Nonlinearity in Structural Dynamics: Detection, Identification and Modelling

NONLINEARITY IN STRUCTURAL DYNAMICS: Detection, Identification and Modelling......Page 1
Contents......Page 5
Preface......Page 12
1.1 Continuous-time models: time domain......Page 17
1.2 Continuous-time models: frequency domain......Page 26
1.3 Impulse response......Page 29
1.4 Discrete-time models: time domain......Page 33
1.5.2 Moving-average (MA) models......Page 37
1.6 Discrete-time models: frequency domain......Page 38
1.7 Multi-degree-of-freedom (MDOF) systems......Page 39
1.8.1 Free, undamped motion......Page 45
1.8.2 Free, damped motion......Page 51
1.8.3 Forced, damped motion......Page 53
2.2.1 Definition of linearity—the principle of superposition......Page 57
2.2.2 Harmonic distortion......Page 62
2.2.3 Homogeneity and FRF distortion......Page 65
2.2.4 Reciprocity......Page 67
2.3.1 Cubic stiffness......Page 68
2.3.3 Piecewise linear stiffness......Page 71
2.3.4 Nonlinear damping......Page 72
2.4 Nonlinearity in the measurement chain......Page 73
2.4.1 Misalignment......Page 74
2.5 Two classical means of indicating nonlinearity......Page 75
2.5.1 Use of FRF inspections—Nyquist plot distortions......Page 76
2.5.2 Coherence function......Page 78
2.6 Use of different types of excitation......Page 81
2.6.1 Steady-state sine excitation......Page 82
2.6.2 Impact excitation......Page 83
2.6.4 Random excitation......Page 84
2.7 FRF estimators......Page 85
2.8.1 Theory......Page 88
2.8.2 Application to Duffing’s equation......Page 92
2.8.3 Experimental approach......Page 94
3.2 Harmonic balance......Page 97
3.3 Harmonic generation in nonlinear systems......Page 104
3.4 Sum and difference frequencies......Page 106
3.5 Harmonic balance revisited......Page 107
3.6 Nonlinear damping......Page 109
3.7.1 Quadratic stiffness......Page 111
3.7.2 Bilinear stiffness......Page 114
3.8 Application of harmonic balance to an aircraft component ground vibration test......Page 117
3.9.1 Nyquist plot: linear system......Page 121
3.9.2 Nyquist plot: velocity-squared damping......Page 123
3.9.3 Nyquist plot: Coulomb friction......Page 124
3.9.4 Carpet plots......Page 125
3.10 Inverse FRFs......Page 127
3.11 MDOF systems......Page 128
3.12.1 The method of slowly varying amplitude and phase......Page 138
3.12.2 Linear damping......Page 140
3.13 Summary......Page 141
4.1 Introduction......Page 143
4.2.1 A relationship between real and imaginary parts of the FRF......Page 144
4.3 Computation......Page 148
4.3.1 The direct method......Page 149
4.3.2 Correction methods for truncated data......Page 151
4.3.2.2 The Fei correction term......Page 152
4.3.2.4 The Simon correction method......Page 153
4.3.2.5 The Ahmed correction term......Page 154
4.3.3 Fourier method 1......Page 158
4.3.4 Fourier method 2......Page 165
4.3.5 Case study of the application of Fourier method 2......Page 169
4.4 Detection of nonlinearity......Page 172
4.4.1 Hardening cubic stiffness......Page 176
4.4.3 Quadratic damping......Page 177
4.4.4 Coulomb friction......Page 179
4.5 Choice of excitation......Page 181
4.6.1 NPR: non-causal power ratio......Page 184
4.6.3 Spectral moments......Page 186
4.7 Measurement of apparent damping......Page 189
4.8 Identification of nonlinear systems......Page 191
4.8.1 FREEVIB......Page 196
4.8.2 FORCEVIB......Page 205
4.9 Principal component analysis (PCA)......Page 206
5.2 Hilbert transforms from complex analysis......Page 218
5.3 Titchmarsh’s theorem......Page 221
5.4 Correcting for bad asymptotic behaviour......Page 223
5.4.1 Simple examples......Page 225
5.4.2 An example of engineering interest......Page 227
