

}

(90)

7. Finite Element Discretization An explicit construction of the eigenfunctions { e l , . . . , eN } of energy operator E~, cannot be obtained in the general case. A finite dimension approximation E~,n of EB, must be introduced and the eigenfunctions { e ~ . . . , e~v} of EB,~ (associated with the N highest eigenvalues) constitute the approximation of { e l . . . , eN}. This finite approximation can be obtained using the finite element method. We consider a finite element mesh of master structure f~ and we introduce the subspace Co,~ C CO of finite dimension n. Let U - ( U I , . . . , U~) be the complex vector of the DOFs which are the values of u at the nodes of the finite element mesh of domain Ft. Since the finite element method uses a real basis for constructing the finite element matrices, the finite element discretization of the variational formulation defined by Eq. (60) yields the complex symmetric matrix equation c

[A(w)] U = ~(w) F(w)

c

,

(91)

in which [A(w)] is the dynamic stiffness matrix of the master structure. It is an invertible symmetric (n x n) complex matrix such that [A(w)] - - w 2 [ M ] + iw [D(w)] + [K(w)]

.

(92)

Mass, damping and stiffness matrices [M], [D(w)] and [K(w)] are positivedefinite symmetric (n • n) real matrices. The finite element discretization of antilinear form f(w ;Su) yields the complex vector ~(w)F(w) E C n. 7.1. Finite element discretization of the energy operator

The finite element discretization of operator E ~ defined by Eq. (76) is the positive-definite symmetric (n • n) real matrix [EB.,n ] such that

[EB,,n]-[G][En][G]

,

(93)

in which the invertible symmetric (nxn) real matrix [G] corresponds to the finite element discretization of the sesquilinear form fa u. 5--u dx defined on

125

CHAPTER VIII. REDUCED MODEL IN THE MF RANGE c

12

CO• C0. The positive-definite symmetric (n•

real matrix [En] is written

as

(94)

[En] -- ]B [en (0J)] dw

in which (n • n) real matrix [en (w)] is given by [en(W)] - 1 2 71"

{[T(w)]*[M][T(w)]}

i (w)l 2

(95)

in which [T(w)] is the symmetric (n • n) complex matrix such that [ T ( w ) ] - [A(w)] -1

,

[T(w)]* - [ T ( w ) ]

.

(96)

In the particular case where f(w ;hu) - ~(w) (g, 5u)n, denoting its finite element discretization as ~(w)F(w), we have (E~ g, g),~ - F(w)* [En] F(w)

(97)

.

It should be noted that the right-hand side of Eq. (97) includes [En], not [EB~,,n]. For the proof of the results of Section 7.1, we refer the reader to Soize, 1997a. 7.2. Finite element discretization of the spectral problem associated with the energy operator

Let P~ = ( P ~ , . . . , P~) be the real vector of the DOFs which are the values of ea (x) at the nodes of the finite element mesh of domain fl. The finite element discretization of Eq. (78) yields [E~,,,n]Pa-An[G]P

The eigenvectors {p1,... ,pn} the orthogonality conditions

a

.

(98)

Eq. (98) form a basis of ~n and satisfy

of

p ~ T [ c ] p Z _ 5~/~ ,

p,~T[Eo~,,n] pZ _ An 5,~/~

.

(99)

7.3. Reduced model adapted to medium-frequency band B~

Let N

126


VIBRATION

9." >- )~v. In the finite element discretization, the reduced model defined by Eq. (90) becomes {0(w), [A~v(W)] ,.T ~ (w) }

,

(101)

in which

[A~v (w)] = [P ]T [A(w)][P]

(102)

9wn (w) = [ P IT F(w)

(103)

.

7.4. Dominant eigensubspace of the energy operator and order of the reduced model N

c

c

Let Co,n C Co,n C CO be the space spanned by {e~..., e~v}. Let u, (~) be c the projection of u(w) on Co,n in which u(w) is the solution of Eq. (60). The energy of vibration un is then written as n

SB~(u,~) - ~

A~ [.T~[2

,

(104)

r N

and the energy of the projection u N of u(w) on C0,~ is written as N

(105) a----1

We then have

.

(19)

From Eqs. (2) and (19), we deduce that

Fj(w) - L g(x,w), ej(x) dx + J~r G(y,w). ej(y) ds(y)

(20)

c

The projection u~(w) of u(w) in Cs, n can be written as n

(21)

u~(~)- }~ uj(~)ej j=l

in which U(w) - (U1 (w),..., U~ (w)) belongs to Cn. Substituting Eqs. (18) and (21) into Eq. (1) and using Eq. (19) yields [A(w)] U(w) - F(w)

,

(22)

in which the elements of (n x n) complex symmetric matrix [A(w)] are such that [A(w)]jk -- a(w ;ek,ej) . (23) Consequently, the discretization of Eq. (5) is U(w) - [T(w)] F(w)

,

(24)

CHAPTER IX.

RESPONSE TO DETERMINISTIC AND RANDOM EXCITATIONS

133

in which for all real w, [T(w)] = [A(w)] -~

.

(25)

c

The projections vn(o)) and w n (w) in Cs, n of velocity field v(w) and acceleration field w(w) can be written as n

,"(x,~)-

n

~vj(~)

~j(x)

,

,v~(x,~)- ~wj(~)

j-'l

~j(x)

(26)

j=l

in which V(w) - (V1 (w),..., Vn(w)) and W(w) = (Wl(w),..., W~(w)) are in C ~ and such that

v(~) - i~ u(~)

,

w ( ~ ) - i~ v(~)

.

(27)

F~(t)) E Rn be the vector such that,

Time domain. Let F(t) - ( F l ( t ) , . . . ,

Vj e { 1 , . . . , n } , Fj(t) - f ( t ; e j ) - < f ( t ) , e j >

.

(28)

From Eqs. (9) and (28), we deduce that Fj (t) = / a g(x, t)- ej (x) dx + 9fr G(y, t). ej (y) ds(y)

(29)

c

The projection un(t) in Cs, n of u(t) (given by Eq (14)) can be written as '

n

.~ (x, t) = ~

vj (t) ~j

(30)

j=l

in which U(t) - (U1 ( t ) , . . . , Un (t)) belonging to Rn is written as U(t) - j f [ h ( t ' ) ] F(t - t') dt'

(31)

The impulse response function [h(t) ] with values in the (n x n) real symmetric matrices satisfies [h(t)]-[0]

,

Vt ~ = Aw dw

(53)

v

in which Y,(w) in the right-hand side of Eq. (53) is given by Eq. (50). Substituting Eq. (50) in Eq. (53) yields mf

~o(-~ zt) ~0(~ zt)*

(54)

TI2=TTt. 1

The square norm of Y~ is obtained by taking the trace of the two members of Eq. (54),

IIV.ll ~ - tr < Yv Y* >,~,,,

-- ~

~

II~0(mAt)ll 2

(55)

1

4. L F a n d M F D e t e r m i n i s t i c Cases: T i m e S q u a r e I n t e g r a b l e Excitation

We directly construct the numerical procedures for calculating the response of the master structure. We then present the developments within the context of the Ritz-Galerkin approximation introduced in Section 2. 4.1. Definition of the excitation

We assume that the body and surface force fields g(x, t) and G(y, t) are such that excitation force vector F(t) defined by Eq. (29), is an ~n-valued square integrable function on ~ with respect to time"

IIFII~- A llF(t)ll2 dt
0, there exists an integer M O

,

Z~(x,-~)

,

= - Z ~ ( x , ~)

-wZ~(x,w)>0

(19)

,

.

(20)

For all x in Fz, real function w ~ Zi(x,w) is continuous on ~ - {0} and lim { - w Zz (x, w) } - a(x) > ~min > 0

w--+0

-

-

)

V X E FZ

,

(21)

in which OLmin is a given real positive constant and x ~ a(x) is a positivevalued function defined on Fz. Eq. (21) means that Zi(x,w) ~

a(x) ~d

as

w-+ 0

.

(22)

CHAPTER X. LINEAR ACOUSTIC EQUATIONS

175

For all x in F z, real function w ~ ZR(X, w) is continuous on ~ and consequently, lim {w ZR(X, O3)} - - 0 V X e Fz . (23) w--+0

From Eq. (20), we deduce that Z(x,~) # 0

,

w e rz

,

w

e e

,

(24)

and from Eqs. (21) and (23), we deduce that {i~ Z(x, ~ ) } ~ = 0 = { - ~ Z~(x, ~ ) } ~ = o - ~(x) > 0

(25)

.

The assumption defined by Eq. (20) corresponds to an acoustic impedance stiffness-controlled over the entire frequency band (see Fig. 5). This can be useful for modeling an acoustic material coating whose thickness is neglected. | i i e |

! i i

ZR

to

/

O3 /ss~ " " " "Zi i # i !

/

e

| i

Fig. 5. Acoustic resistance Zn (solid line) and acoustic reactance Zi (dashed line) for a wall acoustic impedance Z stiffness-controlled over the frequency band

It should be noted that the sign of Zi(x, w) depends on the Fourier transform convention used. With the convention defined by Eq. (7), Zi(x,w) < 0 for w > 0. If the other convention were used, then we would have Zi(x,w) > 0 for w > 0. Acoustic resistance ZR(X,W) does not depend on the Fourier transform convention.

176

STRUCTURAL

ACOUSTICS AND

VIBRATION

Zero pressure condition on a part of the boundary. Let F0 be a part of the b o u n d a r y 0f~ s u b m i t t e d to a zero pressure field. We have p(x,w)--0

,

VxeF0

9

(26)

2.5. Introduction of velocity and displacement potentials Irrotationality and existence of a velocity potential. For w # 0, applying operator ( V >) to the two sides of Eq. (9) yields iw po V x v(x, w) - 0 and gives the irrotationality condition X7 x v(x,w) - 0

.

(27)

Since f~ is a simply connected domain 4 , then Eq. velocity field can be written as v(x,

-

re(x,

(27) shows t h a t the

(28)

,

in which r is the velocity potential defined to within an additive constant. It should be noted t h a t for w - 0 there is a nonunique solution for the velocity field. Among those solutions, we select the solution satisfying Eq. (28) to ensure the continuity of the unique solution of the structuralacoustic equations for w # 0, as w -+ 0. Consequently, Eq. (28) is used for all real w. As r is defined to within an additive constant, we introduce a linear constraint on r ~(r

with

~(1)#0

,

in which ~ is any linear form 5 on the space of functions r we can choose ~(r such t h a t ~(r - 0 is written as fzr

w) dx - 0

.

(29) For instance,

(30)

Introduction of a displacement potential. We introduce the displacement potential ~(x, w) such t h a t r

-

v(x,

.

(31)

4 It should be noted that Eqs. (27) and (28) are equivalent only for a simply connected domain (see Dautray and Lions, 1992). 5 Let C be any real constant, let C T C and r satisfy Eq. (28). Then C=O because s 1 6 2 ~(r ~(1) implies C ~(1)-0, i.e. C--0.


177

Substituting Eq. (31) into Eq. (28) yields u(x,

= V

(32)

(x,

As above, we introduce a linear constraint equation on g(~)-0

,

with

g(1)#0

,

(33)

in which g is any linear form 6 on the space of functions ~. As above, we can choose g(~) such that g(~v) - 0 is written as

/~(

x,w) d x - 0

.

(34)

2.6. Formulation in terms of the pressure and the velocity potential Equations in terms of p and r (10) yields

Substituting Eq. (28) into Eqs. (9) and in

iwpoVr

f~ in

iw p - - p o c~ V 2 r -~- C2 Q

, f~

(35) .

(36)

Eq. (35) can be rewritten as V{iw po r + p} = 0 whose solution is p(x, w) - - i w Po r

w) + lr(w)

in

f~

(37)

,

in which lr(w) depends only on w but not on x (see Ohayon, 1987; Morand and Ohayon, 1992 and 1995). Helmholtz equation in terms of r Substituting Eq. (37) into Eq. (36) yields the Helmholtz equation expressed in terms of velocity potential r w) and modified by the presence of r(w), w2

iw

co

poc~

V 2 r + -5- r + - -

1

lr(w) - -- Q

in

f~

.

(38)

po

Boundary conditions in terms of r We express the boundary condition presented in Section 2.4 in terms of velocity potential r w). 6 Let C be any real constant, let ~o+C and ~o satisfy Eq. (32). Then C - 0 because s l(~o)+C~(1) implies C l ( 1 ) = 0 , i.e. C=0.

178


VIBRATION

1- Neumann boundary condition on r. Substituting Eq. (28) into Eq. (14) yields the Neumann condition on F, or

On in which 0 r

-- i w

Uwall 9n

,

(39)

- W e . n.

2- Neumann boundary condition on rz with wall acoustic impedance. Substituting

Eqs. (28) and (37) into Eq. (16) yields the Neumann condition on Fz, 0r

On

=iwu

wall

iw po -n-~r

r(w)

Z

.

Z

(40)

3- Dirichlet boundary condition on to. From Eq. (26), since p(x, w) is equal to zero on F0, Eq. (37) yields 7r(w) - iw Po r for all x in F0, which shows that r w) is independent of x on F0. Consequently, since r w) is defined to within an additive constant, this constant is removed by choosing r We then deduce that 7r(w) -

on

F0

0 in Eq.

9

(41)

(37) and therefore, p(x,w) =

po r 2.7. Formulation in terms of the pressure and the displacement potential. Equations in terms of p and ~o. Substituting Eq. (31) into Eqs. (35) and

(36) yields --W2 Po V~9 -I-

p -

-#o

Vp = 0 -Q -

+

zw

in in

f~ , f~

(42) .

(43)

EQ. (42) can be written as V { - w 2 po qo + p } - 0 whose solution is p(x, w) - w 2 Po qo(x, w) + 7r(w)

in

~2 ,

(44)

which can also be found by substituting Eq. (31) into Eq. (37). Helmholtz equation in terms of 99. Substituting Eq.

(44) into Eq. (43) yields the Helmholtz equation in terms of displacement potential qo(x, w) modified by the presence of 7r(w), V299+~qo+ co

= poc~

in Po iw

~2 .

(45)

179

CHAPTER X. LINEARACOUSTIC EQUATIONS

B o u n d a r y conditions in terms of ~p. We express the boundary condition

presented in Section 2.4 in terms of displacement potential ~(x, w). 1- Neumann boundary condition on r.

Substituting Eq. (32)into Eq. (15)

yields the Neumann condition on F, ,,

On

=Uwa H . n

.

(46)


Eqs. (32) and Eq. (44) into Eq. (16) yields the Neumann condition on Fz,

O~

iw po

On = U w a n ' n - ~ ~ +

~r(w) iw~

"

(47)

3- Dirichlet boundary condition on to. From Eq. (26), since p ( x , w ) is equal to zero on F0, Eq. (44) yields lr(w) - - w 2 po ~(x, w) for all x in F0, which shows that 9~(x,w) is independent of x on F0. Consequently, since ~o(x, w) is defined to within an additive constant, this constant is removed by choosing ~-0

on

F0

9

(48)

J

We then deduce that r(w) w 2 po ~

0 in Eq.

(44) and therefore, p(x,w) =

(x, w).

3. Dissipative Acoustic Fluid 3.1. Basic assumptions

In the model presented in Section 2, the acoustic fluid is inviscid and consequently there is no acoustic dissipation inside the fluid. In this section, we introduce a dissipation model for the acoustic fluid. Generally, there are two main physical dissipations. The first is an internal acoustic dissipation inside the cavity due to the viscosity and thermal conduction of the fluid. These dissipation mechanisms are assumed to be small. In this model, we consider only the dissipation due to viscosity. This correction introduces an additional dissipative term in the Helmholtz equation (see Eqs. (38) and (45)) without modifying the conservative part. The second dissipation is generated inside the "wall viscothermical boundary layer" of the cavity. This dissipation is not considered in this section, but can possibly be modeled with an equivalent wall acoustic impedance (already introduced in Section 2.4). Acoustic impedance Z(x, w) on boundary F z is useful for

180

STRUCTURAL

ACOUSTICS AND

VIBRATION

modeling an acoustic material coating whose thickness is neglected in the model. We then consider only the acoustic mode (irrotational motion) predominant in the volume. The vorticity and entropy modes which mainly play a role in the "wall viscothermical boundary layer" are not directly modeled. For additional details concerning dissipation in acoustic fluids, we refer the reader to Landau and Lifchitz, 1992b; Lighthill, 1978; Mason, 1971; Pierce, 1989.

3.2. Linearized equations in the time domain Concerning the linearized equations recalled in this section, we refer the reader, for instance, to Landau and Lifchitz, 1992b. We denote as p, p, v, T and s the disturbances of the pressure, mass density, velocity vector, temperature and entropy per unit mass (or specific entropy) of the fluid from their undisturbed values po, po, v0, T, and so corresponding to the fluid at equilibrium in the reference configuration (fluid at rest). The disturbances p, p, T and s satisfy the following classical thermodynamic equations of state 1

porTo

co

Cp

p- ~ p

s

T=--p+--s

,

(49)

,

Po%

(50)

%

in which cp and c~ are the specific-heat coefficients at constant pressure and constant volume respectively, and ~ is the coefficient of thermal expansion. The values of coefficients cp, c~ and/~ correspond to the reference configuration. The linearized mass conservation equation, the linearized Navier-Stokes equation and the linearized general equation of heat propagation (also called the Kirchhoff-Fourier equation or entropy equation) are written as O+p + po V . v = Q

p00+v + v v - ~ v ~ v + poT, o r s -

(+") + g v { v . v}

~ V2T

(51)

,

,

,

(52)

(53)

in which 77 is the dynamic viscosity, u = rl/po is the kinematic viscosity, ( is the second viscosity and ~ is the coefficient of thermal conduction (or thermal conductivity). The values of coefficients rl, ( and ~ correspond to the reference configuration. It should be noted that outside the source region or if there are no distributed monopole sources, Eq.r (51) gives the usual linearized mass conservation equation, Otp + po V . v = O.

CHAPTER X. LINEAR A CO USTIC EQUATIONS

181

3.3. Helmholtz equation with viscosity and thermal conduction As explained in Section 3.1, we are interested in the linear acoustic equations corresponding to irrotational motion Vxv=0

.

(54)

From the identity V 2 v - V { V . v } - V • (V • v) and using Eq. (54), we deduce that V2v - V { V . v}. Replacing V2v by V { V . v} in Eq. (52), and taking the Fourier transform with respect to time of Eqs. (49) to (53) yields 1 po/~To (55) p-~p s , co

cr

T = -f~T, - p + - - s T~ poc,

(56)

,

c,

i w p + po V . v = Q

,

(57)

V2T

.

(59)

i w poT, s -

a

In a first step, p is eliminated in Eq. (57) using Eq. (55). In a second step, T is eliminated in Eq. (59) using Eq. (56) and neglecting the term in W2s. Then, eliminating s between the two resulting equations yields iw p

-

- P o co

"

cp2

co

po

V2p+coQ

(60)

9

Eliminating V - v in Eq. (58) using the equation obtained in the first step and neglecting the term in V s yields i w po v + V p

-

Tc~VQ

- iw TVp

,

(61)

in which T is given by T--

poc

~+r

) >0

.

(62)

In general, T can depend 7 on frequency w. Taking the divergence of Eq. (61) and using Eq. (60), we obtain a Helmholtz equation with a dissipative term and a source term 02 2

p + iw

Ttot V2p + V2p

-

T C~ V 2 Q

- iw

Q

,

(63)

CO S e c o n d v i s c o s i t y ( c a n d e p e n d on w. T o s i m p l i f y t h e n o t a t i o n , we o f t e n w r i t e "1" i n s t e a d of

~(~).

182


VIBRATION

in which Trot is defined by Ttot = r + T~ ,

(64)

where T is the coefficient due to viscosity (defined by Eq. (62)) and T~ is the coefficient due to thermal conduction, ~ 2 It,To T~--

> 0

Cp2 Po

.

~

(65)

Various expressions for the parameter Ttot. Using the classical thermody-

namic identity (see for instance Landau and Lifchitz, 1992b), cp - c~ - T f l

2 2 Cv

co -

(66)

,

c~

and introducing the specific-heat ratio -y defined by "y- ~

Cv

,

(67)

coefficient T~ defined by Eq. (65) can be rewritten as ~ (1 poc~

) _ cp

c~

,7 ( ~ - 1) poc~

(68)

P

'

in which P is the Prandtl number defined by p=77cp

.

(69)

From Eqs. (62), (64) and (68), we deduce the following two expressions,

~tot -

1 {(4

po~o~

-3 ~ + r

) (1 1)} + ~

~

~? {4 ~ ~-1 / r t o t - poc~ 3 +-~? + P

~.

"

,

(70)

(71)

C H A P T E R X.

183

LINEAR ACOUSTIC EQUATIONS

The numerical values of the different coefficients for pure water and air at 20 ~ C are

Po (/~9/m3) co (m/8) ?7 (Pa.s)

7 cp ( J / ( ~ . K ) c~ (J/(kg.K) (W/(m.K))

P

Water

Air

1000 1482 0.001 1.006 4180 4150 0.597 7.00

1.2 340 0.0000181 1.4 1004 717 0.02569 0.707

Dispersion relation. Taking the spatial Fourier transform

p(k, w) - jf~3 p(x, w) exp(i k. x) dx

(72)

of the homogeneous equation associated with Eq. (63), and introducing the wave number k -Ilkll, we obtain the classical dispersion relation k2 _

1

2w

~ _

w2

2

2

(1 + iw "/'tot) CO

CO

iw 3

- - T2 t o t

9

(73)

CO

Eq. (73) shows that the dissipation coefficient varies as w2. 3.4. Linearized acoustic equations in the frequency domain in terms of pressure and velocity

Below, the dissipation due to thermal conduction is neglected. Eqs. (60) and (61) become iw p - - p o co~ V v +9" c o Q . (74) iw po v + V p -- T c ~ W Q -- iw T V p

,

(75)

in which T is given by T = poc~

)>0

71 + ~

(76)

184


3.5. Introduction of the velocity and displacement potentials Velocity potential r potential r

w) and field r

w). Eq. (54) allows a velocity

w) to be introduced such that N

v(x, ~) - re(x, ~)

(77)

It is convenient to introduce a new unknown field r potential r by (Soize et al., 19865) r

w) - (1 + iw T)r

W) -~

w) related to velocity

~-~ Q(x, ~) Po

( 7 8 - 1)

iw

Velocity field v given by Eq. (77) is then written as v(x, w) = (1 + iwT) Vr

+

~ Po

V

(Q(x,~)) zw

(7s- 2)

Since velocity potential r is defined to within an additive constant, field r w) is also defined to within an additive constant. As in Section 2.5, we introduce a linear constraint equation on r s162 = 0

,

with

s

,

in which s is any linear form on the space of functions x ~ r instance, we can choose s162 such that s162 = 0 is written as ar

w) d x - 0

.

(79) w). For

(S0)

It should be noted~ that the constraint introduced on r leads to the following constraint fa r w) dx - TC~/Po fn Q(x, w)/iw dx for velocity potential r

Displacement potential ~(x,w) and field ~(x,w).

As in Section 2.5, we introduce a displacement potential ~(x, w) such that r w) - iw ~(x,w). From Eqs. (77) and (78-1), we deduce that

u(x, ~) = V@(x, ~)

.

(81)

We then introduce an unknown field ~o(x,w) related to displacement potential ~(x,w) by (see Eq. (78-1)) ~(x, w) = (1 + iw T)~O(X,W)

~-~ Q(x, ~) Po

w2

( 8 2 - 1)

CHAPTER X.

185

LINEAR ACOUSTIC EQUATIONS

Displacement field u given by Eq. (81) is written as u(x, w) = (1 + iw T) V ~ ( x , w) -

(82- 2)

~) v ( p Q o (x' ~ ~)

Since ~o is defined to within an additive constant, as above, we introduce a linear constraint equation on ~, g(~)=0

,

with

g(1)r

,

(83)

in which g is any linear form on the space of functions x ~ ~(x, w). For instance, we can choose g(~) such that g(~) = 0 is written as

/~(

x,w) d x - O

.

(84)

It should be noted that the constraint introduced on ~ leads to the following constraint fn ~(x, w) dx - --TC~/Po f~ Q(x, w)/w 2 dx for displacement potential ~. 3.6. Equations in terms of r Equations in terms ofp and r Substituting Eq. (78-2) into Eq. (75) yields (1 + iw T)~7{iw Por + P} = 0 whose solution is

(85)

p = -iw por + r(w) ,

in which lr(w) depends only on w but not on x (see Ohayon, 1987; Morand and Ohayon, 1992 and 1995). Substituting Eq. (78-2) into Eq. (74) yields

iwp = -poc~(l+iw T)V2r -- TC4V2(Q) + c~Q

Dissipative Helmholtz equation in r yields a Helmholtz type equation,

Substituting Eq. (85)into Eq. (86)

w2 V 2 r + iw TV2r -{- ~ r -~- iw ~r(w) co

poco

(86)

.

1 ---Qpo

TC 2

p,

/ok V 2 [ .~--- ]

.

(87)

Boundary conditions in terms of r We express the boundary condition presented in Section 2.4 in terms of field r w).

186


VIBRATION

1- Neumann boundary condition on r. Substituting Eq. (78-2) into Eq. (14) yields the Neumann condition on r , (1 + iwT) 0r

rC~o 0 (_~wI --Po fin

- ~ -- iw Uwall. n

(88)


Eqs. (78-2) and (85) into Eq. (16) yields the Neumann condition on r z,

0r (1 + i W T ) - ~ n

--

iw U w a l l

TC~ 0 (-~W) po On

9n

iw Po r(w) Z r 4- ~Z

(89)

"

3- Dirichlet boundary condition on ro. From Eq. (26), since p(x,w) is equal to zero on F0, Eq. (85) yields r(w) = iw po r for all x in Fo, which shows that r w) is independent of x on Fo. Consequently, since r w) is defined to within an additive constant, this constant is removed by choosing r

on

F0

9

(90)

We then deduce that ~r(w) = 0 in Eq. (85) and consequently, p = -iw Por 3.7. Equations in terms of ~o Equations in terms of p and ~o. Substituting r =

iw ~o into Eq. (85) yields

P = w 2 po~o + Ir(w) ,

(91)

in which It(w) depends only on w but not on x. Substituting r = iw ~o into Eq. (86) yields (92)

Dissipative Helmholtz equation in ~o. Substituting Eq. (91) into Eq. (92)

yields the Helmholtz type equation OJ2

1

Co

poCo

1 Q + TC~ v 2 ( Q ~ po

.

(93)

po

Boundary conditions in terms of ~o. We express the boundary condition presented in Section 2.4 in terms of field ~o(x,w).

187


I- Neumann boundary condition on r.

S u b s t i t u t i n g r = iw ~o into Eq.

(88)

yields the Neumann condition on F,

rc~ 0 (~__2)

(1 + iWT) ~0~0 -- Uwall 9n --IPo - ~

(94)

"

2- Neumann boundary condition on r z with wall acoustic impedance. Substituting

= iw ~ into Eq. (89) yields the Neumann condition on Fz, (1 + iw'r) 0~o

- - ~ - - Uwall 9n +

Po-~

~-~)

Z

- ~

~o4 i w Z

.

