Vibration and Shock Handbook 11

Vibration and Shock Handbook 11 Every so often, a reference book appears that stands apart from all others, destined to become the definitive work in its field. The Vibration and Shock Handbook is just such a reference. From its ambitious scope to its impressive list of contributors, this handbook delivers all of the techniques, tools, instrumentation, and data needed to model, analyze, monitor, modify, and control vibration, shock, noise, and acoustics. Providing convenient, thorough, up-to-date, and authoritative coverage, the editor summarizes important and complex concepts and results into “snapshot” windows to make quick access to this critical information even easier. The Handbook’s nine sections encompass: fundamentals and analytical techniques; computer techniques, tools, and signal analysis; shock and vibration methodologies; instrumentation and testing; vibration suppression, damping, and control; monitoring and diagnosis; seismic vibration and related regulatory issues; system design, application, and control implementation; and acoustics and noise suppression. The book also features an extensive glossary and convenient cross-referencing, plus references at the end of each chapter. Brimming with illustrations, equations, examples, and case studies, the Vibration and Shock Handbook is the most extensive, practical, and comprehensive reference in the field. It is a must-have for anyone, beginner or expert, who is serious about investigating and controlling vibration and acoustics.

11 Wavelets — Concepts

and Applications

Pol D Spanos

Rice University

Giuseppe Failla

Universita` degli Studi

Mediterranea di Reggio Calabria

Nikolaos P Politis

Rice University

11.1 Introduction 11-111.2 Time–Frequency Analysis 11-2

Gabor Transform † Wavelet Transform † Wavelet Families

11.3 Time-Dependent Spectra Estimation of StochasticProcesses 11-1111.4 Random Field Simulation 11-1411.5 System Identification 11-1511.6 Damage Detection 11-1711.7 Material Characterization 11-1811.8 Concluding Remarks 11-19

Summary

Section 11.1 provides a brief introduction to wavelet concepts in vibration-related applications Aspects of time–frequency analysis are discussed in Section 11.2 Specifically, the Gabor and wavelet transforms are outlined.Further, several wavelet families commonly used in vibration-related applications are presented Estimation oftime-dependent spectra of stochastic processes is considered in Section 11.3 Section 11.4 to Section 11.7 discussapplications of wavelet analysis in vibration-related applications In particular, applications in random fieldsimulation, system identification, damage detection, and material characterization are examined Section 11.8provides an overview and concluding remarks on the applicability and usefulness of the wavelet analysis invibration theory To enhance the usefulness of this chapter, a list of readily available references in the form of booksand archival articles is provided

11.1 Introduction

Wavelets-based representations offer an important option for capturing localized effects in many signals.This is achieved by employing representations via double integrals (continuous transforms), or viadouble series (discrete transforms) Seminal to these representations are the processes of scaling andshifting of a generating (mother) function Over a period of several decades, wavelet analysis has been set

on a rigorous mathematical framework and has been applied to quite diverse fields Wavelet familiesassociated with specific mother functions have proven quite appropriate for a variety of problems In thiscontext, fast decomposition and reconstruction algorithms ensure computational efficiency, and rivalclassical spectral analysis algorithms such as the fast Fourier transform (FFT) The field of vibrationanalysis has benefited from this remarkable mathematical development in conjunction with vibrationmonitoring, system identification, damage detection, and several other tasks There is a voluminous body

11-1

of literature focusing on wavelet analysis However, this chapter has the restricted objective of, on onehand, discussing concepts closely related to vibration analysis, and on the other hand, citing sources thatcan be readily available to a potential reader In view of this latter objective, almost exclusively books andarchival articles are included in the list of references First, theoretical concepts are briefly presented; formore mathematical details, the reader may consult references [1–23] Next, the theoretical concepts aresupplemented by vibration-analysis-related sections on time-varying spectra estimation, random fieldsynthesis, structural identification, damage detection, and material characterization It is noted that most

of the mathematical developments pertain to the interval [0,1] relating to dimensionless independentvariables derived by normalization with respect to the spatial or temporal “lengths” of the entire signals

