Cover Page

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

Title Page

List of Figures

Figure 2.1 The ideal HT: the impulse response function (a), the module (b), and the phase (c) of the HT transfer function

Figure 2.2 The quasiharmonic (a) and the square wave (b): the initial signal x(t), the HT pair projection Inline Math, the envelope A(t)

Figure 2.3 The HT projection (1), the real signal (2), the analytic signal (3), and the phasor in complex plain (4)

Figure 2.4 The analytic signal in the complex plain

Figure 2.5 The instantaneous phase: unwrapped (- -) and wrapped (—)

Figure 3.1 The block diagram of the envelope and the IF extraction

Figure 3.2 The block diagram of the synchronous demodulation

Figure 3.3 The HT and the synchronous demodulations: two components of the composition (a), the composition (b, —), the HT envelope (b, ), the synchronous amplitude (b, -·-), the demodulated component (c, —), and the synchronous amplitude (c, -·-)

Figure 3.4 The digital Hilbert transformer as a filter: the impulse characteristics (a); the magnitude (b)

Figure 3.5 The distortion of the envelope and the IF: the amplitude step (a), the frequency step (b), the single spike (c), the random noise (d); the signal (—), the envelope (– –), the IF (), and the noise (·)

Figure 4.1 The random signal and the envelope (a), the signal distribution (b, ), the envelope distribution (b, – –), the IF (c), and its distribution (d)

Figure 4.2 The random signal spectrum (a): the wideband (– –) and narrowband (); the narrowband signal envelope vs. IF plot (b)

Figure 4.3 The damped oscillation () and the envelope (– –). The envelope and the extrema points

Figure 4.4 The damped oscillation: the spectrum (a), the envelope vs. IF plot (b)

Figure 4.5 The frequency sweeping oscillation () and the envelope (a,– –), the IF (b)

Figure 4.6 The frequency sweeping oscillation: the spectrum (a), the envelope vs. IF plot (b)

Figure 4.7 Harmonic frequency modulation: the signal (a), the signal IF (b)

Figure 4.8 Fast harmonic frequency modulation: the spectrum (a), the envelope vs. IF plot (b)

Figure 4.9 An AM with low modulation index: the signal (a), the upper positive envelope (b), the lower negative envelope (c)

Figure 4.10 An AM with low modulation index: the spectrum (a), the envelope vs. IF plot (b)

Figure 4.11 An AM with high modulation index: the signal (a), the upper positive envelope (b), the alternate envelope (c)

Figure 4.12 An AM with high modulation index: the spectrum (a), the envelope vs. IF plot (b)

Figure 4.13 The product of two harmonics: the signal (a), the upper positive envelope (b), the alternate envelope (c)

Figure 4.14 The product of two harmonics: the spectrum (a), the envelope vs. IF plot (b)

Figure 4.15 A harmonic with DC offset: the signal (a) with the envelopes, the constant DC; the upper positive envelope (b)

Figure 4.16 A harmonic with DC offset: the spectrum (a), the envelope vs. IF plot (b)

Figure 4.17 The relative and absolute motion of the sum of two vectors

Figure 4.18 Two tones: the signal (a) with the envelopes, the IF (b)

Figure 4.19 Two tones: the spectrum (a), the envelope vs. IF plot (b)

Figure 5.1 The sum of two in-phase (a) and out-of-phase (b) harmonics: the envelope (- -), the congruent envelope (_), the largest (·), the secondary (-·-) harmonic

Figure 5.2 The sum of two in-phase (a) and out-of-phase (b) harmonics: the IF ( ), the congruent frequency (_), the largest (·), the secondary (-·-) harmonic

Figure 5.3 The triangle signal and its high harmonics: the envelope (a, _) and the congruent envelope (a, - -); the IF (b, _) and the congruent IF (b, - -)

Figure 5.4 The square signal and its high harmonics: the envelope (a, _ the congruent envelope (a, - -); the IF (b, _) and the congruent IF (b, - -)

Figure 6.1 Block diagram of the EMD method

Figure 6.2 The sum of two harmonics: the initial signal (a, —), the upper (-·-) and lower (…) envelope, the top (Δ) and bottom extrema points (∇); the top (b, —) and bottom (b, …) extrema, the mean value between the top and bottom extrema (- -); the EMD decomposed first harmonic (c, - -), the second harmonic (—)

