Hilbert Transform Applications in Mechanical Vibration

Contents......Page 6
List of Figures......Page 12
List of Tables......Page 24
Preface......Page 26
Introduction......Page 29
1.1 Brief history of the Hilbert transform......Page 30
1.2 Hilbert transform in vibration analysis......Page 32
1.3 Organization of the book......Page 33
Part I HILBERT TRANSFORM AND ANALYTIC SIGNAL......Page 35
2.2 The Hilbert transform notation......Page 37
2.3 Main properties of the Hilbert transform......Page 38
2.4 The Hilbert transform of multiplication......Page 39
2.5 Analytic signal representation......Page 40
2.6 Polar notation......Page 41
2.7 Angular position and speed......Page 42
2.8 Signal waveform and envelope......Page 43
2.9 Instantaneous phase......Page 44
2.10 Instantaneous frequency......Page 45
2.11 Envelope versus instantaneous frequency plot......Page 46
2.12 Distribution functions of the instantaneous characteristics......Page 47
2.13 Signal bandwidth......Page 48
2.15 Conclusions......Page 49
3.1 Envelope and instantaneous frequency extraction......Page 51
3.2 Hilbert transform and synchronous detection......Page 52
3.3 Digital Hilbert transformers......Page 55
3.4 Instantaneous characteristics distortions......Page 57
3.5 Conclusions......Page 60
Part II HILBERT TRANSFORM AND VIBRATION SIGNALS......Page 61
4.1 Random signal......Page 63
4.2 Decay vibration waveform......Page 65
4.3 Slow linear sweeping frequency signal......Page 66
4.4 Harmonic frequency modulation......Page 68
4.5 Harmonic amplitude modulation......Page 70
4.6 Product of two harmonics......Page 74
4.7 Single harmonic with DC offset......Page 76
4.8 Composition of two harmonics......Page 78
4.10 Signal level......Page 81
4.12 Narrowband and wideband signals......Page 85
4.13 Conclusions......Page 86
5.1 Monocomponent signal......Page 87
5.2 Multicomponent signal......Page 88
5.3 Types of multicomponent signal......Page 90
5.4 Averaging envelope and instantaneous frequency......Page 91
5.5 Smoothing and approximation of the instantaneous frequency......Page 92
5.6 Congruent envelope......Page 94
5.7 Congruent instantaneous frequency......Page 97
5.8 Conclusions......Page 99
6.1 Empirical mode decomposition......Page 102
6.2 Analytical basics of the EMD......Page 106
6.4 Instantaneous frequency of the largest energy component......Page 120
6.5 Envelope of the largest energy component......Page 121
6.6 Subtraction of the synchronous largest component......Page 122
6.7 Hilbert Vibration Decomposition scheme......Page 123
6.8 Examples of Hilbert Vibration Decomposition......Page 125
6.9 Comparison of the Hilbert transform decomposition methods......Page 143
6.10 Common properties of the Hilbert transform decompositions......Page 144
6.12 Amplitude—frequency resolution of HT decompositions......Page 145
6.13 Limiting number of valued oscillating components......Page 146
6.14 Decompositions of typical nonstationary vibration signals......Page 148
6.15 Main results and recommendations......Page 152
6.16 Conclusions......Page 154
Experience in the practice of signal analysis and industrial application......Page 156
7.1 Structural health monitoring......Page 157
7.2 Standing and traveling wave separation......Page 160
7.6 Multichannel vibration generation......Page 164
7.7 Conclusions......Page 165
Part III HILBERT TRANSFORM AND VIBRATION SYSTEMS......Page 168
8.1 Kramers–Kronig relations......Page 170
8.2 Detection of nonlinearities in frequency domain......Page 172
8.3 Typical nonlinear elasticity characteristics......Page 174
8.4 Phase plane representation of elastic nonlinearities in vibration systems......Page 180
8.5 Complex plane representation......Page 183
8.6 Approximate primary solution of a conservative nonlinear system......Page 185
8.8 Nonlinear damping characteristics in a SDOF vibration system......Page 188
8.9 Typical nonlinear damping in a vibration system......Page 190
8.10 Velocity-dependent nonlinear damping......Page 191
8.12 Combination of different damping elements......Page 195
8.13 Conclusions......Page 196
Identification of the primary solution......Page 198
9.1 Theoretical bases of the Hilbert transform system identificatio......Page 199
9.2 Free vibration modal characteristics......Page 200
9.3 Forced vibration modal characteristics......Page 201
9.4 Backbone (skeleton curve)......Page 203
9.6 Frequency response......Page 204
9.7 Force static characteristics......Page 206
9.8 Conclusions......Page 208
The FREEVIB and FORCEVIB methods......Page 210
10.1 FREEVIB identificatio examples......Page 213
10.2 FORCEVIB identificatio examples......Page 216
10.3 System identificatio with biharmonic excitation......Page 221
10.4 Identificatio of nonlinear time-varying system......Page 229
10.5 Experimental Identificatio of nonlinear vibration system......Page 243
10.6 Conclusions......Page 250
Considering high-order superharmonics. Identificatio of asymmetric and MDOF systems......Page 252
11.1 Description of the precise method scheme......Page 253
11.2 Identificatio of the instantaneous modal parameters......Page 254
11.3 Congruent modal parameters......Page 255
11.4 Congruent nonlinear elastic and damping forces......Page 257
11.5 Examples of precise free vibration identificatio......Page 258
11.6 Forced vibration identificatio considering high-order superharmonics......Page 266
11.7 Identificatio of asymmetric nonlinear system......Page 269
11.8 Experimental identificatio of a crack......Page 273
11.9 Identificatio of MDOF vibration system......Page 277
11.10 Identificatio of weakly nonlinear coupled oscillators......Page 281
11.11 Conclusions......Page 292
Experience in the practice of system analysis and industrial application......Page 296
12.1 Non-parametric identificatio of nonlinear mechanical vibration systems......Page 297
12.2 Parametric identificatio of nonlinear mechanical vibrating systems......Page 298
12.3 Structural health monitoring and damage detection......Page 299
12.4 Conclusions......Page 301
References......Page 304
Index......Page 315