f ðtÞ ¼ 12p g 2

ð1 21

ð1

21Gfðv; t0Þgðt 2 t0Þei v ðt2t 0 Þdv dt0 ð11:2Þwhere g 2¼Ð211 gðtÞ2dt:TheGabor transform(Equation11.1)maybeseenastheprojection off ðtÞontothe family {gðv;t0ÞðtÞ;v; t0[ R} of shifted and modulated copies (atoms) of gðtÞ expressed in the form

gðv;t0ÞðtÞ ¼ ei v ðt2t 0 Þgðt 2 t0Þ ð11:3ÞThese time–frequency atoms, also referred to as Gabor functions, are shown inFigure 11.1for threedifferent values ofv: Clearly, if gðtÞ is an appropriate window function, then Equation 11.1 may be regarded

as the standard Fourier transform of the function f ðtÞ; localized at the time t0: In this context, t0is the

time parameter which gives the center of the window, andv is the frequency parameter which is used tocompute the Fourier transform of the windowed signal.

As intuition suggests, the accuracy of the CGT representation (Equation 11.2) of f ðtÞ depends on thewindow function gðtÞ; which must exhibit good localization properties both in the time and thefrequency domains As discussed in Ref [6], a measure of the localization properties may be obtained bythe average and the standard deviation of the density gðtÞ2in the time domain That is,

stsv$ 1

Thus, high resolution in the time domain (small value of st) may be generally achieved only at theexpense of a poor resolution (bigger than a minimum valuesv) in the frequency domain and vice versa.Note that the optimal time–frequency resolution, that isstsv¼ 1=2; may be attained when the Gaussianwindow

gðtÞ ¼ ffiffiffiffiffiffiffi1

2ps2 t

24st2

the effective support is the same for the three values of the frequency.

to enhance the time resolution for short-lived high-frequency phenomena and frequency resolution forlong-lasting low-frequency phenomena.

11.2.2 Wavelet Transform

The preceding shortcomings of the Gabor transform have been overcome with significant effectivenessand efficiency by wavelets-based signal representation Its two formulations, continuous and discrete, aredescribed in the ensuing sections Because of the numerous applications of wavelets beyond time–frequency analysis, the t-time domain will be replaced by a generic x-space domain For succinctness, theformulation will be developed for scalar functions only, but generalization for multidimensional spaces iswell established in the literature [1–22]

11.2.2.1 Continuous Wavelet Transform

The concept of wavelet transform was introduced first by Goupillaud et al for seismic records analysis[26,27] In analogy to the Gabor transform, the idea consists of decomposing a function f ðxÞ into a two-parameter family of elementary functions, each derived from a basic or mother wavelet,cðxÞ: The firstparameter, a; corresponds to a dilation or compression of the mother wavelet that is generally referred to

as scale The second parameter, b; determines a shift of the mother wavelet along the x-domain Inmathematical terms

An example of wavelet functions is shown in Figure 11.2 for different values of the scale parameter a:

As a result of scaling, all the wavelet functions exhibit the same number of cycles within the x-support ofthe mother wavelet Obviously, the spatial and frequency localization properties of the wavelet transformdepend on the value of the parameter a: As a approaches zero, the dilated wavelet a21=2cðx=aÞ is highlyconcentrated at the point x ¼ 0; the wavelet transform, Wfða; bÞ; then gives increasingly sharper spatialresolution displaying the small-scale/higher-frequency features of the function f ðxÞ; at various locationsb: However, as a approaches þ1, the wavelet transform Wfða; bÞ gives increasingly coarser spatialresolution, displaying the large-scale/low-frequency features of the function f ðxÞ:

For the function f ðxÞ to be reconstructable from the set of coefficients (Equation 11.9), in the form

f ðxÞ ¼ 1

pcc

ð1 0

whereca;bđxỡ Ửa21=2cơđx 2 bỡ=a ; the wavelet function cđởỡ must satisfy the admissibility condition

ccỬđ121

^Cđvỡ2

^Cđvỡ

Subcondition 2, then, implies that ^Cđ0ỡ Ử 0; that is, Đ121cđxỡdx Ử 0: Therefore, for an analyzingwavelet to be admissible, its real and imaginary parts must both be symmetric with respect to the x-axis.From the reconstruction formula (Equation 11.10), it can be shown that

f 2Ử pc1c

đ1 0

đ1

21Wfđa; bỡ2da

Based on Equation 11.14, the square modulus of the wavelet transform (Equation 11.9) is often taken as

an energy density in a spatial-scale domain Extensive use of this concept has been made for spectraestimation purposes, as discussed in Section 11.3