Figure 6.3 The square wave envelope vs. the IF (); the congruent envelope vs. the congruent IF (•)

Figure 6.4 The square wave Hilbert spectrum

Figure 6.5 The vertical position of the local maxima (), the initial signal (– –), the upper envelope (-·-) and the top maxima (Δ): the negative IF (A1 = 1, ω1 = 1,A2 = 0.6,ω2 = 1.8), the positive IF (b) (A1 = 1, ω1 = 1,A2 = 0.25,ω2 = 3.9)

Figure 6.6 The theoretical mean value between the local maxima and minima at the highest maximum position: the envelope of the first harmonic (a), the envelope of the second harmonic (b)

Figure 6.7 The theoretical mean value between the local maxima and minima at the lowest maximum position: the envelope of the first harmonic (a), the envelope of the second harmonic (b)

Figure 6.8 The theoretical boundary of the first harmonic filtering: the highest maximum position (a), the approximation A2/A1 ≤ 1.44 (ω21)- 1.4 (- -), the approximation A2/A1 = ω12 (); the lowest maximum position (b), the approximation A2 / A1 ≤ (ω21)- 2 (- -)

Figure 6.9 The EMD ranges of two harmonics separation: (1) the impossible decomposition for very close frequency harmonics and small amplitude ratio; (2) the decomposition requires several sifting iterations for close frequency harmonics; (3) the single iteration separation for distant frequency harmonics and large amplitude ratio

Figure 6.10 The EMD of two very close harmonics (A1 = 1, ω1 = 1,A2 = 0.6,ω2 = 1.1): the initial signal (a, ), the upper (-·-) and lower (···) envelope, the top (Δ) and bottom maxima points (∇); the initial signal (b, ), the top (b, -·-) and bottom (b, ···) extrema curves, the mean value between them (b, - -); the first harmonic (c, ), the second harmonic (c,- -), the mean value between the top and bottom extrema (c, - -)

Figure 6.11 The EMD of two close harmonics (A1 = 1, ω1 = 1,A2 = 0.9,ω2 = 1.8): the initial signal (a,), the upper (-·-) and lower (···) envelope, the top (Δ) and bottom maxima points (∇); the initial signal (b,), the top (b, -·-) and bottom (b, ···) extrema curves, the mean value between them (b, - -); the first harmonic (c,), the second harmonic (c,- -), the mean value between the top and bottom extrema (c, - -)

Figure 6.12 The EMD of two distant harmonics (A1 = 1, ω1 = 1,A2 = 0.4,ω2 = 2.9): the initial signal (a,), the upper (-·-) and lower (···) envelope, the top (Δ) and bottom maxima points (∇); the top (b,) and bottom (b,) maxima curves, the mean value between them (b, - -); the decomposed first harmonic (c,), the second harmonic (c,- -), the mean value between the top and bottom maxima curves (c, - -)

Figure 6.13 Block diagram of the HVD method

Figure 6.14 The HVD ranges of two harmonics separation: (1) impossible decomposition for very close frequency harmonics; (2) good separation for distant frequency harmonics

Figure 6.15 The nonstationary single-tone amplitude modulated signal (a) and its decomposed superimposed components (b)

Figure 6.16 First three components of a nonstationary single-tone amplitude modulated signal

Figure 6.17 The IF (a) and envelope (b) of a single-tone amplitude modulated signal: the carrier signal component (), the low (- - -) and high (···) modulation component

Figure 6.18 The Hilbert spectrum of the single-tone amplitude modulated signal

Figure 6.19 The overmodulated AM signal (a, ), its envelope (— -□-); the alternate envelope (b, - -)

Figure 6.20 The first three components of a nonstationary overmodulated AM signal

Figure 6.21 The IF (a) and envelope (b) of an amplitude overmodulated signal; the carrier signal component (-·-), the low () and high frequency (- -) modulation component

Figure 6.22 The Hilbert spectrum of an amplitude overmodulated signal

Figure 6.23 The nonstationary square wave: the initial signal (a, - -), the sum of the first five components (a,), and the decomposed superimposed components (b)