5.5 Fourier transform conventions......Page 231
5.6 Hysteretic damping models......Page 233
5.7 The Hilbert transform of a simple pole......Page 239
5.8 Hilbert transforms without truncation errors......Page 240
5.9 Summary......Page 244
6.1 Introduction......Page 246
6.2 Linear discrete-time models......Page 248
6.3.1 Parameter estimation......Page 249
6.3.2 Parameter uncertainty......Page 251
6.4 The effect of noise......Page 253
6.5 Recursive least squares......Page 258
6.6 Analysis of a time-varying linear system......Page 262
6.7.1 Choice of input signal......Page 265
6.7.2 Choice of output signal......Page 267
6.7.3 Comments on sampling......Page 268
6.7.4 The importance of scaling......Page 269
6.8 NARMAX modelling......Page 271
6.9 Model validity......Page 273
6.9.2 Model predicted output......Page 274
6.9.3 Correlation tests......Page 275
6.10 Correlation-based indicator functions......Page 276
6.11 Analysis of a simulated fluid loading system......Page 277
6.12 Analysis of a real fluid loading system......Page 289
6.13.1 Introduction......Page 293
6.13.2 A linear system......Page 298
6.13.3 A nonlinear system......Page 299
6.13.3.2 A nonlinear model......Page 300
7.1 Introduction......Page 301
7.2.1 Basic theory......Page 302
7.2.2 Interpolation procedures......Page 306
7.2.3.1 A linear system......Page 308
7.2.3.2 A Van der Pol oscillator......Page 313
7.2.3.3 Piecewise linear systems......Page 315
7.3.1 Basic theory......Page 321
7.3.2 Some examples......Page 326
7.4.1 Basic theory......Page 331
7.4.2 Display without interpolation......Page 335
7.4.2.1 Sections......Page 336
7.4.2.2 Crawley/O’Donnell surfaces......Page 337
7.4.3 Simple test geometries......Page 338
7.4.3.1 Transmissibility or base excitation......Page 339
7.4.3.2 Mass grounded......Page 340
7.4.4.1 Low excitation tests......Page 341
7.4.4.2 High excitation test......Page 349
7.4.5 Application to measured shock absorber data......Page 350
7.5.1 Basic theory......Page 357
7.5.2 Experiment: linear system......Page 362
7.5.3 Experiment: nonlinear system......Page 366
7.6 System identification using optimization......Page 371
7.6.1.1 Genetic algorithms......Page 372
7.6.1.2 A linear system......Page 374
7.6.1.3 A piecewise linear system......Page 377
7.6.1.4 A hysteretic system......Page 378
7.6.2.1 The hyperbolic tangent model......Page 383
7.6.2.2 Gradient descent parameter estimation......Page 385
7.6.2.3 Results using experimental data......Page 388
7.6.2.4 Discussion......Page 392
8.1 The Volterra series......Page 393
8.2 An illustrative case study: characterization of a shock absorber......Page 396
8.3 Harmonic probing of the Volterra series......Page 402
8.4 Validation and interpretation of the higher-order FRFs......Page 410
8.5 An application to wave forces......Page 420
8.6.1 The FRF......Page 421
8.6.2 Hilbert transform......Page 427
8.6.2.1 Pole–zero form of the Duffing oscillator FRF......Page 428
8.6.2.2 Partial fraction expansion......Page 430
8.7 FRFs and Hilbert transforms: random excitation......Page 432
8.7.1 Volterra system response to a white Gaussian input......Page 434
8.7.2 Random excitation of a classical Duffing oscillator......Page 437
8.8 Validity of the Volterra series......Page 447
8.9 Harmonic probing for a MDOF system......Page 450
8.10 Higher-order modal analysis: hypercurve fitting......Page 454
8.10.1 Random excitation......Page 456
8.10.2 Sine excitation......Page 460
8.11 Higher-order FRFs from neural network models......Page 466
8.11.1 The Wray–Green method......Page 468
8.11.2 Harmonic probing of NARX models: the multi-layer perceptron......Page 471
8.11.3 Radial basis function networks......Page 474
