دانلود کتاب Mechanical Vibrations II

فهرست مطالب :

7. Modal analysis 1
7.1 Modes of vibration 1
7.2 Real undamped natural modes 2
7.2.1 Undamped non-gyroscopic systems 3
7.2.1.1 Normalization of real modal vectors 5
7.2.1.2 Orthogonality of real modal vectors 5
7.2.1.3 Modal matrix 6
7.2.1.4 Free vibration solution 6
7.2.1.5 Undamped forced vibration 8
7.2.1.6 Excitation modal vectors 9
7.2.2 Systems with proportional damping 10
7.2.2.1 Viscous damping 10
7.2.2.2 Structural damping 12
7.3 Complex damped natural modes 14
7.3.1 Viscous damping 14
7.3.2 Structural damping 23
7.4 Forced monophase damped modes 26
7.4.1 Analysis based on the dynamic stiffness matrix 26
7.4.2 Analysis based on the dynamic flexibility matrix 37
7.4.3 Proportional damping 43
7.5 Rigid-body modes 47
7.5.1 Flexibility method 47
7.5.2 Stiffness method 53
7.6 Modal participation factors 57
References 59
8. Eigenvalue solvers 61
8.1 Structural dynamics eigenproblem 61
8.2 Transformation to standard form 62
8.2.1 Cholesky factorization of the mass matrix 62
8.2.2 Shift-and-invert spectral transformation 63
8.3 Determinant search method 64
8.4 Matrix transformation methods 65
8.4.1 The eigenvalue decomposition 66
8.4.2 Householder reflections 67
8.4.3 Sturm sequence and bisection 68
8.4.4 Partial Schur decomposition 69
8.5 Iteration methods 71
8.5.1 Single vector iterations 71
8.5.1.1 The power method 72
8.5.1.2 Wielandt deflation 74
8.5.1.3 Inverse iteration 74
8.5.2 The QR method 76
8.5.3 Simultaneous iteration 78
8.5.4 The QZ method 79
8.6 Subspace iteration methods 80
8.6.1 The Rayleigh-Ritz approximation 80
8.6.2 Krylov subspaces 82
8.6.3 The Arnoldi method 82
8.6.3.1 Arnoldi’s algorithm 83
8.6.3.2 Generation of Arnoldi vectors 83
8.6.3.3 The Arnoldi factorization 85
8.6.3.4 Eigenpair approximation 88
8.6.3.5 Implementation details 90
8.6.4 The Lanczos method 91
8.7 Software 95
References 96
9. Frequency response non-parametric analysis 99
9.1 Frequency response function matrices 99
9.1.1 Frequency response functions 100
9.1.2 2D FRF matrices 101
9.1.3 3D FRF matrices 102
9.2 Principal response analysis of CFRF matrices 102
9.2.1 The singular value decomposition 102
9.2.2 Principal response functions 104
9.2.3 The reduced-rank AFRF matrix 109
9.2.4 SVD plots 111
9.2.5 PRF plots 112
9.2.6 Mode indicator functions 114
9.2.6.1 The UMIF 114
9.2.6.2 The CoMIF 114
9.2.6.3 The AMIF 116
9.2.7 Numerical simulations 119
9.2.8 Test data example 1 127
9.3 Analysis of the 3D FRF matrices 131
9.3.1 The CMIF 131
9.3.2 Eigenvalue-based MIFs 133
9.3.2.1 The MMIF 133
9.3.2.2 The MRMIF 135
9.3.2.3 The ImMIF 137
9.3.2.4 The RMIF 137
9.3.3 Single curve MIFs 140
9.3.4 Numerical simulations 142
9.3.5 Test data example 1 146
9.4 QR decomposition of the CFRF matrices 147
9.4.1 Pivoted QR factorization of the CFRF matrix 148
9.4.2 Pivoted QLP decomposition of the CFRF matrix 150
9.4.3 The QCoMIF 152
9.4.4 The QRMIF 153
9.4.5 Test data example 2 154
References 161
10. Structural parameter identification 165
10.1 Models of a vibrating structure 165
10.2 Single-mode parameter extraction methods 167
10.2.1 Analysis of receptance data 167
10.2.1.1 Peak amplitude method 167
10.2.1.2 Circle fit method 169
10.2.1.3 Co-quad components methods 181
10.2.1.4 Phase angle method 182
10.2.2 Analysis of mobility data 183
10.2.2.1 Skeleton method 183
10.2.2.2 SDOF mobility data 187
10.2.2.3 Peak amplitude method 188
