Computational Methods in Structural Dynamics

Computational Methods in Structural Dynamics
اسم المؤلف
Leonard Meirovitch
التاريخ
3 أغسطس 2019
المشاهدات
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Computational Methods in Structural Dynamics
Leonard Meirovitch
Reynolds Metals Professor
Department of Engineering Science and Mechanics
Virginia Polytechnic Institute and State UniversityContents
Chapter 1. Concepts from linear algebra
1.1 Introduction
1.2 Linear vector spaces
1.3 Linear dependence
1.4 Bases and dimension of a vector space
1.5 Inner products and orthogonal vectors
1.6 The Gram Schmidt orthogonalization process
1.7 Matrices
1.8 Basic matrix operations
1.9 Determinants
1.10 Inverse of a matrix
1.11 Partitioned matrices
1.12 Systems of linear equations
1.13 Matrix norms
Chapter 2. Free vibration of discrete systems 29
2.1 Introduction
2.2 The system equations of motion
2.3 Small motions about equilibrium points
2.4 Energy considerations
2.5 Free vibration and the eigenvalue problem
Chapter 3. The eigenvalue problem
3.1 General discussion
3.2 The general eigenvalue problem
3.3 The eigenvalue problem for real symmetric matrices
3.4 Geometric interpretation of the eigenvalue problem
3.5 Hermitian matrices
3.6 The eigenvalue problem for two nonpositive definite real
symmetric matrices
3.7 The eigenvalue problem for real nonsymmetric matrices
XIContents Contents
6.8 Response of general dynamical systems 217
6.9 Discrete-time model for general dynamical systems
6.10 Stability of motion in the neighborhood of equilibrium 223
Chapter 4. Qualitative behavior of the cigensolution
4.1 Introduction
4.2 The Rayleigh principle
4.3 Rayleigh’s theorem for systems with constraints
4.4 Maximum-minimum characterization of eigenvalues
4.5 The inclusion principle
4.6 A criterion for the positive definiteness of a Hcrmitian
matrix
4.7 Eigenvalues of the sum of two Hcrmitian matrices
4.8 Gerschgorin’s theorems
4.9 First-order perturbation of the eigenvalue problem
Chapter 7. Vibration of continuous systems 229
7.1 Introduction
7.2 Lagrange’s equation for continuous systems. Boundaryvalue problem
7.3 The eigenvalue problem
7.4 Self-adjoint systems
7.5 Non-self-adjoint systems
7.6 Vibration of rods, shafts and strings
7.7 Bending vibration of bars
7.8 Two-dimensional problems
7.9 Variational characterization of the eigenvalues
7.10 Integral formulation of the eigenvalue problem
7.11 The response problem
Chapter 5. Computational methods for the eigensolution 261
5.1 General discussion
5.2 Gaussian elimination
5.3 Reduction to triangular form by elementary row operations
5.4 Computation of eigenvectors belonging to known eigenvalues
5.5 Matrix iteration by the power method
5.6 Hotelling’s deflation
5.7 Wielandt’s deflation
5.8 The Cholesky decomposition
5.9 The Jacobi method
5.10 Givens’ method
5.11 Householder’s method
5.12 Eigenvalues of a tridiagonal symmetric matrix. Sturm’s
theorem
5.13 The QR method
5.14 The Cholesky algorithm
5.15 Eigenvectors of a tridiagonal matrix
5.16 Inverse iteration
Chapter 8. Discretization of continuous systems 285
8.1 Introduction
8.2 The Rayleigh-Ritz method
8.3 The assumed-modes method
8.4 The method of weighted residuals
8.5 Flutter of a cantilever aircraft wing
8.6 Integral formulation of the method of weighted residuals 319
8.7 Lumped-parameter method employing influence coefficients
8.8 System response by approximate methods
Chapter 9. The finite element method 328
9.1 Introduction
9.2 Second-order problems. Linear elements
9.3 Higher-degree elements. Interpolation functions
9.4 Fourth-order problems
9.5 Two-dimensional domains
9.6 Errors in the eigenvalues and eigenfunctions
9.7 Inconsistent mass matrices
Chapter 6. Response of discrete systems
6.1 Introduction
6.2 Linear systems. The superposition principle
6.3 Impulse response. The convolution integral
6.4 Discrete-time systems
6.5 Response of undamped nongyroscopic systems
6.6 Response of undamped gyroscopic systems
6.7 Response of damped systems
Chapter 10. Systems with a large number of degrees of freedom 368
10.1 Introduction
10.2 Static condensation
XII XIIIContents
10.3 Mass condensation
10.4 Simultaneous iteration
10.5 Subspace iteration
10.6 The method of sectioning
Chapter 11. Substructure synthesis
11.1 General discussion
11.2 Component-mode synthesis
11.3 Branch-mode analysis
11.4 Component-mode substitution
11.5 Substructure synthesis
Bibliography 410
Suggested problems 415
Author index
432
Subject index 434
XIVAuthor index
Smolitskiy, K. L., 413
Stahle, C. V., 391, 412
Stewart, G. W., 167, 414
Stoker
, J. R., 371, 414
Strang, G., 266, 336, 365, 367, 414
Wilkinson, J. H„ 142, 149, 152, 155, 163.