Figure 6.24 The first five components of a nonstationary square wave

Figure 6.25 The IF (a) and envelope (b) of each component of the nonstationary square wave: the first signal component ()

Figure 6.26 The Hilbert spectrum of a nonstationary square wave

Figure 6.27 The nonstationary vibration solution (a) and the separated vibration components: the steady state (b), the transient (c)

Figure 6.28 The IF (a) and envelope (b) of a nonstationary vibration solution: the steady state component (), the transient component (- -)

Figure 6.29 The Hilbert spectrum of a nonstationary vibration solution

Figure 6.30 The asymmetric transformation of the signal amplitude

Figure 6.31 An asymmetric signal with a linearly increasing envelope: the signal (), the envelope (-·-), the IF (·)

Figure 6.32 The decomposed components of an asymmetric signal with a linearly increasing envelope

Figure 6.33 An asymmetric signal with a partial linearly increasing envelope: the signal (), the positive partial envelope (–), the negative partial envelope (- - -)

Figure 6.34 The envelope vs. the IF plot of an asymmetric signal with an linearly increasing envelope: the positive partial envelope (Δ), the negative partial envelope (□)

Figure 6.35 An asymmetric signal with a linear increasing frequency: the signal (), the envelope (-·-), the IF (·)

Figure 6.36 The decomposed components of an asymmetric signal with a linear increasing frequency

Figure 6.37 An asymmetric signal with a linear increasing frequency: the signal (), the positive envelope (–), the negative envelope (- - -)

Figure 6.38 The envelope vs. the IF plot of a asymmetric signal with a linear increasing frequency: the positive partial envelope (Δ), the negative partial envelope (□)

Figure 6.39 An asymmetric signal with a linear decreasing envelope and increasing frequency: the signal (), the envelope (-·-), the IF (·)

Figure 6.40 The decomposed components of an asymmetric signal with a linear decreasing envelope and increasing frequency

Figure 6.41 An asymmetric signal with a linear decreasing partial envelope and increasing frequency: the signal (), the positive envelope (–), the negative envelope (- - -)

Figure 6.42 The envelope vs. the IF plot of an asymmetric signal with a linear decreasing envelope and increasing frequency: the positive partial envelope (Δ), the negative partial envelope (□)

Figure 6.43 The largest number of valued oscillating components of the EMD: the frequency ratio is equal to 3 (–), the frequency ratio is equal to 5 (- -)

Figure 6.44 The HVD decomposition of two frequency- crossing components: the initial composition (a), the first decomposed component with its envelope (b), the second decomposed component with its envelope (c), the IF of the both components (d)

Figure 6.45 Extraction of the sweeping oscillations and exhibition of remaining impulses: the initial composition (a), the extracted sweeping oscillation (b), the de-noising impulses with a slow triangle component (c)

Figure 7.1 The HT procedures in signal processing

Figure 7.2 The traveling part of a wave

Figure 7.3 The standing part of a wave

Figure 8.1 The real (a) and imaginary (b) parts of the FRF of a linear vibration system

Figure 8.2 The modulus (a), and phase (b) of the FRF of a linear vibration system and group delay (– –)

Figure 8.3 A Nyquist plot of the FRF: the linear vibration system (a), the nonlinear Duffing system (b); the measured data (– –), the Hilbert transformed data ()

Figure 8.4 A polynomial nonlinear restoring force (a) and the vibration system backbones (b): a hardening spring (), a softening spring (– –)

Figure 8.5 A vibro-impact nonlinear restoring force (a) and the vibration system backbone (b)

Figure 8.6 An elasticity saturation restoring force (a) and the vibration system backbone (b)

Figure 8.7 Restoring force with backlash (a) and the vibration system backbone (b)

Figure 8.8 A precompressed restoring force (a) and the vibration system back-bone (b)

Figure 8.9 A bilinear restoring force (a) and the vibration system backbone (b)

Figure 8.10 The solution (a) and instantaneous oscillation period (b, ) of the Duffing equation: the average period (– –)

Figure 8.11 The Duffing equation (ε = 5) phase plane (– –) and the analytic signal ()

