Hilbert Transform Applications in Mechanical Vibration -- 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 -- 2.12.1 Envelope distribution and average values -- 2.12.2 Instantaneous frequency average values -- 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 -- 3.3.1 Frequency domain -- 3.3.2 Time domain -- 3.4 Instantaneous characteristics distortions -- 3.4.1 Total harmonic distortion and noise -- 3.4.2 End effect of the Hilbert transform -- 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 -- 4.5.1 Envelope and instantaneous frequency of AM signal -- 4.5.2 Low modulation index -- 4.5.3 High modulation index -- 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 -- 4.10.1 Amplitude overall level -- 4.10.2 Amplitude local level -- 4.10.3 Points of contact between envelope and signal -- 4.10.4 Local extrema points -- 4.10.5 Deviation of local extrema from envelope -- 4.10.6 Local extrema sampling -- 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 -- 6.2.1 Decomposition of a harmonic plus DC offset -- 6.2.2 Decomposition of two harmonics -- 6.2.3 Distance between envelope and extrema -- 6.2.4 Mean value between the local maxima and minima Curves -- 6.2.5 EMD as a nonstationary and nonlinear filter -- 6.2.6 Frequency resolution of the EMD -- 6.2.7 Frequency limit of distinguishing closest harmonics -- 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 -- 6.7.1 Frequency resolution of the HVD -- 6.7.2 Suggested types of signals for decomposition -- 6.8 Examples of Hilbert Vibration Decomposition -- 6.8.1 Nonstationary single-sine amplitude modulated signals -- 6.8.2 Nonstationary overmodulated signals -- 6.8.3 Nonstationary waveform presentation -- 6.8.4 Forced and free vibration separation -- 6.8.5 Asymmetric signal analysis.

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 -- 6.12.1 The EMD method -- 6.12.2 The HVD method -- 6.13 Limiting number of valued oscillating components -- 6.13.1 The EMD method -- 6.13.2 The HVD method -- 6.14 Decompositions of typical nonstationary vibration signals -- 6.14.1 Examples of nonstationarity 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 -- 7.1.1 The envelope and IF as a structure condition indicator -- 7.1.2 Bearing diagnostics -- 7.1.3 Gears diagnosis -- 7.1.4 Motion trajectory analysis -- 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 -- 8.3.1 Large amplitude nonlinear behavior. polynomial model -- 8.3.2 Vibro-impact model -- 8.3.3 Restoring force saturation (limiter) -- 8.3.4 Small amplitude nonlinear behavior backlash spring -- 8.3.5 Preloaded (precompressed) spring -- 8.3.6 Piecewise linear spring bilinear model -- 8.3.7 Combination of different elastic elements -- 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 -- 8.10.1 Velocity squared (quadratic, turbulent) damping -- 8.10.2 Dry friction -- 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 -- 9.7.1 Averaging of the instantaneous modal parameters -- 9.7.2 Polynomial scaling technique -- 9.7.3 Selecting extrema and scaling technique -- 9.7.4 Decomposition technique -- 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 -- 10.3.1 Linear system model -- 10.3.2 Nonlinear hardening system -- 10.3.3 Nonlinear softening system -- 10.4 Identification of nonlinear time-varying system -- 10.4.1 Model 1. Modulated elasticity -- 10.4.2 Model 2. Modulated elasticity + Quadratic damping + Swept excitation -- 10.4.3 Model 3. Parametric excitation -- 10.4.4 Model 4. Van-der-Pol + Duffing -- 10.4.5 Model 5. Van-der-Pol + Biharmonic excitation -- 10.4.6 Model 6. Van-der-Pol + Swept excitation -- 10.5 Experimental Identification of nonlinear vibration system -- 10.5.1 The structure under test -- 10.5.2 Free vibration identification -- 10.5.3 Forced vibration identification -- 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 -- 11.3.1 Congruent envelope of the displacement -- 11.3.2 Congruent modal frequency -- 11.3.3 Congruent modal damping -- 11.3.4 Congruent envelope of the velocity -- 11.4 Congruent nonlinear elastic and damping forces -- 11.5 Examples of precise free vibration identification -- 11.5.1 Nonlinear spring identification -- 11.5.2 Nonlinear damping identification -- 11.5.3 Combined nonlinear spring and damping identification -- 11.6 Forced vibration identification considering high-order superharmonics -- 11.7 Identification of asymmetric nonlinear system -- 11.7.1 Asymmetric nonlinear system representation -- 11.7.2 The Hilbert transform identification technique -- 11.7.3 Asymmetric nonlinear system examples -- 11.8 Experimental identification of a crack -- 11.9 Identification of MDOF vibration system -- 11.9.1 Identification of linear coupled oscillators -- 11.9.2 Spring coupling -- 11.9.3 Reconstruction of coupling coefficients -- 11.10 Identification of weakly nonlinear coupled oscillators -- 11.10.1 Coupled nonlinear oscillators with linear coupling -- 11.10.2 Coupled linear oscillators with nonlinear coupling -- 11.10.3 HT decomposition and analysis -- 11.10.4 Modal skeleton curve estimation -- 11.10.5 Mode shape estimation -- 11.10.6 Description of the identification scheme -- 11.10.7 Simulation examples -- 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 -- 12.3.1 Damage detection in structures and buildings -- 12.3.2 Detecting anomalies in beams and plates.

12.3.3 Health monitoring in power systems and rotors.