164, 165, 166, 167, 414
Wilson, W. L., 378, 410
Wright, G. C., 372, 414
Author index Topp, L. J., 328, 414 Ziegler, H., 34, 414
Turner, M. J., 328, 414
Huebner. K. H., 339, 359, 363, 411
Hurty, W. C., 82, 384, 391, 411. 412
Huseyin, K., 34, 412
Bajan, R. L., 391, 410
Bamford
, R. M., 391, 410
Bampton, M. C. C., 390, 410
Bathe
, K. J., 378, 410
Bell, K., 355, 411
Benfield, W. A., 384, 395, 401, 410
Bolotin, V. V., 34, 410
Bowman
, F., 272, 410
Irons
, B., 370, 412
Issid
, N. T., 30, 413
Jaszlics, 1. J., 391, 410
Jennings, A., 373, 374, 412
Jensen, P. S„ 380, 381, 382, 412
Kublanovskaya, V. N., 163, 412
Kuhar, E. J., 391, 412
Carey, G. F., 412
Caughey, T. L., 211, 410
Clint
, M., 373, 374, 410
Clough, R. W., 328, 414
Courant, R., 245, 271, 279, 280, 293, 328,
350
, 410
Craig, R. R., 390, 410
Lancaster, P., 82, 412
Likins. P. W., 34, 412
Martin, H. C., 328, 412
Meirovitch, L., 30, 31, 38, 40, 45, 46, 47,
81, 146, 189, 190, 192, 202, 205, 207,
208, 212, 215, 225, 231, 232, 244, 248,
255
, 265, 269, 271, 274, 280, 299, 384,
402, 403, 405, 407, 409, 412, 413, 414
Melosh, R. J.. 328, 413
Mikhlin, S. G., 413
Miles
, G. A., 372, 414
Mingori, D. L., 34, 413
Murdoch, D. C., 20, 24, 55, 58, 413
Daniel, J. W., 9, 10, 28, 152, 413
Dowell, E. H., 319, 391, 410, 411
Feng, C. G, 391, 410
Finlayson, B. A., 411
Fix, G. J., 266, 336, 365, 367. 414
Francis
, J. G. F., 163, 411
Franklin, J. N.. 55, 67, 100, 121, 129, 411
Friedman, B., 253, 411
Fung, Y. C„ 253, 313, 316, 411
Gallagher, R. H„ 364, 411 Noble, B., 9, 10, 28. 152, 413
Gladwell
, G. M. L., 384, 391, 411
Goldman, R. L., 391, 411
Gould, S. H , 411
Guyan, R. J., 371, 411
Paidoussis, M . P., 30, 413
Przemieniecki, J. S., 371, 414
Ralston, A., 162, 414
Ramsden, J. N., 371, 413
Rayleigh (Lord). 288, 414
Reinsch, C., 167, 414
Rubin, S., 391, 414
Rutishauser, H., 163, 414
Ryland , G., 202, 414
Hale, A. L„ 384, 402. 407, 409, 411, 413
Hilbert, D., 245, 271, 279, 280, 293, 410
Hintz
, R. M., 391 , 411
Holland, I., 355, 411
Hou
, S. N., 391, 411
Hruda, R. F., 384, 395, 401, 410
432
433Subject index
Damping forces, 30
Damping matrix, 35
hysteretic, 212
Deflation (matrix):
Hotelling’s, 128-134
Wielandt’s
, 134-135
Degenerate system, 270, 272
Degrees of freedom, 29
Determinant(s), 16-19
cofactor, 17
Gramian, 26
minor, 17
Dirac delta function, 186
spatial, 282
Discrete-time systems, 192-199, 221-223
Dissipation function;
general, 35
Rayleigh’s, 30
Dual expansion theorem, 72
Dynamic potential, 32
Element stiffness matrix, 335, 341
Energy functions, 249
Energy inner product, 249
Energy space, 249
Equilibrium point:
nontrivial, 32
trivial, 32
Errors (in eigensolution), 364
Expansion theorem:
continuous systems,
non-self-adjoint, 254
self-adjoint, 252
discrete systems,
damped, 69
dual, see non-self-adjoint
Hermitian, 67
non-self-adjoint, 72
real symmetric, 60
Subject index
Adjoint eigenvalue problem :
algebraic, 72
differential
, 253
Adjoint operator, 253
Admissible functions, 249
Amplitude, 44
Approximation in the mean, 247
Area coordinates, 353
Asymptotically stable equilibrium,
Characteristic values, see eigenvalues
Characteristic vectors, see eigenvectors
Cholesky algorithm, 172-176
Cholesky decomposition, 135-138
Circulatory forces, 34
Circulatory matrix, 35