Figure 8.12 The solution of the Duffing equation (a, ), the envelope (a,-·-), the IF (b, ), and the average IF (b, – –)

Figure 8.13 The backbone of the Duffing system (α3 = ε = 5): a precise period solution (), the average natural frequency (·)

Figure 8.14 The turbulent quadratic damping model: the frictional force characteristics (a), the envelope of a free decay (b), and the damping curve (c)

Figure 8.15 The turbulent damping force in time: the quasiharmonic solution (), the damping force (·), the HT of the damping force (-·-), the damping force envelope (), and the force envelope mean value (– –)

Figure 8.16 A dry friction model: the friction force (a), the envelope of a free decay (b), and the damping curve (c)

Figure 8.17 The dry friction force function in time: the quasiharmonic solution (), the square damping force (·), the HT of the damping force (-·-), the damping force envelope (), and the force envelope mean value (– –)

Figure 9.1 Typical nonlinear stiffness force characteristics (a) and backbones (b): hardening (1), softening (2), backlash (3), preloaded (4)

Figure 9.2 Typical nonlinear damping force characteristics (a) and damping curves (b): turbulent (1), dry (2), viscous friction (3)

Figure 10.1 The free vibration of the Duffing equation: the displacement (a,) and the envelope (a, - -) of the solution, the IF of the solution (b)

Figure 10.2 The identified Duffing equation: the backbone (a, ), the FRF (a,- -); the damping curve (b)

Figure 10.3 The identified Duffing equation spring force (a, ), the initial spring force (a,- -); the damping force (b)

Figure 10.4 Free vibration of the system with a backlash and a dry friction: the displacement (a, ), the envelope (a,- -), the IF of the solution (b)

Figure 10.5 The identified system with the backlash and dry friction: the backbone (a, ) and the FRF (a,- -); the damping curve (b)

Figure 10.6 The backlash and dry friction system identified spring force (a, ), the initial spring force (a,- -), the identified damping force (b, ), the initial damping force (b,-·-)

Figure 10.7 The forced vibration of the Duffing equation: the displacement (a,) and the envelope (a, - -); the IF of the swept excitation (b, ), the instantaneous modal frequency (b, - -), the phase shift between an input and an output (b,·)

Figure 10.8 The identified Duffing equation: the backbone (a, ), the FRF (a,- -); the damping curve (b)

Figure 10.9 The Duffing equation identified spring force (a, ), the initial spring force (a,- -); the damping force (b)

Figure 10.10 The forced vibration of the system with the backlash and dry friction: the displacement (a, ), the envelope (a, - -); the IF of the swept excitation (b,), the instantaneous modal frequency (b, - -), the phase shift between an input and an output (b, ·)

Figure 10.11 The identified system with the backlash and dry friction: the backbone (a, ) and the FRF (a,- -); the damping curve (b)

Figure 10.12 The backlash and dry friction identified spring force (a, ), the initial spring force (a,- -); the identified damping force (b, · , the initial damping force (b,-·-)

Figure 10.13 Linear vibration system: the excitation (a), the displacement (b), excitation spectrum (c), displacement spectrum (d)

Figure 10.14 Linear vibration system: the displacement and the envelope of the solution (a), the instantaneous frequencies (b)

Figure 10.15 The linear system identification: skeleton curve (a, ), FRF (a,·); elastic static force (b); damping curve (c); friction force characteristics (d)

Figure 10.16 Nonlinear hardening system: the excitation (a), the displacement (b), excitation spectrum (c), displacement spectrum (d)

Figure 10.17 Nonlinear hardening system: the displacement and the envelope of the solution (a), the instantaneous frequencies (b)

Figure 10.18 The nonlinear hardening system identification: skeleton curve (a, ), FRF (a,·); elastic static force (b); damping curve (c); friction force characteristics (d)

Figure 10.19 The softening system with biharmonics force excitation (a); the displacement solution and the envelope (b); the IF of the swept excitation (), the instantaneous modal frequency (- -), the phase shift between input and output (·)

Figure 10.20 The identified Duffing equation under biharmonics force excitation: the backbone (a, ), the FRF (a,- -); the damping curve (b)