(95)

3- Dirichlet boundary condition on to. As for the Dirichlet boundary condition defined by Eq. (90), we have 9~=0

on

F0

9

(96)

We then deduce that r(w) - 0 in Eq. (91) and consequently p --- w 2 po 9~.


189

CHAPTER XI

Internal Acoustic Fluid Formulation for the LF and MF Ranges

1. Introduction In this chapter we present the formulation for the calculation of the frequency response function of an internal acoustic fluid (gas or liquid) in the LF and MF ranges. The boundary of the acoustic cavity is submitted to a prescribed wall displacement field and there is a wall acoustic impedance on a part of this boundary. In addition, an acoustic source density is given inside the cavity. In Sections 2 and 3, we state the internal acoustic problem and give the boundary value problem. In Sections 4 to 6, we establish the variational formulation, the linear operator equation whose unique solution allows the frequency response function to be constructed and finally the finite element discretization. In Sections 7 and 8, we give the formulation of the acoustic modes of the problem, the corresponding spectral properties and the finite element discretization. Using the methodology presented in Chapter VI, Section 9 gives the method for calculating the frequency response function in the low-frequency range. The construction of the frequency response function is based on its projection on a finite set of acoustic modes including quasi-static correction terms. In Section 10, we present a method for calculating the frequency response function in the medium-frequency range. This method is based on the frequency transform technique presented in Chapter VII. Section 11 deals with the case of an acoustic cavity with a zero pressure condition on a part of the boundary (returning to Sections 2 to 10).

190


VIBRATION

In Section 12, we discuss the particular case of an axisymmetric acoustic cavity. Finally, Section 13 refers to Chapter IX for calculating the response of an acoustic cavity subjected to deterministic and random excitations.

2. Statement of the Internal Acoustic Problem We consider a bounded acoustic cavity f~ filled with a dissipative acoustic fluid (gas or liquid). The boundary 0f~ is written as F U F z (see Fig. 1). n

x

/

l

~x 2

x1 Fig. 1. C o n f i g u r a t i o n of t h e a c o u s t i c cavity

On wall F t.J Fz, a displacement field Uwan (x, w) is prescribed. Wall F z has acoustical properties modeled by an acoustic impedance Z(x, w) satisfying the hypotheses defined by Eqs. (X.18) to (X.25). In addition, an acoustic source density Q(x, w) is given inside ~ and satisfies the hypotheses defined by Eq. (X.8). The problem consists in calculating the FRF of the system in the low- and medium-frequency ranges, whose input is due to Uwan (x, w) and Q(x,w), and whose output is field ~o. The pressure field p(x,w) in cavity f~ can then be deduced. The mathematical model of this physical case is described in Section X.3.

3. Boundary Value Problem Equations in terms of ~o. The equations of the boundary value problem are described in Section X.3.7 and are rewritten as

_~2po C2 ~0 -- SW 7" Po V

1 ~r(w)=

2 ~0 -- Po V 2 ~0 -- ~-2Co

Q ~W

CHAPTER XI.

191

INTERNAL ACOUSTIC FLUID FORMULATION

with the Neumann boundary condition,

po(1 + iwT) ~n 0~o - P~

"n +

0 Q on r , Tc:-onn(-w-ff)

(2)

the Neumann boundary condition with wall acoustic impedance,

po(1 + iw'r) O~ = PoUwan 9n + TC~-~n 0 ( '~ Q) "4-

p~

iw Z

+ po~r(~)

on

iw Z

Fz,

(3)

and the uniqueness condition defined by Eq. (X.84), odx-O

.

'

(4)

The pressure field is given by Eq. (X.91), p(x, w) = w2 Po~(x, w) + Ir(w)

in

(5)

f~ .

Expressionfor ~r(w). Integrating Eq. (1) in domain f~, using Stokes' formula and Eqs. (2)-(3) yields

~(~) = ~(~) {-~po ~(~ ;~) -

~("w~,,) +

~, (~)}

,

(6)

in which fr

~(~; ~) = lr2(u) = f r

~o(x,w ) d s ( x ) ~ i~ Z(x, ~)

( 7 - 1)

u(x,w), n(x)ds(x)

(7-

2)

UFz

__1fa Q(x,w). dx ~ro()_w = po ~w

,

( 7 - 3)

[f~[ is the volume of domain f~, I~1

-- fn dx

(s)

and a(w) is the dimensionless complex number defined by 2

~(w) = p~ Co

lal x

{ 1 + po~O~fr d~(x) }-~ If~l ~ i ~ Z ( x , ~ )

(9)

192


VIBRATION

From Eqs. (X.25) and (X.21), we deduce that in the limit static case w = 0, and n(0) is real and is such that 0 < n(O) < +c~

lim '

amin

n(O)+oo

poc~

(10)

lal

"

It should be noted that the general expression of lr(w) defined by Eq. (6) includes the term -~(w)(032/C 2) fgt ~0dx which is equal to zero due to Eq. (4). For all real w, for Uwan(W) and Q(w) given, the problem consists in finding ~(w) as a solution of Eqs. (1) to (4) in which ~r(w) is expressed by Eqs. (6) to (9). The pressure is then given by Eq. (5). It should be noted that the static problem corresponding to w - 0 is correctly stated.

Problem to be solved.

4. Variational Formulation and Linear Operator Equation 4.1. Admissible function space

The variational formulation is constructed using the test function method presented in Chapter II. We introduce the complex vector space g~ of sufficiently differentiable 1 functions ~o defined on f~ with values in C and the admissible function space g~o which is the subspace of g~ defined by 12

E~~ _ { ~ e $~

;

]~ ~ d x - 0 }

(11)

4.2. Variational formulation c

c

Let qo be in $~. Multiplying Eq. (1) by 5-~ E $~, using Green's formula, taking into account the Neumann boundary condition defined by Eqs. (2) and (3), and since fn 5--~dx - 0, yields the following variational formulation. c

Variational formulation. For all fixed real w, find ~ in $~ such that c

-

1 Space $c is the Sobolev space H 1 (gt).

,

e

,

(12)

CHAPTER

XI.

INTERNAL

ACOUSTIC

FLUID

193

FORMULATION C

C

in which sesquilinear form a(w;~o, 6~o) on ~ • ~ is the restriction to ~ of the following sesquilinear form a(w; ~o, 6(p) on • defined by

C

a(w; ~o, 6~o)--w2m(~o, 6~o)+iw dr(w; ~o, 6~o)+k(~o, 6~o)+Sz(W; ~o, 6~o) . (13)

Sesquilinear form m(~o, 6~o) and operator M. Positive-definite sesquilinear c

form m(~o, 6~o) on E x

~e

is such that

Pof--

m(~o, 6~o) - ~

(p 6(p dx

(14)

t30

and defines the linear operator M such that 2 < M ~o, 6~o > = m(~o, 6~)

(15)

.

Sesquilinear form k(~o, 6~o) and operator K. Sesquilinear form k(~o, 6~o) is defined on ~r215C r by k(~o, 6~o) - po /~ V~o . V6~o dx and is positive semidefinite (k(~o, 6~o) - 0 if ~o is equal to a constant). The restriction of k(~o, 6~o) to the admissible function space E~ x E~ is positivedefinite. Sesquilinear form k(~o, 6~o) defines the linear operator 3 K such that < K ~ , 6~ > = k(~, 6~) . (17) c

c

Sesquilinear form d~-(w ; ~o, 6~o) and~ c operator D~(w). Positive-semidefinite c sesquilinear form d~-(w ; ~o, 6~o) on x ~ is such that dT(W ; ~0 , 6~0) = T(W) k(~o, 6(p) ,

(is)

and defines the linear operator Dr (w) such that < Dr (w) ~o, 6~o > = d,(w ;~o, 6~o) .

(19)

2 Sesquilinear f o r m m(tp,6~o) is c o n t i n u o u s on H C x H c w i t h HC=L2(ff/) a n d is also c o n t i n u o u s on ~CxEC, M is a real c o n t i n u o u s o p e r a t o r from C c into its a n t i d u a l space E c ' , a n d t h e angle b r a c k e t s in Eq. (15) d e n o t e the a n t i d u a l i t y p r o d u c t b e t w e e n ~c, and E c . 3 Sesquilinear f o r m k(~o ,5~o) is c o n t i n u o u s on C c x~: c , K is a c o n t i n u o u s o p e r a t o r from E c into its a n t i d u a l space E c' a n d t h e angle brackets in Eq. between s ~c into ~ c ,

9

(17) d e n o t e t h e a n t i d u a l i t y p r o d u c t

a n d Ec. C o n t i n u o u s linear o p e r a t o r K is positive semi-definite H e r m i t i a n f r o m c T h i s o p e r a t o r is coercive on ~

194


VIBRATION

Sesquilinear form d~(w;qo, ~o) has the same properties as k(~o,~o) and consequently is positive-definite on g~o x g~o. c

c

Sesquilinear form Sz(W ; ~o, ~o) and operator Sz(w ). Sesquilinear form Sz(W ; qo, ~o) defined on E~ • is such that

~(~ ;~, ~ ) = - p ~ ' [

~ ~ z-~ a~ + ~ ( ~ ) JrzZco~

p~ ~(~; ~) ~(~; ~ ) , (20)

and defines the linear operator Sz(w ) such that 4

< s~(~) ~, ~ >= ~(~ ;~, ~ )

(21)

Antilinear form f (w ; 6qo) and excitation vector f(w). Antilinear form ~qo f(w ; ~o) on E ~ can be written as

f (co ;6~o) -- fQ (co ;~q0) + f%~. (co ;6qO) ,

(22)

in which (using Green's formula for the term containing V2Q), - - dx+po to(w) Irq(w) rl (w ;gq0), jVQ.V~v

fQ (w; 5q0)- /n ~Q--zw ~ dx t rc~ w2 f

(23)

and where fUw~,,(w ; ~qo) is such that

f"w~" (~ ; ~ ) - P

~r "w~." n ~ d~ - p0 ~(~) ~("w~.) ~ (~ ;~)" (24)

Antilinear form f(w ; ~qo) defines f(w) such that 5

< f(~), ~ > = / ( ~ ; ~ ) 4

(25)

For all w fixed in R, since function xs-~{iwZ(x,w)} -1 is b o u n d e d on F z (see Eq. (X.18)), we d e d u c e t h a t Sz(W ;q0,6qo) is a c o n t i n u o u s sesquilinear form on ECxE c . This sesquilinear f o r m defines a c o n t i n u o u s o p e r a t o r Sz(w ) from Ec into its a n t i d u a l space Ec ,, a n d t h e angle b r a c k e t s in Eq. (21) d e n o t e t h e a n t i d u a l i t y p r o d u c t b e t w e e n E c ' a n d Ec. It is a s s u m e d t h a t Z is such t h a t w - l ~ e { - i S z (W;qO,~O)} = w - l ~ m { Sz (W;qo,~o)} >_O.

5 For all w fixed in R, it is a s s u m e d t h a t f u n c t i o n xs-+Q(x,w)/w 2 belongs to E C = H l ( f l ) a n d c x~-~Uwall(x,w)'n(x) belongs to Eon--H1/2(Ofl)CL2(Ofl). Since f u n c t i o n x~+{iwZ(x,w)} -1 is b o u n d e d on F z (see Eq. (X.18)), we t h e n d e d u c e t h a t 6~o~-+f(w ;6qo) is a c o n t i n u o u s a n t i l i n e a r f o r m on E c . T h i s a n t i l i n e a r form defines an e l e m e n t f(w) belonging to t h e a n t i d u a l space E c' of E c a n d t h e angle b r a c k e t s in Eq. (25) d e n o t e t h e a n t i d u a l i t y p r o d u c t b e t w e e n E c ' a n d E c.

195

CHAPTER XI. INTERNAL ACOUSTIC FLUID FORMULATION

4.3. Linear operator equation Using Eqs. (15), (17), (19), (21) and (25), the linear operator equation associated with Eqs. (12) and (13) is written as 6 A(w)~o=f(w)

,

c

~e~

,

(26)

in which operator A(w) is defined by 7 A(w) = - w 2 M + iw D~(w) + K + Sz(w )

.

(27)

5. Frequency Response Function The operator-valued frequency response function is defined by the linear operator which associates each f with field ~o as the solution of the variational formulation defined by Eq. (12). It can be proved s that, for all fixed real w and f(w) given, the variational formulation defined by Eq. (12) has a unique solution x ~ ~(x, w) in $~. Consequently, linear operator A(w) is invertible on ~ and Eq. (26) yields c

c

~(w) = T(w)f(w)

(28)

,

in which w ~ T(w) = A(w) -~ is the operator-valued frequency response function 9.

6. Finite Element Discretization Since the finite element method uses a real basis for constructing the finite element matrices, finite element discretization of sesquilinear forms 6 Continuous sesquilinear form a(w ;~,8~) on from s

C

into s

Cl

C

C

s163

defines a continuous linear operator A(w)

by =a(w ;~o,6~o). Eq. (26) is an equality in s c

C!

.

ct

7 From footnote 6, linear operator A(w) is continuous from g~ into g~ . The operator on the c

right-hand side of Eq. (27) is the restriction to g~o of an operator defined on go. 8 For w>O, r ( w ) > 0 (see Eq. (X.76)) and Ne{-iSz(W;~o,~o)}>__O(see footnote 4). The existence c and uniqueness of the solution in g~ is then obtained by applying the Lax-Milgram theorem to ac(W;~o,5~o):fc(W;5~o)in which ac(w;~o,5~o)-'-ia(w;~o,5~o) and fc(W;~o):-i f(w;~o). In effect, ~eac(w;~o,~o)=wdr(w;~o,~o)+~e{-iSz(W;~o,~o)} is coercive because dr(w;~o,~o) is coerc cive on g~, and Ne{-iSz(W;~o,~o)}?_O. The proof is similar for w__1, spanned by the finite family {~ol,..., ~ON} of acoustic modes ~o~ defined in Section 7, for which A~ > 0. The projection ~oN on $~,N of the unique solution ~o of Eq. (12) can be written as {2

C

c

N

~(x.~)

= ~

q~(~)~(x)

.

(51)

t~--1

in which q -- ( q l , - . . ,

qN)

(52)

is a complex vector of the generalized coordinates. From the orthogonality conditions defined by Eqs. (42) and (43), we deduce that q is the solution of the following matrix equation of dimension N [A(w)] q(w) - 9r(w)

.

(53)

From Eq. (13), we deduce that the (N x N) complex symmetric matrix [A(w)] is written as [A(~)] - _~2 [ M ] + i~ [v~(~)] + [ ~ ] + [s~(~) ]

.

(54)

where [A4], []C] and IDa(w)] are (N x N) positive-definite diagonal real matrices and where [Sz(W)] is a dense (N x N) symmetric complex matrix such that [M ] ~ - - ~ ( ~ , ~ ) - ~ ~ , (55) [ ~ ] . ~ - k(v~. V.) - , . w.2 t~a~

, 2 t~t~~

[s~(~)]~ - ~(~ ;~. ~)

.

(56) (57)

(58)

in which sesquilinear form s z is defined by Eq. (20). It should be noted that dense matrix [Sz(w)] is due only to the presence of an acoustic impedance Z(x, w) on boundary Fz and consequently uses only the values of the acoustic modes on Fz. Component 9r~ of the generalized force vector

~ " - (~'~,... ,~'N)

,

(59)

7~(~) - f(~ ; ~ )

,

(60)

is written as

201

CHAPTER XL INTERNAL ACOUSTIC FLUID FORMULATION

in which antilinear form f(w ;6~o) is defined by Eqs. (22) to (24). From Eq. (53) and the invertibility of matrix [A(w)], we deduce that q(w) - [T(w)] ~'(w)

,

(61)

in which [T(w)] is an (N x N) complex symmetric matrix such that I T ( w ) ] - [A(w)] -~

,

IT(w)] T - [ T ( w ) ]

.

(62)

.

(63)

From Eqs. (51) and (61), we deduce that N

~oN (x, w) - ~

N

~

[7"(w)]., y~(w) ~o.(x)

a = l B=I

9.2. Projection of the FRF on the acoustic modes Operator-valued frequency response function T(w) is an intrinsic operator characterizing the dynamics of the system independently of the values of excitation vector f(w). We can then choose a particular f(w) to construct the projection of W(w), and we define element f(w) by f(w ; 6~) - < f ( w ) , 6~>

- / - g(x, w)6~o(x) dx Jfl

(64)

in which x ~+ g(x, w) is any complex-valued square integrable lo function on f~. Consequently, Eqs. (60) and (64) yield (65)

~#(w) - / n g(x, w)~o#(x) dx c

Let TN(w) be the projection o n ~o,N of operator-valued frequency response function W(w) defined by Eq. (28). ~ o m EQ. (63), we deduce that N

N

TN(w) f(w) -- y ~ E

(66)

[T(w)]~ Y~(w) ~

a=l/3=1 10 Since x~--}g(x,w) belongs to HC=L2(~), the antiduality product between s equal to the usual inner product (qo ,6~O)He=fn ~o(x) 6qO(X)dx in H e.

{21

C

and E~ is

202


VIBRATION

Substituting Eq. (65) into Eq. (66) yields, for all x in ~2, (TN(w) f(w))(x) = / n T~(W;X,X') g(X',W)dx'

(67)

in which rN(w; x, x') is such that N

N

-r',,,(w;x, x') = ~ ~

[T(w)],~/~qo~(x)qo/~(x')

(68)

a = l ~=I

Eq. (67) shows that linear operator ~ T N(w) is an integral operator with kernel TN(W; X, X') satisfying the following property -,-,,(,,,; x, x') =

(,,,; x', x)

.

(69)

Consequently, for all fixed real w, we have 12 T N ( W ) * -- T N ( W )

9

(70)

9.3. Finite element discretization of the FRF projection on acoustic modes

Consider the finite element discretization introduced in Section 6. Let {~a },~=l,...,N be the set of N eigenvectors defined in Section 8. The matrix [T~(w)] of operator IN(w) obtained by finite element discretization can then be written as N

N

[T. (w)] = ~ ~ [T(w)].~~ ~

(71)

a = l ~=1

in which [T(w)] is the (N • N) symmetric complex matrix defined by the first Eq. (62). Matrix [TN(w)] is an (n • n) complex matrix such that _

.

(72)

This property corresponds to Eq. (70). Ii Integral operator TN(w) considered as an operator in HC=L2(f~) can be written as T N ( w ) = 12

T N(w)* is the adjoint operator of bounded operator T~(w) (see Mathematical Notations in the appendix).

CHAPTER XI.

203

INTERNAL ACOUSTIC FLUID FORMULATION

9.4. Introduction of quasi-static correction terms Construction of the quasi-static correction terms. Let N >_ 1 be the integer

such that 0 < r

-

Antilinear form ~o ~-+ f(w ; ~o) on

~c

can be written as (127)

f (w ;59~) -- fQ (w ; ~ ) + fUwall (W ; (~(~) , in which fQ(w;5(p)=-f~

--~ Q -~ dx + Tc~ fn ~-~VQ. 1

V~dx

( 2s)

(129)

Uwall 9n ~--~ds

fUwall (W ; (~(~) -- Po /

JFUFz

and defines the element f(w) such that 16 < f(w), ~ > = f(w ;J~o)

(130)

.

Linear operator equation. Using Eqs. (15), (17), (19), (126) and (130), the linear operator equation associated with Eqs. (123) and (124) is written as

17

A(w)~o-f(w)

,

c

~oeC O ,

(131)

in which operator A(w) is defined by IS A(w) - - w 2 M + iw DT(w) + K + Sz(W )

.

(132)

15 The m a t h e m a t i c a l properties of operator Sz(w) are those given in footnote 4. 16 The m a t h e m a t i c a l properties of f(w) are those given in footnote 5.

c

17 Sesquilinear form {~o ,5~o}~-+a(w ; ~o,5~o) is continuous on Eo • C

CI

A(w) continuous from ~:o into ~o

c

and defines a linear operator Cl

and Eq. (131) is an equality in Eo 9

c

cl

18 Since linear o p e r a t o r A(w) is continuous from Eo into Eo , then the operator in the right-hand c side of Eq. (132) is the restriction to ~o of the operator defined on Ec.

212


VIBRATION

11.4. Frequency response function (FRF)

The operator-valued frequency response function is defined by the linear operator which associates each f with field ~o as the solution of the variational formulation defined by Eq. (123). It can be proved 19 that, for all fixed real w and f(w) given, the variational formulation defined by Eq. (123) has a unique solution x ~ ~o(x,w) in $o. Consequently, linear operator A(w) defined on $0 is invertible and Eq. (131) yields c

~o(w) = T(w)f(w)

,

(133)

in which w ~ T(w) - A(w) -~ is the operator-valued frequency response function 20. 11.5. Finite element discretization

The finite element discretization of Eq. (123) yields [A(w)] 9 - F(w)

,

(134)

in which 9 = ( ~ 1 , . . . , On) denotes the complex vector of the DOFs which are the values of qo at the nodes of the finite element mesh of domain Ft. Vector F = ( F 1 , . . . , F n ) is the complex vector corresponding to the discretization in $ 0 of antilinear form f ( w ; h q o ) defined by Eq. (130). The (n x n) complex matrix [A(w)] is symmetric and is defined by [A(w)] - - w 2 [M] + iw IDa(w)] + [K] + [Sz(w)]

,

(135)

where the real symmetric positive-definite matrices [M], [K], IDa(w)] correspond to the finite element discretization in CO and the symmetric complex matrix [Sz(w)] is such that w - l . ~ m [ S z ( w ) ] is positive semidefinite. For all fixed real w, Eq. (134) has a unique solution which is written as c

O(w) - [T(w)]F(w)

(136)

,

in which the (n x n) symmetric complex matrix [ T ( w ) ] - [A(w)]-1 is the matrix-valued frequency response function. The corresponding (n • n) real matrix-valued impulse response function t ~ [h(t)] is such that (see EQ. (V.72)), [T(w)] - / e - i ~ t [h(t)] dt J~

(137)

.

19 The proof of existence and uniqueness is similar to the proof given in footnote 8. 12

Cl

20 Since linear o p e r a t o r A(w) is continuous and invertible from go into g o , T(w)=A(w) -1 is c!

continuous from go

12

into $o-

213

CHAPTER XI. INTERNAL ACOUSTIC FLUID FORMULATION

11.6. Acoustic modes of the acoustic cavity

The spectral boundary value problem. As in Section 7 and from Eqs. (117) to (120), the equations of this classical spectral problem are the Helmholtz equation 03 2

V2~o+~o=0

in

~t

,

(138)

Co

with the Neumann boundary condition,

On

= 0

on

0f~\Fo

,

(139)

and the Dirichlet condition,

~o-0

on

F0

9

(140)

Variational formulation of the spectral boundary value problem. From Section 11.3, we deduce the variational formulation of the spectral boundary value problem defined by Eqs. (138) to (140). Find A - w2 and ~o in go such that k(~o, 5~o) - ,X m(~o, 5~o) , V 5~o ~ go 9 (141) c

c

Properties of the eigenvalues and eigenfunctions. They are similar to those given in Section 7.2. There is a countable number of positive eigenvalues, the family of eigenfunctions is a complete set in E0 (see Eq. (40)) and the orthogonality conditions with respect to m and k are satisfied (see Eqs. (42) and (43)). c

Finite element discretization and generalized symmetric matrix eigenvalue problem. The matrix equation of the generalized symmetric eigenvalue problem corresponding to the finite element discretization of Eq. (141) is written as [K]O-A[M]r , (142) in which [M ], [K] and 9 are defined in Section 11.5.

11.7. FRF calculation in the LF and MF ranges The results are similar to those presented in Sections 9 and 10. It is sufficient to replace the admissible function space $~ by $o. It should be noted that the constraint defined by Eq. (87) must be removed. c

c

214


VIBRATION

12. Particular Case of an Axisymmetric Acoustic Cavity We consider the boundary value problem defined in Sections 2 and 3 for which f~ is an axisymmetric domain and F and Fz are axisymmetric parts of boundary 0f~. The wall acoustic impedance Z(x,w) on Fz is assumed to have the same axisymmetry property. Consequently, using the analysis presented in Section III.8.2, p(x, w) and ~o(x,w) can be written as -~-00

p(r, 0, z) - p0(r, z) + E {P+ (r, z) cos nO + p~ (r, z) sin nO}

(143)

n--1 -~-00

~o(r,O,z) - ~o0(r, z ) + ~-~{~+(r, z) cosn0 + ~On(r, z ) sinn0}

(144)

n=l

From Eq. (5), we deduce that p0( , z) =

po

z) +

,

(145)

and for all n >_ 1,

(140) These equations show that lr(w) = 0 for n >_ 1. Consequently, in the boundary value problem, for n >_ 1, we set ~r(w) - 0 in Eqs. (1) and (3), c Eq. (4) must be suppressed, admissible function space E~ is replaced by subspaces C: + and t~:- of t~~, the second term in the right-hand side of Eq. (20) must be suppressed and finally, in Eqs. (23) and (24), the terms containing % (w) and 7r2(Uwau) must also be suppressed.

13. Response to Deterministic and Random Excitations In the boundary value problem defined in Section 3 or 11, the excitations are wall displacement field UwaH(x, t) and internal acoustic source density Q(x, t). For calculation of the reponse of the acoustic cavity subjected to these deterministic or random excitations, we refer the reader to Chapter IX.

215

CHAPTER XII

External Acoustic Fluid. Boundary Integral Formulation for the LF and MF Ranges

1. Introduction This chapter is devoted to the external acoustic problem in the frequency domain related to the Helmholtz equation. We present an appropriate boundary element method to model the external acoustic fluid in structural-acoustic problems. Various methods can be used for solving this external acoustic problem. For simple shapes and geometries, we refer the reader to Bowman et al., 1969; Crighton et al., 1992; Junger and Feit, 1993. For methods based on the use of a finite element mesh of the external fluid, see Everstine and Yang, 1984; Givoli and Keller, 1989; Pinsky and Abboud, 1989; Harari and Hughes, 1994, and for the infinite element application, see Bettess, 1993 and Zienkiewicz et al., 1985. For methods based on the use of asymptotic approximations, concerning the slender body theory, see Coupry and Soize, 1984; Chabas and Soize, 1986; concerning the doubly asymptotic approximation theory (DAA), see Belytschko and Geers, 1977; Geers and Felippa, 1983; Geers and Zhang, 1994. Herein, the external acoustic fluid problem is solved using integral equation techniques. This kind of approach is well suited to this problem and various methodologies are proposed in the literature (Maue, 1949; Panich, 1965; Mitzner, 1966; Schenck, 1968; Burton and Miller, 1971; Jones, 1974; Belytschko and Geers, 1977; Brebbia, 1978; Angelini and Hutin, 1983; Everstine and Yang, 1984; Brebbia et al. 1984; Mathews, 1986; Kress, 1989; Amini and Harris, 1990; Chen et al., 1990; Amini et al., 1992; Brebbia and Dominguez, 1992; Chen and Zhou, 1992; Colton and Kress, 1992; Dautray and Lions, Vol 4, 1992). In this chapter, we present an appropriate

216


VIBRATION

symmetric boundary integral formulation valid for all real values of the frequency, and the corresponding boundary element method for constructing the acoustic impedance boundary operator and the radiation impedance operator in the LF and MF ranges. In Section 2, we introduce the external acoustic problem in the frequency domain, consisting in finding the pressure field in the external domain ~E and on its boundary F~ for a given velocity field on FE, an acoustic source density in ~E and an incident plane wave. In Section 3, we present the basic exterior Neumann problem related to the Helmholtz equation expressed in terms of the velocity potential field, for a given normal velocity field on boundary FE. This problem, which includes the outward Sommerfeld radiation condition at infinity, has a unique solution for all real values of the frequency. In Section 4, we introduce the acoustic impedance boundary operator which relates the normal velocity field on F~ to the pressure field on F E. This fundamental operator is related to the basic exterior Neumann problem related to the Helmholtz equation and will be used for coupling an external acoustic fluid with the structure in the LF and MF ranges. The symmetry and positivity properties of this operator are proved. We then introduce another fundamental operator, the so-called radiation impedance operator, which allows calculation of the near field and far field pressure. Section 5 deals with the free-space Green's function, used for constructing the integral boundary representations on F~. In Section 6, due to the linearity of the problem, we show that the solution of the above basic exterior Neumann problem allows the construction of the equations for radiation by an elastic boundary, scattering of an acoustic source density by a rigid boundary and scattering of an incident plane wave by a rigid boundary. Section 7 contains a brief review of the potential theory introducing the single- and double-layer potentials used for constructing the boundary integral equations related to the solution of the basic exterior Neumann problem. Section 8 is devoted to Helmholtz integral representations and their normal derivatives for the external domain and the internal domain. In Section 9, we introduce two basic boundary integral equations and their variational formulation for the exterior Neumann problem related to the Helmholtz equation. These basic formulations do not allow construction of a unique solution of the physical problem in the case of a sequence of real frequencies called spurious or irregular frequencies. Various methods are

CHAPTER XII.

