[M.haase Et Al.,2003] Damage Identification Based on Ridges and Maxima Lines of the Wavelet Transform

Damage identification based on ridges and maxima linesof the wavelet transform

M. Haase a,*, J. Widjajakusuma b

a Institut f€uur Computeranwendungen (ICA II), University of Stuttgart, D-70569 Stuttgart, Germanyb Institut f€uur Mechanik (Bauwesen), University of Stuttgart, D-70569 Stuttgart, Germany

Abstract

The paper analyses the transient vibration behaviour of structures using the continuous wavelet trans-

form (CWT), which provides effective tools for detecting changes in the structure of the material. The

advantage of the CWT over commonly used time-frequency methods like the Wigner–Ville and the Gabor

transform is its ability to decompose signals simultaneously both in time (or space) and frequency (or scale)

with adaptive windows. The essential information is contained in the maxima of the wavelet transform.

From the ridges, the modal parameters of the decoupled modes can be extracted and the signal can be

reconstructed. From the maxima lines, defects can be localized. This paper presents a new approach for the

calculation of wavelet transform ridges and maxima lines, which is based on a direct integration of dif-ferential equations. The potential of the method is demonstrated by the analysis of the impact vibration

response of different bars.

� 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Parameter identification; Nondestructive testing; Wavelet transform; Maxima lines; Ridges

1. Introduction

Damage can be defined as the deterioration of the material properties due to the presence ofmicrocracks, microvoids and other microdefects. As a result of damage, the function and theworking condition of engineering structures and designs such as buildings, bridges, platforms,aircraft and machines are affected. Furthermore, as a result of excessive service loads, impact andfatigue, the damage continuously accumulates within the structures or designs during their service

*Corresponding author. Tel.: +49-711-685-3700; fax: +49-711-685-3758.

E-mail address: [email protected] (M. Haase).

0020-7225/03/$ - see front matter � 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0020-7225(03)00026-0

International Journal of Engineering Science 41 (2003) 1423–1443www.elsevier.com/locate/ijengsci

mail to: [email protected]

period. Therefore, in order to avoid the failure of the structure, which may lead to accidents andcost human lives, the structure�s damage should be detected as early as possible.Damage detection by nondestructive testing is carried out in most cases by visual inspection.

However, this is unreliable for complex structures because certain microdefects may occur ininaccessible areas. Therefore, in addition to visual inspection, other more sophisticated methodssuch as thermal methods, radiography, eddy currents, liquid penetrants, ultrasonic methods andvibration measurements [1] have been developed and are applied to detect the damage.The advantage of vibration-based methods is that they can detect damage in a global sense even

when the location of the damage is inaccessible. Due to the damage, the dynamic behaviour ofstructures is changed and can therefore be used to identify and quantify the structural damage.The change of the dynamic behaviour can be expressed in terms of the variation of dampingparameters and the change of the vibration frequencies. Furthermore, as long as the defects arelarge enough, they can be localized by analysing and interpreting sonic echo traces [2].In order to estimate the modal parameters of a structure, it is useful to analyse its free response

to a short-term impulse excitation. This excitation merely influences the amplitude of the resultingvibration. During the transient phase, the structure exhibits oscillations at its natural frequencies.The decay of these oscillations determines the damping behaviour [14].Different techniques have been proposed for the estimation of modal parameters; most of them

work either in the time or the frequency domain. For general nonlinear multi-degree-of-freedom(MDOF) systems, there are limitations to all these methods (for details see [3,4] and the referencestherein). For systems which display changes in their instantaneous frequencies as time evolves,combined time-frequency approaches are used. Among the most commonly used time-frequencymethods are Wigner–Ville and Gabor representations. These approaches also suffer from severaldrawbacks. The Wigner–Ville transform shows spurious interference phenomena due to thequadratic form of the transformation. These artifacts can only be removed at the expense of areduction of the time-frequency resolution. The Gabor transform, although taking advantage ofan optimal time-frequency localization according to Heisenberg�s uncertainty principle [11], suf-fers from the drawback of having fixed windows, a disadvantage common to all windowedFourier transforms. Time-frequency methods are often used together with the Hilbert transform[5–7]. In this case, one first has to decouple the modes using the band pass filtration of the signalwhich, however, may fail in nonlinear situations.Following recent advances in wavelet analysis, new techniques in signal processing have been

developed. The main advantage of the continuous wavelet transformation (CWT) is its ability toprovide information simultaneously in time and scale with adaptive windows. The CWT offerspromising tools for the estimation of modal parameters and new perspectives for damage iden-tification within the structures. There is a growing number of publications reporting on systemidentification using wavelet techniques, see, for example, [2,4,8,9,25]. Two main features make theCWT particularly attractive. Firstly, the vibration modes can be automatically decoupled in mostcases where the natural frequencies are not too close, which allows for an accurate extraction ofthe instantaneous frequencies and damping parameters. Secondly, the essential information iscontained in a small subset of the CWT, namely in the maxima lines and ridges. Therefore, incontrast to other studies, we propose in this paper to restrict the calculation of the CWT from thebeginning only to these special lines, which are obtained by direct tracing. For this purpose, wederive sets of ordinary differential equations which are directly integrated in physical space leading