Figure 10.21 The identified Duffing equation under biharmonic force excitation: the spring force (a); the damping force (b)

Figure 10.22 Model 1: the excitation (a), the displacement (b), excitation spectrum (c), displacement spectrum (d)

Figure 10.23 Model 1: the displacement and envelope of the solution (a), the IF (b)

Figure 10.24 Model 1: the identified damping curve (a), the friction force characteristics (b)

Figure 10.25 Model 2: the excitation (a), the displacement (b), excitation spectrum (c), displacement spectrum (d)

Figure 10.26 Model 2: the displacement and the envelope of the solution (a), the IF (b)

Figure 10.27 Model 2: the identified damping curve (a), the friction force characteristics (b)

Figure 10.28 Model 3: the excitation (a), the displacement (b), excitation spectrum (c), displacement spectrum (d)

Figure 10.29 Model 3: the skeleton curve (a), the elastic static force (b), the damping curve (c), the friction force characteristics (d)

Figure 10.30 Model 4: the excitation (a), the displacement (b), excitation spectrum (c), displacement spectrum (d)

Figure 10.31 Model 4: the displacement and the envelope of the solution (a), the IF (b)

Figure 10.32 Model 4: the skeleton curve (a), the elastic static force (b), the damping curve (c), the friction force characteristics (d)

Figure 10.33 Model 5: the excitation (a), the displacement (b), excitation spectrum (c), displacement spectrum (d)

Figure 10.34 Model 5: the displacement and the envelope of the solution (a), the instantaneous frequencies (b)

Figure 10.35 Model 5: the skeleton curve (a), the elastic static force (b), the damping curve (c), the friction force characteristics (d)

Figure 10.36 Model 6: the excitation (a), the displacement (b), excitation spectrum (c), displacement spectrum (d)

Figure 10.37 Model 6: the displacement and the envelope of the solution (a), the instantaneous frequencies (b)

Figure 10.38 Model 6: the skeleton curve (a), the elastic static force (b), the damping curve (c), the friction force characteristics (d)

Figure 10.39 The experimental stand: mass (1), ruler springs (2), actuator (3), tension mechanism (4), LVDT sensor (5)

Figure 10.40 The measured time histories: the repeated interrupted force excitation (a), the output free vibration displacement (b)

Figure 10.41 The measured time histories: the input sweeping force excitation (a), the output displacement (b)

Figure 10.42 The experimental skeleton curves and frequency response: the skeleton curves of free vibrations (-·-), the frequency response functions (- -), the skeleton curves of forced vibrations (···)

Figure 10.43 The experimental stiffness static force characteristics

Figure 10.44 The experimental damping curve (a) and the friction static force characteristics (b)

Figure 11.1 The nonlinear spring free vibration: the displacement (—), the envelope (), the congruent envelope of the displacement ()

Figure 11.2 The high harmonics of the instantaneous natural frequency of the nonlinear spring free vibration. The time history of two first harmonics (a); the Hilbert spectrum (b)

Figure 11.3 The identified parameters of the nonlinear spring free vibration; the skeleton curve (a): instantaneous (—), averaged (- -), identified with high harmonics (); the spring static force characteristics (b): initial (-·-), identified with high harmonics (); the damping curve (c): initial (-·-), averaged (- -), identified with high harmonics (); the friction static force characteristics (d): initial (-·-), identified with high harmonics ()

Figure 11.4 Nonlinear friction free vibration: the displacement (—), the envelope (), the congruent envelope of the displacement ()

Figure 11.5 The high harmonics of the instantaneous damping curve of the nonlinear friction free vibration: The time history of four first harmonics (a); the Hilbert spectrum (b)

Figure 11.6 The identified parameters of the nonlinear friction free vibration: the skeleton curve (a): instantaneous (—), averaged (- -), identified with high harmonics (); the spring static force characteristics (b): initial (-·-), identified with high harmonics (); the damping curve (c): initial (-·-), averaged (- - -), identified with high harmonics (); the friction static force characteristics (d): initial (-·-), identified with high harmonics ()

Figure 11.7 Combined nonlinear spring and damping free vibration: the displacement (—), the envelope (), the congruent envelope of the displacement ()

