Structural Acoustics and Vibration

437 Pages·1997·14.95 MB·English

by  Roger Ohayon

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 Office National d'Etudes et de
Arts et M6tiers (CNAM) Recherches A6rospatiales (ONERA)
Paris, France Chatillon, France
ACADEMIC PRESS
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Copyright (cid:14)9 1998 ACADEMIC PRESS LIMITED
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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 ..................................................... xi
Chapter I. A Strategy for Structural-Acoustic Problems .............. 1
I. Introduction, 1
2. Structural-acoustic master system, 2
3. Concept of fuzzy structure and structural-acoustic fuzzy system, 2
4. LE MF and HF frequency ranges, 3
5. Strategy used for the LF and MF ranges, 4
6. Excitations and responses, 5
7. Organization of the book, 6
Chapter II. Bask Notions on Variational Formulations ............... 9
1. Introduction, 9
2. Boundary value problem, 9
3. Strong solution of the boundary value problem, 10
4. Variational formulation and weak solution of the boundary value
problem, 11
5. Converse, 12
6. Associated linear operator equation, 13
7. Ritz-Galerkin approximation and finite element method, 14
8. 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. Introduction, 51
2. Notation for the Fourier transform, 52
3. Damping model with frequency-independent coefficients, 52
4. Model with frequency-dependent coefficients based on the linear theory of
viscoelasticity, 53
5. Summary, 57
Chapter V. Master Structure Frequency Response Function .......... 61
1. Introduction, 61
2. Equations in the frequency domain, 62
3. Variational formulation, 65
4. Linear operators and algebraic properties, 66
5. Frequency response function, 70
6. Finite element discretization, 75
7. Boundary impedance operator, 76
Chapter VI. Calculation of the Master Structure Frequency Response Func-
tion 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 Func-
tion 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 frequency-
dependent 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. Introduction, 109
2. Definition of a narrow MF band, 110
3. Energy properties of a simple-linear-oscillator response, 110
4. Variational formultaion of the master structure in the MF range, 117
5. 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 ..... 129
1. Introduction, 129
2. Operator-valued FRF and associated linear filter in the time domain, 129
3. LF and MF deterministic cases: time-periodic excitation, 134
4. LF and MF deterministic cases: time square integrable excitation, 138
5. LF and MF random cases: time stationary stochastic excitation, 141
6. 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. Introduction, 189
2. Statement of the internal acoustic problem, 190
3. Boundary value problem, 190
4. Variational formulation and linear operator equation, 192
5. Frequency response function, 195
6. Finite element discretization, 195
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. Introduction, 215
2. Statement of the external acoustic problem, 217
3. Exterior Neumann problem related to the Helmholtz equation, 218
4. Acoustic impedance boundary operator and radiation impedance
operator, 221
5. Free-space Green's function and solution of the inhomogeneous Helmholtz
equation, 224
6. Response to prescribed wall displacement, acoustic source density and
incident plane wave, 226
7. Results of potential theory: single- and double-layer potentials, 231
8. Helmholtz integral representations and their normal derivatives, 233
9. Boundary integral equations for the exterior Neumann problem related to
the Helmholtz equation, 237
10. Acoustic impedance boundary operator construction: appropriate symmet-
ric boundary integral method valid for all real values of the fre-
quency, 244
11. Construction of the radiation impedance operator and asymptotic formula
for the radiated pressure field, 246
12. Symmetric boundary element method valid for all real values of the fre-
quency, 249
13. Case of a free surface, 253
Chapter XIII. Structural-Acoustic Master System in the LF range .... 255
1. Introduction, 255
2. Statement of the structural-acoustic problem in the frequency domain, 256
3. Boundary value problem of the structural-acoustic master system, 259
4. Variational formulation of the structural-acoustic problem, 261
5. Finite element discretization, 269
6. Symmetric reduced matrix model in the LF range, 272
7. FRF calculation in the LF range, 278
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. Introduction, 307
2. Statement of the structural-acoustic problem in the frequency domain, 308
3. Boundary value problem of the structural-acoustic master system, 311
4. Variational formulation of the structural-acoustic problem, 313
5. Finite element discretization, 320
6. FRF calculation in the MF range, 323
7. Case of a master structure coupled with an external acoustic fluid, 329
8. 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 ........................... 343
1. Introduction, 343
2. Statement of the problem, 344
3. Fuzzy structure theory stated as an inverse problem, 348
4. Random equation of the master structure coupled with fuzzy substructures
in terms of the displacement field of the master structure, 353
5. Homogeneous model of a fuzzy substructure, 355
6. Construction of a homogeneous fuzzy impedance law for a fuzzy substruc-
ture and for a local direction, 358
7. Solving method for the random equation of the master structure coupled
with fuzzy substructures, 372
8. Ritz-Galerkin approximation and finite element discretization, 377
9. Identification method for the parameters of type I and type II fuzzy imped-
ance laws, 381
10. 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