1424 M. Haase, J. Widjajakusuma / International Journal of Engineering Science 41 (2003) 1423–1443

either to maxima lines or ridges. Hence, time-consuming calculations of the complete CWT andcumbersome chaining techniques are avoided.The outline of the paper is as follows. In Section 2, we briefly discuss the CWT. Two families of

wavelets, the real Gaussian and the complex Morlet family of wavelets with an increasing numberof vanishing moments, are presented in Sections 3.1 and 3.2, respectively. For both families, wederive recursive relations and partial differential equations for the wavelet transform, which arethe basis for a direct integration of maxima lines and ridges in the time or space domain. Twoexamples are given in Section 3.3 to describe the special fields of applications for the Gaussian andthe Morlet family of wavelets. Section 4 reviews the potential of complex progressive wavelets forthe analysis of asymptotic signals which are characterized by slowly varying amplitudes and phasevariations. In Section 5, we derive two systems of ordinary differential equations for the directintegration of maxima lines and ridges, respectively. In Section 6, the method is demonstrated ona two-degree-of-freedom system. In Section 7, we apply the analysis to the experimental data ofthe impact vibration response of bars with and without joint connections [23]. The results of thewavelet analysis are compared with those of the vibration analysis system Medusa [24]. Finally,conclusions are drawn in Section 8 and future work is discussed.

2. Continuous wavelet transform

Recently, wavelet analysis has attracted much attention since it allows signals to be unfolded intime and scale. According to the definition

Wf ða; bÞ ¼ 1

a

Z þ1

�1f ðtÞw t � b

a

� �dt ða; b 2 R; a > 0Þ; ð1Þ

the CWT decomposes the signal f ðtÞ 2 L2ðRÞ hierarchically in terms of elementary componentsw ðt � bÞ=að Þ which are obtained from a single mother wavelet wðtÞ by dilations and translations.Here, wðtÞ denotes the complex conjugate of wðtÞ, a the scale and b the shift parameter. Thecrucial point is to choose wðtÞ so that it is well localized both in physical and Fourier space.A unique reconstruction of the signal is ensured if wðtÞ (resp. its Fourier transform wwðxÞ)

satisfies the admissibility condition

Cw ¼Z þ1

0

jwwðxÞjx

2

dx < 1; ð2Þ

which reduces for wðtÞ 2 L1ðRÞ to the simple zero mean condition for wðtÞZ 1

�1wðtÞdt ¼ 0: ð3Þ

There is an infinite number of possible choices for the mother wavelets. For example, some ofthem are especially suitable for detecting and characterizing irregularities in the signal or even in

M. Haase, J. Widjajakusuma / International Journal of Engineering Science 41 (2003) 1423–1443 1425

its N th derivative. For this purpose, we require wðtÞ to be orthogonal to polynomials up to orderN . Hence,

Z þ1

�1tkwðtÞdt ¼ 0 06 k6N : ð4Þ

Complex analytic wavelets, often denoted as progressive wavelets, i.e. wavelets wðtÞ such thatwwðxÞ ¼ 0 for negative values x, are used to extract instantaneous frequencies. In the next section,we describe the properties of two such families of wavelets.The signal f ðtÞ can be uniquely recovered by the inverse wavelet transform

f ðtÞ ¼ 1

Cw

Z þ1

�1

Z 1

0

Wf ða; bÞw t � ba

� �daa2db: ð5Þ

Note that, analogously to the Fourier transform, the CWT is a linear integral transformationand therefore according to Parseval�s theorem conserves energy, i.e. inner products and norms inL2ðRÞ, up to a factor 2p [10,11]. Consequently, the CWT may as well be performed in Fourierspace reading

Wf ða; bÞ ¼ 1

2p

Z þ1

�1ff ðxÞeibxwwðaxÞdx: ð6Þ

Here, we used the following convention for the Fourier transform ff ðxÞ of a function f ðtÞ 2 L2ðRÞ

ff ðxÞ ¼Z þ1

�1f ðtÞe�ixt dt; t 2 R: ð7Þ

3. Two families of wavelets

In this section, two families of wavelets, each of specific use for different purposes, are intro-duced. The first is the Gaussian family which consists of real wavelets. This family is obtained asderivatives of the Gaussian function. They are very efficient in detecting sharp signal transitionsand irregularities. The second is the Morlet family which consists of complex progressive wavelets.Using Morlet wavelets means the vibration modes can be decoupled and the temporal evolutionof frequency transients and damping coefficients can be measured.

