Mechanical Vibrations II

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