EXTERNAL ACOUSTIC FLUID FORMULATION

217

proposed in the literature to overcome this mathematical difficulty (Panich, 1965; Schenck, 1968; Burton and Miller, 1971; Mathews, 1986; Amini et al.; 1990 and 1992; Colton and Kress, 1992). It appears that the most popular and robust formulation is due to Burton and Miller, 1971. Nevertheless, it should be noted that this formulation is not symmetric. In Section 10, we present a method initially developed by Angelini and Hutin, 1983, based on an appropriate use of the two basic boundary integral equations presented in Section 9, leading to an appropriate symmetric boundary integral method valid for all real values of the frequency. Finite element discretization based on a variational formulation leads to a boundary element method which is numerically stable and very efficient. This method does not require introducing additional degrees of freedom in the numerical discretization for treatment of irregular frequencies. This method was validated by comparisons with analytical and experimental results in the LF and MF ranges (Angelini and Hutin, 1983; Soize et al., 1986b and 1992; Petitjean, 1992) and then extended to the Maxwell equations (Angelini and Soize, 1989 and 1993). In Section 11, we construct the radiation impedance operator and we give an asymptotic formula for the radiated pressure field. Section 12 is devoted to the boundary element method corresponding to finite element discretization of the method presented in Section 10 and 11. Finally, in Section 13, using the image method, we show that the case of an unbounded domain with a free surface is equivalent to the case of an unbounded domain without a free surface.

2. Statement of the External Acoustic Problem We consider an external acoustic fluid (gas or liquid) occupying an unbounded open three-dimensional simply connected domain 1 f~E whose complement f~i - ~3\(f~E t_J FE) is an open bounded domain (see Fig. 1). The acoustic fluid is assumed to be inviscid. A displacement field Uwan (y, w) is prescribed on the boundary r E -- 0f~E -- 0f~i which is assumed to be smooth. We denote the outward unit normal to F s as n (see Fig. 1). In addition, an acoustic source density Q(x,w) and an incident plane wave defined by the velocity potential r (x, w) = r e x p ( - i k.x} are given inside f~E. The problem consists in calculating the pressure field p~(y,w) on boundary F~ and the pressure field p~(x,w) in the unbounded 1 A domain ft is called simply connected if any continuous closed curve in f / c a n be continuously shrunk to a point without leaving f/.

218


domain ~'~E in the low- and medium-frequency ranges, for given Uwan (y, w), Q(x, w) and r (x, w). For this problem, we need to solve the basic problem corresponding to the Helmholtz equation in external unbounded domain f~E with an arbitrary Neumann boundary condition on FE. This problem is called the exterior N e u m a n n problem related to the Helmholtz equation. The linearity of the equations then allows us to construct the solution for a given acoustic source density and a given incident plane wave.

~E X

~/~i ,/ / / nc(X,~0)

ny

Fig. 1. Geometry of the external unbounded domain

3. Exterior Neumann Problem Related to the Helmholtz Equation 3.1. Helmholtz equation and Neumann boundary condition This basic problem corresponds to the case for which w is a given real number and there is neither a source nor an incident plane wave inside f~E. Using Section X.2, we deduce the Helmholtz equation for velocity potential r w) and we write the Neumann boundary condition for a given normal velocity field v(y) defined on F E with values in C.

Expression of the pressure in terms of the velocity potential. The region at infinity is at rest, which means that for all fixed real w, the pressure field p(x,w) and the velocity field v(x,w) converge to zero as the distance Ilxll

CHAPTER XII.

219


from the origin approaches infinity 2. We then have p(x, o.,) -----> o

as

Ilxll ~

+00

r

as

Ilxll ~

+~

~

o

,

(1) 9

(2)

From Eqs. (1), (2) and (X.37), we deduce that lr(w) is equal to zero and consequently, p(x, w)

-

- i w PE r

w)

in

fie

(3)

,

in which p~ is the constant mass density of the external fluid at equilibrium. Helmholtz equation in terms of the velocity potential.

For all real w, we

introduce k defined by W k

-

--

(4)

,

C~

in which c~ is the constant speed of sound in the external fluid at equilibrium. Since Q and ~r(w) are equal to zero, Eq. (X.38) yields the Helmholtz equation in terms of velocity potential r w) V2r

2r

in

f~S

9

(5)

Neumann boundary condition on F~. This condition is written as

0r

w) COny

=v(y)

on

FE

,

(6)

in which the normal derivative of r on FE is defined by 0r

w) 0ny

= {Vxr

ny

,

y E FE

9

(7)

2 We will see below that in the frequency domain, the uniqueness of the exterior Neumann problem related to the Helmholtz equation is obtained by introducing the Sommerfeld conditions at infinity. These conditions imply that r Vr and p(x,w)--~0 as Ilxll-+-I-c~.


220

3.2. Introduction of the outward Sommerfeld radiation condition at infinity For all fixed real k r 0 and for v given on FE, the problem defined by Eqs. (5) and (6) does not have a unique solution. In order to make the solution unique, it is necessary to introduce the outward Sommerfeld radiation condition at infinity which ensures that we only consider outward traveling waves at infinity, 1

0r

-~-R + i k r

1

,

(s)

as R - ][xl] + ~ , in which O/OR denotes the derivative in the radial direction from the origin (see Fig. 1).

3.3. Existence and uniqueness of the boundary value problem Admissible function spaces. Let Ca E be the admissible function space 3 of functions x ~+ r defined on f~E with values in C. Let CrE be the function space constituted by the traces 4 y ~ Cr~(Y) - r on F E of functions x ~ r belonging to C~ E. We introduce the space 5 C~E of the normal derivative y ~ 0r on r E of functions x ~ r belonging to Cn~. Finally, we introduce the space HrE -- L2(FE) of the square integrable complex functions on FE and we have CrE C Hr~ C C~E. Existence and uniqueness of the boundary value problem. For all real k r 0 and for all y ~-~ v(y) in C~E , the exterior Neumann problem related to the Helmholtz equation defined by Eqs. (5), (6) and (8), has a unique solution x ~-~ r in Ca E whose trace x ~ erE(X, W) on boundary r E belongs to CrE (see for instance Dautray and Lions, 1992).

3 Space Cf~E=H~oc(-~E); H~oc(~E ) is the set of functions from ~ E into C such t h a t , for all b o u n d e d subsets OC-~E, the restriction of r to O belongs to H I ( o , C ) . 4 Vector space

CrE-'H1/2(FE).

5 Vector space Cr' E is the dual space of CrE for the dual brackets

C~E,CrE in

which

vEC~E and C r E ECrE and which is linear in v and linear in Cr E (and not antilinear in Cr E ). We have C~E--H-1/2(FE ). Here we use the dual space instead of the antidual space as is done in all the other chapters of the book.

CHAPTER XII.

221


4. Acoustic Impedance Boundary Operator and Radiation Impedance Operator Acoustic impedance boundary operator. From Section 3.3, there exists a unique linear operator BrE(w/c~) such that 6

Cr~ (w) - BrE (W/CE) V

r~

on

.

(9)

From Eqs. (3) and (9), we deduce that PrE(W)=Zr~(W)V

rs

on

,

(10)

in which PrE(W) is the trace on F E of p(w), and ZrE(W) is the acoustic impedance boundary operator defined by 7 Zrs(W) = --iWpEBrE(W/C~)

(11)

9

Radiation impedance operator in the external domain. From Section 3.3, there exists a unique linear operator R(X,W/CE) such that s r

,

X e f~E

,

X E ~E

(12)

9

From Eqs. (3) and (12), we deduce that p(X,W) -- Zrad(X,W) V

(13)

,

in which Zrad(X, w) is the radiation impedance operator defined by 9 Z~ad(X,W)=--iwpER(x,w/C~)

,

Xe~2E

9

(14)

Symmetry property of the acoustic impedance boundary operator.

The transpose 10 of operator BrE(W/CE) is the linear operator tBrE(W/CE) such that, for all v and 5v in C ~ ,

< 5v. Br. (w/c.) v >%~.c~ = < tBrE (W/CE) 5V, V > CrE , ,Or E

(15)

Mapping v~-CCrE is linear and defines a unique continuous linear operator BrE(O.)/CE) f r o m C~E into CrE. 7

Linear operator ZrE(w) is continuous from C~E into CrE.

8 For all x fixed in FtE, mapping v~-~r R(X,W/CE) from C~E into C.

is linear and defines a unique linear operator

9 Linear operator Zrad (x,w) is defined from C~E into C. 10 Transpose tBrE (W/CE) is a continuous linear operator from C~E into (C~E)'=CrE (see footnote 5).

222


VIBRATION

We then have the following symmetry property tBr~ (w/c,) - Br~ (w/c,)

(16)

,

and from Eq. (11), we deduce that, for w ~t 0, (17)

~Zr~ (~) = Zr~ (~)

Let us prove Eq. (16) for v and 6v in the space HrE C C~E of the square integrable complex function on F~. We introduce CrE and ~r in CrE CHrE such that Cr~ - Br~ (w/c~) v and 6r ~ - Br~ (w/c~) 6v. Consequently, we can write 11 < ~v, Br ~ (w/c~) v > ~ ~ ,c~ --< 6V, r

(18)

>-- fr 6VCrs ds E

< Br~ (w/c~) ~v, v > ~

, - R0,

fr

(6v

r

-

6r

v) ds -

-

L B(O,R)(--~-n 0~r r

0r ) ds

6r ~

-

(21)

.

Considering R --+ +co and using the Sommerfeld condition defined by Eq. (8), the right-hand side of Eq. (21) converges to zero and yields Eq. (20).

Positivity of the real part of the acoustic impedance boundary operator. Operator i w Zr~ (w) can be written as i w Zr~ (w)

( ~ / M + i~ Dr~ (~/M

= - w 2 Mr~

,

(22)

in which M r s (w/c~) and Dr~(wlce) are two linear operators such that w Mr~ (~/M

-

~,',,

Zr~ (~)

Dr~ (w/c~) - ~e ZrE (w)

(23)

,

(24)

.

Let v be in H r s . Trace r on F E is associated with v using Eq. (9). We have the following positivity property of the real part Dr~(w/c~) of the acoustic impedance boundary operator 12 Vu,,6~

,

.

(25)

ds

(26)

>0

The proof is carried out as follows. First, we have

- w p ~ m ~ r ~ r FE

because Eqs. (9), (11) and (22) yield =

i

wpE

1 p~

,

(27)

and because MrE(w/c~) and Dr~(w/c~) are linear operators which have symmetry properties due to Eq. (17). Secondly, from Eq. (26), we deduce that Eq.(25) holds if and only if

Vw 6 [R ,

f

w.~m] ~ r

ds >_O .

(28)

JFE 12 Since CrECHrE=L2(FE), the angle brackets in Eq. (25) are such t h a t = f r where v and w are in H r E . The inner product in Hr E is (v,w)~-~.

E

vwds

224


We then have to prove Eq. (28). Introducing the bounded domain f~R (see Fig. 2) and from Eq. (5), we deduce that

/f (~72~ + k2 ~) r dx - 0 R

(29)

Using Green!s formula in domain f~R and taking the imaginary part, we deduce that

.~m

r ds fo~R or On

0 .

(30)

Since Of~R -- FE U OB(O, R) and using Eq. (6), Eq. (30) yields JF~

JOB(O,R)

-ff~ ds

.

(31)

The right-hand side of Eq. (31) is transformed using the identity

2k .~m f JOB(O,R) r - ~ ds - ~OB(O,n)

~-~

0r -

fOB(O,R)

ds

+ k 2 Ir

2

-~n + i k r

a~

.

(32)

The first term on the right-hand side of Eq. (32) is positive or equal to zero and the second term converges to zero as R --+ +oc due to the Sommerfeld condition defined by Eq. (8). From Eqs. (31) and (32), we deduce Eq. (28).

5. Free-Space Green's Function and Solution of the Inhomogeneous Helmholtz Equation Free-space Green's function. The free-space Green's function G ( x - x') is a fundamental solution of the Helmholtz equation

V~a(x-

x') + k: C ( x - x') - ~x, (x)

,

Vx e ~

,

(33)

in which V 2 - A is the Laplacian operator with respect to x, x' is any point fixed in ~3, k is defined by Eq. (4) and 5x,(x) is the Dirac generalized function at point x'. The outward fundamental solution of the Helmholtz equation in ~3 is the fundamental solution satisfying Eq. (33) and the outward Sommerfeld radiation condition at infinity OG

9

1 O (~--ff)

,

(34)

CHAPTER Xll. EXTERNALACOUSTIC FLUID FORMULATION

225

as R - [Ixll -+ +oc, in which O / O R denotes the derivative in the radial direction from the origin. It can be proved (Morse and Feshbach, 1953; Schwartz, 1965) that, for all x' fixed in R3, Eqs. (33) and (34) have a unique solution which is written as

a(x-

x') -- g ( l l x - x'll)

1 e -ikr 41r r

g ( r ) --

,

r -

(35)

, .

[ l x - x'll

(36)

In addition, it can easily be proved that

V x G ( X - x') - - V x , C ( x -

x')

.

(37)

Solution of the inhomogeneous Helmholtz equation. Consider the inhomogeneous Helmholtz equation V 2r

1 (X, w) -- -p~- Q(x, w)

w2 - r (X, w) -~- -c~

in

R3

,

(38)

in which, for all fixed real w, the inhomogeneous part x ~ Q ( x , w ) i s a bounded function with compact support ~3 K Q included in ~tE. Consequently, we have

La IQ(x,~)1 dx -

f

/

JKQ

IQ(x, ~)1 dx < +oc

(39)

Introducing the outward Sommerfeld radiation condition at infinity,

0r

_~_i k r OR

1 - O(~)

,

(40)

the unique solution ~4 of Eqs. (38) and (40) is written as

r

(x' w)--j~afe(x-xt) %/9.Q(xl'w) dX! : / K e(g-xl) %[9.Q(x', w ) d x ' .

13 Compact support KQC~E means that Q(x,w)-O for all x~KQ. 14 This unique solution satisfies Eq. (38) in the generalized function sense.

(41)

226


VIBRATION

6. Response to Prescribed Wall Displacement, Acoustic Source Density and Incident Plane Wave 6.1. Radiation by an elastic boundary We consider the boundary value problem defined by the Helmholtz equation in f~s w2 V2r (X, W) + ~ r (X, W) -- 0 , (42) with the Neumann condition on F~, considered as an elastic boundary, 0r

(y, W) -- i w

Ony

(43)

Uwall(y, w). n(y)

in which Uwan is a given wall displacement field and with the outward Sommerfeld radiation condition at infinity, [r

1 I -- 0 ( ~ )

O~rad OR + i k Cr~d

1

.

(44)

Referring to the basic problem defined in Sections 3 and 4, for all real w, the boundary value problem defined by Eqs. (42) to (44) has a unique solution. The velocity potential on FE is written as (see Eqs. (9)),

Crad ]r E(W) --

i w n FE (W/Ce){Uwall (W) " n }

,

(45)

and the pressure field on F E is such that (see Eq. (10))

Prad[r E (W)

-- iW

Z FE (W){Uwall (W)" n}

(46)

,

in which Prad[rE(W) denotes the trace of Prad(W) on FE, and Zr~(w) is the acoustic impedance boundary operator related to Br~(w/ce) by Eq. (11). At any point x fixed in f~E, the radiated pressure is given by (see Eq. (13)),

Brad (X, W)

-- i w

Zrad (X, W) {Uwall(W)" n}

,

(47)

in which Zrad(X,W ) is the radiation impedance operator defined by Eq. (14). The linear mappings Uwan(w) ~ Prad[r~(W)and Uwan(w) ~-+ Prad(X,W) define the frequency response functions to be calculated.

227

CHAPTER Xll. EXTERNAL ACOUSTIC FLUID FORMULATION 6.2. Scattering of an acoustic source density by a rigid boundary

We consider the boundary value problem defined by the Helmholtz equation in ftE w2

v~r

1

~) + -5- r

~) = - Q(x, ~)

e

,

fie

(48)

in which, for all fixed real w, acoustic source density x ~+ Q(x,w) is a bounded function with compact support t5 KQ included in ftE, with the Neumann boundary condition on F E, considered as a rigid boundary, 0r

w)

Ony

= 0

,

(49)

and with the outward Sommerfeld radiation condition at infinity,

Ir

or -5-~ + i k r

1

1 = o(~)

.

(50)

The solution of Eqs. (48) to (50) can then be written as ~3 -- ~Pinc,Q "+" ~)rig

9

(51)

The field r is the incident velocity potential induced by the acoustic source density. This field satisfies Eqs. (38) and (40) whose solution is given by (see Eq. (41)), r

w) = / g

G(x - x') 1 Q(x', w) dx' Q

(52)

Pe

The incident pressure field in ftE U F E is then given by Pinc,Q (x, w) -- - i w f K Q

(53)

G ( x - x') Q(x', w) dx'

The field Crig is the velocity potential scattered by FE considered as a rigid boundary. This field is a solution of the boundary-value problem w2 V2~/~rig(X, w) -+- C--rE~brig(X, W) = 0

in

15 Compact support KQCflE means that Q(x,w)=0 for all x~KQ.

fRs IQ(x,~)ldx=fK Q IQ(x,~)ld~C~E. E=

'

' rE

.

(121)

We then have the following symmetry property ~S~(w/c~) - S~(w/c~)

(122)

.

Let us prove Eq. (122). From Eq. (120), we deduce that <ST(&d/CE)r

6r

"

- - L ~ [02G(x-Y)

]rE 0nx0ny erE(Y) 6r

(123)

From the equality 0 2 G ( x - y)/0n(x)0n(y) = 0 2 G ( y - x)/0n(y)0n(x) (see Section 5), we deduce Eq. (121).

Variational formulation in Cr~ of the second integral equation on FE. Using Eqs. (116) and (120), the variational formulation of the second boundary integral equation (119) is written as follows. For all w fixed in ~ and for v given in C~E the trace r--r E in CrE of the unique solution r of Eqs. (104) to (106) is such that, for all ~r E in CFE,

=< rF E ~

1 ti _ tS D(w/c~))

F E 'CF E

v 6r

C!

(124)

F E 'CF E

in which tI denotes the identity operator 31 9 The corresponding operator equation is 32

Ds~ -- (1 ti

t

(125)

29 Denoting the ball with center x and radius ~ as B(xjT), the Cauchy principal value is defined by f r E e r e (y) a2 G(x-y) / an, any dsy=lim,._,o fr E \ B(x,,) e r e (y) a2 G(x-y) / anxany dsy. 30 Space Cr' E is the dual space of Cr E for the dual brackets C~E 'CrE in which v6C~E

(and not antilinear in Cr~ )" The

transpose tST(W/CE) is an operator from (C{~E)'--CrE into C~E. 31 The identity operator tI from C'rE into C'rE is defined by the bilinear form (v ,6r E )~-+

32 Eq. (125) is an equality in H - 1 / 2 ( F E ) .

CHAPTER XII.

241


From Section 3.3, for all real w, the boundary value problem defined by Eqs. (104) to (106) has a unique solution r176 Below, we will see that operator ST(w/ce) is singular for a countable set of values of w (called the spurious or irregular frequencies). Consequently, for these spurious frequencies, the trace ~b~~ (w) on FE of the solution r (w) cannot be constructed by Eq. (125) and therefore does not allow the construction of Brs (w/ce) for all real w. Transformation of the bilinear form associated with ST(w/ce). The bilinear form <S~(w/ce)Crs, 6r >%.Crs defined by Eq. (123) contains the non-

integrable singular term 02G(x - y)/0nx0ny. This term can be transformed to eliminate this second derivative. It can be proved (see Maue, 1949; Mitzner, 1966; Nedelec, 1978) that, for sufficiently differentiable complex functions r and 6r E belonging to Crs,

< ST(W/CE)r 6r + ~

/ E

'

Cr~ --k2/F / G ( x - y ) n x " EJFE

ny CrE(y) 6~rs(X ) dsy dsx

G(x-y){nyX VYCrE(Y)}. {nxX Vx6r

dsydsx

.

(126)

FE

9.3. Null spaces of integral operators ST (w/ce) and 89 - SD (w/ce)

To discuss the singularity of operator ST(w/CE) used in Eq. (125), we have to study the eigenvalues and eigenvectors of the interior Neumann boundary value problem related to bounded domain fli (see Fig. 5-b) for operator -~72, - V 2 r = Ar in f~i , (127)

or 0ny

= 0

on

n r

FE

,

(128)

.

(129)

i

The last equation implies that A - 0 is not an eigenvalue. This classical eigenvalue problem has an increasing sequence of positive eigenvalues 0 < )~1 < )k2 < . . . < )~a < . . . . Each eigenvalue >,~ has a finite multiplicity na and we denote the corresponding eigenvectors as r cn~. Finally, we define the countable set 8 g such that sN=

,

1,2,...}

,

(130)


242 in which

(131) For all j in { 1 , . . . , n~ }, the pair {k~, c J } satisfies V2r162 0r = 0 0ny

on

in

ai

F~

.

,

(132) (133)

From Eqs. (97), (98), (100) and (102), we deduce that, for all x in FE, the trace r of c J on FE verifies

~r

-

OG(x - y) dsy oqny

r E

-

~r E

r

0 2 G ( x - y) dsy - 0

(Y)0nx0ny

,

,

(134)

(135)

in which G ( x - y) is given by Eqs. (35)-(36) with k = ka. Using the definition of operators So(w/c~) and ST(W/CE) given by Eqs. (109) and (120), Eqs. (134) and (135) can be rewritten as

1

('~ I-- SD(Wa/CE)) r

= 0

ST (wa/c~) r

.

E - 0

,

(136) (137)

Consequently, for all w/c~ in S N, the null spaces of operators ST(w/cE) and 12 1 - S D (w/c~) are not reduced to zero. In addition, for all wa/c~ in S g operators ST(wa/cE) and. 891 - So(wa/c~) have the same null space which is the real vector space C~,r~ of finite dimension n~, N C~,r~ = null

1 - null {21 - S , ( ~ : / ~ , ) }

(138)

defined by N

C~,r~ - span{r

,..., r

n(~

}

(139)

243

CHAPTER XII. EXTERNAL ACOUSTIC FLUID FORMULATION

9.4. Existence of a solution of the second boundary integral equation

For all real w the trace ~bTM (w) on F E of the unique solution r ~r E

of Eqs.

(104) to (106) satisfies Eq. (125). Conversely, we discuss below the existence and the expression of solutions Cr~ (w) of

s ~ ( ~ / ~ ) % (~o) - (~1 ti - ~so(~o/~)) ~

(140)

{W/Cg} ~ $ N . F r o m Section 9.3, the null space of operator ST(W/CE) is reduced to zero and consequently ST(W/CE) is invertible. There-

First case:

fore Eq. (140) has a unique solution Cr~ (w) = rr r Vw6[R

and

w~S g

E

(w)

CrE ( w ) - r 1r 7r 6E

(141)

9

{W/CE} 6 8 N. Consider W/CE -- Wa/CE in 8 N . F r o m Section N 9.3, the null space Ca,r~ of operator ST(Wa/CE), defined by EQ. (138) or (139), is not reduced to zero and is of finite dimension na. Consequently, Eq. (140) will have at least one solution if and only if the right-hand side b(w,~) of Eq. (140),

Second case:

1t b(toa) - (-~ I -tSD(We,/CE)) V

is orthogonal to null space CON~ r E

(142)

~

< b ( w a ) , C J , r E >C~E 'crE

-0

,

j-1,...,na

,

(143)

which is the case 33 due to Eq. (136). Consequently, Eq. (140) has a family of solutions which are written as no~

{~.lc.} e s N in which u i , . . .

,

% (~.) _ r

--F E

(~.) + ~

~j r

rE

'

(144)

Un~ are arbitrary complex numbers and where r- - r E (wa) is

the trace on rE of the unique solution r176

33

j=l

of Eqs. (104) to (106) with

We haveC~E,Cr E 0 and j(u(0), u(0)) > 0 for all u(0) r 0 in Celas, and since ~(0) > 0 (see Eq. (6-2)), we deduce that kS(u(0), u(0)) +~(0) j (u(0), u(0)) > 0 for all u(0) r 0 in Celas. Consequently, Eq. (38) has a unique solution u(0) in Celas- Since m(p'(O), p'(0)) > 0 for all p'(0) r 0 in gs we deduce that Eq. (40) has a unique solution p'(O) - 0

.

(42)

Substituting Eq. (42) into Eq. (39) yields k(~(0), 5~) - ~(0;u(0), 5~)

,

c

VS~ e E,

(43)

,

c

which has a unique solution in E~ because k(~o(0), ~o(0)) > 0 for all ~o(0) ~ 0 in g~. From Eqs. (1) to (6-1), (X.8) and (X.25), we deduce that c

p(x, O) - -~(0) fr

UFz

u(x, 0)-n(x)ds(x)

,

Vxef~

.

(44)

Symmetric variational formulation for w ~= 0. In order to obtain a symmetric matrix finite element model or a symmetric reduced matrix model, valid for all w r 0, Eq. (31) is multiplied by w2 and Eqs. (23) and (36) 4 From Eqs. (XI.20), (XI.10), (X.21) and (X.25), we deduce that Sz(0;~o,5~o)--0 and consequently, Eq. (XI.21) yields Sz(0)=0. c 5 k s(u(O) ,u(0)) and j(u(0),u(0)) have the same null space which is Crig (space spanned by the set of rigid body displacements).