Classical modal analysis, 201
for damped systems, 210
Collocation method, 304-306
Comparison functions, 247
Complete set, 247
in energy, 250
Complex stiffness matrix, 213
Component modes, 385
Condensed eigenvalue problem, 370
Condensed mass matrix, 372
Configuration space, 31
Conservation of energy, 41
Conservative forces, 41
Conservative systems, 40
gyroscopic, 45
nongyroscopic, 41
Consistent mass matrix, 366
Constraint damping forces, 34
Convergence:
in energy, 250
in the mean
, 247
Convolution integral, 189
Convolution sum, 197
Coriolis forces, 30
Coriolis matrix, 35
Critical behavior, 225
225, 226
Finite element method, 328-367
Finite elements:
one-dimensional,
cubic, 342
linear, 332, 339
quadratic. 339
two-dimensional,
quadrilateral, 359
rectangular, 359
triangular, 351
Flapping mode, 263
Flutter
, 312-319
Free vibration:
continuous systems, 239
discrete systems, 29-49
distributed systems, see continuous systems
Frequency equation:
continuous systems, 241
discrete systems, 42
distributed systems, see continuous systems
Back substitution
, 112
Bandwidth of matrix, 336
Bessel inequality, 246
Bilinear elements, 359
Biorthogonality:
of functions, 253
of vectors, 71
Biorthonormality:
of functions, 254
of vectors, 71
Bisection method
, 160
Boundary conditions, 234
dynamic, 238
essential, 238
geometric, 237
natural, 238
Boundary-value problem 234
Branch modes
, 392
Eigenfunctions, 242
Eigensolutions, 43
Eigenvalue problem:
continuous systems, 241
integral form, 280
discrete systems,
conservative gyroscopic, 45
conservative nongyroscopic, 42
damped nongyroscopic, 47
general dynamical, 48
distributed systems, see continuous systems
Eigenvalue problem (algebraic):
adjoint, 72
damped systems, 68-70
general, 50
for Hermitian matrices
, 65-68
for real nonsymmetric matrices, 70-72
for real symmetric matrices, 56-65
self -adjoint, 72
Eigenvalues, 42. 50, 242
multiplicity, 51
repeated, 46, 51
shift in. 52
Eigenvectors, 42, 50
left
, 70
right, 71
Element mass matrix, 335. 341
Centrifugal forces, 30
Characteristic determinant, 42, 51
Characteristic equation:
continuous systems, 241
discrete systems, 42, 51
distributed systems, see continuous
Characteristic functions, see eigenfunctions
Characteristic polynomial, 51
Characteristic value problem, see eigenvalue
problem
Galerkin equations, 289
Galerkin method, 303-304
Gaussian elimination
, 1 11-115
Gauss-Jordan reduction, 114
Generalized coordinates, 30
Generalized forces
, 31
Generalized momenta
, 31
Damping:
constraint, 34
proportional, 211
structural
, 212
viscous, 30
Damping coefficients, 30
434 435r
Subject index Subject index
Generalized velocities, 30
Generalized virtual displacements, 31
Gerschgorin theorems, 99-101
Givens’ method, 146-151
Global mass matrix, 336
Global stiffness matrix, 336
Gram-Schmidt orthogonalization, 8-11
modified
,
9
Green’s function, 279
Group property, 221
Gyroscopic coefficients, 34
Gyroscopic forces, 30
Gyroscopic matrix, 35
nonnatural systems, 30
Kinetic energy density, 231 Modal Modal truncation vectors, 44, 201
Moments, method of, 308
normal, 66
null, or zero, 13
nullity of, 24
orthonormal, 58
partitioned, 21
permutation, 16, 116
positive definite, 36
positive semidefinite, 36
rank of, 23
rectangular, 11
rotation, 139
row, 13
sign-variable, 36