3.1. Gaussian family of wavelets

We define Gaussian wavelets of nth order wðtÞ as

wnðtÞ ¼d

dtwn�1ðtÞ ðn 2 NÞ; ð8Þ


where

w0ðtÞ ¼ e�t2=2: ð9Þ

By induction, it can be shown that the following relation holds

ðnþ 1ÞwnðtÞ þ twnþ1ðtÞ þ wnþ2ðtÞ ¼ 0: ð10Þ

For n > 0, the functions wnðtÞ fulfil the admissibility condition Eq. (2) and can thus be used asmother wavelets. Although wnðtÞ has an infinite support, the function as well as its Fouriertransform decay rapidly to zero. Therefore, in practice, it can be considered to be well localized intime and frequency. The second derivative w2ðtÞ is called Mexican hat and was first used incomputer vision to detect multiscale edges [21]. Fig. 1 shows the Mexican hat together with itsFourier transform.For the CWT of a signal f ðtÞ using wnðtÞ as a kernel, we use the following abbreviation

Wf ða; bÞ ¼ 1

a

Z þ1

�1f ðtÞwn

t � ba

� �dt ¼: W nf : ð11Þ

The special properties of Gaussian wavelets can be used to derive a partial differential equationfor W nf (for details, see [22])

ao2

ob2

�� o

oaþ na

�W nf ¼ 0; ð12Þ

which is subsequently used for a direct integration of the maxima lines.An important point is that, for discretely sampled functions f ðtÞ, it is possible to derive explicit

expressions for the convolution integral Eq. (11). It is then possible to evaluate W nf at any

-4 -2 0 2 4-0.1

0

0.1

0.2

0.3

0.4

0.5

(a)

-6 -4 -2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(b)

ψ 2t()

t

ψ 2ω(

)

ω

Fig. 1. (a) Mexican hat wavelet w2ðtÞ, (b) corresponding Fourier transform cw2w2ðxÞ.


arbitrarily chosen point ða; bÞ separately which proves to be essential for an efficient direct inte-gration of the maxima lines presented in Section 5. This method contrasts with the commonly

used method to evaluate the integral Eq. (11) as an inverse Fourier transform of ff ðxÞwwðaxÞaccording to Eq. (6) where the full wavelet transform has to be calculated for each fixed scale a.

3.2. Morlet family of wavelets

In contrast with real wavelets, the complex wavelets can separate amplitude and phase, en-abling the measurement of instantaneous frequencies and their temporal evolution.Let us define a family of complex wavelets which are obtained as derivatives of the classical

Morlet wavelet

W0ðtÞ ¼ e�t2=2eix0t ð13Þ

which marks the starting point of the development of wavelet analysis. Initially, it was employedby the geophysicist Jean Morlet in the late 1970s for oil exploration. The Morlet wavelet does notfulfil the admissibility condition (2) in a strict sense. However, for practical purposes, because ofthe fast decay of its envelope towards zero we can consider the Morlet wavelet W0ðtÞ to be ad-missible for x0 P 5. Contrary to W0ðtÞ which depends on x0, all derivatives of W0ðtÞ are waveletsin a strict sense

WnðtÞ ¼d

dtWn�1ðtÞ ðn 2 NÞ: ð14Þ

The main importance of the usefulness of this family of wavelets is the fact that all members areprogressive (or analytic) meaning

WWnðxÞ ¼ 0 for x < 0: ð15Þ

This condition is easily understood if we consider the wavelet transform of f ðtÞ ¼ cosx1t. Byinserting its Fourier transform ff ðxÞ ¼ p½dðx þ x1Þ þ dðx � x1Þ into Eq. (6), one obtains

Wf ða; bÞ ¼ 1

2eibx1WWðax1Þh

þ e�ibx1WWð�ax1Þi; ð16Þ

which means that in general jWf ða; bÞj is oscillating in b-direction. However, for progressivewavelets satisfying Eq. (15), these oscillations disappear, resulting in the relation

jWf ða; bÞj ¼ 1

2jWWðax1Þj: ð17Þ

For the Morlet wavelet, the Fourier transform reads WW0ðxÞ ¼ffiffiffiffiffiffi2p

pe�ðx�x0Þ2=2 and one observes

a perfect localization of energy around the line a ¼ x0=x1 [11]. In Fig. 2, the Morlet wavelet isplotted together with its Fourier transform.


In analogy to the Gaussian family, both a recursive relation

ðnþ 1ÞWnðtÞ þ ðt � ix0ÞWnþ1ðtÞ þ Wnþ2ðtÞ ¼ 0 ð18Þ

and a partial differential equation

ao2

ob2

�� o

oa� ix0

o

obþ na

�W nf ¼ 0 ð19Þ

can be derived for the Morlet family [12]. These relations enable us to evaluate the wavelettransform for discretely sampled functions f ðtÞ explicitly in the physical space. As mentionedbefore, this is proven to be crucial for direct integration of ridges described in the next section.

3.3. Two examples

The CWT based on Gaussian wavelets is especially suitable for detecting unexpected events insignals. For example, these may be caused by reflection of waves at cracks or boundaries leadingto discontinuities in the derivatives [2].

Fig. 2. Morlet wavelets: (a) W0ðtÞ and (b) W0ð2t � 2Þ; (c, d) Modulus of the corresponding Fourier transforms.