Note that the reconstruction wavelet in Equation 11.10 can be different from the analyzing waveletused in Equation 11.9 That is, under some admissibility conditions onxđxỡ [1], the original function

f đxỡ may be reconstructed as

f đxỡ Ử c1

cx

đ1 0

đ1

wherexa;bđxỡ Ử a21=2xơđx 2 bỡ=a and ccxis a constant parameter depending on the Fourier transforms

of both cđxỡ and xđxỡ: This property, referred to as redundancy in mathematical terms, may beadvantageous in some applications for reducing the error due to noise in signal reconstruction [28,29],but highly undesirable for signal coding or compression purposes [1] Further, under certain conditions[1], the following simplified reconstruction formula holds

Use of this formula has been made, in a discrete version, in the approximation theory of functional spaces[1] and also in structural identification applications, as discussed in Section 11.5

11.2.2.2 Discrete Wavelet Transform

For numerical applications, where fast decomposition or reconstruction algorithms are generallyrequired, a discrete version of the CWT is to prefer In this sense, a natural way to define a discrete wavelettransform (DWT) is

Wfð j; kÞ ¼ 1ffiffiffi

a0j

21f ðxÞcða2j0 x 2 kb0Þdx; j; k [ Z ð11:18ÞEquation 11.18 is derived from a straightforward discretization of the CWT (Equation 11.9) byconsidering the discrete lattice a ¼ a0j; a0 1; b ¼ kb0a0j; b0– 0: In developing Equation 11.18,however, the main mathematical concern is to ensure that sampling the CWT on a discrete set of pointsdoes not lead to a loss of information about the wavelet-transformed function f ðxÞ: Specifically, theoriginal function f ðxÞ must be fully recoverable from a discrete set of wavelet coefficients That is,

f ðxÞ ¼ Xj;k[Z

where cj;kðxÞ ¼ a2j=20 cða2j0 x 2 kb0Þ Another crucial aspect in Equation 11.18 involves selecting thewavelet functionscj;kðxÞ such that Equation 11.19 may be regarded as the expansion of f ðxÞ in a basis,thus eliminating the redundancy of the CWT

This issues are addressed by using the theory of Hilbert space frames, introduced in 1952 by Duffin andSchaeffer in context with non-harmonic Fourier series [30] In general, if hlðxÞ [ L2ðRÞ andL is acountable set, a family of functions {hlðxÞ;l [ L} constitutes a frame, if for any f ðxÞ [ L2ðRÞ

multiresolution analysis requires that

1 The subspaces Vj’s are closed and embedded, that is

· · · , V2, V1, V0, V21, V22, · · · ð11:23Þwhere V2m! L2ðRÞ for m ! 1 and f [ Vm, f ð2·Þ [ Vm21:

2 A scaling functionfðxÞ [ L2ðRÞ exists, such that, for each j; the family of functions

spans the subspace Vj and constitutes a Riesz basis for Vj; that is, there exists 0 , C0# C00, 1such that

C0Xk

ck2#ð121

Xk

can be developed by appropriate algorithms For this, Mallat has used the frequency response of a pass filter [35], while Daubechies has devised a systematic approach to build orthonormal wavelet baseswith compact support in the x-domain [36] Specifically, for each even integer 2M; the Daubechiesscaling functionfðxÞ can be computed as

information contained in the subspace Vj and lost when “moving” to the subspace Vjþ1: Therefore,

dkj¼2M21Xl¼0

Similarly, the reconstruction algorithm can be implemented by the formula

ckj21¼Xl

hk22lþ2cljþ gk22lþ2dlj ð11:34ÞThe reconstruction algorithm described by Equation 11.34 lends itself to interpretation as a scalelinear system [37,38] Based on this concept, applications have also been developed for random fieldsimulation [39]

11.2.3 Wavelet Families

A great number of wavelet families with various properties are available Selecting an optimal family for aspecific problem is not, in general, an easy task and there are properties that prove more important tocertain fields of application For instance, symmetry may be of great help for preventing dephasing inimage processing, while regularity is critical for building smooth reconstructed signals or accuratenonlinear regression estimates Compactly supported wavelets, either in the time or in the frequencydomain, may be preferable for enhanced time or frequency resolution The number of vanishingmoments, M; that is the highest integer m for which the equation

A brief description of the most-used families is given below A distinction is made between real andcomplex wavelets, and the most relevant properties for application purposes are discussed A moreexhaustive review may be in found in Ref [15].