266


are preserved. The symmetric variational formulation is written as follows. For all fixed real w # 0 , find u in C c , ~o in E~" and p~ in $v' such that, aS(o); U,

6U) -- 0)2 aE

(0)ICE;U,

- L (w; 6u)

6U) ~- tr j (U, 6U) -- 0)2 C(0); ~p, 6U)

~

V6u

C c

(45)

-w2~d(w ;u, 5~o) + iw 3 dr(w; ~o, 5~o) + w2k(~o, 5~o) + W2Sz(W; ~o, 5~o)

~(p', 6~) -

__0)2 1

~ 2L (~ ; 6 ~ ) ,

v6~ e E ~ ,

_0)2 ~2 1 ~ (@', ~) + p~,m(p', ~p') - 0 , v~p' e c~, c

.

(46)

(47)

Co

4.3. Linear operator equation For w # 0, the linear operator equation corresponding to the symmetric variational formulation defined by Eqs. (45) to (47) is

{u,~,p'} e

C c

C

C

x~'~xq,,

,

(48)

in which A(w) is the linear operator 6 defined by

As ( ~ ) - ~ A s ( ~ / ~ ) + ~ ( ~ ) J A(w) =

_w2

0

-~C(~)

t~(w) i0)3i~.(W)+0)2KWw2Sz(w) o

- ~w 2

09 2

-~CI

. (49)

tC

The dynamic stiffness linear operator As (w) is such that ~

< As(~,) u, 6u > = a s(~, ;u, 6u)

,

(50)

6 Linear o p e r a t o r A(w) defined by Eq. (49) is continuous from CCxECxHC into CC'xEC'xHC' c c c in which HC'-'HC--L2(f~). The restriction of this linear operator to C x E ~ x E v, is continuous from

c cl X c lX c i. CCxE:xgv, into C $~ $ v,

7 Sesquilinear form a s (w; u ,6u) is continuous on C~x C~, As (w) is a continuous operator from C ~ into its antidual space Cc' and the angle brackets in Eqs. (50) to (51-4) denote the antiduality p r o d u c t between C c' and C c . For all real w e 0 , linear operator As (w) is invertible. For w--0, c the restriction of continuous linear operator As (w) to Cs is invertible (see Eq. (V.56) for the c

definition of Cs).

CHAPTER XIII.

267

STRUCTURAL-ACOUSTIC SYSTEM IN THE LF RANGE

and can then be written as A s (w) - - w 2 M s + iw D s (w) + K s

,

( 5 1 - 1)

where M s, DS(w) and K s are the mass, damping and stiffness operators of the master structure defined by ,

< M s u , ~u > - m s (u , ~u)

( 5 1 - 2)

(51- 3) (5~-4)

< D s (w) u , Ju > = d S (w;u, Ju)

=k Linear operator

AE

(w/ce)

.

s(u,Ju)

is such that s

< A~ (~/~) u. ~u > = . ~ (~/c~; u. ~u)

(52)

.

Linear operator J is such that 9

<Ju,~u>=

j(u,~u)

(53)

.

Coupling operator C(w) is such that 10 < c ( ~ ) ~..~u > = c(~; ~. ~u)

( 5 4 - 1)

.

and can be written as

c(~) = Co + c~(~)

,

( 5 4 - 2)

in which operators Co and C~(w) are defined by

=

co(~o,6u)

( 5 4 - 3)

, m

< c ~ ( ~ ) ~o, ~ . >= -p,, ~(~) ~ ( ~ ; ~) ~ ( ~ . )

( 5 4 - 4)

.

8 For all w fixed in R, sesquilinear form a E(w/cO ; u,(~u) is c o n t i n u o u s on CCxC c (see f o o t n o t e 3) a n d t h e n AE(w/cE) is a c o n t i n u o u s linear o p e r a t o r f r o m C c into its a n t i d u a l space C c~. T h e angle b r a c k e t s in Eq. (52) d e n o t e t h e a n t i d u a l i t y p r o d u c t b e t w e e n C c~ a n d C c. 9 Sesquilinear f o r m j ( u , 6 u ) is c o n t i n u o u s on C C•

c a n d t h e n J is a c o n t i n u o u s linear o p e r a t o r

f r o m C c into C c~. T h e angle b r a c k e t s in Eq. (53) d e n o t e t h e a n t i d u a l i t y p r o d u c t b e t w e e n C c~ and Cc . 10 For all w fixed in R, sesquilinear form c(w ;~,Su) is c o n t i n u o u s on s a c o n t i n u o u s linear o p e r a t o r from s c into C c~. T h e angle b r a c k e t s in Eqs. d e n o t e t h e a n t i d u a l i t y p r o d u c t b e t w e e n C c~ a n d C c.

a n d t h e n C(w) is (54-1) to (54-4)

268

STRUCTURAL

ACOUSTICS

AND

VIBRATION

Coupling operator tC(w) is such that ll
-

;u, 6 o)

,

(55)

and we have < tC(w) u, ~o > = < u, C(w) ~o >. Linear operators K, Dr (w) and Sz(w ) are defined by Eqs. (XI.17), (XI.19) and (XI.21)respectively. Linear operator C1 is such that 12 < Clp', 5~o> - c~(p', ~o) and linear operator

,

(56)

.

(57)

< Mp' , @' >= m(p' , @') ,

(58)

is such that 13

tc 1

Ns

u Ns (x, w) -- ~ in which

qS

--

(q~

q~ (w)u~ (x)

,'"

(80)

s

'qNs)

is a complex vector of generalized coordinates. The projection c of ~o E g~o is written as

(81) ~'qo,N c

(/9g o n

N

~o~(x, ~o) = ~ q~(~) ~o~(x) ,

(a2)

~'--1

in which q = (ql,... ,qN)

(83)

is a complex vector of generalized coordinates. Finally, the projection p~ c c on g~o,N of p~ E gp, is written as N

(84)

~o(~o)~o~(x) ,

p~(x, ~o) - ~ c~'--I

in which

(85)

r = ( ~ , . . . ,~N) is a complex vector of generalized coordinates.

6.4. Symmetric reduced matrix model Using Eqs. (80), (82) and (84), the projection of the variational formulation defined by Eqs. (45) to (47) yields

-

,

in which complex matrix [~4(w)] is symmetric and is deanna by

[,a S ( ~ ) ] - ~ [ A ~ ( ~ / ~ ) ] + ~ ( ~ ) [ J ]

- ~ [c(~)] ~ [0]

-~[c(~)]

~[~r

[o] -~[cl] 0.) 2

w2

~ [c~]T

1[.~]

(87)

CHAPTER XIII. STRUCTURAL-ACOUSTICSYSTEM IN THE LF RANGE

275

in which [As(w)] is an (Ns • Ns) symmetric complex matrix written as

[A s (,.,)] - - . = [ M s] + i,., [~s (~,)] + [lcs]

.

(88)

where [A4 s] is an (Ns x Ns) symmetric positive-definite real matrix and [7:)S (w)] and [/Cs] are (Ns x Ns) symmetric positive-semidefinite real matrices. Generalized mass and stiffness matrices [A4 s] and [K:s] are diagonal and are such that [A4s]a/~ _ #s 6at~ , (891 [/(:Slafl = #as As 6af~ .

(90)

Generalized damping matrix [Ds (w)] is dense and is such that [Ds (w)]af~ = d S (w ;ut~ , u~)

.

(911

The (Ns x Ns) symmetric complex matrix [AF'(w/cE)] is such that

[ut~(~/~)]., - ~ ( ~ / ~ ; = -P~ f r

~ , u~) (ate" ns) Br~(w/c~){u~. n s} ds

(92)

E

Matrix [3"] is an (Ns x Ns) symmetric positive-semidefinite real matrix such that [ 3" ]aft = ~r2 (u~) lr2 (ul~) , (93) in which ~r2(u) is defined by Eq. (4). Matrices [A4], [K:] and [D~(w)] are (N x N) diagonal positive-definite real matrices and [Sz(W)] is a dense (N x N) symmetric complex matrix such that [M ]a/~ = #a 6a/3 , [lC].z = u. ~.5.~

(94) (95)

.

(96) [s~(~)].~ = ~z(~ ; ~ , ~.)

(97)

,

in which sesquilinear form s z is defined by Eq. (XI.20). It should be noted that dense matrix [$z(W)] is due only to the presence of acoustic impedance Z(x,w) on boundary Fz. Coupling rectangular matrix [C(w)] is an (Ns • N) complex matrix which can be written as [ c ( ~ ) ] ~ e = [Co]~ + [ c . ( ~ ) ] ~

,

(98)

276


where [Co] is the (Ns •

VIBRATION

real matrix such that =

u.)

Po ]

~o~ n. u,~ ds

(99 - 1)

Jr, uFz

and where matrix [C~ (w)] is an (Ns x N) complex matrix such that (,,

=

.

( 9 9 - 2)

Matrix [C~] is an (N x N) diagonal positive-definite real matrix such that

Po

(100)

Po

Generalized forces .Ts (w) and .TQ (w) a r e C NS- and CN-vectors respectively, such that .Ts (w)~ = fs (w ; u , ) , (101) .TQ (w)~ -- fq (w; ~o~) .

(102)

6.5. Construction of the reduced model using the finite element approximation Step 1. This step concerns the finite element calculation of the structural modes of master structure ~ s in vacuo (see Section III.9). The finite element discretization Ua of structural mode ua is a solution of the (ns• ns) generalized symmetric real matrix eigenvalue problem [K s ] U - As [M s ] U

,

(103)

which gives {#as, Aas, Ua}a=l,...,NS and consequently which allows the construction of diagonal matrices [M s] and [Es]. We introduce the ( n s x N s ) real matrix of the Ns structural modes Ua, [u]

=

.

(104)

Solution AS - 0 corresponds to the six rigid body modes U 1,..., U6. Step 2. This step corresponds to the finite element calculation of the acoustic modes of internal acoustic cavity ~ with a rigid wall (see Section XI.8). The finite element approximation (I)a of acoustic mode ~oa is a solution of the (n x n) generalized symmetric real matrix eigenvalue problem

[K] (I) = AIM] 9

,

(105)

CHAPTER XIII. STRUCTURAL-ACOUSTICSYSTEM IN THE LF RANGE

277

which gives {#a, :ha, (I)a}cz=l,...,N and consequently which allows the construction of diagonal matrices [,A/i], [K:], [Dr(w)] and [(71]. We introduce the (n x N) real matrix of the N acoustic modes (I)a,

[0] = [ r 1 6 2

.

(106)

Step 3. The third step is the finite element calculation of the nondiagonal matrices of the reduced model. For matrix [7:)s(w)], the finite element discretization leads to [79s(w)] ~_ [ u ]T [D s (w)] [U]

.

(107)

For matrix[3"], we obtain

[ 3"] ___ (n~r[ u ])T (n2r[ u ])

,

(108)

which uses only the values of the structural modes at the nodes belonging to boundary F U Fz. For matrix [$z(W)], we have

[Sz(~,)] -- [r ]T [Sz(~,)] [,z,]

(109)

,

which uses only the values of the acoustic modes at the nodes belonging to boundary F z, and for matrix [C(w)] we have [C(w)] _~ [U] T [C0(w)] [(I)] - poa(w)(IIT[U]) T (IIl(w)T[o])

,

(110)

which uses only the values of the structural modes and acoustic modes at the nodes belonging to boundary F U F z. For matrix [AE (wick)], the finite element discretization yields [A~ (~/~)] -~ - p ~ [ e r ~ ( ~ / ~ ) ]

(111)

,

in which the (Ns x Ns) symmetric complex matrix [Br~ (wick)] is given by

[~

(~/~)] - [ v ~ ] r [ B ~ (~/~)] [ V ~ ]

,

(112)

with [Urn] the (hE x Ns) real matrix such that [Ur~] = [@] [U]

.

(113)

The (nE X hE) symmetric dense complex matrix [Brs (w/ce)] is defined by Eq. (XII.164) and the (hE X ns) real matrix [O] is defined by Eq. (69).

278


The right-hand side of Eq. XII.12.3.

(112) is calculated as presented in Section

7. FRF Calculation in the LF Range In this section, for w E ~LF (w ~ 0), we construct the projection of the frequency response function using the symmetric reduced model presented in Section 6. 7.1. Definition of the FRF in the LF range

For all w E ~LF' since M is an invertible operator, we eliminate field p~ in Eq. (48) using the third equation. For all w E [~LF' the resulting operator is invertible and gives the operator-valued frequency response function T(w) such that ~7

[;]- T(.,)[~r r~(,,)J1

.

(114)

Once ~o(w) is known, pressure field p(w) is calculated using Eq. (62). 7.2. Response to an excitation vector and projection of the FRF in the LF range

The projection of FRF T(w) is then obtained by eliminating r in the symmetric reduced model defined by Eq. (86). We then have

L,.,.,~~ (,.,.,)j

(115)

N

in which complex matrix [A(w)] is symmetric and is defined by

I [.as (~,)1- ~ [A~(~,/r +,~(~) [y ] -~,~[c(~)] I" --W2 [~(W)]T iW3[~).r(W)lq-w2[~]q-W2[gz(W)]--W4[d~ ]

(116)

N

For all w E BLF (w

7'= 0), matrix [j[(w)] is invertible and consequently,

[qS]-[T(w)][~2~%(.~)] r ~s(,.,,) 1 ,

(117)

17 For all w6~3LF (w#0), the operator resulting from the elimination of p' in Eq. (48) is a cf

continuous and invertible operator from Ccxg: into Cc~ 1

(128)

[T,~'~(~)]~ [T,,~'~(~1] L~- r~ (~)j

in which U NS - ( V N s ) ' ' " ) U UNs) is the complex vector of the DOFs which S are the values of u Ns at the nodes of the finite element mesh of domain ~ s , O N - ( 0 ~ , . . . , 0 ~ ) is the complex vector of the DOFs which are the values of ~og at the nodes of the finite element mesh of domain ~ and, vectors Fs(w ) E C as and Fq(w) E Cn are defined in Section 5. Matrix [T~S'N(w)] is an ( n s x n s ) symmetric complex matrix, [T~S'N(w)] is an (n x n) symmetric complex matrix and [T~S'N(w)] is an ( n s x n) complex matrix such that, Ns

Ns

[T,~ ~(~)1- ~ ~ [~1(~)1~, u~ u~,

(129)

a=l cd=l

Ns

[T,~'~(~)] - ~

N

~

[~(~)1~ u~ o~

(130)

cz=l ,B=I N

N

(131) f~=l f~'=l Calculation of the pressure field p(x, w) inside ~2. The complex vector pNS.N of the values of pressure field p(x, w) at the nodes of the finite element mesh of domain ~ is calculated using Eq. (73),

c~ [M]pNS,N -- w2{ c o2[ M] ~N - tc(W)poII1(w) T ,I~N L } p~ - ~(~)n~

u N~ L +

,~(~1~ (~1 L

,

(132)

in which U Ns and (I)N a r e given by Eqs. (128) to (131). As explained at the end of Section 5, an alternative to Eq. (132) can be used in order to avoid having to solve a linear matrix equation of order n.

CHAPTER XIII.

281


Calculation of the pressure field on boundary FE and of the resultant pressure at any point of external domain fiE. Finite element approximation PrE (w) of pressure field PElrE(W) on FE is calculated by Eq. (XII.173) in which V(w) - l U r E ] qS(w) where [U rE] is defined by Eq. (113). At any point of external domain f~E, the finite element discretization of resultant pressure pE(x,w) is given by EQ. (XII.174) in which V(w) is defined above. 7.4. Frequency-by-frequency construction of the FRF in the LF range The numerical procedure for constructing an approximation of w ~ T(w) on low-frequency band [~LF is based on the use of projection TNs'N(w) of T(w) introduced in Sections 7.2 and 7.3. In this case, no quasi-static correction terms are introduced. Convergence must be controlled by increasing the number N s of structural modes of the master structure in vacuo and the number N of the acoustic modes of the internal acoustic cavity with a rigid wall. This procedure requires the construction of matrix [T(w)] (see Eqs. (117) and (118)). In the low-frequency range, N s and N are small. In the context of a finite element discretization with n s DOFs for the master structure and n DOFs for the internal fluid, we therefore have N s 0 for all u(0) # 0 in Celas , Eq. (169) has a unique solution u(O) in Cel~s. c

10.2. Linear operator equation

The linear operator equation corresponding to the variational formulation defined by Eq. (165) is A(w) U - f s ( W )

,

uGC ~ ,

(170)

(W/CE) ,

(171)

in which A(w) is the operator defined by A(w) = As (w)

-

w 2 A E

in which the dynamic stiffness linear operator AS (w) of the master structure is defined by Eq. (50) and is written as As ( w ) - - w 2 M S + i w D s ( w ) + K s

,

(172)

where M S, D S (w) and K s are the mass, damping and stiffness operators of the master structure defined by Eqs. (51-2), (51-3) and (51-4) respectively. Linear operator As (W/CE)is defined by Eq. (52) and element fs (w) is defined by Eq. (59). Pressure in the external acoustic fluid. Pressure field PE[rE(w) on FE due to the external acoustic fluid is given by Eq. (7). At any point x of unbounded external acoustic fluid f~E, resultant pressure field pE(X, w) is given by Eq. (8). 10.3. Finite element discretization

We consider a finite element mesh of master structure f~s. The finite element mesh of surface F E is the trace of the finite element mesh of f~s (see Fig. 5).

290


~2E

r

Fig. 5. Finite element mesh of the master structure We use the finite element method to construct the discretization of the various sesquilinear forms of the problem. For the master structure, we introduce subspace Cns C cC of finite dimension ns. Let U - (U1 ~ ' ' ' ~ Uns) be the complex vector of the DOFs which are the values of u at the nodes of the finite element mesh of domain f~s. Since the finite element method uses a real basis for constructing the finite element matrices, the discretization of the variational formulation defined by Eq. (165) yields the symmetric complex matrix equation [A(w)] U = F s (w)

.

(173)

The complex matrix [A(w)] is symmetric and is defined by [A(w)] - [As (w)] - w 2 [As (wick)]

,

(174)

where [As (w)] is the dynamic stiffness matrix of the master structure which is an (as x ns) symmetric complex matrix (invertible for all real w r 0 and not invertible for w = 0 due to the presence of the rigid body modes) written as [AS(w)]- -w2[M s] + iw [DS(w)] + [K s] . (175) Mass, damping and stiffness matrices [MS], [D s (w)] and [K s] are (us x us) symmetric real matrices. Matrix [M S ] is positive definite and matrices [D s (w)] and [K s] are positive semidefinite. The finite element discretization of sesquilinear form aE(w/CE;U,(~U) yields the (ns x ns) symmetric complex matrix [As (wick)] related to the nodal values of u, which is written as [AE (wick)] - -p~ [ O ]T [Sr~ (w/c~)][ O ] , (176)

291

CHAPTER XIII. STRUCTURAL-ACOUSTIC SYSTEM IN THE LF RANGE

where [Br~ (w/c~)] is the ( n s x n ~ ) symmetric dense complex matrix defined by Eq. (XII. 164) and related to the nodal values of the normal displacement field on FE. Rectangular matrix [O] is an (ns x n s ) sparse real matrix defined by Eq. (69). Equations (167) and (176) show that matrix [AE (w/ce)] has zero rows and zero columns corresponding to the structural nodes which are not located on interface F~. Finally, the finite element discretization of antilinear form fs (w ; he) yields the complex vector F s (w) in C ns . 10.4. Symmetric reduced matrix model in the LF range

We consider the low-frequenc,y case for which w belongs to frequency band [~LF defined by

BLF =1 0, ~ , ~ , ~

] .

(177)

The methodology for constructing the symmetric reduced matrix model consists in using the Ritz-Galerkin projection of the variational formulation on a finite dimension subspace spanned by a set of structural modes of the master structure in vacuo. 1- Finite dimension subspace for the projection. We introduce subspace CNs of C~, of dimension N s >__ 1, spanned by the finite family {Ul,...,UNs} of structural modes defined in Section 6.1. The projection u Ns on CNs of u E C~ is written as c

c

Ns

(178) o~=1

in which qS = (qls ' ' ' ' ' q Nss ) " 2- Symmetric reduced matrix model. Using Eq. (178), the projection of the variational formulation defined by Eq. (165) yields [A(w)] qS = .Ts (w)

,

(179)

in which complex matrix [A(w)] is symmetric and is defined by [A(~o)] - [.4 s(~o)] - ~o~[.4 ~(~o/c,)]

(180)

,

in which [A s (co)] is an ( N s x N s ) symmetric complex matrix written as

[A s (~)] - - ~ [ M s] + i~ [~s (~)] + [~s]

,

(181)

292

STRUCTURAL

ACOUSTICS

AND

VIBRATION

where [A4 s] is an (Ns x Ns) symmetric positive-definite real matrix and [D s (w)] and [/Cs] are (Ns • Ns) symmetric positive-semidefinite real matrices. Generalized mass and stiffness matrices [A4 s] and [/Cs] are diagonal and are defined by Eqs. (89) and (90) respectively. Generalized damping matrix [DS(w)] is dense and is defined by Eq. (91). The (Ns x Ns) symmetric complex matrix [ritE (wick)] is defined by Eq. (92). Generalized force ~s (w) is a cgs-vector defined by Eq. (101). 3- Construction of the reduced model using the finite element approximation. The

first step is the finite element calculation of the structural modes of master structure f~s in vacuo as explained in Step 1 of Section 6.5. The second step corresponds to calculation of the nondiagonal matrices of the reduced model. Finite element discretization leads to the construction of matrix [Ds(w)] given by Eq. (107) and to the construction of matrix [A~(w/c~)] given by Eqs. (111) to (113). 10.5. F R F Calculation in the LF Range

In this section, we construct the projection of the frequency response function using the symmetric reduced model presented in Section 10.4 and we introduce the quasi-static correction terms. 1- Definition of the FRF in the LF range. For all w 6 [~LF' the operator-valued frequency response function T(w) is such that la u(w) = T(w) fs (w)

.

(182)

2- Response to an excitation vector and projection of the FRF in the LF range. For

all w 6 BLF (w r 0), matrix [,4(w)] is invertible and consequently, qS _ [Y(w)]

(w)

,

(183)

in which [T(w)] is an (Ns x Ns) symmetric complex matrix such that [T(w)] = [A(w)]

,

[T(w)] T = IT(w)]

.

( 8a)

18 From C h a p t e r III, we have C c = C rci g $ C ecl a s . For all WGBLF (we0), from Eq. (XII.25), we c

c

deduce t h a t , for all u r Celas, ~ m { < A ( w ) u , u > } r and then A(w) is invertible on Celas. c (2 For all u r ~e{}r and then A(w) is invertible on Crig. Consequently, for all w 6 [3LF, A(w) is continuous and invertible from C c into Cc' and T ( w ) = A ( w ) -1 is a continuous operator from Cc' into C c.

CHAPTER Xlll.

293


Substituting Eq. (183) into Eq. (178) yields the response of the structuralacoustic master system, Ns Ns

E Er()loo,

uo(x)

(185)

cz=l cz'--i

Consequently, the projection TNS(w) of the operator-valued frequency response function T(w) is such that u gs (w) - T Ns(w) fs (w)

,

(186)

where projection TNS(w) is such that Ns

I Ns(w) fs (w) - E

Ns

E

(187)

fs (w ;ua, ) ua

cz=l cz'=l

3- Finite element discretization of the projection of the FRF.

The finite element

approximation of Eq. (186) is then written as U Ns - IT ~s(w)] F s (w)

,

(188)

in which U N s -- ( U N s ~ ' ' " ~ U,US N s ) is the complex vector of the DOFs which are the values of u gs at the nodes of the finite element mesh of domain f~s and [TNS(w)] is an (ns x ns) symmetric complex matrix which can be written as Ns

Ns

IT NS(w)] - E E [T(w)]~, U~ U T, cz=l a'=l

.

(189)

4- Calculation of the pressure field on boundary FE and of the resultant pressure at any point of external domain f~E. Finite element discretization PrE(W) of

pressure field PElrE(W) on FE is calculated (see Section XII.12.5) by Pr E(w) - - i w p~ [Br ~ (w/c~)][Ur E] qS (w) + Pgiven(W)

,

( 9o)

in which [Ur~] is defined by Eq. (113). At any point in external domain f~E, the finite element discretization of resultant pressure ps(x, w) is given by (see Section XII.12.5),

pE(X, ~3) -- [grad (X, 03)] [ UFE] QS (03) + Pgiven (X, ~d) , in which Pgiven (X, W) is given by Eq. (XII.81).

(191)

294


VIBRATION

5-Introduction of quasi-static correction terms. We use the method presented in Section VI.5. Let Ns >__1 be an integer such that

0=

f-

"'"

< --'''--

Ns

- a s (w ;u, 5u) , (43) and can be written as A S (w) = - w 2 M s + iw D s (w) + K s (w)

,

( 4 4 - 1)

where M s, D s (w) and K s (w) are the mass, damping and stiffness operators of the master structure defined by < M S u , Su > - m s (u , Su)

< D s (w) u, 5u > - d s(w ;u, 6u)

,

( 4 4 - 2)

,

( 4 4 - 3)

< K s (w) u, 5u > - k s (w ;u, 6u)

( 4 4 - 4)

Linear operator AE (wick) is such that 9

= ae

; u,

.

7 L i n e a r o p e r a t o r A(w) defined by Eq. (42) is c o n t i n u o u s from cCx$ c into CC'x~ c'. r e s t r i c t i o n to C c •162 c of this o p e r a t o r is c o n t i n u o u s from cc• c into C c, x ~ rc ,.

(45) The

8 Sesquilinear f o r m aS(w;u,Su) is c o n t i n u o u s on CC• c, AS(w) is a c o n t i n u o u s o p e r a t o r f r o m C c into C c' a n d t h e angle b r a c k e t s in Eq. (43) d e n o t e t h e a n t i d u a l i t y p r o d u c t b e t w e e n C cr c a n d C c . For all real w e 0 , linear o p e r a t o r As (w) is invertible. For w----O, t h e r e s t r i c t i o n to Cs c of c o n t i n u o u s linear o p e r a t o r As (0) is invertible (see Eq. (V.56) for t h e definition of Cs). 9 For all w fixed in ~, sesquilinear form aE(w/co ;u,Su) is c o n t i n u o u s on CC• c (see f o o t n o t e 3) a n d t h e n AE(W/CE) is a c o n t i n u o u s linear o p e r a t o r from C c i n t o C c ' . T h e angle b r a c k e t s in Eq. (45) d e n o t e t h e a n t i d u a l i t y p r o d u c t b e t w e e n C c' a n d C c.

318


Linear operator J is such that 10

<Ju, ~u>= j(u,~u)

(40)

.

Linear operator AF (ca) related to the internal fluid is such that 11

< AF@) r

>- a

;r

,

(47)

and can be written as AF ( w ) -

-w2M

+ iW])r(W) -{- K +

Sz(w )

,

(48)

in which linear operator M is such that 12

< M r ~r > = re(C, ~r

,

(49)

,

(50)

linear operator K is such that 13

= ~(~ ;r ~u)

(53)

,

and can be written as c(,,,) = Co + ,~(~)Cz(~)

(54)

,

in which operators Co and Cz(w) are defined by ,

(55)

< Cz(~) r ~u > - ~z(~ ;r ~u)

(56)

< Co r

5u > = co(r 5u)

Coupling operator tC(w) is such that 16 < ~ ( ~ ) u, ~r > = ~(~ ;u, ~r

(57)

,

and we have < tC(w) u, 5r > = < u, C(w) 5r >. Finally, elements fs (w) and fQ (w) are such that 17 < L (~), ~u > = f s ( ~ ;~u)

,

(58)

< fQ (w), 5r > = fQ (w ;5r

.