10.2.2.4 Circle-fit method 189
10.2.3 Base excited systems 190
10.3 Multiple-mode parameter extraction methods 194
10.3.1 Phase separation method 194
10.3.2 Residues 197
10.3.3 Modal separation by least squares curve fit 199
10.3.4 Elimination of the modal matrix 200
10.3.5 Multipoint excitation methods 203
10.3.6 Appropriated excitation techniques 204
10.3.7 Real frequency-dependent characteristics 208
10.3.7.1 Characteristic phase-lag modes 208
10.3.7.2 Best monophase modal vectors 216
10.3.7.3 Eigenvectors of the coincident FRF matrix 217
10.4 Time domain methods 227
10.4.1 Ibrahim time-domain method 227
10.4.2 Random decrement technique 230
References 232
11. Dynamic model reduction 237
11.1 Reduced dynamic models 237
11.1.1 Model reduction philosophy 238
11.1.2 Model reduction methods 240
11.2 Physical coordinate reduction methods 242
11.2.1 Irons-Guyan reduction 242
11.2.1.1 Static condensation of dynamic models 242
11.2.1.2 Practical implementation of the GR method 245
11.2.1.3 Selection of active DOFs 247
11.2.2 Improved Reduced System (IRS) method 249
11.2.3 Iterative IRS method 252
11.2.4 Dynamic condensation 258
11.2.5 Iterative dynamic condensation 259
11.3 Modal coordinate reduction methods 261
11.3.1 Definitions 261
11.3.2 Modal TAM and SEREP 262
11.3.3 Improved Modal TAM 265
11.3.4 Hybrid TAM 269
11.3.5 Modal TAMs vs. non-modal TAMs 269
11.3.6 Iterative Modal Dynamic Condensation 271
11.4 Hybrid reduction methods 275
11.4.1 The reduced model eigensystem 275
11.4.2 Exact reduced system 276
11.4.3 Craig-Bampton reduction 278
11.4.4 General Dynamic Reduction 279
11.4.5 Extended Guyan Reduction 280
11.4.6 MacNeal’s reduction 282
11.5 FRF reduction 283
References 284
12. Test-analysis correlation 287
12.1 Dynamic structural modeling 287
12.1.1 Test-analysis requirements 288
12.1.2 Sources of uncertainty 290
12.1.3 FRF based testing 291
12.2 Test-analysis models 293
12.3 Comparison of modal properties 299
12.3.1 Direct comparison of modal parameters 299
12.3.2 Orthogonality criteria 300
12.3.2.1 Test Orthogonality Matrix 301
12.3.2.2 Cross Orthogonality Matrix 301
12.3.3 Modal vector correlation coefficients 302
12.3.3.1 Modal Scale Factor 302
12.3.3.2 The Modal Assurance Criterion 302
12.3.3.3 Normalized Cross Orthogonality 306
12.3.3.4 The AutoMAC 306
12.3.3.5 The FMAC 306
12.3.4 Degree of freedom correlation 311
12.3.4.1 Coordinate Modal Assurance Criterion 311
12.3.4.2 Enhanced CoMAC 312
12.3.4.3 Normalized Cross Orthogonality Location 312
12.3.4.4 Modulus Difference 313
12.3.4.5 Coordinate Orthogonality Check 314
12.3.5 Modal kinetic energy 314
12.4 Comparison of FRFs 314
12.4.1 Comparison of individual FRFs 315
12.4.2 Comparison of sets of FRFs 316
12.4.2.1 Frequency Response Assurance Criterion 317
12.4.2.2 Response Vector Assurance Criterion 318
12.4.2.3 Frequency Domain Assurance Criterion 319

12.5 Sensor-actuator placement 320
12.5.1 Selection of active DOFs / Sensor placement 320
12.5.1.1 Small stiffness / large inertia criterion 320
12.5.1.2 Effective independence method (EfI) 321
12.5.1.3 Sensor location with Arnoldi and Schur vectors 326
12.5.1.4 Selection of the candidate set of sensors 333
12.5.2 Exciter placement 334
12.5.2.1 Preselection by EfI method 334
12.5.2.2 Use of synthesized FRF data 334
12.5.2.3 Final selection using MMIF 335
12.5.3 Input/output test matrix 337

References 340
Index 343