As a first example, let us consider the CWT of the function

f ðtÞ ¼ c1e�kðt�t1Þ2 þ c2jt � t2jh; ð20Þ

(c1 ¼ 0:2, c2 ¼ �0:6, k ¼ 312:5, h ¼ 0:5, t1 ¼ 0:2, t2 ¼ 0:8, 06 t6 1). Fig. 3a shows the graph ofthe function. Apart from the irregular cusp at t2, the function is everywhere C1. In Fig. 3b, theCWT based on the Mexican hat wavelet w2ðtÞ is plotted where a logarithmic scale is used for a andsmall scales are at the top. It can be shown [17,18] that the CWT scales like

jWf ða; t0Þj � ahðt0Þ for a ! 0 ð21Þ

provided there is no oscillating singularity (like for instance f ðtÞ ¼ sin½1=ðt � t0Þ at t0) and themother wavelet has nw > hðt0Þ vanishing moments. The exponent hðt0Þ is denoted as H€oolder ex-ponent and characterizes the strength of the irregular behaviour at t0: the faster jWf ða; t0Þj tends tozero as a ! 0, the more regular the function is [17]. If hðt0Þ > nw, then

jWf ða; t0Þj � anw for a ! 0: ð22Þ

Fig. 3. (a) Signal f ðtÞ with cusp irregularity, (b) modulus of the CWT using the Mexican hat wavelet, (c) maxima lines,(d) scaling behaviour along maxima lines.


This scaling information is completely contained in the maxima lines of the CWT defined by

o

objWf ða; bÞj ¼ 0 and

o2

ob2jWf ða; bÞj < 0; ð23Þ

see Fig. 3c. From the log–log plot displayed in Fig. 3d, the scaling exponent at t2 ¼ 0:8 can be readoff as h ¼ 0:5 according to the signal given in Eq. (20), while the scaling exponent at t1 ¼ 0:2 justreflects the number of vanishing moments nw ¼ 2 of the Mexican hat w2ðtÞ. Moreover, only at theirregular point t2 do the maxima lines converge towards t2, thus identifying its location.The extraction of irregularities in the signal or in its derivatives can be used for the interpre-

tation of experimental vibration data recorded for damage detection. Transient events are oftenburied in the vibration signals but can be highlighted and located with the aid of wavelets. Ex-amples, among others, are the nondestructive testing of foundation piles [2], tooth-fault detectionin gear-boxes [19] or detection of cracks that open and close during vibration [20]. Another in-teresting area would be the interpretation of vibration frequency spectra where wavelets couldhelp to estimate the location of peaks better, to detect additional frequencies or their shiftsstemming from deteriorations.As a second example, let us demonstrate the difference in use between real and complex

wavelets. We consider a nonstationary signal with increasing frequency, a so-called linear chirp

f ðtÞ ¼ sinf2p½t þ ðt � 256Þ2=8000g: ð24Þ

The graph of f ðtÞ is plotted in Fig. 4a. We calculate the modulus of the CWT of f ðtÞ using boththe real wavelet w1ðtÞ and the complex progressive waveletW1ðtÞ as mother wavelet (Fig 4b and c).Clearly, the CWT using the real wavelet repeats all oscillations of the signal, the increase infrequency can be seen only qualitatively. In contrast, the modulus of the CWT using the complexwavelet W1ðtÞ shows a concentration near a line, the so-called ridge, which traces the instanta-neous frequency; note that again a logarithmic scale is used for the parameter a.From these examples, it is easy to understand that complex progressive wavelets are very ef-

ficient in filtering out the natural frequencies and the damping behaviour of the modes of vi-brating systems, while real wavelets are used to detect abrupt changes and to characterizeirregular behaviour.

Fig. 4. Linear chirp: (a) graph of f ðtÞ, (b) modulus of the wavelet transform using the real wavelet w1ðtÞ, (c) modulus ofthe wavelet transform using the complex progressive wavelet W1ðtÞ.


4. Wavelet transform of asymptotic signals

An arbitrary real signal f ðtÞ can always be written in the form

f ðtÞ ¼ sin ~AAðtÞ cos½~UUðtÞ: ð25Þ

However, such a representation is not unique. To achieve uniqueness, it is common to intro-duce the analytic function

zðtÞ ¼ f ðtÞ þ iHf ðtÞ ð26Þ

with f ðtÞ as the real part and the Hilbert transform Hf ðtÞ as the complex part, defined as

Hf ðtÞ ¼ 1

p

Z þ1

�1

f ðsÞs � t

ds: ð27Þ

By definition, zðtÞ is entirely characterized by its real part and its Fourier transform zzðxÞ is zerofor negative frequencies. The so-called canonical representation of f ðtÞ

zðtÞ ¼ AðtÞeiUðtÞ ð28Þ

is then unique if we assume AðtÞP 0 and UðtÞ 2 ½0; 2p, see [11]. This allows one to introduce theinstantaneous frequency

xðtÞ ¼ U0ðtÞ; ð29Þ

where the sign 0 denotes the derivative with respect to time t. The physical significance of xðtÞmight be doubtful in specific cases unless we restrict consideration to asymptotic signals with AðtÞand U0ðtÞ slowly varying [16]. The envelope AðtÞ and xðtÞ then have a physical meaning. Using awavelet of the Morlet family, the CWT can be approximated by the first term of an asymptoticexpansion using the stationary phase argument [15,16]