(59)

Once {u, r is known, internal pressure field p(x, w) in if/can be deduced from the variational formulation of Eq. (1) which is such that, for all 5p in H ~,

/o-

/o-

p 5p dx - - i w po

r 5io dx + lr(w ; u ; r

/o-

5p dx

(60)

15 Sesquilinear form c(w ;r operator from E c into C c'. antiduality product between 16 Sesquilinear form Nc(w ;5r

is continuous on s and then C(w) is a continuous linear The angle brackets in Eqs. (53), (55) and (56) denote the cC' and C c . is continuous on ECx cC and then tC(w) is a continuous linear operator from C c into E c'. The angle brackets in Eq. (57) denote the antiduality product between s and Ec. 17 Antilinear forms fs(w;Su) and fQ (w;5r are continuous on cc and E c respectively,and define elements fs (w) in C c' and fQ (w) in $c, respectively. The angle brackets in Eq. (58) denote the antiduality product between C c' and C c . The angle brackets in Eq. antiduality product between $c, and s

(59) denote the

320


We then deduce that internal pressure field p(x, w) is the unique solution of the linear operator equation

c~ Mp - -iw potClr + r(w ;u ;r l po

(61)

in which I is the antilinear form defined on H ~ such that -

~

m

~pdx

(62)

and linear operator tC1 is such that is -/~

r

(63)

Pressure field PEIrE(W) on FE due to the external acoustic fluid is given by Eq. (9) and at any point x of unbounded external acoustic fluid ~E, resultant pressure field pS(x,w) is given by Eq. (10).

5. Finite Element Discretization We consider a finite element mesh of master structure f~s and a finite element mesh of internal acoustic fluid f~. We assume that the two finite element meshes are compatible on interface F U F z. The finite element mesh of surface FE is the trace of the mesh of f~s (see Fig. 2).

rE

Fig. 2. E x a m p l e of s t r u c t u r e and internal fluid finite element meshes 18 Sesquilinear form (r

L

r /5pdx is continuous on E Cx H

c

and then tC1 is a continuous

linear operator from Ec into HC'--H c. The angle brackets in Eq. product in H c.

(63) denote the inner

321

CHAPTER XIV.. STRUCTURAL-ACOUSTIC SYSTEM IN THE MF RANGE

We used the finite element method to construct the discretization of the various sesquilinear forms of the problem. For the master structure, we introduce subspace Cns C cC of finite dimension ns. Let U - (U1 ~ ' ' " Uns) be the complex vector of the DOFs which are the values of u at the nodes of the finite element mesh of domain ~ts. For the internal acoustic fluid, we introduce subspace E~ C E ~ of finite dimension n. Let 9 - ( ~ 1 , . . . , ~n) be the complex vectors of the DOFs which are the values of r at the nodes of the finite element mesh of domain f~. Since the finite element method uses a real basis for constructing the finite element matrices, the finite element discretization of the variational formulation defined by Eqs. (22) and (32) yields the symmetric complex matrix equation (corresponding to the operator equation defined by Eq. (41),

(64) with the constraint LTg ' -- 0

.

(65)

Eq. (65) corresponds to the finite element discretization of linear constraint f~ r dx - 0 in admissible function space Cr defined by Eq. (21), in which L is a vector in ~n. Complex matrix [A(w)] is symmetric and is defined by [A(w)] = I [As (w)] - w2[A[C(w)] T(w/c~+ )] ~(w)[J]

1

(66)

where [As (w)] is the dynamic stiffness matrix of the master structure which is an (nsx ns) symmetric complex matrix (invertible for all real w ~- 0 and not invertible for w -- 0 due to the rigid body modes) written as

[As (w)] - -w2[M s] + iw [Ds (w)] + [K S (w)]

.

(67)

Mass, damping and stiffness matrices [MS], [Ds (w)] and [K S (w)] are (ns x ns) symmetric real matrices. Matrix [M S] is positive definite and matrices [D s (w)] and [K S (w)] are positive semidefinite (due to the presence of the rigid body modes). The finite element discretization of sesquilinear form a E (w/ce ; u, 5u) yields the (ns x ns) symmetric complex matrix [AE(w/c~)] related to the nodal values of u, written as

[AE (micE)] = -p~ [ e ]T [Br~

[e]

,

(68)

322


VIBRATION

where [Br~(w/cE)] is the (nE x nE) symmetric dense complex matrix defined by Eq. (XII.164) and related to the nodal values of the normal displacement field on FE. Rectangular matrix [O] is an (nE X ns) sparse real matrix corresponding to the finite element discretization 5wT [ O ] U of the sesquilinar form (u, 5w) ~ O(u, 5w) = Jfr 5w u. n s ds

(69)

E

defined on CCx Cry. Eqs. (68) and (69) show that matrix [AE(w/cE)] has zero rows and zero columns corresponding to the structural nodes which are not located on interface FE. Matrix [J] is an (ns x ns) symmetric real matrix which can be written as I J]

-

II2II

T

,

(70)

in which I I 2 is a vector in ~ns such that IIT corresponds to the finite element discretization of the linear form 7r2(u) defined by Eq. (4). Matrix [ J ] has zero rows and zero columns corresponding to the structural nodes which are not located on interface F LJ F z. Matrix [AF(w)] corresponding to the finite element discretization of sesquilinear form a F ( w ; r 5r is an (n x n) symmetric complex matrix (invertible for all real w # 0) written as

[AF(w)] = --w2 [M] + iw [D~(w)] + [K] + [Sz(w)] ,

,(71)

in which matrices [M], [K], IDa(w)] and [Sz(w)] correspond to the finite element discretization of the sesquilinear forms defined by Eqs. (36), (34), (35) and (37). The (n x n) real matrix [ M ] i s symmetric positive definite, the (nx n) real matrices [ g ] and [D~(w)] are symmetric positive semidefinite, with rank equal to n - 1 and the null space of which is constituted by the vector c 1 where c is any real constant and 1 = ( 1 , . . . , 1) 6 ~n. The (nx n) complex matrix [Sz(w)] is symmetric and such that w-l.~m[Sz(w)] is positive semidefinite (see footnote 6). Coupling rectangular matrix [C(w)] corresponding to the finite element discretization of coupling sesquilinear form c(w ;r 5u) is an (us x n) complex matrix which is written as =

[Co] +

,

(72)

in which [Co] is the (nsx n) real matrix corresponding to the finite element discretization ofsesquilinear form c0(r 5u) and [Cz(w)] is the (nsxn) complex matrix corresponding to the finite element discretization of sesquilinear form cz(w;r 5u) and such that

[cz(w)]--ponzn (w) T

,

(73)

CHAPTER XIV

STRUCTURAL-ACOUSTIC SYSTEM IN THE MF RANGE

323

where II~ (w) is a vector in C n such that II~ (03) T corresponds to the finite element discretization of the linear form ~rl (w ; ~). Matrix [C(w)] has zero rows and zero columns corresponding to the structural and fluid nodes which are not located on interface F U Fz. Finally, the finite element discretization of antilinear forms fs(w;hu) and fq (w ; 5r yields the complex vectors F s (w) and FQ (w)in Cus and Ca respectively.

Calculation of the pressure inside f~, Let P = ( p l , . . . , Pn) be the complex vector of the DOFs which are the values of p at the nodes of the finite element mesh of domain f~. Then the finite element discretization of Eq. (60) is written as c~ [ M ] P -

Po

-iwc~[M]~ + a(w){iwpoIIl(w)T'~--IITU+~rq(W)} L, (74)

in which L E ~n is the finite element discretization of antilinear form l defined by EQ. (62) (which is also used in Eq. (65)) and where ~rq(w) is defined by Eq. (5). It should be noted that the construction of P requires solving a linear matrix equation with nondiagonal matrix (c~/Po) [M ]. An alternative discretization of Eq. (60) can be used to recover p locally element by element. Since p(w) belongs to H ~, the pressure can be taken, for instance, as a constant over each element. In this case, no linear matrix equation of order n has to be solved.

Calculation of the pressure on boundary F~ and of the resultant pressure at any point of external domain f~E. The finite element discretization Pr E(w) of pressure field PElr~(w) on FE is given by Eq. (XII.173) in which V is the vector in CnE such that V = [O]U. At any point of external domain f ~ , the finite element approximation of resultant pressure p~(x, w) is given by Eq. (XII.174) in which V is defined above. 6. F R F C a l c u l a t i o n in the M F R a n g e Let [~MF be the MF broad band defined by ~MF -- [ 03MF,init' WMF,final ]

"

(75)

The MF narrow band B~ is defined by

(76) where Ft~ > 0 is the center frequency of band B~ and Aw is its bandwidth. We use the finite element discretization introduced in Section 5 to construct

324


VIBRATION

the FRF in the MF range by the method described in Chapter VII. For all w in 03MF, we have to construct the FRF [ T ( w ) ] - [A(w)]-1 where [A(w)] is defined by Eq. (66). We then have to solve for the class of MF narrow band excitations 0~(w)B introduced in Section VII.4,

O.(w)B

[A(w)] Y(w) -

,

w E B~

with

Y(w)-

[U(w)l L~(~)j

(77) ,

with the constraint LT~(w)--O

,

wEB~

.

(78)

Symmetric complex matrix [A(w)] can be written a s [~(~)] - - ~ : [ ~ ( ~ ) ]

+ i~ [D(~)] + [~(~)]

,

(79)

where [M(w)], [D(w)] and []((w)] are the symmetric complex matrices such that [N(~)] -

I [Ms]+[AE(w/c~)] [0]

[D(w)]

--

,

(80)

-[M]

I[Ds(w)]

[c(~)]

[C(~)] T

- I D a ( w ) ] - [S:(w)]

[0((w)] - [ [KS(w)] + a(w)[J] [

[0] 1

[0]

1

col 1

(81)

(82)

- [ K] - [S: (w)]

in which we have written (83)

where real matrices IS: (w)] and [S: (w)] are such that [ s : (~)] - ~ [s~(~)]

,

~ [s:(~)] - .~m [s~(~)]

(84)

For construction of the frequency response function, we take O~(w) - 1 for w E 03,, and B represents the vectors of the canonical basis of [~ns+n. Concerning the methods for calculating the response to various types of excitations, we refer the reader to Chapter IX.

325

CHAPTER XIV. STRUCTURAL-ACOUSTIC SYSTEM IN THE MF RANGE

6.1. Direct use of the MF method

The MF method presented in Chapter VII uses an approximation of Eq. (77) due to the frequency-dependent matrices [M(w)], [D(w)] and [0((w)] consisting in replacing Eq. (77) by

(-w2[M~,]+iw[Dv]+[0 ti .

,

(87) (88)

with the constraint LT~0(t)=0

,

Vt>t i

,

(89)

in which Y0(t) is such that

Yo(t)

=

[u~ L~o(t) J

(9o)

,

N

N

and where symmetric complex matrices [Dr] and [0(v] are such that N

[D,,,] = [D,.,] + 2 i ~ v

[M,.,]

,

(91) (92)

326


Once Y0 is obtained by solving Eqs. (87) to (89) in the time domain, the expression of Y~(w) in MF narrow band By is given by mf Y.(w) ~_ 1B.(w) At E Yo(mAt) e -imat(o~-~.) m-'712.

,

Vw E IR , (93)

1

in which le.(w ) - 1 if w E [By and - 0 if w ~ [[3,, and where integers m i and mf are defined in Section VII.6.4.

6.2. Special procedure From Eq. (37), we deduce that matrix [Sz(w)] can be rewritten as

[s,(~)] = [&(~)] + [&(~)]

(94)

,

in which [$1 (w)] is an (nxn) symmetric sparse complex matrix corresponding to the finite element discretization of the sesquilinear form (r 6r

~ _ p ~ w 2 ~ r z i w1Z r -61b - ds

,

(95)

and [S2(w)] is a partially dense (n x n) symmetric complex matrix corresponding to the finite element discretization of the sesquilinear form

(r ~r ~ ~2 ~(~) p~ ~. (~ ;r

.~(~ ;~-~)

(96)

Matrix [S2(w)] can then be written as

[&(~)] = ~ ~(~) d n~ (~) n~(~)T

(97)

where IIz(w) is a complex vector such that IIl(tfl) T corresponds to the finite element discretization of linear form r ~ r l (w ;t/;). The right-hand side of Eq. (97) shows that the block submatrix corresponding to the fluid nodes which are located on interface Fz is dense and not sparse (the other elements of this matrix are zero). Similarly, matrix [C(w)] defined by Eq. (72) is the sum of sparse real matrix [Co] with partially dense matrix [Cz(w)] defined by Eq. (73). Finally, matrix [J] defined by Eq. (70) is partially dense. Consequently, matrix [A(w)] defined by Eq. (79) appears as the sum of a sparse matrix with a partially dense matrix. This property can be put to advantage by using the following algebraic result.

327


Algebraic result. Let [A] be an (N x N) invertible complex matrix, r be a vector in C N, n be a complex number and F be a vector in C N. If the linear matrix equation

([A l + n r r ~) x = v

(98)

has a unique solution X, then this solution can be written as nrTv X = V - 1 + n rTW W

,

(99)

in which Y and W are the solutions of the linear matrix equations [A]V = F

,

(100)

[A]W=I'

.

(101)

It should be noted that Eqs. (100) and (101) correspond to the same matrix equation with two different second members. Application to the FRF calculation in the MF range. Complex matrix [$1 (w)] appearing in Eq. (94) is written as

[s~(~)] - IS[ (~)] + ~ [s'~(~)]

,

(102)

in which symmetric sparse real matrices [S~ (w)] and [S~I(w)] are such that [s~ (~)] = ~r [s~ (~)1

,

~ [s'~(~)] = ~ m [s~(~)]

(103)

.

The approximation of Eq. (77) defined by Eq. (85) is then replaced by the new approximation ([A~(w)] + n(w) r ( ~ ) r ( ~ )

~) v ~ ( ~ ) = 0 ~ ( ~ ) B

,

~ e ~

,

(104)

with the constraint LT~v(w)--0

,

weBv

,

(105)

in which Yv(w) = (U~(w), ~ ( w ) ) . In Eq. (104), symmetric sparse complex matrix [A~(w)] is defined by [A. (w)] = - w 2 [M~] + iw [D~] + [K~]

,

(106)

328


VIBRATION

in which frequency-independent matrix [El,] is defined by Eqs. (86) and (80). In Eq. (106), frequency-independent matrices [D~] and [gv] are such that [DS(a~)] [Co] 1 [D~] , (107) [Co]T -[D~(f~)] - [S~ (f~)]

I

[K~] -

I [KS(f~v)]

[01

1

[0]

- [ K ] - [S~ (a~)]

,

(108)

in which [Co] is the matrix used in Eq. (72). In Eq. (104), r(w) is the complex vector such that r(w) -

- i w po HI

(w)

'

(109)

in which II1 (w) and II2 are the vectors used in Eqs. (r3) and (to). Using Eqs. (98) and (99), the solution of Eqs. (104) and (105) can then be written as

~(~)r(~)rv~(~) Yu(w) - V ~ ( w ) - 1 +

~(w)F(w)TW~,(w)Wv(w) ,

(110)

in which V(w) and W(w) are the solutions of the linear matrix equations, [A,.,(w)]V,.,(w)-

O~,(w) B

,

ca E B~,

,

(111)

with the constraint LTV~(w)-0

,

weBv

,

(112)

and [A~(w)] W~(w) - r(~)

,

wEB~

,

(113)

with the constraint LTwv(w)-0

,

wE03,,

,

(114)

in which we have introduced the notation

v~(~)-

v~(~)l

'

w~(~)J

Eqs. (110) and (112) and Eqs. (113) and (114) are solved using the direct MF method presented in Section 6.1.

329


0

Case of a Master Structure Coupled with an External Acoustic Fluid

In this section, we summarize the model, equations and solving methods for a master structure coupled with an external acoustic fluid and without an internal acoustic fluid. The following results are deduced directly from Sections 2 to 6. 7.1. Variational formulation For all fixed real w, the variational equation related to the master structure is

aS(w; u, d;u) - w 2a E (wick; u, d;u)

fs (w; 6u)

V~u e

(116)

in which a s (w ; u, ~u) is the dynamic stiffness sesquilinear form of the master structure defined on C~x C ~ by

a s ( w ; u , 6u) -- - w 2 m s(u, ~;u) + iw d s(w ;u, 6u) + k s(w ;u, ~u)

, (117)

where mass, damping and stiffness structural sesquilinear forms m s, d s and k S are defined by Eqs. (23-2), (23-3) and (23-4) respectively. In Eq. (116), aE(w/c~; u ~;u) is the sesquilinear form defined on c c x C c related to the presence of the external acoustic fluid, such that

--w2aE(w/CE;U, d;U) -- w2p~jr (6--u.n s) BrE(wlc~){u.n s }

ds

(118)

E

Antilinear form fs(w;~u) on

cc

due to external forces is written as

f s (w ;&u) - f (w ;&u) + Agiven (W ;t~U) in which antilinear form f(w ;6u) is defined by Eq.

,

(119)

(30) and antilinear

form fpgiven (o3;t~u) is defined by Eq. (31). For all fixed real w r 0, the variational formulation is written as follows. Find u in C~ such that Eq. (116) is satisfied, for all 6u in C~. 7.2. Linear operator equation The linear operator equation corresponding to the variational formulation defined by Eq. (116) is A(w) U - f s ( W )

,

ueC ~ ,

(120)

330


VIBRATION

in which A(w) is the operator defined by A ( w ) = A s (co) - w 2 A s (w/c~)

(121)

,

in which the dynamic stiffness linear operator As (w) of the master structure is written as A s (co) - - w 2 M s + iw D s (w) + K s (w)

,

(122)

where M s, D s (w) and K S (w) are the mass, damping and stiffness operators of the master structure defined by Eqs. (44-2), (44-3) and (44-4) respectively. Linear operator AE(w/cE) .is defined by Eq. (45) and element fs (w) is defined by Eq. (58). Pressure in the external acoustic fluid. Pressure field PE[rE(W) on FE due

to the external acoustic fluid is given by Eq. (9). At any point x of unbounded external acoustic fluid ~E, resultant pressure field p E ( x , w ) is given by Eq. (10). 7.3. Finite Element Discretization

We consider a finite element mesh of master structure ~s. The finite element mesh of surface F s is the trace of the finite element mesh of f~s (see Fig. 3). f~E

rE

r

Fig. 3. Finite element mesh of the master structure

We used the finite element method to construct the discretization of the various sesquilinear forms of the problem. For the master structure, we cC ~''" ~ introduce subspace Cns C of finite dimension ns. Let U = (U1 Uns) be the complex vector of the DOFs which are the values of u at the nodes of the finite element mesh of domain ~s. Since the finite element method uses a real basis for constructing the finite element matrices, the discretization

CHAPTER XIV.


331

of the variational formulation defined by Eq. (116) yields the symmetric complex matrix equation [A(w)] U = F s (w)

.

(123)

The complex matrix [A(w)] is symmetric and is defined by -

[A

-

[A

,

(124)

where [AS (w)] is the dynamic stiffness matrix of the master structure which is an (ns x ns) symmetric complex matrix (invertible for all real w ~ 0 and not invertible for w - 0 due to the presence of the rigid body modes) written as

[As (w)] = -w2[M s] + iw [Ds (w)] + [K s (w)] ,

(125)

where [MS], [DS(w)] and [KS(w)] are the mass, damping and stiffness (ns x ns) symmetric real matrices. Matrix [M s] is positive definite and matrices [DS(w)] and [KS(w)] are positive semidefinite. The discretization of sesquilinear form aE(w/cE ; U, 5U) yields the (ns x ns) symmetric complex matrix [AE (w/cg)] related to the nodal values of the vector-valued displacement field of the structure, such that

[A~ (wick)] - -p~ [O ]T [Br~ (~/~)] [ e ] ,

(126)

where [Br~ (w/cE)] is the (nEX ns) symmetric dense complex matrix defined by Eq. (XII.164) and related to the nodal values of the normal displacement field on FE. Rectangular matrix [O] is an (hE x ns) sparse real matrix defined by Eq. (69). Eqs. (118) and (126) show that matrix [AE(w/c~)] has zero rows and zero columns corresponding to the structural nodes which are not located on interface F E. Finally, the finite element discretization of antilinear form fs(w;hu) yields the complex vector F s (w) in C us. Calculation of the pressure on boundary FE and of the resultant pressure at any point of external domain f~E. Finite element discretization Pre(W) of pressure field p~lr~(W) on F~ is given by Eq. (XII.173) in which V is the vector in C nE such that V = [O]U. At any point of external domain f~E, the finite element approximation of resultant pressure pE(X, w) is given by Eq. (XII.174) in which V is defined above.

332


VIBRATION

7.4. FRF calculation in the MF range Let [~MF be the MF broad band defined by

~MF = [ WMF,init' O2MF,final ]

(127)

"

The MF narrow band fl3v is defined by Eq. (VII.9),

= [u . - a

/2, u. +Aw/2 ]

(128)

,

where f ~ > 0 is the center frequency of band G3~ and Aw is its bandwidth. We use the finite element discretization introduced in Section 7.3. For all w in BMF , we have to construct the FRF I T ( w ) ] - [A(w)]-1 where [A(w)] is defined by Eq. (124). We then have to solve for the class of MF narrow band excitations 0.(w)B introduced in Section VII.4, [A(w)]U(w)=0u(w)B

,

welB~,

.

(129)

Symmetric complex matrix [A(w)] can be written as [A(w)] - -w2[M(w)] + iw [D s(w)] + [K S(w)]

,

(130)

where [M(w)] is the symmetric complex matrix such that [M(w)]- [M s] + [A~ (wick)]

.

(131)

For construction of the frequency response function, we take O~(w) = 1 for w E ~3u and B represents the vectors of the canonical basis of ~ns. Concerning the methods for calculating the response to various types of excitations, we refer the reader to Chapter IX. The MF method presented in Chapter VII uses an approximation of Eq. (129) due to the frequencydependent matrices [M(w)], lOS(w)] and [KS(w)] consisting in replacing Eq. (129) by (-w2[~4~] + iw [D S] + [KS]) U~(w) - O~(w)B

,

w e [13,, ,

(132)

in which the frequency independent complex matrices IMp], [D S] and [K S] are such that [M~]- [M(a~)]

,

[ D f f ] - [DS(ft~)]

,

[K~s] - [ K S ( f ~ ) ]

.

(133)

Functions w ~ [M(w)], [DS(w)] and [KS(w)] can be considered as slowly varying functions on MF narrow band ~3~ (function w ~ [AE(w/cE)] has

CHAPTER XIV.

333


a nonresonant behavior and varies slowly on ~ ; functions w ~-+ [DS(w)] and [K s (w)] are related to the viscoelastic coefficients of the master structure and vary slowly). We then apply the frequency transform technique presented in Section VII.6 to obtain the associated LF equation which is written in the time domain as

[Mu] [~Jo(t)-{-[~S] Uo(t) -{-[gS]Uo(t) Uo(t i ) - O

,

Oo(t) n

1Uo(ti ) - O

,

(134)

V t > ti ,

(135)

,

and where complex symmetric matrices [/9~] and [K~] are such that [b~] - [D S] + 2ift~ [M~] ,

(136)

[K~] - - f ~ [M~] +if~v [D S] + [K S]

(137)

.

Once U0 is obtained by solving Eqs. (134) and (135) in the time domain, the expression of U.(w) in MF narrow band By is given by

mf

u0(. zt) m=m.

--imAt(~--a,,)

,

VW E ~

,

(138)

1

in which ls~(w) - 1 if w E B~ and - 0 if w ~ B~ and where integers m i and mf are defined in Section VII.6.4.

Q

Structure Coupled with an External and an Internal Acoustic Fluid. Case of a Zero Pressure Condition on Part of the Internal Fluid Boundary

In this section, we summarize the model, equations and solving methods for a master structure coupled with an external and an internal acoustic fluid for which a zero pressure condition exists on a part of the internal fluid boundary. In this case, quantity 7r(w ; u ; r = 0 and the results presented in this section are directly deduced from Sections 2 to 6 in which all the terms related to 7r(w ; u ; r vanish.

8.1. Statement of the structural-acoustic problem We consider the problem described in Section 2 for which boundary 0[2 of f~ is written 0f~ = F U F0 U Fz in which Fo is submitted to a zero pressure field (see Fig.4).

334

STRUCTURAL ACOUSTICS

nS

VIBRATION

F E

~2 E

rE

AND

rz a- Gas or liquid configuration

b- Liquid with free surface

Fig. 4. Configurations of the structural-acoustic master system

There is no gravity effect. Consequently, boundary F in Section 2 is replaced by boundary F U F0 in this section. 8.2. Internal and external acoustic pressure fields

Concerning the pressure field in internal acoustic fluid f~, we saw in Sections X.2.6-3 that lr(w ; u ; r - 0. Consequently, ~rl (w ;r ~r2(u) and lrQ (w) do not appear in the calculation (this is achieved formally by eliminating all the terms containing a(w)). At any point x in f~ U0f~, pressure field p(x, w) is then given by p(x, w) = -iw po r w) . (139) Pressure field p~lr~(W) on F~ and resultant pressure field ps(x,w) at any point of unbounded external acoustic fluid f~s are calculated using Eqs. (9) and (10). 8.3. Boundary value problem of the structural-acoustic master system

We use the fact that l r ( w ; u ; r = 0 and from Section 3, we deduce the equations of the boundary value problem. Concerning the master structure, Eqs. (11) to (13) become --032 Ps Ui -- a i j , j (U) --

gi

in

f~s

,

(u) nj

on

aij(u) njS = G i _ i w p o C n i

on

FtJFz

FE

,

( 40) (141) (142)

335


Concerning the internal acoustic fluid, Eqs. (15) to (20) become _w2 Po ~ r

~) - po v ~ r

~1 - i~ ~ p0v~r

~1 = i~ g~

in f~ , (143)

with the Dirichlet boundary condition r

onFo

,

(144)

and the Neumann boundary conditions po(l +iwT) 0 r = iw Po u . n + iw GQ

o~

po(1 "+-iWT) 0 r 2 w 2Por + iw G Q --~ = iw Po u . n + iw Z

r

,

o.rz

(14~) ,

(146)

in which 9Q is defined on f~ by

g~ (x, ~) =

1 1 ~,2 iwQ(X,W ) - TC~ ~ Q(x,w)

(147)

GQ is defined on F U Fz by

1 0 Q ( x , w)

(148)

GQ(X'W) = TC~~w--5 On(x)

8.4. Variational formulation of the structural-acoustic problem The variational formulation of the boundary value problem defined by Eqs. (140) to (148) is deduced from Section 4. Admissible function spaces. The admissible function space of displacement field u of master structure f~s is the complex vector space C~ defined in Section 4.1. Concerning internal acoustic fluid f~, we introduce the complex vector spaces ~0,r such that c

c

Eo,r 1 6 2 in which s

~c

;

r

is defined in Section 4.1 and H ~ -

Variational formulation in terms of {u, r the master structure yield

aS(w; u,~u)

on

w2a ~(W/c~;u, ~u) + iw co(r

Fo}

(149)

L2(~).