Figure 11.8 The identified parameters of combined nonlinear spring and damping vibrations: the skeleton curve (a): instantaneous (—), averaged (- -), identified with high harmonics (); the spring static force characteristics (b): initial (-·-), identified with high harmonics (); the damping curve (c): initial (-·-), averaged (- -), identified with high harmonics (); the friction force characteristics (d): initial (-·-), identified with high harmonics ()

Figure 11.9 Combined nonlinear spring and damping forced vibration: the displacement (—), the envelope (), the congruent envelope ()

Figure 11.10 The identified parameters of combined nonlinear spring and damping forced vibrations: the skeleton curve (a): instantaneous (—), averaged (- -), identified with high harmonics (); the spring static force characteristics (b): initial (-·-), identified with high harmonics (); the damping curve (c): initial (-·-), averaged (- -), identified with high harmonics (); the friction force characteristics (d): initial (-·-), identified with high harmonics ()

Figure 11.11 The free vibration of the asymmetric bilinear system

Figure 11.12 The estimated characteristic of a bilinear system: backbone (a), spring force characteristic (b), damping curve (c), damping force characteristics (d)

Figure 11.13 The free vibration of the asymmetric system with two cubic stiffnesses

Figure 11.14 The estimated characteristic of a system with two cubic stiffnesses: backbone (a), spring force characteristic (b), damping curve (c), damping force characteristic (d)

Figure 11.15 The estimated force characteristic of a crack and notch structure: backbone (a), spring force characteristic (b), damping curve (c), damping force characteristic (d)

Figure 11.16 Influence of the coupling coefficient η on the skeleton curves of the first ω1 and the second ω2 mode of a nonlinear 2DOF vibration system: ω1, η = 1 (- -); ω1, η = 0.5 (); (-·-); ω2, η = 0 (·); ω2, η = 0.5 (+), ω2, η = 1 (⋄)

Figure 11.17 The free vibration of two coupled Duffing oscillators: the first ε (t) and second ξ(t) spatial coordinate vibration (a), the HT decomposed normal coordinates (b), the mode shapes of two modes (c)

Figure 11.18 The skeleton curves of the first hardening stiffness mode (a): the modal (- -), the identified spatial (), the initial spatial (···); the skeleton curves of the second softening mode (b): the modal (- -), the identified spatial (), the initial spatial (···); the coupling of nonlinear modes (c): the first equation identified spatial (- -), the second equation identified spatial (), the initials (-·-)

Figure 11.19 The free vibration of two oscillators with nonlinear couplings: the first φ (t) and second ξ(t) spatial coordinate vibration (a), the HT decomposed normal coordinates (b), the mode shapes of two modes (c)

Figure 11.20 The skeleton curves of the first mode with nonlinear couplings (a): the modal (- -), the identified spatial (), the initial spatial (-□-); the skeleton curves of the second softening mode (b): the modal (- -), the identified spatial (), the initial spatial (-□-); the coupling of nonlinear modes (c): the first equation identified spatial (- -), the second equation identified spatial (), the initials (-·-)

Figure 11.21 Self-excited vibration of two coupled van der Pol oscillators: the first φ(t) and second ξ(t) spatial coordinate vibration (a), the HT decomposed normal coordinates (b), mode shapes of two modes (c)

Figure 11.22 Skeleton curves of the first mode (a): the modal (- -), the initial spatial (-□-); skeleton curves of the second mode (b): the modal (- -), the initial spatial (-□-); the identified nonlinear friction force of the first van der Pol oscillator (c); the identified nonlinear friction force of the second van der Pol oscillator (d)

Figure 12.1 HT procedures for the identification of vibration systems.

List of Tables

Table 4.1 Typical examples of the central frequency and the spectral bandwidth of random vibration (Feldman, 2009b)

Table 6.1 Application of the HT decompositions for typical vibration signals

Table 10.1 Extreme and mean values of the envelope and the IF of the biharmonic signal

Table 10.2 Model parameters

Preface

The object of this book, Hilbert Transform Applications in Mechanical Vibration, is to present a modern methodology and examples of nonstationary vibration signal analysis and nonlinear mechanical system identification. Nowadays the Hilbert transform (HT) and the related concept of an analytic signal, in combination with other time--frequency methods, has been widely adopted for diverse applications of signal and system processing.