Wzða; bÞ ¼ AðbÞeiUðbÞWWðaU0ðbÞÞ þ OjA0jjAj ;

jUU00jU02

!: ð30Þ

For a monocomponent signal Eq. (28), the modulus of the wavelet transform is concentrated inthe neighbourhood of a curve, called ridge of the wavelet transform satisfying the condition

a ¼ arðbÞ ¼x�

U0ðbÞ ; ð31Þ

where x� denotes the peak frequency of WnðxÞ [16]. By inserting this relation into (30), we see thatthe wavelet transform along the ridge is approximately proportional to the analytic signal zðtÞgiven in Eq. (28) with a constant factor


C ¼ WWðaU0ðbÞÞ ¼ WWðx�Þ: ð32Þ

In this paper, the investigated signals are vibrations of MDOF systems which in general can bewritten as a superposition of monochromatic components

f ðtÞ ¼XMj¼1

AjðtÞ cosUjðtÞ ð33Þ

which can be assumed to have slowly varying amplitudes AjðtÞ and phases U0jðtÞ. The corre-

sponding wavelet transform, being a linear operation, may be written in the form

Wf ða; bÞ ¼ 1

2

XMj¼1

AjðbÞ cosUjðbÞWWðaU0jðbÞÞ þ rða; bÞ; ð34Þ

where

rða; bÞ � OjA0

jjjAjj

;jUjU

00j j

U02j

!:

If the signal contains several components whose frequencies are sufficiently apart, the singlecomponents can be reconstructed again from the ridges of the wavelet transform. For interactingridges or close frequencies, the analysis and reconstruction of a k-frequency shifted version of f ðtÞare recommended to analyse and reconstruct in order to obtain a better frequency resolution (fordetails see [15,4]).

5. Direct tracing of maxima lines and ridges

One of the most valuable features of the wavelet transform is that it allows a very preciseanalysis of the regularity properties of a signal. Even for sampled functions given byff1; f2; . . . ; fng, one can filter out very precisely those points where the signal or one of its de-rivatives displays abrupt changes. This is possible by analysing the scaling behaviour, Eq. (21),along some special lines, the so-called maxima lines where the modulus of the CWT is concen-trated.Another main feature of the wavelet transform is its capacity to decompose vibrations into

natural components according to their natural frequencies, see Eqs. (33) and (34). If the Fouriertransform of an analysing progressive wavelet wðtÞ concentrates near a fixed frequency x�, themodulus of the CWT concentrates near a series of curves called ridges.The CWT Wf ða; bÞ is a two-dimensional unfolding in scale and time of a one-dimensional

signal f ðtÞ and therefore highly redundant. We have seen that the most relevant information iscontained in a much smaller set of ða; bÞ-values, namely the maxima lines and the ridges, see Figs.3 and 4. The signal may even be reconstructed accurately from these special lines; an example is


given in the next section. In the following, we describe how maxima lines and ridges can be de-termined directly without having to calculate the full CWT first.Commonly, the calculation of maxima lines and ridges is performed by first evaluating the

CWT, which is done, as for other convolutions, by a fast Fourier transform. Hence, the CWT isobtained for a discrete mesh of points ðai; bjÞ. Local maxima are next determined either for aconstant scale ai or a constant time bj for maxima lines and ridges, respectively. An additionalchaining stage is then necessary to obtain connected lines, which might be difficult in the case ofbifurcating lines.We do not follow this line but propose instead a method for calculating the CWT continuously

only along the relevant lines, thus avoiding unnecessary calculations and difficult chaining al-gorithms. The special properties of the Gaussian and Morlet family of wavelets allow the deri-vation of a set of ordinary differential equations for the parameterized lines aðsÞ, bðsÞ which maybe integrated numerically. Such a procedure is only effective if the calculation of the CWT can beperformed for each point ða; bÞ separately in the time domain, which is possible for discretelysampled functions. This method has another advantage over the common scheme, where thewavelet transform is evaluated in the Fourier domain, which is based on the assumption that thesignal is periodic. The advantage is that spurious wrap-around effects are avoided (see [2,25]).