Eqs. (140) to (142) related to

Is(w; ~u)

V~u e

(150)

336


AND

VIBRATION

in which a S (w ;u, diu), a E (w/ce; u, 5u) and co(r 5u) are defined by Eqs. (23), (24) and (27) respectively. In nq. (150), fs(w; 5u) is the antilinear form on C~ defined by (151)

fs (w ;Su) - f (w ; 5u) -4- fpgiven (o3 ; (~11) ,

in which antilinear forms f(w ;~u) and fpgiven (W ;~11) are defined by Eqs. (30) and (31) respectively. Eqs. (143) to (148) related to the internal acoustic fluid yield

iwco(5r

F(co;r162162

in which the sesquilinear form

a F (w ; r 1 6 2

aF

-w2m(r162

c

,

(w ; r

VhCeCo, r

,

(152)

(~r is given by

dr(w; r162162162

(co;r162

(153)

where k(r 5r d~(w ; r 5r and m(r 5r are defined by Eqs. (34), (35) and (36) respectively and where sesquilinear form sl (w ; r 5r defined on E~x E c is deduced from Eq. (37) and is such that

81(CO; r ' (~r

--P~

2

2Jr

z i w1Z r -~r

( 54)

ds

In Eq. (152) , fq (w ;~r is the antilinear form defined on gc deduced from Eq. (40) and is such that

s

-

__

dx +

uo

VQ.V

(155)

r dx

For all fixed real w # 0, the variational formulation is written as follows. Find u in (7r and r in Co,r such that, for all 5u in C~ and ~r in go,c, Eqs. (150) and (152)are satisfied. c

c

Linear operator equation. Considering Eqs. (150) and (152), Eqs. (41) and (42) are modified as follows, ~c

fq (W)]

{u,r

E

c

x ~0,r

,

(156)

in which A(w) is the restriction to (7r x C0,r of the linear operator

A(w) -

I

AS( ) iw tC 0

Co 1 - A F (w)

(157)

337


Linear operators AS(w), AE(w/c~) and Co are defined by Eqs. (43), (45) and (55) respectively. Linear operator A F(W) defined by sesquilinear form a F(w ; ~),5~b) can be written as AF (w) - -w2M +

iwD~-(w) + K + S~ (w) ,

(158)

in which operators M, Dr(w) and K are defined by Eqs. (49), (51) and (50) respectively. Linear operator Si (w) is such that i9 < Si (w) ~b, (~r > -

si (w ; r

(~b)

(159)

,

in which sesquilinear form si ( w ; r 5r is defined by Eq. (154). Finally, elements fs (w) and fQ (w) are defined by Eqs. (58)and (59). 8.5. Finite element discretization

We consider the finite element mesh of master structure ~ts and the finite element mesh of internal acoustic fluid ~t introduced in Section 5. Eq. (64) becomes

iWFQ(W)J

'

(160)

with the constraint 9j - 0

for nodes

j

belonging to

F0

9

(161)

corresponding to the finite element discretization of the Dirichlet conditions defined by Eq. (144). Complex matrix [A(w)] is symmetric and is defined by

I [AS (w)] - w2[AE(w/c~.)] iw[Co] l

[A(w)] --

,

iw [Co]T

(162)

-[A F (w)]

in which matrices [AS (w)], [A~ (w/cs)], [Co] are defined by Eqs. (67), (68), (72) respectively. Matrix [AF (w)] is defined by

[AF (w)] -

- w 2 [ M] +

iw [Dr (w)] +

[ K ] + [Si (w)]

,

(163)

in which matrices [M ], [K] and [DT(w)] are defined in Section 5 and matrix [Si (w)] corresponds to the finite element discretization of the sesquilinear form defined by Eq. (154). Finally, vector F s (w) is defined in Section 5 19 Sesquilinear form si(w ;r ,5r is continuous on s163 c (see also footnote 6) and defines a continuous operator Si(w) from s into s The angle brackets in Eq. (159) denote the antiduality product between s and $c.

338


VIBRATION

and vector FQ (w) correspond to the finite element discretization of the antilinear form defined by Eq. (155). Calculation of the pressure inside f~. Let P = ( P l , . . . , Pn) be the complex vector of the DOFs which are the values of p at the nodes of the finite element mesh of domain ft. From Eq. (139), we deduce that P = -iwpo@

(164)

.

Calculation of the pressure on boundary FEand of the resultant pressure at any point of external domain fiE. The finite element discretization Pr~(W) of pressure field pE[r~(W) on FE is given by Eq. (XII.173) in which V is the vector in C nE such that V = [O]U. At any point of external domain f~E, the finite element approximation of resultant pressure pE(x, w) is given by Eq. (XII.174) in which V is defined above. 8.6. FRF calculation in the MF cange

Let

[~MFbe

the MF broad band defined by ~MF -- [ WMF,init' ~dMF,final ]

(165)

"

The MF narrow band ~ is defined by

= [

Aw/2,

+Aw/2 ]

(166)

,

where f~. > 0 is the center frequency of band B~ and Aw is its bandwidth. We consider the finite element discretization introduced in Section 8.5. For all w in [BMF, we have to construct the FRF [T(w)] - [A(w)] -1 where [A(w)] is defined by Eq. (162). We then have to solve for the class of MF narrow band excitations O r ( w ) B introduced in Section VII.4,

[A(w)] Y(w) =

Or(w)B

'

w E IB~ with

Y ( w ) = [U(w)] [@(w)J'

(167)

with the constraint 9j = 0

for nodes

j

belonging to

F0

9

(168)

Symmetric complex matrix [A(w)] can be written as -

+

+

,

(169)

CHAPTER XIV.

339


where [M(w)], [D(w)] and [D

(1)

~

(2)

.~

LF LF

. . . . . . . . . .

~

-

_~.~

MF MF

~..~

HF

2 ~ . . ~

HF

Fig. 3. Effects of fuzzy s u b s t r u c t u r e s on t h e m a s t e r s t r u c t u r e - First case. ( Q u a l i t a t i v e drawing.) ...... R e s p o n s e of t h e m a s t e r s t r u c t u r e uncoupled with t h e fuzzy s u b s t r u c t u r e s . R e s p o n s e of the m a s t e r s t r u c t u r e coupled with the fuzzy s u b s t r u c t u r e s . (1) D y n a m i c a l behavior of t h e u n c o u p l e d m a s t e r s t r u c t u r e . (2) D y n a m i c a l behavior of t h e coupled m a s t e r s t r u c t u r e (fuzzy s t r u c t u r e ) .

Second case. This case is illustrated by Fig. 4. It corresponds to the situation where the first eigenfrequency of the fuzzy substructures lies in the MF range of the master structure uncoupled with the fuzzy substructures. Fig. 4 shows a small shift of the LF resonances and small modifications of the dynamical response at the beginning of the MF range. The dotted line represents the response of the master structure uncoupled with the fuzzy substructures and the solid line represents the response of the master structure coupled with the fuzzy substructures which only induce an added-mass effect below the first eigenfrequency of the fuzzy substructures. In this case, the LF range of the master structure coupled with the fuzzy substructures coincides with the LF range of the master structure uncoupled with the fuzzy substructures. In addition, the MF range of the master structure coupled with the fuzzy substructures coincides with the MF range of the master structure uncoupled with the fuzzy substructures.

348


VIBRATION

The presence of fuzzy substructures induces an "apparent strong damping" in the master structure in the MF range. FRF (dB)

Frequency (Hz) >

(1) ..~

LF

MF

(2) -~

LF

MF

HF

Fig. 4. Effects of fuzzy substructures on the master structure - Second case. (Qualitative drawing.) ...... Response of the master structure uncoupled with the fuzzy substructures. Response of the master structure coupled with the fuzzy substructures. (1) Dynamical behavior of the uncoupled master structure. (2) Dynamical behavior of the coupled master structure (fuzzy structure).

3. F u z z y Structure Theory Stated as an Inverse P r o b l e m By construction, the fuzzy structure theory belongs to the class of inverse problems and is not a classical structural-dynamics problem with random uncertainties. In order to clarify this fundamental difference, the first section below is devoted to a short review of structural-dynamicsproblem with random uncertainties.

3.1. Structural-dynamics problem with random uncertainties stated as a direct problem Here we recall the definition of a classical structural-dynamics problem with random uncertainties. Let us consider the boundary value problem for a structure made of two substructures, studied in Section V.7 (see Fig. 5). To model the uncertainties in this structural-dynamics problem, we assume that the dynamic-stiffness operator of substructure ~2 depends on an ~m-valued parameter y related to the uncertainties and belonging to a subset s of ~m, yeLC~ m (1) .

349

CHAPTER XV. FUZZY STRUCTURE THEORY

For instance, the components of y are mechanical coefficients such as the elastic and damping coefficients of the constitutive equation of the viscoelastic material, the mass density and geometrical parameters (except 2

Fig. 5. Structure decomposed into two substructures

interface ~ which is assumed to be known). This operator is then written as A2(w ;y). Consequently, for all real w, mapping y ~ A2(w ;y) defined on s is known. The boundary value problem consists in finding (u 1, u 2) in Cn such that, for all (6u 1 6u 2) E Cn (see Equation (V.89)) c

< A I (w) u 1, 6u I > 4- < A 2 (w ;y) u 2, 6u 2 > -- < fl (w), 6u I >

,

(2)

c

in which the admissible function space C~ is defined by Eq. (V.85). It is proved in Section V.7.3 that the problem defined by Eq. (2) is equivalent to a problem posed in subdomain ~1 by introducing boundary dynamic stiffness operator A s (w) defined by Eqs. (V.101) and (V.106) (see Fig. 6). As seen in Section V.7.3, Eq. (2) is equivalent to the following operator equation (Al(w) + A~.(w ;y))ul(w) = fl(w) , (3) Cc

c

in which ul(w) E n~ where C~ is defined in Section V.7.2. In terms of impedance operators, Eq. (3) can be rewritten as iw (zl(w) + Z~. (w ;y)) u 1(w) - fl (w)

,

(4)

in which impedance operator zl(w) related to domain ~1 is defined by iw Zl(w) - Al(w)

,

(5)


350

and boundary impedance operator Z~. (w ;y) related to interface ~Eis defined by iw Zr, (w ;y) = Ar (w ;y) . (6)

f~ = Ax(co, y) u x

Z

Fig. 6. S t r u c t u r e f~z u f~2 e q u i v a l e n t to s u b s t r u c t u r e f~x with a b o u n d a r y i m p e d a n c e o p e r a t o r on interface E

Concerning the corresponding finite element discretization, we refer the reader to Sections V.7.4 and V.7.5. In the context of random uncertainties, a probabilistic model of parameter y must be introduced. To do so, parameter y is modeled by an ~m-valued random variable denoted as Y. The probability distribution z of random variable Y is denoted as Pv(dy) and, since parameter y belongs to L, the support 2 of Pv(dy) is E C Rm. Consequently, Z~. (w ; Y) is a random variable 3 with operator values. Equation (4) is then rewritten as

iw (ZZ(w) + Z~ (w ; Y)) U z (w)

-

fl (w)

,

(7)

in which Z~. (w;Y) is a random operator and the unknown u 1(w) is a random variable denoted as ul(w) with values in admissible function space 1 A probability distribution Pv(dy) of an ~m-valued random variable is a positive bounded measure on ~m such that ~" m Pv(dy)--l. If this probability distribution has a density with JR respect to the Lebesgue measure dy on ~m, then Pv(dy)--pv(y)dy and the probability density function pv(y) is a positive valued function defined on ~m which is integrable and such t h a t ~p.m pv(y)dy--1. Concerning probability theory and random variables, see Guikhman and Skorokhod, 1979; Halmos, 1976; Kree and Soize, 1986. 2 If

EC~,

part s

TM

is the support of probability distribution Pv(dy), then f L P v ( d y ) = l and for any TM

such that s163

we have f L

Pv(dy)=0. 0

3 We assume that mapping y~-~Z~ (w;y) is measurable and consequently Z~. (w;Y) is a random variable.

CHAPTER XV.

FUZZY STRUCTURE THEORY

351

c

C a . Equation (7) corresponds to a structural-dynamics problem with random uncertainties for which appropriate methods exist for constructing the random solution. This problem is considered as a direct problem because, (1)- the uncertain vector-valued parameter is identified, (2)- its probabilistic model is known, (3)- mapping y ~ Z~ (w ;y) is known. 3.2. Fuzzy structure theory stated as an inverse problem

In the above problem, if mapping y ~ Z~ (w;y) is unknown, i.e. cannot be constructed due to the complexity of mechanical subsystem Ft2, which is partially unknown from a geometrical and mechanical point of view, then the above direct approach cannot be applied. If substructure ~'~2 is a fuzzy substructure, then by the definition given in Section 2.1, mapping y ~-+ A2(w ;y) cannot be constructed and therefore mapping y ~-~ Z~. (w;y) which is deduced from A2(w ;y) cannot be constructed either and is thus unknown. The purpose of the proposed fuzzy structure theory is to solve this problem as an inverse problem. Consequently, since A2(w;y) cannot be constructed, the stated inverse problem consists in directly constructing a random boundary impedance operator Zfu~(W) related to interface E to represent the effects of the fuzzy substructure on the master structure. The objective of the fuzzy structure theory is to directly construct Zfu~(W) which depends on mechanical parameters (interface E is assumed to be known) and to propose a procedure for identifying these parameters. 3.3. Basic requirements of the construction of the fuzzy-structure-theory

(1)- The objective of the fuzzy structure theory is to predict the effects of fuzzy substructures on the local dynamical response of the master structure, not to predict the dynamical response of fuzzy substructures. Consequently, the fuzzy structure theory is used to predict the frequency response functions in the master structure coupled with fuzzy substructures. More precisely, this construction gives the real and imaginary parts of the frequency response function, i.e. the modulus and the phases. Therefore, such an approach allows vibration propagations inside the master structure to be predicted and allows an extension to the case of structural-acoustic fuzzy systems (fuzzy structure coupled with external and internal acoustic fluids, see Chapter I). (2)- Since a fuzzy substructure is made of a large number of secondary subsystems inaccessible to conventional modeling, a statistical approach is used and a probabilistic mechanical model is proposed for describing the boundary impedance operator which models the effects of the fuzzy substructures on the master structure. Consequently, the model is based

352


VIBRATION

on the construction of a random boundary impedance operator Zfu,.(w) of the fuzzy substructures. The model of the fuzzy structure is then written as +

=

,

(8)

in which Z(w) is the impedance operator of the master structure, Zfuz(W) is the random boundary operator introduced in Section 3.2 to represent the effects of the fuzzy substructures on the master structure, U(w) is the random displacement field of the master structure and f(w) represents the external forces applied to the master structure. Modeling of the fuzzy structure corresponding to Eq. (8) does not introduce any additional unknown field with respect to the model of the master structure. (3)- A requirement of the fuzzy structure theory is that it must construct random operator Zfuz(W) in such a way that Eq. (8) can be solved using an efficient solver, especially in the MF range. This requirement is consistent with the construction of an inverse problem. FRF (dB)

Frequency (Hz)

Fig. 7. F R F in the LF and MF ranges. (Qualitative drawing.) Response of the coupled master structure (experimental results). Mean response of the coupled master structure (fuzzy structure theory). - - - Upper and lower values associated with a confidence interval (fuzzy structure theory).

(4)- Random displacement field U(w) of the master structure, as the solution of Eq. (8), is described by its probabilistic quantities such as mean value, second-order moments, etc. For instance, Fig. 7 shows the mean value and the upper and lower values associated with a confidence interval of a component Uj(x,w) of random displacement field U(w) at a point x located in the master structure obtained by the fuzzy structure theory,

353


with respect to the fuzzy structure experimental result in the LF and MF ranges.

4. Random Equation of the Master Structure Coupled With Fuzzy Substructures in Terms of the Displacement Field of the Master Structure The geometry of the fuzzy structure and the notations are defined in Fig. 8.

Fig. 8. Geometrical configuration of a fuzzy structure

The master structure occupies the open bounded domain f~ of ~3 referred to a Cartesian reference system {i,j,k}. The boundary of domain f~ is written as 0f~ - F tA Z such that F N Z - 0. This master structure is coupled with L fuzzy substructures. Fuzzy substructure ~ is attached to the master structure by boundary Z~ and there is no mechanical connection between any two fuzzy substructures. The part Z - U~= I~Ee is the common boundary between the fuzzy substructures and the master structure. The outward unit normal to 0f~ is denoted n and the two-dimensional surface area element is denoted ds. We then have fr~ ds(x) - [ ~ e ] , where [~el is the area of surface Ze. We are interested in the dynamical response of the master structure coupled with the fuzzy substructures in the MF broad band defined by [~MF -" [~gMF,init ' WMF,final] C [R+

,

with

~gMF,init > 0

.

(9)

354

STRUCTURAL

ACOUSTICS AND

VIBRATION

Of course, as explained in Section 2.2, the master structure uncoupled with the fuzzy substructures can have an LF dynamical response for w in BMF (see Fig. 3). Consequently, we need an LF model and an MF model for the master structure.

Modeling the master structure. The master structure is assumed to be a free structure submitted to a given body force field g(x, w) defined in f~ and surface force field G(x, w) defined on F. These forces are assumed to be in equilibrium for each w in BMF (see Eq. (V.49)). The model of the master structure is described in Chapter V. The admissible function space 4 C~ of the displacement fields u(x,w) = (Ul(X,W),U2(X,W), U3(X,(.O)) is defined in Section V.3. When the master structure is not coupled with the fuzzy substructures, its operator equation on C c is given by (see Eqs (V.43) and (V.47)), i~ z ( ~ ) u ( ~ ) = f(~) , (10) in which Z(w) is the impedance operator on c ~ of the master structure defined by iw Z(w) - - w 2 M + iw D(w) + K(w) , (11) u(w) E C~ is the displacement field of the master structure and f(w) represents the given forces and is defined by Eqs. (V.38) and (V.39). We recall that in the LF range, stiffness operator K(w) is independent of w but in the MF range, K(w) depends on w. From Section V.5.3 (footnote 15), for all w E ~MF' since w r 0, Eq. (10) has a unique solution in C~.

Modeling the fuzzy substructures. As explained in Section 3.3-(2), the effects on the master structure of fuzzy substructure ~ attached to the master structure by boundary Ee are represented by a random boundary impedance operator Z~u~.(w) defined on C~. The effects of all the fuzzy substructures on the master structure are then represented by a random boundary impedance operator, denoted as Zfuz(~d ) defined on cc~ and such that L

Zfuz(W)- ~

Zfuz(W)

9

(12)

~--1

Random equation of the fuzzy structure in terms of the master structure displacement field. We deduce that the random equation of the master structure coupled with the L fuzzy substructures is written as +

4 Space C c is the Sobolev space (Hl(f~)) 3.

-

,

(13)

CHAPTER XV.

355

F U Z Z Y STRUCTURE THEORY

in which Zfuz(W) is given by Eq. (12) and U(w) is the master structure random displacement field with values in (ft. The construction of random operator Ztuz(W) in the present context of the fuzzy structure theory stated as an inverse problem is described in Section 5.

5. Homogeneous Model of a Fuzzy Substructure As explained in Section 3.2, the effects of fuzzy substructure s on the master structure are represented by a random boundary impedance operator Z~u~.(w) which has to be constructed in the context of an inverse problem.

5.1. Integral representation of the random boundary impedance operator of a fuzzy substructure For this construction, we use the following integral representation of random boundary impedance operator Zf~uz(w). For all deterministic functions u and 6u in C~, this operator is written as 5

(14) l

in which, for all x and x' in ~e, the kernel [Ze(w;x,x')] of the integral representation is a random (3 x 3) complex matrix which has the following algebraic symmetry property,

[z ~(~ ;x, x')] - [z ~(~ ;x', x)] r

,

and the following property which corresponds to the fact that this kernel must be a random (3 • 3) real matrix in the time domain,

[ z ~ ( - ~ ;x,x')] - [zt(~ ;x,x')]

.

(16)

In addition, for all w i n [~MF and for all deterministic functions u in C~, random kernel [7/e(w ;x,x')] must be such that random operator Zfluz(W) defined by Eq. (14) satisfies

E{l[ 2} < +c~

,

(17)

5 In Eq. (14), the brackets denote the antiduality product between Cc~ and Cc in which C.c~ is the antidual space of CC.

356

STRUCTURAL

ACOUSTICS

AND

VIBRATION

and random operator Zfuz(W) defined by Eqs. (12) to (17) is such that, for all deterministic elements 6 5f, E{J < 5f, (Z(w) § Zruz(W))-zSf> j2} < §

(18)

.

5.2. Choice of an algebraic representation of random kernel [Z~(w ; x, x')]

This choice is mainly guided by the fact that we want to construct a model which is defined by a small number of significant mechanical parameters to satisfy the constraints defined by Eqs. (15) to (18). It is assumed (see Fig. 9) that there exists an orthogonal (3 • 3) real matrix lOt(x)] T for transition from canonical basis {i,j,k} to a local orthonormal basis {e~ (x), ee2(x), e~(x)} related to boundary El at point x e Ee such that 7 = [r

x1

[r

,

(19)

1 x2 Fig. 9. Local basis r e l a t e d to b o u n d a r y ~

in which [Zdeiag(x, x', w)] is a random (3 • 3) complex diagonal matrix whose elements are written as [7/~iag(X,X/,w)ljj,

- - ~ j ( W ) ( ~ E t (X-- X / ) ( ~ j j '

,

(2o)

in which 7/~ (w), 7~2(w) and 7/t3(w) are second-order random variables with values in C. Local vector basis ee3(x) coincides with normal n(x) introduced 6 Element 5f belongs to the antidual space C c~ of C c . In Eqs. denote the antiduality product between C c~ and C c .

(17) and (18), the brackets

7 It is assumed t h a t mapping x~-~[~t(x)] is a bounded function on S t .

357


in Section 4. In Eq. (20), 6jj, is the Kronecker symbol and 6 ~ ( x - x ' ) is the Dirac function related to boundary Ee which is such that, for all complex functions ~o(x) continuous on Ee and x ~ E Ee,

~o(x)~r,~(x-x')ds(x)--9~(x' )

/

(21)

l

Substituting Eq. (19)with Eq. (20)in Eq. (14) and using Eq. (21) yields < Zfeuz(W)u, 6u>--/~![~e(x')] [Ze(w)] [~e(X')]T U(X')}-~U(X') ds(x'), (22)

in which second-order random complex matrix [Ze(w)] is diagonal and is defined by [Ze(w)]jj, -- Zj(w)~jj, . (23) For fuzzy substructure ~ and for local direction j, since Z}(w) is a secondorder random variable with values in C then Eq. (17) is satisfied. In addition, we assume that zj(

)

.

(24)

Consequently, Eq. (16) is satisfied. It should be noted that Eqs. (19) and (20) imply that Eq. (15) is satisfied.

Terminology. Since for j = 1, 2, 3, random impedance Z~ (w) is independent ofx and x', the model defined by Eqs. (19) and (20) is called a homogeneous model of fuzzy substructure ~. The construction of Z~ (w) is described in the next section introducing a parametric model. Zj(w) is called the homogeneous fuzzy impedance law or the homogeneous probabilistic impedance law of fuzzy substructure ~ related to local direction j at a given frequency. 5.3. Definition of a resonant frequency of fuzzy substructure ~ for local direction j As explained in Section 2.2, fuzzy substructure ~ is a dynamical system having a countable number of resonant frequencies. A resonant frequency in local direction j is associated with a displacement field on El, which has a nonzero contribution in local direction j.

358


VIBRATION

6. Construction of a Homogeneous Fuzzy Impedance Law for a Fuzzy Substructure and for a Local Direction In this section, we construct homogeneous fuzzy impedance law Zj(w) for fuzzy substructure ~ and for local direction j at a given frequency w. Two types of homogeneous fuzzy impedance law are constructed, defined as type I and type H (type I is a particular case of type II). For additional details concerning this construction, we refer the reader to Soize, 1986 for the type I construction and Soize, 1993b for the type II construction. All the fuzzy impedance law coefficients introduced depend on local direction j at each point x of 53e but are independent of this point x. In order to simplify the notations, index j and superscript ~ are omitted in all of Section 6. In particular, 7/j(w) is written as 7(w).

6.1. Choice of representation for random variable Z(w) The choice of the representation of second-order complex-valued random variable Z(w) is guided by the requirement introduced in Section 3.3-(3) which leads to introducing a linear random dependence of the random fluctuation part around its mean value, d -

(25)

+ k----1

in which d is a positive integer, Z(w) is the mean value of random variable Z(w) - E { Z ( w ) }

(26)

,

Z1 ( w ) , . . . , Zd(w) are complex numbers and ~ 1 , . . . , ~d are mutually independent, second-order, real-valued random variables. Each random variable ~k has a uniform probability distribution P~k (dx) - P~k (x) dx

,

(27)

where the probability density function P~k (x) with respect to dxk is written as (see Fig. 10) 1 P~k (X)= 2V/~ 11[_V~,~/~](X) , (28) in which ]LB(x) is the indicator function of set B (see Mathematical Notations in the appendix). The support S~ k of this probability distribution

CHAPTER XV.

359


is then Sxk - [ - v ~ , v/3]. Consequently, Xk is a second-order centered random variable and its variance is equal to 1 (normalized random variable), E{Xk}-0,

E{X2}-I

.

(29)

P~k( x )

i

i

i l I

o

x

Fig. 10. Probability density function px k (x)

From Eqs. (25) and (29), we deduce that effectively a second-order random variable. construction of fuzzy impedance law Z(w) deterministic quantities Z(w) and Zk(w). algebraic representation defined by Eq. (25) the construction of an inverse problem.

Z(w) defined by Eq. (25) is The problem related to the then consists in constructing It should be noted that the is legitimate in the context of

6.2. Underlying deterministic model for the construction of

Z(w)

The underlying deterministic model of random variable 7/(w) is denoted as ~(w) and depends on parameters f~c, It(w), ~(w), lip(W) and a(w) defined below.

Expression for the dynamic stiffness below a cutoff frequency. We introduce a positive deterministic cutoff frequency of fuzzy substructure g related to local direction j and denoted as f~c. This cutoff frequency is such that for w < f~c, dynamic stiffness iw ~(w) behaves like a pure mass (see Section 2.2) and can be written as =

,

(30)

in which #(w) is a positive parameter which has the dimension of a mass per unit area (kg/m 2) and which is related to fuzzy substructure ~ for local direction j.