11.2.3.1 Real Wavelets

1 Daubechies orthonormal wavelets — A family of bases, each corresponding to a particular value ofthe parameter M in Equation 11.27 and Equation 11.28 [36] Closed-form expressions forfðxÞ inEquation 11.27 are available only for M ¼ 1; to which the well-known Haar basis corresponds Inthis case, the scaling function and the mother wavelet are

is equal to the segment ½0; 2M 2 1 : Also, M is equal to the number of vanishing moments of thewavelet function Note that most Daubechies wavelets are not symmetric; regularity andharmonic-like shape increases with M:

2 Meyer wavelets — Families of wavelets [10], each defined for a particular choice of an auxiliaryfunction vðvÞ which appears in the following expression for the Fourier transform of the motherwavelet:

^CðvÞ ¼

1ffiffiffiffi2p

p ei v =2sin p

2v

32p v 2 1 ;

2

3p # v # 43p;

1ffiffiffiffi2p

p eiv=2cos p

2v

34p v 2 1 ;

Meyer wavelets are suitable for both CWT and DWT Unlike Daubechies wavelets, they arecompact in the frequency domain but not in the x-domain Because of their fast decay, however,

an effective x-support [28,8] is generally assumed Appealing features of Meyer wavelets areorthogonality, infinite regularity, and symmetry

3 Mexican Hat wavelets — A family of wavelets in the x-domain [15] related to a mother functionthat is proportional to the second derivative of the Gaussian probability density function

4 Biorthogonal wavelets — Families of wavelets derived by generalizing the ordinary concept ofwavelet bases, and creating a pair of dual wavelets, say ðcðxÞ; ~cðxÞÞ; satisfying the followingproperties [41,42]:

ð1

21cj;kðxÞ ~cj 0 ;k 0ðxÞdx ¼djj 0dkk 0 ð11:44Þwhere the symboldmn denotes the Kronecker delta One wavelet, saycðxÞ; may be used forreconstruction and the dual one, ~cðxÞ; for decomposition Therefore, Equation 11.18 andEquation 11.19 can be rewritten as

Wfð j; kÞ ¼ 22j=2ð1

21f ðxÞcj;kð22jx 2 kÞdx; j; k [ Z ð11:45Þ

f ðxÞ ¼ Xj;k[Z

Biorthogonal wavelets support both CWT and DWT The properties of a biorthogonal basisare specified in terms of a pair of integers ðNd; NrÞ: These integers, in analogy with theDaubechies wavelets, govern the regularity and the number of vanishing moments Nd ofthe decomposition waveletcðxÞ; and the regularity and the number of vanishing moments Nrofthe reconstruction wavelet ~cðxÞ: Obviously, this feature allows a greater number of choices forsignal decomposition and reconstruction Both wavelet functionscðxÞ and ~cðxÞ are symmetric.11.2.3.2 Complex Wavelets

5 Harmonic wavelets — A Family of bases defined in the frequency domain by the formula[16,43,44]:

^

Cm;nðvÞ ¼

12p ðn 2 mÞ; mp # v # np;

cm;n;kðxÞ ¼ exp in2p x 2 n 2 mk 2 exp im2p x 2 n 2 mk

the wavelet’s center is relatively low and proportional to x21: Further, they satisfy relevantorthogonality properties [16].