5.1. Differential equations for the maxima lines

For a fixed scale a0 the local maximum of jW nf ða0; bÞj is defined as

jW nf ða0; bÞjb ¼ 0 and jW nf ða0; bÞjbb < 0; ð35Þ

where we used the abbreviation ðoF =obÞ ¼ Fb. Connected lines of local maxima are called maximalines. For simplicity, let us restrict ourselves here to the Gaussian family of wavelets. An extensionto complex wavelets is straightforward. Instead of Eq. (35), we use the following simpler condi-tions

ðW nf ða0; bÞÞb ¼ 0 and ðW nf ða0; bÞÞbb < 0; ð36Þ

which lead to maxima lines of ðW nf ða0; bÞÞ2. Describing the maxima lines in a parametric formfaðsÞ; bðsÞg, we may approximate the change of ðW nf Þb along the line by the first terms in a Taylorexpansion yielding

d

dsðW nf Þb ¼ ðW nf Þba

dads

þ ðW nf Þbbdbds

¼ 0: ð37Þ

The differential equations for maxima lines can thus be written in the form

dads

¼ �cðW nf Þba;dbds

¼ cðW nf Þbb ð38Þ

with an arbitrary constant c regulating the parameterization. Explicit expressions for theGaussian and the Morlet family can easily be obtained using Eqs. (8)–(10) and (12), and Eqs. (13,


14) and (18, 19), respectively. Strictly speaking, Eq. (38) is a set of two integro-differentialequations. However, the integrals on the right hand side can be explicitly expressed for theGaussian and Morlet family of wavelets of each point ða; bÞ in terms of the sampling valuesff1; f2; . . . ; fng. The procedure works as follows. First, one calculates the b-values fb1; b2; . . . ; bMgof all maxima of ðW nf Þ2 on the smallest scale amin. Each pair ðamin; biÞ ði ¼ 1; . . . ;MÞ is then usedas an initial condition for the differential equations (38), which can be integrated numerically. Themaxima lines are obtained following this procedure (Fig. 3c).

5.2. Differential equations for the ridges

The transient vibration behaviour of structures can be described by a function f ðtÞ in the formof Eq. (33), i.e. as a superposition of asymptotic components with slowly varying amplitudes andphase variations. If we use wavelets WnðtÞ of the Morlet family, the modulus of the CWT showshigh concentrations along a series of curves denoted as ridges and given by Eq. (31). For thedetermination of the ridges, it is in general sufficient to determine local maxima of jW nf ða; b0Þj fora fixed time b0 [11]. Hence, the conditions for ridges can be written in the form

jW nf ða; b0Þja ¼ 0 and jW nf ða; b0Þjaa < 0 ð39Þ

in analogy to Section 5.1. For practical calculations, we use

jW nf ða; b0Þj2a ¼ 0 and jW nf ða; b0Þj2aa < 0: ð40Þ

As before, we derive differential equations for the ridges, which are written in a parametric formfaðsÞ; bðsÞg. Since we use complex wavelets here, they have a slightly different form

dads

¼ �CGbða; bÞdbds

¼ CGaða; bÞ; ð41Þ

where Gða; bÞ ¼ R½ðW nf ÞaW nf and C is an arbitrary constant.For the free response of a structure to a short time impulse, we calculate the local maxima

fa1; a2; . . . ; aMg for a fixed time b0 and use the pairs ðai; b0Þ ði ¼ 1; . . . ;MÞ as initial conditions forthe integration of the differential equations (41).As an example, let us consider the free vibration of a Duffing oscillator with a nonlinear hard

spring described by the equation of motion for the deflection yðtÞ [13]

y00 þ cy 0 þ k1y þ k3y3 ¼ 0 ð42Þ

with c ¼ 0:08, k1 ¼ 1, k3 ¼ 0:14. The equation is integrated for 06 t6 90 using the initial con-ditions y0 ¼ 3, y 00 ¼ 0. The graph of yðtÞ is plotted in Fig. 5a. By inspection, one can see that thefrequency decreases as time evolves and the vibration is damped. Fig. 5b shows the modulus of theCWT using the Morlet wavelet. It can be seen that the modulus is highly concentrated along a


curved sharp ridge displaying the increase in scale, i.e. the decrease of the instantaneous fre-quency.

6. Application to system identification

To illustrate the method, we analyse the simple case of a linear 2 degrees-of-freedom system anddemonstrate how instantaneous frequencies and damping parameters can be extracted from theinformation provided by the CWT along the ridges alone. We also show that the signal can beaccurately reconstructed in a simple way.Let us analyse the following impulse response of a two-degree-of-freedom system, which can be

formulated as [4]

f ðtÞ ¼X2j¼1

AjðtÞ cosUjðtÞ; ð43Þ

where AjðtÞ ¼ aje�fjxjt and UjðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� f2j

qxjt þ uj; xj are the natural frequencies, fj the damping

ratios, aj the amplitudes for t ¼ 0 and uj the phase shift of the jth mode ða1 ¼ 0:5, a2 ¼ 3:0,f1 ¼ 0:03, f2 ¼ 0:045, x1 ¼ 40p ¼ 125:66, x2 ¼ 156p ¼ 490:09, u1 ¼ u2 ¼ p=2Þ. In Fig. 6a, thegraph of the impulse response function is shown for 06 t6 1.As the analysing wavelet for the calculation of CWT, we have chosen the first derivative