similar, 53
singular, 18
skew symmetric, 13
square, 12
order of, 12
symmetric, 13
trace of, 56
transpose of, 12
tridiagonal, 12
unit
,
or identity, 12
unitary, 66
Matrix(ces), special:
circulatory, 35
Coriolis, see gyroscopic
damping, 35
gyroscopic, 35
inertia, 35
mass, 35
stiffness, 35
Matrix diagonalization, 140
Matrix iteration (power method), 123-128
Maximum-minimum theorem:
continuous systems, 277
discrete systems, 89
Maxwell’s reciprocity theorem, 279
Mean square error, 246
Minimax theorem, 90
Modal analysis:
continuous systems, 281-284
discrete systems,
damped systems, 210-217
general dynamical systems, 217-221
undamped gyroscopic systems, 204-210
undamped nongyroscopic systems, 199-
204
Modal matrix
, 62
Lagrange’s equations:
discrete systems, 31
distributed systems, 233
Lagrangian, 31
Lagrangian density, 232
Laplace operator, 268, 271
Least squares method, 306-308
Linear combination:
of functions, 245
of vectors, 3
Linear transformation, 52
Linearized system, 224
Linearly dependent:
functions, 245
vectors, 4
Linearly independent:
functions, 245
vectors, 3
Local basis
, 332
Local coordinates
, 334
Lumped-parameter method, 321-322
Natural coordinates, 201
Natural coordinates (in finite element
method), 334, 352
Natural frequencies, 44, 242
Natural modes, 44, 242
Natural systems, 40
Newton-Raphson method, 162
Nodal vector, 336
Nodes
, 271
Nodes (in finite element method), 332
external, 340
internal, 340, 342
Nonconservative forces, 41
Nonnatural system, 30
Non-self-adjoint systems:
continuous, 252-255, 312-319
discrete, 316
Norm:
energy, 249
Euclidean, 7
of functions, 245
of vectors, 7
Normal coordinates, 201, 283
Normal modes, 61, 242
Normalization:
of functions, 242
of vectors, 7, 58, 61
Half-bandwidth, 358
Hamiltonian
, 40
Hermite cubics
, 347
Hotelling deflation, 128-134
Householder method, 151-157
Hysteretic damping matrix, 212
Impulse response, 187
Inclusion principle, 90-94
Inconsistent mass matrix, 366
Inertia coefficients
, 33
Inertia matrix, 35
Influence function
, 279
Initial value problem, 234
Inner product:
energy, 249
of functions, 244
of vectors, 6
Integral operator, 279
Intermediate structure, 407
Interpolation functions:
cubic, 342
linear, 335, 339
quadratic, 339
Inverse iteration
, 177-182
Isoparametric elements, 364
Mass coefficients
, 33, 288
Mass condensation
, 370
Mass matrix
, 35, 289
Matrix(ces):
adjoint, 13
adjoint of, 20
augmented, 24
banded, 19
block-diagonal, 22
cofactor, 19
column, 13
determinant of, 16, 56
diagonal, 12
dimensions of, 12
Gramian
, 26
Hermitian, 13
Hessenberg, 12
identity, or unit, 12
inverse of
, 19
negative definite, 36
negative semidefinite, 36
nonsingular, 18
norm of, 27
Euclidean
, 28
Orthogonal functions, 245
Orthogonal matrices, 58
Orthogonalization:
of functions, 293
of vectors, 8-11
Orthonormal functions, 245
Orthonormal vectors, 8
Perturbation of eigenvalue problem, 102-
Jacobi integral, 40
Jacobi method, 138-146
Jordan form, 55
109
Phase angle, 44
Positive definite system:
continuous, 250
discrete, 44
distributed, see continuous
Positive, semidefinite system:
continuous, 250
Kinetic energy:
discrete systems, 30
distributed systems, 231
natural systems, 40
436 437Subject index
Subject index
Positive, semidefinite system: (Corct.)
