Contents

Cover Page

Title Page

Copyright

List of Figures

List of Tables

Preface

1: Introduction

1.1 Brief history of the Hilbert transform

1.2 Hilbert transform in vibration analysis

1.3 Organization of the book

Part I: Hilbert Transform and Analytic Signal

2: Analytic signal representation

2.1 Local versus global estimations

2.2 The Hilbert transform notation

2.3 Main properties of the Hilbert transform

2.4 The Hilbert transform of multiplication

2.5 Analytic signal representation

2.6 Polar notation

2.7 Angular position and speed

2.8 Signal waveform and envelope

2.9 Instantaneous phase

2.10 Instantaneous frequency

2.11 Envelope versus instantaneous frequency plot

2.12 Distribution functions of the instantaneous characteristics

2.13 Signal bandwidth

2.14 Instantaneous frequency distribution and negative values

2.15 Conclusions

3: Signal demodulation

3.1 Envelope and instantaneous frequency extraction

3.2 Hilbert transform and synchronous detection

3.3 Digital Hilbert transformers

3.4 Instantaneous characteristics distortions

3.5 Conclusions

Part II: Hilbert Transform and Vibration Signals

4: Typical examples and description of vibration data

4.1 Random signal

4.2 Decay vibration waveform

4.3 Slow linear sweeping frequency signal

4.4 Harmonic frequency modulation

4.5 Harmonic amplitude modulation

4.6 Product of two harmonics

4.7 Single harmonic with DC offset

4.8 Composition of two harmonics

4.9 Derivative and integral of the analytic signal

4.10 Signal level

4.11 Frequency contents

4.12 Narrowband and wideband signals

4.13 Conclusions

5: Actual signal contents

5.1 Monocomponent signal

5.2 Multicomponent signal

5.3 Types of multicomponent signal

5.4 Averaging envelope and instantaneous frequency

5.5 Smoothing and approximation of the instantaneous frequency

5.6 Congruent envelope

5.7 Congruent instantaneous frequency

5.8 Conclusions

6: Local and global vibration decompositions

6.1 Empirical mode decomposition

6.2 Analytical basics of the EMD

6.3 Global Hilbert Vibration Decomposition

6.4 Instantaneous frequency of the largest energy component

6.5 Envelope of the largest energy component

6.6 Subtraction of the synchronous largest component

6.7 Hilbert Vibration Decomposition scheme

6.8 Examples of Hilbert Vibration Decomposition

6.9 Comparison of the Hilbert transform decomposition methods

6.10 Common properties of the Hilbert transform decompositions

6.11 The differences between the Hilbert transform decompositions

6.12 Amplitude—frequency resolution of HT decompositions

6.13 Limiting number of valued oscillating components

6.14 Decompositions of typical nonstationary vibration signals

6.15 Main results and recommendations

6.16 Conclusions

7: Experience in the practice of signal analysis and industrial application

7.1 Structural health monitoring

7.2 Standing and traveling wave separation

7.3 Echo signal estimation

7.4 Synchronization description

7.5 Fatigue estimation

7.6 Multichannel vibration generation

7.7 Conclusions

Part III: Hilbert Transform and Vibration Systems

8: Vibration system characteristics

8.1 Kramers–Kronig relations

8.2 Detection of nonlinearities in frequency domain

8.3 Typical nonlinear elasticity characteristics

8.4 Phase plane representation of elastic nonlinearities in vibration systems

8.5 Complex plane representation

8.6 Approximate primary solution of a conservative nonlinear system

8.7 Hilbert transform and hysteretic damping

8.8 Nonlinear damping characteristics in a SDOF vibration system

8.9 Typical nonlinear damping in a vibration system

8.10 Velocity-dependent nonlinear damping

8.11 Velocity-independent damping

8.12 Combination of different damping elements

8.13 Conclusions

9: Identification of the primary solution

9.1 Theoretical bases of the Hilbert transform system identification

9.2 Free vibration modal characteristics

9.3 Forced vibration modal characteristics

9.4 Backbone (skeleton curve)

9.5 Damping curve

9.6 Frequency response

9.7 Force static characteristics

9.8 Conclusions

10: The FREEVIB and FORCEVIB methods

10.1 FREEVIB identification examples

10.2 FORCEVIB identification examples

10.3 System identification with biharmonic excitation

10.4 Identification of nonlinear time-varying system

10.5 Experimental Identification of nonlinear vibration system

10.6 Conclusions

11: Considering high-order superharmonics. Identification of asymmetric and MDOF systems

11.1 Description of the precise method scheme

11.2 Identification of the instantaneous modal parameters

11.3 Congruent modal parameters

11.4 Congruent nonlinear elastic and damping forces

11.5 Examples of precise free vibration identification

11.6 Forced vibration identification considering high-order superharmonics

11.7 Identification of asymmetric nonlinear system

11.8 Experimental identification of a crack

11.9 Identification of MDOF vibration system

11.10 Identification of weakly nonlinear coupled oscillators

11.11 Conclusions

12: Experience in the practice of system analysis and industrial application

12.1 Non-parametric identification of nonlinear mechanical vibration systems

12.2 Parametric identification of nonlinear mechanical vibrating systems

12.3 Structural health monitoring and damage detection

12.4 Conclusions

References

Index