W1ðtÞ ¼ ð�t þ ix0Þe�t2=2eix0t with x0 ¼ 5. In Fig. 6b and c, two views of the modulus of the CWTare plotted. It can be seen that the CWT decouples the vibration modes automatically.However, for our method, it is not necessary to calculate the full CWT. We only have to

evaluate the CWT for a fixed time b0 (Fig. 6d) and estimate the scales a1, a2 of the maxima yielding

Fig. 5. Duffing oscillator: (a) graph of vibration f ðtÞ, (b) modulus of the wavelet transform using the Morlet wavelet

W0ðtÞ.


a1 ¼ 4:159� 10�2 and a2 ¼ 1:046� 10�2. The corresponding frequencies xj and damping pa-rameters fj are obtained from the equations

ajffiffiffiffiffiffiffiffiffiffiffiffiffi1� f2j

qxj ¼ x�; fjxj ¼ �mj: ð44Þ

The first equation is obtained from Eq. (31), where, for the peak frequency x� of WW1ðxÞ we haveto insert the value x� ¼ 1=2ðx0 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix20 þ 4

pÞ. The slope mj in the second equation of (44) is ob-

tained by fitting a straight line to the log-plot of jW 1f ðaj; bÞj versus b along the ridge (see Fig. 7).The following values are obtained: m1 ¼ �3:76, m2 ¼ �21:34 and hence x1 ¼ 124:91, x2 ¼496:88, f1 ¼ 0:0301, f2 ¼ 0:0429.

0.001

0.01

0.1

1

10

0 0.4 0.8

m1 ζ1ω1–=

m2 ζ2ω2–=W1 f

a ib,

()

ln

b

Fig. 7. Identification of damping parameters from ridges.

Fig. 6. Free vibration: (a) graph of f ðtÞ; (b, c) modulus of the wavelet transform using W1ðtÞ; (d) extraction of scalesand natural frequencies.


In the next step, we demonstrate how the signal can be reconstructed from the CWT along theridges. In Eq. (5), the general reconstruction formula is given. A simpler reconstruction rule hasbeen proposed in [11] reading

f ðtÞ ¼ 1

Cw

Z 1

0

Wf ða; bÞdaa: ð45Þ

In this case, different wavelets are used for the analysis and for the synthesis; in the particularcase, a Dirac mass is used for reconstruction.With Eq. (34) as a basis, it is even simpler to recover the signal for the Morlet family of

wavelets. Considering that aU0ðbÞ ¼ x� holds along ridges, Eq. (34) can be rewritten in the form

Wf ða; bÞ � 1

2CXMj¼1

AjðbÞ cosUjðbÞ; ð46Þ

where C ¼ WWðax�Þ is a common constant to all ridges. Together with Eq. (33), the signal canimmediately be read off as a superposition of the wavelet transform components along the ridges.In Fig. 8b and c the two modes resulting from the ridges are plotted together with their su-

perposition shown in Fig. 8a. Apart from small edge effects, the reconstructed function matchaccurately with the signal plotted in Fig. 6a.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1

0

1

2

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.2

0

0.2

0.4Mode 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1

0

1

2Mode 2

(a)

(b)

t

(c)

f 1t()

t

t

f 2t()

ft()

Fig. 8. (a) Reconstructed signal f ðtÞ as a superposition of two modes: (b, c) mode 1 and mode 2.


The simplicity of the scheme becomes even more pronounced if we analyse nonlinear systemswith time-dependent frequencies.

7. Application to real structure

In analogy to the system identification described in the previous section, we apply the wave-let identification technique to experimental data obtained from laboratory tests [23]. To inves-tigate the influence of a bolted joint on the dynamic behaviour of steel rods of case hardened steel16 Mn Cr 5 (diameters 40 mm) the vibrations of a homogeneous rod (length 729.70 mm) and abolted rod (length 731.86 mm) are compared. The bolted rod is centrically connected by a

Fig. 10. Log-plot of the wavelet transform modulus along ridges vs time for homogeneous steel bar. Estimation of

damping coefficients using linear regression.

Fig. 9. (a) Free vibration of a homogeneous steel bar. Ridges (b) of the wavelet transform modulus (c) using W1ðtÞ. (d)Identification of natural frequencies.


threaded bolt M 12. Its contact surfaces have been machined by turning. To protect themfrom fretting, a polyester washer (thickness 50 lm) is embedded between the surfaces. Thebolted joint connection has been tightened by applying a torque of 34.9 Nm. Finally, to mini-mize the influence of external bearings, the rods are suspended by plastic ropes at 3/7 and 4/7of their overall length. On one side, the system is excited by means of an impact hammer,whereas on the other side, the velocity is measured using a laser vibrometer (sample rate D ¼196608 1/s).We apply the wavelet analysis described in the previous section to the velocity signal in the time

domain in order to determine the eigenfrequencies and the damping ratios of the natural vibra-tions. Due to the free-free suspension of the rods the hammer blow causes rigid body motions inaddition to the vibrations of the first eigenmodes. For comparison, these modal parameters arealso determined by applying the analysing software Medusa [24].For the CWT analysis we chose the first derivative W1ðtÞ ¼ ð�t þ ix0Þe�t2=2eix0t as the analysing