Expression for the dynamic stiffness above cutoff frequency f~c. Above the cutoff frequency, i.e. for w fixed in [f~c, +c~[, dynamic stiffness /w ~(w)

360


VIBRATION

behaves like a weakly damped resonant system (see Section 2.2) described with four positive parameters #(w), ~(w), f~p(w) and ~(w) and is written as

iw'~(w)

-

a(w)

-

a(w) a(w)2 b(~)

(31)

'

in which a(w) is a dimensionless parameter belonging to interval [0,1] and where complex numbers a(w) and b(w) are defined by

~(~) - 2/~ ~(~) , ( ~ ) ~ ( ~ ) + , ( ~ ) a~(~)~ b(~) = - ~

,

(32)

~(~) + 2i~ ~(~) ~(~) ~ ( ~ ) + ~(~) ~ ( ~ ) ~

(33)

When c~(w)- 1, Eqs. (31) to (33) yield i~(~)

= -~

~(~)(2i~ ~(~)~(~) + ~(~)~) f~p(W) 2 - w 2 + 2iT ~(w) f~p(W)

"

(34)

The underlying deterministic model defined by Eq. (34) is used to construct homogeneous fuzzy impedance 7/(w), and is called the type I homogeneous ]uzzy impedance law. It corresponds to the dynamic stiffness iTS(W) of the elementary dynamic model constituted by a linear oscillator excited by its support. For 0 _ c~(w) < 1, the underlying deterministic model defined by Eqs. (31) to (33) is used to construct homogeneous fuzzy impedance 7/(w), and is called the type H homogeneous ]uzzy impedance law. For a(w) - 1, the type II law gives the type I law. Dynamic stiffness iw'~(w) defined by Eq. (31) can be rewritten as iw'~(w) - a(w) - c~(w)a(w) b(w)-1 a(w). This algebraic expression is deduced from the general expression of a reduced boundary-dynamic-stiffness-matrix model defined by Eq. (V. 110-3). Concerning the proof that c~(w) can be chosen as a real number belonging to the interval [0,1], we refer the reader to Soize, 1993b. From a modeling point of view, if fuzzy substructure ~ is composed of a large number of independent dynamical subsystems, then the type I fuzzy impedance law should be used (see Fig. 11). The type II fuzzy impedance law should be used for other cases (see Fig. 12).

Fig. 11. Fuzzy substructure corresponding to type I fuzzy impedance law

CHAPTER XV. FUZZYSTRUCTURETHEORY

361

Fig. 12. Fuzzy substructure corresponding to type II fuzzy impedance law Dimensionless parameter c~(w) is called the equivalent coupling factor because it allows the type of attachment between the master structure and the secondary subsystems constituting fuzzy substructure ~ to be modeled for local direction j. The case c~(w) = 1 corresponds to discrete attachments and the case 0 < c~(w) < 1 corresponds to continuous attachments. In Eqs. (31) to (33), dimensionless parameter ~(w) is related to the rate of internal damping of fuzzy substructure s for local direction j and is assumed to be small (0 < ~(w) f~c, f~p(w) represents the eigenfrequency of a weakly damped resonant system whose dynamic stiffness is iw "~(w). The dependence of f~p(w) on w is introduced for the following reason. For w fixed, w does not necessarily coincide with a resonant frequency of fuzzy substructure ~ for local direction j (see Section 5.3). We therefore introduce the probability that eigenfrequency f~p of the underlying deterministic model of the fuzzy impedance law is in a neighborhood of w. 6.3. Construction of a probabilistic model of the parameters

For fuzzy substructure ~ and for local direction j, we introduce six parameters which are f~c, #(w), ~(w), n(w), c~(w) and f~p(w). Cutoff frequency f/c is considered as a deterministic parameter. The four parameters #(w) > 0, 0 < ~(w) 0 and 0 < a(w) < 1 are modeled by random variables whose mean values are called the mean coefficients and whose deviations (or "spread" or "variability") are controlled by given deterministic parameters Al(w), ~2(W), ~3(Cd) and ),4(w) which are called the deviation coefficients. Parameter f~p (w) is modeled by a random variable whose probability distribution is constructed using modal density parameter n(w). In conclusion, four random parameters/L(w), ~(w), n(w) and c~(w) remain and consequently, in the representation of random variable 7/(w) defined by Eq. (25), we have d= 4 . (35) Introduction of the modal density related to fuzzy substructure ~ for local direction j. The modal density of fuzzy substructure ~ for local direction j is denoted as n(w), which means that the number of resonant frequencies of

362


VIBRATION

fuzzy substructure g for local direction j (see Section 5.3) in a neighborhood 5w of w is n(w)5w. The distance, denoted as 2e(w), between two resonant frequencies of fuzzy substructure g for local direction j in the neigborhood 5w of w is then given by 5w 1 2 e ( w ) - n(w) 5w = n(w)

"

(36)

Fuzzy impedance law mean coefficients as the mean values of the random parameters. The mean values of random parameters #(w), ~(w), n(w), t~(w)

are defined by

t t ( w ) - E { # ( w ) ) , ~_(w)-E{~(w)), n ( w ) - E { n ( w ) } , ~_(w)-E{a(w)), (37) and are called the fuzzy impedance law mean coefficients (or simply, the mean coefficients). It is assumed that these mean coefficients satisfy the following property

#(-w)-tt(w),

~(-w)-~(w),

n(-w)-n(w),

a(-w)=a(w).

(38)

(1)- The mean value it(w) > 0 of random parameter it(w) which has the dimension of a mass per unit area (kg/m 2) is described by a dimensionless mean coefficient _v(w) > 0 which is such that l U(W) -- ~(0)) mref -

levi

(39) '

in which IZtl is the area of surface E~ and rare f l is a reference mass related to fuzzy substructure g. For instance, if the total mass of fuzzy substructure g can be estimated, then rare f can be chosen as this total mass. Mean coefficient _v(w) is called the mean coefficient of the participating mass of fuzzy substructure g for local direction j. (2)- Dimensionless mean coefficient ~_(w) is the mean rate of internal damping of fuzzy substructure g for local direction j and is such that 0 < ~ (w) 0 is the mean modal density (measured in seconds) related to fuzzy substructure g for local direction j. (4)- Dimensionless mean coefficient _a(w) is the mean equivalent coupling factor of fuzzy substructure g for local direction j and is such that 0 _< a_(w) _< 1. For _a(w) - 1, the model corresponds to the type I homogeneous

CHAPTER XV.

363


fuzzy impedance law and for 0 < a(w) < 1, the model corresponds to the type II homogeneous fuzzy impedance law. Deviation coefficients and construction of the probability distributions of random variables ~(w), ~(w), n(w), a(w). To control the deviation of random

variables ~(w), ~(w), n(w) and a(w), we introduce the ~4-valued function

W ~-~ X(W) ----()~1(W),)~2(W), )~3(W), ~4(W))

,

(40)

defined on [~MF and such that Ak(W) G [0,1[

,

Vke{1,2,3}

(41)

,

and for k - 4,

0_ 0

,

Vu E C~ .

(125)

In addition, master structure mass operator M is symmetric positive definite on ~ , (see Section V.4.1) It can then be proved 9 that, for all w in [~MF' operator iw Zdet (W; A ) is i n v e r t i b l e . Definition of )~max and introduction of random operator T(w; A). For all w fixed in BMF , deterministic operator iw Zdet (w; A) is invertible. Let T(w; A) be the random operator such that T(w; A) -

--Zdet(W;

A) -1Zrand(W; A )

.

(126)

Substituting Eq. (113)in Eq. (126) yields M

T(w; A) = - ~

Xi

Zdet(W; A ) - 1 Z i ( w ; A )

(127)

i=1

in which, for each i - 1 , . . . , M, operator Zdet(W; A) -1Zi(w; A) is a deterministic operator from C* into C*. Since operator Zdet(W; 0) is invertible and due to Eq. (120), we deduce that T(w;A)-+0

as

A--+0

.

(128)

9 Let us consider w fixed in I~MFC~+*. Due to the construction of operator -w2Rfuz(W;A), we can write-w2Rfuz(W;A)---w2Mfuz(w;A)+K_fuz(W;A) in which Mfuz(W;A ) and _Kfu,(W;A) are positive-semidefinite operators. Consequently, Eq. (124) can be rewritten as iw Zdet (W ;k) -" --w 2 (M+M___fuz (w;A))+iw(D(w)+I fuz (w;A))+K(w)+_K fuz (w;A). We recall that Cc =Crig(~ C:las c (see Eq. (V.52)). Operator iw Zdet(w;A) is invertible on Celas because D(w)+Ifu,.(w;A) is positive definite. Operator iw Zde t (w;A) is invertible on trig because M+M__fuz (w;A) is positive definite. We then deduce that operator iw Zde t (W;/~k) is invertible on Cc .

375


Due to Eq. (128), there exists )ima x > 0 independent of w such that, for all w in BMF and for all A(w) satisfying [IA(w)l]

IIin(t;v) =

,

,

(189)

(190)

which represent instantaneous random powers. For all fixed t in R, we denote the mathematical expectation of random variable H(t; v) as < II(~_) > - E{H(t; ~_)} ,

(191)

which is independent of time t due to the stationarity of stochastic process {H(t; ~_),t e R}. From Eq. (186) to (191), we deduce that L < II~on~(~_)> + < IIdiss (~)> + E < nmast-fuz (~-) > -- < IIin (v)> s

9 (192)

From Eqs. (118) and (168), and since K(-w) - K(w), we deduce that operator Acons(W;~_) defined by L

(193) ~=l jeJ~

12 The brackets on the right-hand side of Eqs. (187) to (191) denote the antiduality product between CC~ and Co.

387

CHAPTER XV.. FUZZY STRUCTURE THEORY

is a real operator such that

(194)

Acons (-co; y_y_)- Acons (co;y.y_) In addition, for all u and ~u in C~, we have 13 < Acons(W;_Y) u, (~u >--< u, Acons(W; Y_) ~ U >

.

(195)

Since U(t,F_) is a stochastic field with values in ~3 (and not in C 3) and from Eqs. (193) and (194), we deduce that

< IIcons (~) >----0

.

(196)

Substituting Eq. (196) in Eq. (192) yields, for A - O, the mean power flow equation related to frequency band Bm for the master structure coupled with the fuzzy substructures, L

< IIdiss(_Y)> + ~

< IIm~t_fuz(~_)> = < IIi~(~_)>

( 97)

~----1 in which < IIdiss (Y_)> is the mean power dissipated in the master structure, l < IImast_fuz(y_) > is the mean value of the net transmitted power flowing from the master structure to fuzzy substructure ~ and < IIin (Y_) > is the mean input power from external excitation forces applied to the master structure. 9.5. Estimation of mean powers of the real structure using a statistical energy approach

The real structure is constituted by the master structure coupled with L substructures, each substructure ~ being constituted by secondary subsystems which are complex and which have a fuzzy character. This master structure is submitted to a time-stationary stochastic excitation force defined in Section 9.4. For frequency band Bin, a statistical energy approach "S~A ~s used to calculate a rough estimate of the mean power < Hdiss > dissipated in the master structure, the mean value of the net transmitted power __SEA,I l[mast_fuz > flowing from the master structure to substructure ~ across 9

13 The brackets on the left-hand side of Eq. (195) denote the antiduality product between C c~ and C c, and the brackets on the right-hand side of Eq. (195) denote the antiduality product

between CC and CC'.

388

STRUCTURAL

ACOUSTICS AND

VIBRATION

__SEA

interface Ze and the mean input power < ]]in ~ from the external excitation forces applied to the master structure. These mean power quantities are used only in the identificationprocedure (it should be noted that they do not allow us to calculate the local dynamical response of the master structure and a fortiori,the local dynamical responses of the substructures). These mean powers satisfy the mean power balance equations for the master structure and for the substructures. One of these mean power balance equations represents the mean power flow equation for the master structure coupled with the substructures, L

SEA

~'~

SEA,s

SEA

( l'Idiss ) -t- ~ < 1-Imast_fu z ) -- < ]-Iin l=l

)

(198)

9

For instance, these mean power quantities can be estimated using the Statistical Energy Analysis method (see for instance Lyon and DeJong, 1995). 9.6. Estimate of ~ as the solution of a nonlinear constrained optimization problem

An estimate of y_ E [~M is obtained by solving a nonlinear constrained optimization problem based on the use of the mean power flow equation for the master structure coupled with the fuzzy substructures. We then introduce the cost function J(~_) from ~ M into ~+ L

(

J (Y-) - E

s A,

,

< nmast-fuz (E) > -- < IImast-fuz >

)2

'

(199)

g--1

and the space ~)M C [~M of constraints such that, for all ~_ in :DM, /

v~_>O,

~e{1,...,L),

jeffe

,

(200)

SEA

< IIdiss (Y_.) > -- < 1-Idiss >---- 0 < 1-Iin (Y) > -- < IliSnEA > - - 0

, .

(201)

(202)

The estimate of y_ is then obtained by minimizing J(~_) on 7)M, i.e. by solving the following nonlinear constrained optimization problem min J(~_)

.

(203)

~_~)M

The nonlinear constrained optimization problem defined by Eq. (203) is justified as follows. Subtracting the mean power flow equation (198) associated with the statistical energy approach from the mean power flow

389


equation (197) associated with the fuzzy structure theory approach, and using the constraints defined by Eqs. (201) and (202), we deduce that, for all y_ in T)M, we have

L

L

Y~ < IIm~st_fu~.(~_1> = ~ s

SEA ,s

(204)

< IIma~t_fu~.>

s

which justifies the choice of the cost function.

9.7. Solving procedure for the nonlinear constrained optimization problem For y_ fixed in ~)M, we have to calculate the value of cost function J(y_) and the constraints defined by Eqs. (201) and (202), i.e., mean power l quantities < IIdiss(Y_) >, < IImast_fuz(Y_) > and < IIin(Y_) >. Consequently, a finite dimension approximation of the problem is required. For this we can use the Ritz-Galerkin method presented in Section 8.1 or the finite element discretization presented in Section 8.2. For the sake of brevity, we present the Ritz-Galerkin method (the corresponding equations for the finite element discretization are similar).

Ritz-Galerkin projection. Using the notations introduced in Section 8.1, the projection of time-stationary stochastic field U(t; y__)defined in Section 9.4 is written as

N U N (x, t; y) = E

(205)

Qa (t; ~) ba (x)

Then, using the classical spectral analysis formula for linear filtering of stationary stochastic processes (see Chapter IX), we obtain the following approximations

N (~) >-- 2 f w2tr{ [Tdet(W; ~_)]* [V(W)] [Tdet(W; ~)] [S~'(W)] } dw, (206) < IXdiss JBm

< IImast_fuz(~_)>-- 2 jeJl

"~

P} 0)2 tr{[Tdet(W;Y_)]* [~uz,j(W)] [Tdet(W;Y_)][S~(w)]} < IIig(~_)>=

2 ~e/B iw tr{[Tdet(W;y__)][Sy(w)]}dw m

dw,

(207)

(20s)

390


VIBRATION

in which [Tdet(W;F_)] is an (N x N) complex symmetric matrix defined by

[Tdet (0J; Y_)]

- -

[r

(CO;y)]-i

,

(209)

where the (N x N) complex symmetric matrix [.Adet(Cz;Y_)] is such that

[r

Y_)]/~a ----< Adet(OJ; Y_)ba, b~ >

,

(210)

Zt with Adet(W; Y__)given by Eq. (174) , [:D(w)] and [_fu~.,j(w)] are ( N x N ) real positive-semidefinite symmetric matrices defined by

[T>(w)]~a - < D ( w ) ba,bz >

,

[Z~uz,j(w)] -

(211) ,

(212)

and finally, [S~(w)] is an (N x N) complex Hermitian matrix such that [S~(w)]~a = Sb~b, (W), i.e. using Eq. (185), [S~(w)]za = f2o ][B,~o~, (w) / a X(x) dx

(213)

which represents the matrix-valued spectral denSity function of secondorder stationary stochastic process .T(t) - ( : ~ l ( t ) , . . . , ~ g ( t ) ) in which ~'a (t) = < F(t),b~>.

Optimization algorithm. Let y__~ G(F_) - (GI(F_),..., GM+2(Y_))be the function from [~M into [~M+2 defined by

G(~_) -

I

N )2 ]'Idiss (Y) ~ -- < lldiss __SEA (<X'[iN(y) > < ]]in >)2 _ _ S ~ . A

(
7 1, the mean coefficients ~_~l(w), ~(w), n_~3(w), __d3(w) and the associated deviation coefficients A~3,1(w), A~3,2(w), A~3,3(w), A~3,4(w) of the fuzzy impedance law for fuzzy substructure s and for local direction j - 3 are defined below (see Chabas, Desanti and Soize, 1986). We have a~3(w)- 1

,

_

Al3,4(w) - 0 ,

(219)

which corresponds to a type I law. Mean coefficient ~_g_~(w)of fuzzy substructure s for local direction j = 3 is such that

~_~l(w)

I ~/Nolsc mosc I ~ l '

(220)

and Eq. (39) can be rewritten as

~__~((Z) ---- //13(6g) mref

(221)

Then, substituting Eqs. (217) and (220) in Eq. (221) yields the dimensionless mean coefficient u_~3(w)of the participating mass of fuzzy substructure ~ for local direction j = 3, _u~a(a~) ~

1 ~/Notsc

.

(222)

The dimensionless mean rate ~ (w) of internal damping of fuzzy substructure s for local direction j - 3 is such that

~ ( ~ ) - ~osc ~

9

(223)

The mean modal density n_n~(w) related to fuzzy substructure s for local direction j - 3 is such that ~A(~)-

~/Ns162 ~ 9

(224)

Deviation coefficient A~3,1(w) associated with ~_g_~(w), deviation coefficient A~3,2(0)) associated with ~(w) and deviation coefficient A~3,3(w) associated with nl3(w) are given by

A~ 3,k(~) ~ , vf3~

VNo~r

,

V k - 1,2,3

(225)

From Eqs. (220), (224) and (217), we deduce that

l mref For instance, the product

~_.~(w)n_~(w)appears

(226) in Eq. (109).

393

APPENDIX

Mathematical Notations

1. Euclidean Space ~

and Hermitian Space C~

Let 0< be the set ~ of all real numbers or the set C of all complex numbers. T h e m o d u l u s of z = a + ib e C is denoted as [z I = (z~) z/2 = (a 2 q- b2) 1/2 in which -5 - a - ib is the conjugate. Let u = ( u z , . . . , u ~ ) be a vector of ~(n. Vectors are usually d e n o t e d by boldface characters. We identify vector u with the (n • 1) c o l u m n m a t r i x of its uj components. A vector u in C a can be w r i t t e n as u - u R + i u z in which u R = ~ e u and u I - C~m u are the real and i m a g i n a r y p a r t s of vector u. T h e real space ~n, called Euclidean space, is equipped with the usual scalar n (or inner) p r o d u c t (u, v) ~ u . v = ~ j = z u j v j and t h e associated n o r m (u.u) z/2. T h e complex space C n, called H e r m i t i a n space, is equipped n with the H e r m i t i a n scalar (or inner) p r o d u c t (u, v) ~-~ u.Y = ~-~j=z uj~'~ and Ilu[I

-

the associated n o r m Ilvll - (v. V)z/2, where V - v R - i v I is the conjugate o f v = v R + i v I i n C ~.

2. Vector Product Let u and v be two vectors in 0 x,x., is defined by V(x,x*) E X x X*

,

< x , X * > x , x . = b(x,x*)

.

We then identify X* with the space of all the linear forms on X, and X* is called the algebraic dual of X.

4.3. Algebraic transpose of a linear operator on complex vector spaces Let X and Y be complex vector spaces and X* and Y* their algebraic duals. Let B be a linear operator from X into Y. The algebraic transpose of B is the linear operator tB from Y* into X* such that

VxEX

,

Vy* EY*

,

< B x , y* ~y,y.

< x , tBy* ~X,X*

A linear operator B from X into Y is transposable if tB maps Y* into X* and we have the following diagram

B X

>

Y

tB X*

~

Y*

396


VIBRATION

5. Linear and Bilinear Forms and Linear Operators on Complex mlbert Spaces 2 5.1. Complex Hilbert space and dual space Let V be a complex Hilbert space equipped with the inner product (u, v)v and the associated norm

II ull~ = (u, u) ~/" V This inner product is a positive-definite Hermitian (so-called symmetric) sesquilinear form on V x V which is linear with respect to the left argument u and antilinear (or semilinear) with respect to the right argument v, i.e. for any complex number A,

(~ u,

V)v - ~ (u, v)~

,

(u, ~ v)~ - ~ (u, v)~

A linear form u ~ f (u) on V is such that, for all complex number A, y(~ u) = ~ y(u) A linear form u ~-+ f(u) is continuous (or b o u n d e d ) on V if

l:(u)l_w,,w =b(u,v)

,

VuEV

,

VvEW

,

in which the angle brackets < . , . >w,,w denote the duality product between W' and W.

6. A n t i l i n e a r a n d S e s q u i l i n e a r F o r m s and L i n e a r O p e r a t o r s on Complex Hilbert Spaces 3

6.1. Complex Hilbert space, continuous antidual forms and antidual space Let V be a complex Hilbert space equipped with the inner product (u, v)v and the associated norm

II ull

= (n,

V

3 For more details concerning this section, we refer the reader to Reed and Simon (Vol. I), 1980; Kato, 1966; Dautray and Lions, 1992; Yosida, 1966; Dunford and Schwartz, 1967.

398


VIBRATION

This inner product is a positive-definite Hermitian (so-called symmetric) sesquilinear form on V x V which is linear with respect to the left argument u and antilinear (or semilinear) with respect to the right argument v, i.e. for any complex number A, (~u,V)v

- ~(u,v)v

,

(u,~v)~

- ~(u,v)~

An antilinear form u ~ f(u) on V is such that, for any complex number A, y ( ~ u) = ~ y(u) An antilinear form u ~ f(u) is continuous (or bounded) on V if lY(u)l~llullv

,

VueV

,

in which c is a positive real constant. The antidual space V' of V is the set of all the continuous antilinear forms on V. Any continuous antilinear form u ~ f(u) on V can be written as

,

f(u)=v,.v

Vu E V

,

with f E V ~ and in which the angle brackets < . , . >v,,v denote the antiduality product between V ~ and V, i.e. which is linear with respect to the left argument and antilinear with respect to the right argument (if there is no possible confusion in the notation, < . , . >v,,v can be simply denoted as < . ~ . >). 6.2. Bounded operators on complex Hilbert spaces Bounded operators from V into W. Let V and W be two complex Hilbert spaces. A bounded operator B from V into W is a linear mapping from V into W such that IIBullw ~ ~llullv

,

Vu e V

,

in which c is a positive real constant. The set of all the bounded (continuous) operators from V into W is denoted as s W). If W = V, then /:(V, V) is denoted as s If B E s W), then the norm of B is defined by Ilnl[sup Ilnullw 9 Ilullv _<x Adjoint of a bounded operator from V into W. Let V and W be two complex Hilbert spaces, and V ~ and W' their respective antidual spaces.

APPENDIX

- MATHEMATICAL

399

NOTATIONS

Let B E E(V, W) be a bounded operator from V into W. The adjoint operator (or the transpose) of B is the bounded operator tB E E(W', V') from W ' into V' such that

w,,w=v,v

, VuEV

,

VvEW t

,

in which the angle brackets < . , . > v , v (or < " , - - > w , w ) denote the antiduality product between V ~ and V (or W ~ and W ) . I f V (and W) is identified with its antidual space V ~ (and W~), i.e. V = V' (and W = W~), then the adjoint operator of B E E(V, W) is denoted as B*. Operator B* E E(W, V) and is such that (Bu,v)w=(u,B*v)v

,

VuEV

,

VvEW

.

Continuous sesquilinear form on V • V and associated bounded operator from V into V ~. Let V be a complex Hilbert space and V ~ its antidual

space. Let (u, v) ~ b(u, v) be a continuous sesquilinear form on V x V, i.e. which is linear with respect to the left argument u and antilinear with respect to the right argument v. Then there exists a bounded linear operator B E E(V, V') defined by <Bu,v>v,,v-b(u,v)

,

VuEV

,

VvEV

,

in which the angle brackets < . , . >v,,v denote the antiduality product between V ~ and V. The adjoint operator of B is the bounded linear operator tB E E(V, V') such that <Bu,v>v,,v-v,

V,

,

Vu E V

,

Vv E V

.

Continuous Hermitian sesquilinear form and associated self-adjoint operator. Let H be a complex Hilbert space identified to its antidual space

H ' (H = H'). Let (u, v) ~-~ re(u, v) be a continuous sesquilinear form on H • H. Then there exists a bounded linear operator M E E(H) defined by (Mu,v)n =m(u,v)

,

VuEH

,

VvEH

.

Let M* E E(H) be the adjoint operator of M which is such that (Mu,v)~

=(u,M*v)~

,

VuEH

,

VvEH

.

Sequilinear form re(u, v) on H x H is Hermitian (so-called symmetric) if m (u, v) = m (v, u)

,

Vu

H

,

Vv

H

.

400


VIBRATION

If continuous sesquilinear form m(u, v) is Hermitian, then the associated bounded operator M E s is self-adjoint (or Hermitian), i.e. M*

---- M

9

6.3. Unbounded operators on complex Hilbert spaces

Let H be a complex Hilbert space identified to its antidual space H ~ (H -H~). An unbounded operator A in H with domain D(A) dense in H is a linear operator defined on a subset D(A) of H (dense in H) with range into H A 9 D(A) C _ _ ~ H - - + H . The notation D(A) c- ~ H means that D(A) C H and that the embedding (injection) f r o m D(A) into H is dense in H. By definition, A is not a bounded operator (A~ s and Au E H , Vu E D(A). Adjoint of an unbounded operator in H. The adjoint of unbounded operator A in H with domain D(A) is the unbounded operator A* in H with domain D(A*) such that (Au,v).-(u,A*v).

,

VuED(A)

,

VvED(A*)

.

Symmetric unbounded operator in H. Unbounded operator A in H with domain D(A) is symmetric if (Au,v).

=(u,Av).

,

VueD(A)

,

VveD(A)

.

Self-adjoint unbounded operator in H. Let A be an unbounded operator in H with domain D(A) and let A* be its adjoint which is an unbounded operator in H with domain D(A*). Unbounded operator A is self-adjoint if D(A*) = D(A) and

(Au.v).-(u.Av).

.

VueD(A)

.

VvED(A)

.

that is to say A* - A It should be noted that for an unbounded operator A, if A is selfoadjoint, then A is symmetric, but the converse is not true. An unbounded operator A can be symmetric and not self-adjoint (if D(A*) r D(A)).

APPENDIX

- MATHEMATICAL

401

NOTATIONS

6.4. Bounded and unbounded operators on a triplet of complex Hilbert spaces

Let V and H be two complex Hilbert spaces such that the embedding (injection) V c ; H from V into H is dense and continuous. Let V ~ and H ~ be the antidual spaces and we identify H with H ~ (H ~ - H). We then have the triplet of complex Hilbert spaces V c-

H-

H' r

V'

the embeddings being dense and continuous. Let (u, v) ~ a(u, v) be a continuous sesquilinear form on V • V which is coercive (V-elliptic) u)}

c Ilull v

,

Vu e V

,

and Hermitian (so-called symmetric) a (u, v) - a (v, u)

,

VueV

,

(then ~e{a(u, u)} - a(u, u) _ 0). Let A e operator associated with a(u, v) such that v,,v-a(u,v)

,

s

VvsV

W) be

VuEV

,

,

the bounded linear

VvEV

,

in which the angle brackets < . , . >v,,v denote the antiduality product between V ~ and V . Let AH be the restriction of A E s W) to H, defined on the domain D(AH) such that D(AH)-{uEV;

AuEH}

.