What makes the HT so unique and so attractive?

The information obtained can be further used in design and manufacturing to improve the dynamic behavior of the construction, to plan control actions, to instill situational awareness, and to enable health monitoring and preventive surplus maintenance procedures. Therefore, the HT is very useful for mechanical engineering applications where many types of nonlinear modeling and nonstationary parametric\break problems exist.

This book covers modern advances in the application of the Hilbert transform in vibration engineering, where researchers can now produce laboratory dynamic tests more quickly and accurately. It integrates important pioneering developments of signal processing and mathematical models with typical properties of mechanical construction, such as resonance, dynamic stiffness, and damping. The unique merger of technical properties and digital signal processing provides an instant solution to a variety of engineering problems, and an in-depth exploration of the physics of vibration by analysis, identification, and simulation. These modern methods of diagnostics and health monitoring permit a much faster development, improvement, and economical maintenance of mechanical and electromechanical equipment.

The Hilbert Vibration Decomposition (HVD), FREEVIB, FORCEVIB, and congruent envelope methods presented allow faster and simpler solutions for problems -- of a high-order and at earlier engineering levels -- than traditional textbook approaches. This book can inspire further development in the field of nonlinear vibration analysis with the use of the HT.

Naturally, it is focused only on applying the HT and the analytic signal methods to mechanical vibration analysis, where they have greatest use. This is a particular one-dimensional version of the application of HT, which provides a set of tools for understanding and working with a complex notation. HT methods are also widely used in other disciplines of applied mechanics, such as the HT spectroscopy that measures high-frequency emission spectra. However, the HT is also widely used in the bidimensional (2D) case that occurs in image analysis. For example, the HT wideband radar provides the bandwidth and dynamic range needed for high-resolution images. The 2D HT allows the calculation of analytic images with a better edge and envelope detection because it has a longer impulse response that helps to reduce the effects of noise.

HT theory and realizations are continually evolving, bringing new challenges and attractive options. The author has been working on applications of the HT to vibration analysis for more than 25 years, and this book represents the results and achievements of many years of research. During the last decade, interest in the topic of the HT has been progressively rising, as evidenced by the growing number of papers on this topic published in journals and conference proceedings. For that reason the author is convinced that the interest of potential readers will reach its peak in 2011, and that this is the right time to publish the book.

The author believes that this book will be of interest to professionals and students dealing not only with mechanical, aerospace, and civil engineering, but also with naval architecture, biomechanics, robotics, and mechatronics. For students of engineering at both undergraduate and graduate levels, it can serve as a useful study guide and a powerful learning aid in many courses such as signal processing, mechanical vibration, structural dynamics, and structural health monitoring. For instructors, it offers an easy and efficient approach to a curriculum development and teaching innovations.

The author would like to express his utmost gratitude to Prof. Yakov Ben-Haim (Technion), Prof. Simon Braun (Technion), and Prof. Keith Worden (University of Sheffield) for their long-standing interest and permanent support of the research developments included in this book.

The author has also greatly benefited from many stimulating discussions with his colleagues from the Mechanical Engineering Faculty (Technion): Prof. Izhak Bucher, Prof. David Elata, Prof. Oleg Gendelman, and Prof. Oded Gottlieb. These discussions provided the thrust for the author's work and induced him to continue research activities on the subject of Hilbert transforms.

The book summarizes and supplements the author's investigations that have been published in various scientific journals. It also reviews and extends the author's recent publications: Feldman, M. (2009) “Hilbert transform, envelope, instantaneous phase, and frequency”, in Encyclopedia of Structural Health Monitoring (chapter 25). John Wiley & Sons Ltd; and Feldman, M. (2011) “Hilbert transform in vibration analysis” (tutorial), Mechanical Systems and Signal Processing, 25 (3).

The author is very grateful to Donna Bossin and Irina Vatman who had such a difficult time reading, editing, and revising the text. Of course, any errors that remain are solely the responsibility of the author.

Michael Feldman