wavelet, where x0 ¼ 25 was chosen for a better resolution of frequencies. The measured dis-placements for the bar without joint (case I) and the bolted bar with joint (case II) are displayed inFigs. 9a and 11a, respectively. Comparison of the two plots of the wavelet transform modulus(Figs. 9c and 11c) reveals the strong damping effect on the odd modes, which is caused by the jointlocated in the center of the bar. Note, that for the determination of the modal parameters it iscompletely sufficient to calculate the CWT only along the ridges. We calculate the wavelettransform modulus for a fixed time t0 (see Figs. 9d and 11d) and extract the maxima, whichidentify the frequencies wi ¼ 2pfi. These maxima are then used as initial conditions for the directintegration of ridges according to Eq. (41). The ridges are plotted in Figs. 9b and 11b. The jointleads to a weak nonlinear behaviour giving rise to a slight increase of the frequency of the 1stmode as time evolves (see Table 1). The damping coefficients of the different modes are extractedfrom log-plots of jW 1uðfi; tÞj along the ridges (Figs. 10 and 12). The estimation of slopes fromfitting with straight lines can be performed with high accuracy by restricting to those time in-

Fig. 11. (a) Free vibration of a homogeneous steel bar with joint connection. Ridges (b) of the wavelet transform

modulus (c) using W1ðtÞ. (d) Identification of natural frequencies.


tervals where the wavelet transform decreases exponentially. It should be emphasized, that con-trarily to other wavelet algorithms [25] our results do not suffer from any wrap-around effect nearthe edges of the time interval. This effect might impair the damping coefficients. The reason is, thatwe calculate the CWT completely in physical space in contrast to other algorithms which areperformed on the basis of the Fast Fourier Transform [2,4,8,25].The natural frequencies and the damping coefficients of the two bars resulting from the

Medusa analyzer and the wavelet based identification are summarized in Tables 1 and 2,

Table 2

Estimated damping coefficients [Hz] of vibrating bar without (case I) and with joint (case II)

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

Case I

Medusa 2.14� 10�3 2.31� 10�3 2.13� 10�3 2.13� 10�3 2.26� 10�3Wavelets 2.14� 10�3 2.30� 10�3 2.13� 10�3 2.15� 10�3 2.26� 10�3

Case II

Medusa 9.21� 10�2 2.89� 10�3Wavelets 9.17� 10�2 2.81� 10�3 1.24� 10�1 4.43� 10�3

Results from Medusa analyzer and wavelet analysis.

Fig. 12. Log-plot of the wavelet transform modulus along ridges vs time for homogeneous steel bar with joint con-

nection. Estimation of damping coefficients using linear regression.

Table 1

Estimated natural frequencies [Hz] of vibrating bar without (case I) and with joint (case II)

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

Case I

Medusa 3563.8 7121.7 10675.1 14217.1 17745.5

Wavelets 3564.0 7122.0 10676.0 14212.0 17744.0

Case II

Medusa 3437.0 7126.2

Wavelets 3432.0–3438.0 7126.0 10094.0 14220.0

Results from Medusa analyzer and wavelet analysis.


respectively. The results from both approaches are in good agreement. Obviously, the joint con-nection with the centric joint leads to a change in the natural frequencies and damping propertiesof the bar.

8. Conclusions

In this paper, we have illustrated on pedagocial examples as well as on experimental mea-surements, how the CWT can be used to analyse the free vibration of structures. The essentialinformation is contained in the skeleton of maxima lines and ridges. From the ridges, the modalparameters can be extracted and the signal can be reconstructed. From the maxima lines, defectscan be located [2]. This is an important issue for damage detection.A new approach is presented which allows to determine maxima lines and ridges directly from

integration of two ordinary differential equations in real space. The advantage of this approach isthat it does not suffer from wrap-around effects, which usually occur in FFT-based wavelet al-gorithms. In addition, the calculation of the full CWT is avoided and thus the computationaleffort can be reduced. For nonlinear systems with time varying frequencies (cf. Figs. 4 and 5), thisreduction is even more pronounced.We have applied our technique to real systems (bars with and without joint). The estimated

modal parameters using wavelet analysis are in good agreement with those using commercialsoftware (Medusa). Moreover, we have reconstructed the original signal by superposing thewavelet transform components along the ridges (cf. Fig. 8). This reconstruction method may beused as the departure point to model a system on the basis of measured data [8].Further investigations are needed to study the capability of the method and are being carried

out for more complex structures. Among these are the study of modes with close frequencies[4,15], the influence of noise on the integration of the differential equations for the ridges and theextraction of backbones to characterize the nonlinear behaviour [8,13].

Acknowledgements

The authors would like to thank Stefan Oexl and Dr. Nils Wagner (Institute A of Mechanics,University of Stuttgart) for fruitful discussions. In particular, we are grateful to Stefan Oexl forsupplying the experimental data.

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Documents

[M.haase Et Al.,2003] Damage Identification Based on Ridges and Maxima Lines of the Wavelet Transform