Hilbert Transform Applications in Mechanical Vibration

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ISBN: 9780470978276 | 0470978279
Cover: Hardcover
Copyright: 3/28/2011

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Hilbert Transform Applications in Mechanical Vibration addresses recent advances in research and applications of the modern Hilbert transform to vibration engineering, through which laboratory dynamic tests can be produced more quickly and accurately. The author integrates important pioneering developments in signal processing and mathematical models with typical properties of mechanical constructions such as resonance, dynamic stiffness and damping. This unique merger of technical properties and digital signal processing allows the instant solution of a variety of engineering problems and in-depth exploration of the physics of vibration by analysis, identification and simulation.Hilbert Transform Applications in Mechanical Vibration employs the author's pioneering applications of the Hilbert Vibration Decomposition method characterized by high frequency resolution, and provides a comprehensive account of the main applications, covering dynamic testing and extraction of the modal parameters of nonlinear vibration systems including the initial elastic and damping force characteristics.

List of Figures	p. xiii
List of Tables	p. xxv
Preface	p. xxvii
Introduction	p. 1
Brief history of the Hilbert transform	p. 2
Hilbert transform in vibration analysis	p. 4
Organization of the book	p. 5
Hilbert Transform and Analytic Signal
Analytic signal representation	p. 9
Local versus global estimations	p. 9
The Hilbert transform notation	p. 9
Main properties of the Hilbert transform	p. 10
The Hilbert transform of multiplication	p. 11
Analytic signal representation	p. 12
Polar notation	p. 13
Angular position and speed	p. 14
Signal waveform and envelope	p. 15
Instantaneous phase	p. 16
Instantaneous frequency	p. 17
Envelope versus instantaneous frequency plot	p. 18
Distribution functions of the instantaneous characteristics	p. 19
Envelope distribution and average values	p. 19
Instantaneous frequency average values	p. 20
Signal bandwidth	p. 20
Instantaneous frequency distribution and negative values	p. 21
Conclusions	p. 21
Signal demodulation	p. 23
Envelope and instantaneous frequency extraction	p. 23
Hilbert transform and synchronous detection	p. 24
Digital Hilbert transformers	p. 27
Frequency domain	p. 27
Time domain	p. 28
Instantaneous characteristics distortions	p. 29
Total harmonic distortion and noise	p. 30
End effect of the Hilbert transform	p. 31
Conclusions	p. 32
Hilbert Transform and Vibration Signals
Typical examples and description of vibration data	p. 35
Random signal	p. 35
Decay vibration waveform	p. 37
Slow linear sweeping frequency signal	p. 38
Harmonic frequency modulation	p. 40
Harmonic amplitude modulation	p. 42
Envelope and instantaneous frequency of AM signal	p. 43
Low modulation index	p. 44
High modulation index	p. 44
Product of two harmonics	p. 46
Single harmonic with DC offset	p. 48
Composition of two harmonics	p. 50
Derivative and integral of the analytic signal	p. 53
Signal level	p. 53
Amplitude overall level	p. 54
Amplitude local level	p. 55
Points of contact between envelope and signal	p. 55
Local extrema points	p. 55
Deviation of local extrema from envelope	p. 56
Local extrema sampling	p. 56
Frequency contents	p. 57
Narrowband and wideband signals	p. 57
Conclusions	p. 58
Actual signal contents	p. 59
Monocomponent signal	p. 59
Multicomponent signal	p. 60
Types of multicomponent signal	p. 61
Averaging envelope and instantaneous frequency	p. 62
Smoothing and approximation of the instantaneous frequency	p. 63
Congruent envelope	p. 65
Congruent instantaneous frequency	p. 68
Conclusions	p. 70
Local and global vibration decompositions	p. 73
Empirical mode decomposition	p. 73
Analytical basics of the EMD	p. 77
Decomposition of a harmonic plus DC offset	p. 78
Decomposition of two harmonics	p. 78
Distance between envelope and extrema	p. 79
Mean value between the local maxima and minima Curves	p. 80
EMD as a nonstationary and nonlinear filter	p. 83
Frequency resolution of the EMD	p. 85
Frequency limit of distinguishing closest harmonics	p. 89
Global Hilbert Vibration Decomposition	p. 91
Instantaneous frequency of the largest energy component	p. 91
Envelope of the largest energy component	p. 92
Subtraction of the synchronous largest component	p. 93
Hilbert Vibration Decomposition scheme	p. 94
Frequency resolution of the HVD	p. 95
Suggested types of signals for decomposition	p. 95
Examples of Hilbert Vibration Decomposition	p. 96
Nonstationary single-sine amplitude modulated signals	p. 96
Nonstationary overmodulated signals	p. 97
Nonstationary waveform presentation	p. 101
Forced and free vibration separation	p. 103
Asymmetric signal analysis	p. 107
Comparison of the Hilbert transform decomposition methods	p. 114
Common properties of the Hilbert transform decompositions	p. 115
The differences between the Hilbert transform decompositions	p. 116
Amplitude?frequency resolution of HT decompositions	p. 116
TheEMDmethod	p. 116
The HVD method	p. 117
Limiting number of valued oscillating components	p. 117