discrete, 44
distributed, see continuous
Potential energy:
discrete systems, 30
distributed systems, 235
Potential energy density, 231
Principal coordinates, 201
Proportional damping, 211, 327
Pyramid functions, 350
QL method, 167
OR method, 162-172
Quadrilateral elements, 359
Sampler, 194
Sampling period, 194
Sectioning, method of , 380
Self-adjoint eigenvalue problem:
algebraic, 72
differential, 248
Self -adjoint systems:
continuous, 248
discrete, 72
distributed
, see continuous
Shape functions, 335
Shift in eigenvalues, 52, 166
Significant behavior, 225
Similarity transformation, 53
Simultaneous iteration, 373
Space:
configuration, 31
phase, 32
state, 32, 45
vector, see vector space
Spatial Dirac delta function, 282
Square summable functions, 245
Stability of motion, 223-228
Stable equilibrium, 225, 226
State vector
, 32, 45
Static condensation, 369
Stiffness coefficients:
elastic, 34, 288
geometric, 34
Stiffness matrix, 35, 289
complex, 212
Structural damping, 212
Structural damping factor, 212
Sturm sequence, 158
Sturm theorem, 159
Subdomains, method of, 308
Subspace iteration, 377
Superposition principle, 185
Sylvester’s criterion, 95
Synchronous motion:
continuous systems, 239
discrete systems, 41
System:
conservative, 41
damped, 47
gyroscopic, 40
linear, 35
linearized, 33
natural, 40
linearly independent, 3, 52
modal, 44
orthogonal, 8
orthonormal, 8
space(s), see vector space(s)
state
,
32
unit, 7
Vector space(s), 1-11
basis for, 5
column, 23
dimension of, 5
generating system of, 5
null, 24
row, 23
spanned, 4
standard basis for, 6
subspace, 3
Virtual work, 31
Virtual work density, 231
Viscous damping, 30
nonconservative, 41
nonlinear
, 38
nonnatural
, 30
unrestrained, 44
Test functions, 301
Test space, 302
Transformation:
linear, 52
orthonormal, 64
similarity, 53
unitary, 67
Transition matrix
, 220, 222
Trial functions, 274, 301
Trial space, 302
Triangular decomposition , 118
Rayleigh dissipation function, 30
Rayleigh energy method , 292
Rayleigh principle, 74-84
Rayleigh quotient:
continuous systems, 274
discrete systems, 74
conservative gyroscopic, 81
conservative nongyroscopic, 80
distributed systems, see continuous
Rayleigh-Ritz method, 286-298
Rayleigh theorem, 84-88
Rectangular elements, 359
Response:
continuous systems, 281-284
discrete systems,
conservative gyroscopic, 204-210
conservative nongyroscopic, 199-204
damped nongyroscopic, 210-217
general dynamical, 217-221
discrete-time systems, 192-199, 221-223
discretized systems,
damped nongyroscopic, 326-327
undamped gyroscopic, 324-326
undamped nongyroscopic, 322-324
distributed systems, see continuous systems
Unit impulse, 186
Unit step function, 187
Unitary transformation, 67
Unstable equilibrium, 225, 226
Weighted residuals method, 301-312
integral formulation, 319-321
Weighting functions, 301
Weyl’s theorem, 97
Wielandt deflation, 134-135
Vector(s):
characteristic, 42
configuration, 35
displacement, see configuration
force, 35
linearly dependent, 4 Zero-order hold, 195
single-degree-of-freedom systems, 186-
192
Ritz system, 289
eigenfunctions, 290
eigenvalues, 290
Rigid-body modes, 44
Roof functions, 332
Rotary inertia, 248
Rotation matrix
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