8.11.4 Scaling the HFRFs......Page 476
8.11.5.1 Results from MLP network......Page 478
8.11.5.2 Results from radial basis function networks......Page 479
8.11.5.3 Discussion......Page 480
8.12 The multi-input Volterra series......Page 482
8.12.1 HFRFs for a continuous-time MIMO system......Page 483
8.12.2 HFRFs for a discrete-time MIMO system......Page 489
9.1 An encastre beam rig......Page 493
9.1.1 Theoretical analysis......Page 494
9.1.2 Experimental analysis......Page 497
9.1.2.1 Linear analysis......Page 498
9.1.2.2 FRF distortion and Hilbert transform......Page 499
9.1.2.3 Static test......Page 501
9.1.2.4 Higher-order FRFs......Page 503
9.1.2.5 Direct parameter estimation (DPE)......Page 507
9.2 An automotive shock absorber......Page 509
9.2.1 Experimental set-up......Page 510
9.2.2.1 Front absorber (F)......Page 517
9.2.2.2 Rear absorber (R)......Page 521
9.2.3 Polynomial modelling......Page 523
9.2.4 Conclusions......Page 526
9.3 A bilinear beam rig......Page 527
9.3.1 Design of the bilinear beam......Page 528
9.3.2 Frequency-domain characteristics of the bilinear beam......Page 534
9.3.3 Time-domain characteristics of the bilinear beam......Page 539
9.3.4 Internal resonance......Page 542
9.3.5 A neural network NARX model......Page 546
9.4 Conclusions......Page 547
A.1 Basic definitions......Page 549
A.2 Random variables and distributions......Page 550
A.3 Expected values......Page 553
A.4 The Gaussian distribution......Page 557
Appendix B: Discontinuities in the Duffing oscillator FRF......Page 559
C.2 Energy conservation......Page 562
C.3 Commutation with differentiation......Page 563
C.4 Orthogonality......Page 564
C.5 Action as a filter......Page 565
C.6 Low-pass transparency......Page 566
Appendix D: Frequency domain representations of Æ(t) and  (t)......Page 568
E.1 Orthogonal least squares......Page 570
E.2 Singular value decomposition......Page 576
E.3.1 Normal equations......Page 578
E.3.4 Recursive least squares......Page 579
F.1 Biological neural networks......Page 582
F.1.1 The biological neuron......Page 583
F.1.2 Memory......Page 585
F.2 The McCulloch–Pitts neuron......Page 586
F.2.1 Boolean functions......Page 587
F.2.2 The MCP model neuron......Page 589
F.3 Perceptrons......Page 595
F.3.1 The perceptron learning rule......Page 597
F.3.2 Limitations of perceptrons......Page 598
F.4 Multi-layer perceptrons......Page 599
F.5.3 Uniqueness of solutions......Page 602
F.6 Radial basis functions......Page 603
G.1 Minimization of a function of one variable......Page 606
G.1.1 Oscillation......Page 607
G.2 Minimizing a function of several variables......Page 608
G.3 Training a neural network......Page 611
H.1 Definitions and orthogonality relations......Page 617
H.2 Recurrence relations and Clenshaw’s algorithm......Page 618
H.3 Chebyshev coefficients for a class of simple functions......Page 620
H.4 Least-squares analysis and Chebyshev series......Page 621
Appendix I: Integration and differentiation of measured time data......Page 623
I.1.1 Low-frequency problems......Page 624
I.1.2 High-frequency problems......Page 630
I.2 Frequency characteristics of integration formulae......Page 632
I.3 Frequency-domain integration......Page 635
I.4 Differentiation of measured time data......Page 638
I.5 Time-domain differentiation......Page 640
I.6 Frequency-domain differentiation......Page 642
Appendix J: Volterra kernels from perturbation analysis......Page 643
K.1 Random vibration of an asymmetric Duffing oscillator......Page 647
K.2.1 The MDOF system......Page 649
K.2.2 The pole structure of the composite FRF......Page 650
K.2.3 Validation......Page 652
Bibliography......Page 657