TheEMDmethod	p. 117
The HVD method	p. 119
Decompositions of typical nonstationary vibration signals	p. 119
Examples of nonstationarity vibration signals	p. 120
Main results and recommendations	p. 123
Conclusions	p. 125
Experience in the practice of signal analysis and industrial application	p. 127
Structural health monitoring	p. 128
The envelope and IF as a structure condition indicator	p. 128
Bearing diagnostics	p. 129
Gears diagnosis	p. 130
Motion trajectory analysis	p. 131
Standing and traveling wave separation	p. 131
Echo signal estimation	p. 135
Synchronization description	p. 135
Fatigue estimation	p. 135
Multichannel vibration generation	p. 135
Conclusions	p. 136
Hilbert Transform and Vibration Systems
Vibration system characteristics	p. 141
Kramers-Kronig relations	p. 141
Detection of nonlinearities in frequency domain	p. 143
Typical nonlinear elasticity characteristics	p. 145
Large amplitude nonlinear behavior, polynomial model	p. 146
Vibro-impact model	p. 147
Restoring force saturation (limiter)	p. 147
Small amplitude nonlinear behavior backlash spring	p. 149
Preloaded (precompressed) spring	p. 149
Piecewise linear spring bilinear model	p. 150
Combination of different elastic elements	p. 151
Phase plane representation of elastic nonlinearities in vibration systems	p. 151
Complex plane representation	p. 154
Approximate primary solution of a conservative nonlinear system	p. 156
Hilbert transform and hysteretic damping	p. 159
Nonlinear damping characteristics in a SDOF vibration system	p. 159
Typical nonlinear damping in a vibration system	p. 161
Velocity-dependent nonlinear damping	p. 162
Velocity squared (quadratic, turbulent) damping	p. 163
Dry friction	p. 164
Velocity-independent damping	p. 166
Combination of different damping elements	p. 166
Conclusions	p. 167
Identification of the primary solution	p. 169
Theoretical bases of the Hilbert transform system identification	p. 170
Free vibration modal characteristics	p. 171
Forced vibration modal characteristics	p. 172
Backbone (skeleton curve)	p. 174
Damping [curve]	p. 175
Frequency response	p. 175
Force static characteristics	p. 177
Averaging of the instantaneous modal parameters	p. 178
Polynomial scaling technique	p. 178
Selecting extrema and scaling technique	p. 178
Decomposition technique	p. 179
Conclusions	p. 179
The Freevib and Forcevib methods	p. 181
FREEVIB identification examples	p. 184
FORCEVIB identification examples	p. 187
System identification with biharmonic excitation	p. 192
Linear system model	p. 194
Nonlinear hardening system	p. 196
Nonlinear softening system	p. 199
Identification of nonlinear time-varying system	p. 200
Model 1. Modulated elasticity	p. 203
Model 2. Modulated elasticity + Quadratic damping + Swept excitation	p. 205
Model 3. Parametric excitation	p. 205
Model 4. Van-der-Pol + Duffing	p. 207
Model 5. Van-der-Pol + Biharmonic excitation	p. 209
Model 6. Van-der-Pol + Swept excitation	p. 211
Experimental Identification of nonlinear vibration system	p. 214
The structure under test	p. 215
Free vibration identification	p. 216
Forced vibration identification	p. 219
Conclusions	p. 221
Considering high-order superharmonics. Identification of asymmetric and MDOF systems	p. 223
Description of the precise method scheme	p. 224
Identification of the instantaneous modal parameters	p. 225
Congruent modal parameters	p. 226
Congruent envelope of the displacement	p. 226
Congruent modal frequency	p. 227
Congruent modal damping	p. 227
Congruent envelope of the velocity	p. 228
Congruent nonlinear elastic and damping forces	p. 228
Examples of precise free vibration identification	p. 229
Nonlinear spring identification	p. 229
Nonlinear damping identification	p. 232
Combined nonlinear spring and damping identification	p. 235
Forced vibration identification considering high-order superharmonics	p. 237
Identification of asymmetric nonlinear system	p. 240
Asymmetric nonlinear system representation	p. 241
The Hilbert transform identification technique	p. 241
Asymmetric nonlinear system examples	p. 242
Experimental identification of a crack	p. 244
Identification of MDOF vibration system	p. 248
Identification of linear coupled oscillators	p. 249
Spring coupling	p. 249
Reconstruction of coupling coefficients	p. 251
Identification of weakly nonlinear coupled oscillators	p. 252
Coupled nonlinear oscillators with linear coupling	p. 252
Coupled linear oscillators with nonlinear coupling	p. 255
HT decomposition and analysis	p. 256
Modal skeleton curve estimation	p. 256
Mode shape estimation	p. 257
Description of the identification scheme	p. 257
Simulation examples	p. 258
Conclusions	p. 263
Experience in the practice of system analysis and industrial application	p. 267
Non-parametric identification of nonlinear mechanical vibration systems	p. 268
Parametric identification of nonlinear mechanical vibrating systems	p. 269
Structural health monitoring and damage detection	p. 270
Damage detection in structures and buildings	p. 270
Detecting anomalies in beams and plates	p. 271
Health monitoring in power systems and rotors	p. 272
Conclusions	p. 272
References	p. 275
Index	p. 287
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