Then, AH is an unbounded (closed) operator in H with domain D(AH) dense in H and A H is self-adjoint A~ --AH

9

Generally, in order to simplify the notation the subscript H is omitted. 6.5. Case of real Hilbert spaces

If the Hilbert spaces V, W and H are real, then the terms antilinear form, antidual space, antiduality product, sesquilinear form, Hermitian sesquilinear form, are replaced respectively by linear form, dual space, duality product, bilinear form, symmetric bilinear form. For instance, if H is a

402


VIBRATION

real Hilbert space, a bilinear form m(u, v) on H x H is symmetric on H if

re(u, ~) = m(~, u). 7. Gradient, Divergence, Curl and Laplacian Operators Let x ~-+ f(x) be a real- or complex-valued function defined on ~3 and x ~ v(x) be a function defined on ~3 with values in ~3 or C 3. We denote the gradient operator with respect to x as V,

of

( V f ) j = Oxj

'

je{1'2'3}

Then, V f(x) is a vector in ~3. The Laplacian operator is written as V 2 = $7. W. The divergence and the curl of function x ~ v(x) with respect to x are written as V - v and V x v respectively.

8. Fourier Transform of Functions 4 In this section, we recall some important results concerning the Fourier transform used throughout the book.

Spaces LI(~) and L2(~). Let LI(~) be the set of all integrable functions from ~ into ~ or C. A function t ~ h(t) belonging to L ~(~) is such that

IlhllL' -- Z

Ih(t)l dt < +c~

Let L2(~) be the set of all square integrable functions from ~ into ~ or C. A function t ~ h(t) belonging to L2(N) is such that Ilhll~ = Z Ih(t)12 dt < +c~ From a mathematical point of view, these sets are defined in the context of measure and integration theory and consequently functions h are only defined almost everywhere on R (see for instance Halmos, 1976; Reed and Simon, 1980). 4 For more details concerning this section, we refer the reader to Halmos, 1976; Dautray and Lions, 1992; Schwartz, 1966; Soize, 1993a.

403

APPENDIX- MATHEMATICAL NOTATIONS

Fourier transform in LI(~). The Fourier transform of a function h belonging to L~(~) is a continuous function w ~ h(w) from ~ into C (using the customary abusive notation) such that, for all real w,

h(w) - j f e -i~t h(t) dt and such that lim h(w) = 0 Iwl~+oo Consequently, h does not generally belong to L 1(~) but is a bounded function on ~. If h belongs to L 1(~), then there is an inverse Fourier transform such that, for all real t in R,

h(t) -

1 j f eiWt h(w) dw

If h does not belong to L 1(~), but if h is, in addition, a bounded variation function on any closed and bounded interval of ~, we then have, for all real t (the Fourier-Dirichlet theorem), lim

1 1 / f f e i~t h(w)dw - ~ ( h ( t + ) + h(t_))

a- +Too ~

a

Fourier transform in L2([R). The Fourier transform of a function h belonging to L2(~) is a function w ~ h(w) from ~ into C, belonging to L2(~), such that, for almost all real w,

h(w) - j f e -iWt h(t) dt If t ~ h(t) belongs to L2(~), then w ~ h(w) belongs to L2(~) and consequently its inverse Fourier transform is in L2(~) and such that, for almost all real t,

h(t) - ~1 j f eiWt h(w) dw

,

and we have the P lancherel equality Ih(t)l 2 dt - ~

Ih(w)l 2 dw

404


VIBRATION

Fourier transform of the derivatives of functions on ~. If the first and second derivatives (in the usual or generalized sense) of h with respect to t are functions belonging to LI(R) or L2(R), we then have h(w) = iw h(w)

,

h(w) = - w 2 h(w)

,

in which h(w) and h(w) denote the Fourier transform of functions t ~ and t ~ h(t) respectively. oo

h(t)

Fourier transform in L2(R"). Let t - (tl,... ,t,) and ~ 0 - (wl,... ,w,) be in Rn. We denote the Euclidean scalar product of ~o with t as to. t = wit1 + . . . + W n t n

,

and the Lebesgue measure on Rn related to variables t and to as dt = d t l . . , dtn and d~o = d w l . . , dwn respectively. Let L2(~ n) be the set of all the square integrable functions from Rn into R or C. A function t ~ h(t) belonging to L 2 (Rn) is such that

Ih(t)l 2 dt < + ~

Ilhll , =

The Fourier transform of a function h belonging to L2(~ n) is a function to ~-~ h(to) from Rn into C, belonging to L2(Rn), such that, for almost everywhere to in Rn, h(~o) = j r . e -i ~'t h(t)dt If t ~ h(t) belongs to L2(~n), then w ~ h(w) belongs to L2(~ ~) and consequently its inverse Fourier transform is in L2(~ n) and such that, for almost all t in ~n, 1

h(t)-(27r)~

jf

.

e i ~. t h(to) dt~

and we have the P lancherel equality

Ih(t)l 2 d t -

1 (27r)n

= Ih(to) i

Remarks concerning Fourier transforms. In general, we use either an abusive notation consisting in using the same symbol for a quantity and its

APPENDIX - MATHEMATICALNOTATIONS

405

Fourier t r a n s f o r m , i.e. denoting the Fourier t r a n s f o r m as h(w), or a n o t h e r symbol if convenient, i.e. denoting the Fourier t r a n s f o r m of t ~ h(t) as T(w).

9. Indicator Function aB Let X be any set and let B be any subset of X. T h e indicator function x ~ aiB(X) from X into ~ is such t h a t liB(X) = 1 if X E B a n d liB(X) = 0 i f x 6 B.

10. Order Symbols o and 0 Let f a n d g be a r b i t r a r y real functions of x defined in the vicinity of x = 0. We are interested in the behavior of function f (x) if x --+ 0. We use symbols O a n d o which are defined as follows. T h e expression

f = o(g) m e a n s t h a t function I f ( x ) / g ( x ) ] is b o u n d e d by a finite positive c o n s t a n t if x ~ 0. Therefore the function O(g) converges to a positive c o n s t a n t which is not zero. For instance, f = O(1) means t h a t f is a b o u n d e d function in the vicinity of x = 0. We can see t h a t sin x = O(x) because for x --+ 0, (sin x ) / x - 1 + x 2/6 + .... T h e expression

f

= o(g)

m e a n s t h a t function f ( x ) / g ( x ) --+ 0 if x --+ 0. Therefore o(g) --+ 0 if x --+ 0. For instance, o(1) means t h a t f ( x ) ~ 0 if x --+ 0, a n d 1 - c o s x - o(x) because (1 - c o s x ) l x = x / 2 - x318 + ....


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417

Subject Index

Acoustic - D i s s i p a t i v e a c o u s t i c fluid, ls3 fluid, 169 impedance boundary operator, 221,246

- I n t e r n a l a c o u s t i c d i s s i p a t i o n , 179 - Internal acoustic boundary value p r o b l e m , 190 - I n v i s c i d a c o u s t i c fluid, 173 - m a t e r i a l c o a t i n g , 175 - m o d e s , 197 Radiation impedance operator, 221,247

- r e a c t a n c e , 174 - r e s i s t a n c e , 174 Scattering of an acoustic source, 227

Boundary dynamic stiffness operator, 81,349

Boundary element method, 249 Boundary impedance operator - E x t e r n a l a c o u s t i c fluid, 221,246 - S u b s t r u c t u r e , 76

Boundary integral equation - First boundary integral equation, 237

- Second boundary integral equat i o n , 239 - E x t e r i o r N e u m a n n p r o b l e m , 237

Boundary layer

- Wall viscothermical boundary layer, 179

Boundary value problem

- s o u r c e d e n s i t y , 173 - W a l l b o u n d a r y c o n d i t i o n , 174 - Wall.impedance boundary c o n d i t i o n , 174

- C o n s e r v a t i v e s t r u c t u r e , 23 - Coupled conservative substruct u r e s , 41 - Coupled dissipative substructures,

-

-

Admissible function space, 11

C o n s e r v a t i v e s t r u c t u r e , 23,24 C o u p l e d s u b s t r u c t u r e s , 77 E x t e r n a l a c o u s t i c fluid, 220 I n t e r n a l a c o u s t i c fluid, 192,210 M a s t e r s t r u c t u r e , 65,72,73 Structural-acoustic master system i n t h e LF_ r a n g e , 261,299 - Structural-acoustic master system in t h e M F r a n g e , 313,335 Antisymmetric moae, 34,35

Axisymmetric

- a c o u s t i c c a v i t y , 214 - s t r u c t u r e , 34 structural-acoustic master system, 304,340

Band - M F n a r r o w b a n d , 99 - M F b r o a d b a n d , 106

Bandwidth, 99 Boundary condition - D i r i c h l e t , 64,178,186 - N e u m a n n , 10,64,178,186,219 W a l l b o u n d a r y c o n d i t i o n , 174 W a l l i m p e d a n c e , 174

79

D i s s i p a t i v e m a s t e r s t r u c t u r e , 64 E x t e r n a l a c o u s t i c fluid, 219 I n t e r n a l a c o u s t i c fluid, 190,209 Structural-acoustic master system i n t h e LF_ r a n g e , 259,297 - Structural-acoustic master system in t h e M F r a n g e , 311,334 Cauchy initial condition, 21

Causal linear filter, 112 Condensation - G u y a n c o n d e n s a t i o n , 48 - S t a t i c c o n d e n s a t i o n , 4s

Conservative

- e l a s t o d y n a m i c p r o b l e m , 20 - s t r u c t u r e , 21

Constitutive equation - Dissipative constitutive equation, 51

- E l a s t i c c o n s t i t u t i v e e q u a t i o n , 21 Viscoelastic constitutive equation, 53,55,58

Correlation function - C r o s s - c o r r e l a t i o n f u n c t i o n , 142, 145,154,162


418 - Matrix-valued autocorrelation f u n c t i o n , 143,144,155,162 C u t o f f f r e q u e n c y , 359 Cyclic symmetry, 36,250

Damping - D i a g o n a l i z e . d o p e r a t o r , 91 - H y s t e r e t i c d a m p i n g , 95 - m a t r i x o f t h e m a s t e r s t r u c t u r e , 67 model with frequency-dependent c o e f f i c i e n t s , 55,58 - model with frequency-independe n t c o e f f i c i e n t s , 53,58 operator of the master structure, 69 - R a y l e i g h d a m p i n g , 93 - T e n s o r of t h e d a m p i n g coefficients, 55

Deterministic

- D i a g o n a l i z e . d o p e r a t o r , 91 H y s t e r e t i c . d a m p i n g , 95 m a t r i x o f t h e m a s t e r s t r u c t u r e , 67 Deviation coefficients of the f u z z y impedance l a w , 363 Dirichlet boundary condition, 64,178, 186 Dispersion relation, 183

Displacement potential, Dissipation

176

- I n t e r n a l a c o u s t i c d i s s i p a t i o n , 179

Dissipative - a c o u s t i c fluid, 183 - c o n s t i t u t i v e e q u a t i o n , 58 - Helmholtz equation, lsl

Double-layer potential, Dynamic

233

- s t i f f n e s s o p e r a t o r , 70 - s u b s t r u c t u r i n g , 41

Eigenfrequency

- I n t e r n a l a c o u s t i c fluid, 198 - S t r u c t u r a l - a c o u s t i c , 283 - S t r u c t u r e , 23

Eigenmode, 23 Eigenvalue problem

- I n t e r n a l a c o u s t i c fluid, 197,213 - S t r u c t u r a l - a c o u s t i c , 283,304 - S t r u c t u r e . 23,28,40 Elastic coefficients - T e n s o r of t h e elastic coefficients, 21 Elastic structural mode, 30 Elastodynamic equation, 20,63 External acoustic fluid, 217 - w i t h a f r e e s u r f a c e , 253

Filter -

Causal Master Simple Stable

Energy

l i n e a r filter, 112 s t r u c t u r e . 73 linear oscillator, ll2 l i n e a r filter, i l 3

-

VIBRATION

K i n e t i c e n e r g y , 120 Mechanical energy, li4 o p e r a t o r , 120 Vibration, ll5

Entropy - e q u a t i o n , 180 - S p e c i f i c e n t r o p y , 180

Finite element discretization,

15 - Acoustic impedance operator, 249 - A c o u s t i c r a d i a t i o n o p e r a t o r , 251 - Boundary impedance operator, 249 - D y n a m i c stiffness o f t h e m a s t e r s t r u c t u r e , 75 - E n e r g y o p e r a t o r , 124 - F u z z y s t r u c t u r e , 379 - Internal acoustic eigenvalue probl e m , 199,213 - Internal acoustic fluid FRF, 196,202,205 - M a s t e r s t r u c t u r e F R F , 75,90 - Structural-acoustic master system i n t h e LF. r a n g e , 270,276,291,302 - Structural-acoustic master system in t h e M F r a n g e , 320,331,337 - S t r u c t u r a l e i g e n v a l u e p r o b l e m , 40

Fluid -

A c o u s t i c fluid, 169 D i s s i p a t i v e a c o u s t i c fluid, 183 I n v i s c i d a c o u s t i c fluid, 173 T h e r m a l c o n d u c t i o n o f t h e fluid, 179 - V i s c o s i t y , 179

Free structure, Free surface

22,25

- E x t e r n a l a c o u s t i c fluid, 253 - I n t e r n a l a c o u s t i c fluid, 209

Frequency range

- High-frequency range, 3 - Low-frequency range, 3 - Medium-frequency range, 4

Frequency response function (FRF) - Finite element discretization, 75,90,196,202,205 - I n t e r n a l a c o u s t i c fluid, 195,201 - M a s t e r s t r u c t u r e , 71 - M a s t e r s t r u c t u r e in t h e L F r a n g e , 86 - M a s t e r s t r u c t u r e in t h e M F r a n g e , 98,120 - Simple linear oscillator, ll2 - S t r u c t u r a l - a c o u s t i c s y s t e m in t h e L F r a n g % 278,293,302 - S t r u c t u r a l - a c o u s t i c s y s t e m in t h e M F r a n g e , 325,332,338

Fundamental solution, Fuzzy

224

- Homogeneous fuzzy impedance law, 357

419

SUBJECT INDEX - i m p e d a n c e law d e v i a t i o n coefficients, 370 i m p e d a n c e law m e a n coefficients, 370

- S t r u c t u r a l - a c o u s t i c fuzzy s y s t e m , 3

- structure, 2 - s t r u c t u r e c o n c e p t , 344 s t r u c t u r e t h e o r y , 345 s u b s t r u c t u r e , 2,345

Generalized - c o o r d i n a t e s , 14,87,200 e i g e n v a l u e p r o b l e m for t h e internal a c o u s t i c fluid, 199 e i g e n v a l u e p r o b l e m for t h e struct u r e , 29,33 - forces, 88 - S t r u c t u r a l g e n e r a l i z e d m ~ s , 29 - s y m m e t r i c m a t r i x eigenvalue p r o b l e m , 40

Green's formula, 11,18 Green's function, 224 Guyan condensation, 48 Kirchhoff-Fourier equation, ls0 Helmholtz equation - Dissipative Helmholtz equation, 185

- for a n e x t e r n a l a c o u s t i c p r o b l e m , 219

- for a n inviscid fluid, 177 - w i t h viscosity a n d t h e r m a l cond u c t i o n , 181

Helmholtz integral representation - in a n e x t e r n a l d o m a i n , 234 - in a n i n t e r n a l d o m a i n , 235

HF, 3

- range, 3

High-frequency range, 3 Hysteretic damping, 95 Impedance - A c o u s t i c i m p e d a n c e b o u n d a r y ope r a t o r , 221,246 - B o u n d a r y i m p e d a n c e o p e r a t o r for a s u b s t r u c t u r e , 76 F u z z y i m p e d a n c e law d e v i a t i o n coefficients, 370 F u z z y i m p e d a n c e law m e a n coefficients, 370 - H o m o g e n e o u s fuzzy i m p e d a n c e law, 357 - m a t r i x of t h e m a s t e r s t r u c t u r e , 75 - o p e r a t o r of t h e m a s t e r s t r u c t u r e , 70,86,98

- Radiation impedance operator, 221,247

- R a n d o m b o u n d a r y i m p e d a n c e ope r a t o r . 354 - T y p e f fuzzy i m p e d a n c e law, 368 - T y p e II fuzzy i m p e d a n c e law, 368 - Wail i m p e d a n c e b o u n d a r y condi-

tion, 174

Impulse response function - I n t e r n a l a c o u s t i c fluid, 196 - M a s t e r s t r u c t u r e , 73 - S i m p l e oscillator, 112 Incident plane wave, 217,226

Initial condition

- C a u c h y initial c o n d i t i o n , 21

tegrai representation for the Helmholtz equation - in a n e x t e r n a l d o m a i n , 234 - in a n i n t e r n a l d o m a i n , 235 Internal acoustic problem, 190 - w i t h a zero p r e s s u r e c o n d i t i o n , 209

Inviscid acoustic fluid, 173 Irregular frequency, 244 LF, 3 - range, 3

Low-frequency range, 3 Mass - c o n s e r v a t i o n e q u a t i o n , 172 - F l u i d m a s s d e n s i t y at e q u i l i b r i u m , 171

- S t r u c t u r a l g e n e r a l i z e d mass, 29 - S t r u c t u r a l m a s s m a t r i x , 40,75 - S t r u c t u r a l m a s s o p e r a t o r , 26,67 Master structure, 2

Mean

- coefficient of t h e p a r t i c i p a t i n g m a s s of a fuzzy s u b s t r u c t u r e , 362 e q u i v a l e n t c o u p l i n g f a c t o r of a fuzzy s u b s t r u c t u r e , 362 f u n c t i o n of a n o n s t a t i o n a r y r a n d o m r e s p o n s e , 161 - m o d a l d e n s i t y of a fuzzy s u b s t r u c t u r e , 362 - p o w e r b a l a n c e e ~ u a t i o n s , 388 - p o w e r flow e q u a h o n , 386 - r a t e of internal d a m p i n g of a fuzzy s u b s t r u c t u r e , 362

Medium-frequency range, MF, 4

4

- b r o a d b a n d , 106 - n a r r o w b a n d , 99 - range, 4 s o l v m g m e t h o d for t h e fuzzy s t r u c t u r e , 377 " - solving m e t h o d for t h e i n t e r n a l a c o u s t i c fluid, 205 - solving m e t h o d for t h e m a s t e r s t r u c t u r e , 102 - solving m e t h o d for t h e s t r u c t u r a l a c o u s t i c m a s t e r s y s t e m , 332,338

Mode -

A c o u s t i c m o d e , 197,213 A n t i s y m m e t r i c m o d e , 34 Cyclic s y m m e t r y , 36 E l a s t i c s t r u c t u r a l m o d e , 29,33 E n t r o p y m q d e , 18o F i x e d - i n t e r f a c e m o d e , 43,48


420 -

R i g i d b o d y m o d e , 29,33 S t r u c t u r a l m o d e , 21,33 S t r u c t u r a l - a c o u s t i c m o d e , 283,304 _ S y m m e t r i c m o d e , 34 V o r t i c i t y m o d e , 180 Narrow MF band, 99

Navier-Stokes equation, 18o Neumann - b o u n d a r y c o n d i t i o n , 10,64,178,186, 219 - E x t e r i o r N e u m a n n p r o b l e m , 218 Normal derivative, 10

Operator - A c o u s t i c b o u n d a r y i m p e d a n c e ope r a t o r , 221,246 - B o u n d a r y d y n a m i c stiffness o p e r a t o r , 81,349 - Boundary impedance operator o f a s u b s t r u c t u r e , 7'8 - E n e r g y o p e r a t o r , 120 - L i f t i n g o p e r a t o r z 44 - Radiation impedance operator, 221,248 - R a n d o m b o u n d a r y i m p e d a n c e ope r a t o r , 354 - S t r u c t u r a l d y n a m i c stiffness o p e r a t o r , 70,86,98 - Structural impedance operator, 7'0,86,98

Orthogonality conditions - A c o u s t i c m o d e s , 198 - S t r u c t u r a l m o d e s , 28,31 Plancherei formula, 403

Plane wave - I n c i d e n t p l a n e w a v e , 226

Potential - D o u b l e - l a y e r p o t e n t i a l , 232 - S i n g l e - l a y e r p o t e n t i a l , 232 _ - t h e o r y , 231

"Power - d i s s i p a t e d , 116 - e q u a t i o n , 116 I n p u t p o w e r , 116 T o t a l i n p u t p o w e r , 116 T o t a l p o w e r d i s s i p a t e d , 116

Pressure -

F l u i d p r e s s u r e a t e q u i l i b r i u m , 17'1 Incident 227 R a d i a t e d ppr er es ss suur ree' , 226,230 R e s u l t a n t p r e s s u r e , 228,230 S c a t t e r e d p r e s s u r e , 228

Probabilistic - m e c h a n i c a l model, 351 - Homogeneous probabilistic _ i m p e d a n c e l a w 2 357' Quasi-static correctmn term, 94,151,203 Radiated pressure, 226,230

Radiation - c o n d i t i o n , 220 - i m p e d a n c e o p e r a t o r , 221,247'

VIBRATION

- b y a n e l a s t i c b o u n d a r y , 226

Random - E x t e r n a l r a n d o m p r e s s u r e field exc i t a t i o n , 306 - b o u n d a r y i m p e d a n c e o p e r a t o r , 354 Nonstationary random excitation, 153 - Nonstationary random response, 160 - Time stationary random excitat i o n , 141 - Time stationary random response, 144,150,152 - Turbulent boundary layer excitat i o n , 305 - u n c e r t a i n t i e s , 348

Resonant frequency - o.f a f u z z y s u b s t r u c t u r e for a l o c a l d i r e c t i o n , 357 - of t h e s t r u c t u r a l - a c o u s t i c m a s t e r s y s t e m , 282 Rayleigh damping, 93

Reduced matrix model -

C o n s e r v a t i v e s u b s t r u c t u r e , 46 D i s s i p a t i v e s u b s t r u c t u r e , 82 I n t e r n a l a c o u s t i c fluid, 251 Structural-acoustic master system in t h e L F r a n g e , 274,303 Reduced model in the MF range, 123

Rigid body - d i s p l a c e m e n t field, 27 - m o d e s , 30,33

Ritz-Galerkin - a p p r o x i m a t i o n m e t h o d , 14,87, 200,378,389 - c o n v e r g e n c e , 16 cattered pressure field, 228,230

Scattering - b y a n a c o u s t i c s o u r c e , 227 - b y a r i g i d boun_dary, 227,229 - o f a n i n c i d e n t p l a n e w a v e , 229 Single-layer potential, 232

Solution - S t r o n g s o l u t i o n , 10 - W e a k s o l u t i o n , 12

Sommerfeld radiation condition, 220 Spectral density function - Cross-spectral density function, 142,146 - Matrix-valued spectral density f u n c t i o n , 143,145

Static - c o n d e n s a t i o n , 48 - f u n c t i o n of a s u b s t r u c t u r e ,

44

Stiffness - D y n a m i c stiffness o p e r a t o r , r0,7'5 - S t r u c t u r a l stiffness m a t r i x , 40,75 - S t r u c t u r a l stiffness o p e r a t o r , 26 Stokes's formula, 24

Strain tensor, 21

SUBJECT INDEX

421

Stress tensor - D a m p i n g p a r t , 57 - E l a s t i c p a r t , 57

Structural - acoustic fuzzy system, a - acoustic master system, 2 - a c o u s t i c m o d e , 283,304 a n t i s y m m e t r i c m o d e , 34 complexity, 2 E l a s t i c s t r u c t u r a l m o d e , 29 - m o d e s , 23,33 m o d e s of a n a x i s y m m e t r i c s t r u c t u r e , 34 - s p e c t r a l p r o b l e m , 23 s y m m e t r i c m o d e , 34

Structure -

A x i s y m m e t r i c s t r u c t u r e , 34 Complex structure, 2 C o n s e r v a t i v e s t r u c t u r e , 21 D i s s i p a t i v e s t r u c t u r e , 51 F u z z y s t r u c t u r e , 2,344 Master structure, 2 Primary structure, 2 w i t h cyclic s y m m e t r i e s , 36 w i t h a p l a n e of s y m m e t r y , 33

Substructure - F i x e d - i n t e r f a c e m o d e , 43 - F u z z y s u b s t r u c t u r e , 2,345 - R e d u c e d m a t r i x m o d e l , 46

Substructuring - D y n a m i c s u b s t r u c t u r i n g , 41

ymmetric mode, Symmetry

34

- C y c l i c s y m m e t r y , 36 - P l a n e of s y m m e t r y , 33

Test-function method, 11 Thermal conduction of the fluid, 179 Thermodynamic equations of state, is0 Turbulent boundary layer - External excitation, 305 Variational formulation, 11 - Coupled conservative substruct u r e s , 43 - Coupled dissipative substructures, 77,81

- First boundary integral equation, 239

- internal acoustic boundary value . p r o b l e m , 192,210 - Internal acoustic spectral proble m , 197 - M a s t e r s t r u c t u r e , 66 - Second boundary integral equat i o n , 240 - Structural-acoustic master system i n t h e LF_ r a n g e , 266,288,300 - Structural-acoustic master system i n t h e M F r a n g e , 316,329,335 - S t r u c t u r a l s p e c t r a l p r o b l e m , 25 Velocity potential, z76,184 Viscoelastic constitutive equation, 55

Viscoelasticity - w i t h m e m o r y , s3 _- w i t h o u t m e m o r y , 2 Viscosity, 170

Wall - b o u n d & r y c o n d i t i o n , 174 imped&nee boundary condition, 174

- v i s e o t h e r m i e & l b o u n d a r y layer, 170


423

Symbol Index

a i j k h , 21,57 bijkh, 57

CE, 219 CO, 172

fq, 194 p, 173 p', 260 Pgiven, 231 Pinc, 229 Pinc,Q, 227 Prad, 226 P E , 217

PrE, 221 damp q , 57,64 Sz, 194 v, 218 ec~, 122 g, 62,63 h(t), 74 u, 20,62 uelas, 30,72 Urig, 27 Uwall , 174 ua, 28,31 v, 173 x, 20, 62 E, 141 G, 224 H , 262 HFE, 220 L2(rs), 222 c

O, 173,21~ Z, 174

BF~, 221 EBb,n, 121 G, 62,63 I, 269 L, 272 P', 270

Pgive.(w), 252 P~(~), 252 n(x, ~/~), 221 U, 75 V, 249

Zr.d (X, W), 221 Zr~(~), 221 [Br~ (wick)I, 250 [Vr~], 251 [Zr.d (X, ~)], 252 [Zr ~ (~)], 250 6kh, 21,64

~:, 18o r/, 180 ~;, 191

~(~), 17z r l , 191 ~z2, 191

%, 191

p, 21,63 PE, 219 Ps, 259 Po, 171 a i j , 21,64 a q , j , 22,64 aelas ij , 57,64

424 r, 181

~o, 176

~, 184 r IZ6,218 r 217

STRUCTURAL ACOUSTICS AND c

C,65 Cel~, 172 CS, 73 C~, 77 c

c

c

Crad, 226 ~rig, 227 erE, 220

CO, 65

184

~ , 192

r

FE, 217 Aw, 99 ~E, 217 ~v, 99 IIz, 207 II2, 271 ,I,, 196 9 , 321 ~ r ~ , 249

c

c

~p,, 261 ~ , 192 E~, 313 c

c

c

Eo, 21o Co,p, 299 Eo,,, 299 ~o,r 335 c

c

C, 23

~LF' 95 ~MF' 106

Crig , 27

[By, 99 [Bu, II0

C~,=, 30 Cry, 220 C' 220 CnE, 220 CO, 24

[Bo, 99 C m, 393 Rm 393

VIBRATION

Structural Acoustics and Vibration

Structural Acoustics and Vibration

Computational aspects of structural acoustics and vibration