From Equation 11.48, it is seen that the real part of the wavelet is even, while the imaginary part

is odd For signal processing, this ensures that harmonic components in a signal can be detectedregardless of the phase Note that this feature cannot be achieved by real wavelets such as the Meyerwavelets, which are all self-similar, being derived from a unique mother wavelet by scaling andshifting Also, note that orthonormal basis of real wavelets can be generated by considering eitherthe real or the imaginary parts only of Equation 11.48 For instance, the well-known Shannonwavelets correspond to the imaginary parts of Equation 11.48, for m; n ¼ 1; 2; 2; 4; 4; 8; … :Harmonic wavelets are used in many mechanics applications such as acoustics, vibrationmonitoring, and damage detection [45–49]

6 Complex Gaussian wavelets — Families of wavelets, each corresponding to a pth order derivative of

a complex Gaussian function That is,

cpðxÞ ¼ Cp dp

dxpðe2ixe2x 2 =2Þ; p ¼ 1; 2; … ð11:49Þwhere Cpis a normalization constant such that cðxÞ2¼ 1: Complex Gaussian wavelets supportthe CWT only They have no finite support in the x-domain, although the interval [25,5] isgenerally taken as effective support Despite their lack of orthogonality, they are quite popular inimage-processing applications due to their regularity [1]

7 Complex Morlet wavelets — Families of [50], each obtained as the derivative of the classical Morletwaveletc0ðxÞ ¼ e2x 2 =2ei v 0 x: That is,

cpðxÞ ¼ dp

dxpðe2x 2 =2ei v 0 xÞ; p ¼ 1; 2; …: ð11:50ÞExcept forc0ðxÞ; which does not satisfy the admissibility condition (Equation 11.11) in a strictsense, all the other members of the family are proper wavelets For practical purposes, however,

c0ðxÞ is generally considered admissible forv0$ 5: Complex Morlet wavelets support the CWTonly and are not orthogonal However, they are all progressive, that is, they satisfy the conditionposed by Equation 11.36 Further, for the Morlet waveletc0ðxÞ; there exists a relation between thescale parameter a and the frequencyv at which its Fourier transform focuses That is,

Appropriate description of such phenomena is obviously crucial for design and reliability assessments

In an early attempt, concepts of traditional Fourier spectral theory were generalized to provide spectralestimates, such as the Wigner–Ville method [25,54] or the CGT of Equation 11.1 However, it soon

became clear that the extension of the traditional concept of a spectrum is not unique, and proposedtime-varying spectra could have contradictory properties [6,55].

Wavelet analysis is readily applicable for estimating time-varying spectral properties, and a significanteffort has been devoted to formulating “wavelet energy principles” that work as alternatives to classicalFourier methods Measures of a time-varying frequency content were first obtained by “sectioning,” atdifferent time instants, the wavelet coefficients mean square map [49,56–58] Developing consistentspectral estimates from such sections, however, is not straightforward From a theoretical point of view, iteither requires an appropriate wavelet-based definition of time-varying spectra, or it must relate to well-established notions of time-varying spectra From a numerical point of view, it involves certaindifficulties in converting the scale axis to a frequency axis, especially when the wavelet functions are notorthogonal in the frequency domain; that is, when the frequency content of wavelet functions at adjacentscales do overlap

Early investigations on wavelet-based spectral estimates may be found in references such as[44,59–64], where wavelet analysis was applied in the context of earthquake engineering problems In aparticular approach, a modified Littlewood Paley (MLP) wavelet basis can be introduced, whose motherwavelet is defined in the frequency domain by the equation

^CðvÞ ¼

1ffiffiffiffiffiffiffiffiffiffiffiffiffi2ðs 2 1Þp

f ðtÞ ¼Xi; j

KDb

aj Wfðaj; biÞca j ;b iðtÞ ð11:53Þwhere aj¼sj; Db is a time step, and K is a constant parameter depending ons:

In many instances, Equation 11.53 can be construed as representing realizations of a stochastic process,and in this case, the following estimate of its instantaneous mean-square value of f ðtÞ has beenconstructed

In this context, linear-response statistics, such as the instantaneous rate of crossings of the zero level orthe instantaneous rate of occurrence of the peaks, have been estimated with considerable accuracy.Analysis of nonlinear systems has also been attempted by an equivalent statistical linearizationprocedure [61,66]

Wavelet analysis for spectral estimation has also been pursued by Kareem et al., who have used thesquared wavelet coefficients of a DWT to estimate the PSD of stationary processes [56] To improve thefrequency resolution of the DWT, where only adjacent octave bands can be accounted for, a CWT can beimplemented based on a complex Morlet wavelet basis The latter is preferable due to the one-to-one