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Mechanical Systems and Signal Processing (1997) 11(2), 207218

NATURAL FREQUENCIES AND DAMPINGS

IDENTIFICATION USING WAVELETTRANSFORM: APPLICATION TO REAL DATA

M. R, A. F, L. G B. P

Dipartimento di Meccanica, Politecnico di Torino, Turin, Italy

(Received May 1996, accepted October 1996)

The wavelet transform is used as a timefrequency representation for systemidentification purposes. It is shown that wavelet analysis of the free response of a systemallows the estimation of the natural frequencies and viscous damping ratios. The

advantages of the wavelet transform in the analysis of the free decay of the system areunderlined and a comparison with previous techniques is made. The accuracy of thismethod is confirmed by applying it to a numerical example and the acceleration responsesfrom a real bridge under ambient excitations (the Queensborough Bridge in Vancouver,Canada). The results obtained agree with those previously obtained on the dynamicbehaviour of the Queensborough Bridge.

7 1997 Academic Press Limited

1. INTRODUCTION

The aim of this paper is to show how the estimation of the modal parameters of a vibrating

system is possible using the wavelet transform (WT) of the systems free response. The

analysis of the impulse response of a system has been used already for modal parameters

estimation [13]. The estimation procedure presented by these authors is based on the

approximation of the actual response with its Hilbert transform (HT). This approximation

is exact only if the analysed system is conservative; errors occur when the considered

structure is damped [2]. Furthermore, the HT allows the identification only if the analysed

signal is monocomponent; when multi-degree of freedom (mdof) systems are considered,

band-pass filtering of the free response is necessary in order to separate each mode.

The WT analysis for the estimation of the modal damping ratios has already been

applied [4], where the WT as a timefrequency representation is used in the analysis of

mdof systems for modal decoupling purposes.

Besides this decoupling capability, the present paper shows how WT analysis of the freeresponse of the system represents a consistent improvement of the well-known estimation

HT technique. The WT, in fact, directly supplies a complex valued signal, where

information on the decay rate and on the phase variation in time are available, without

making any approximation.

Finally, WT allows the accurate extraction of the natural frequencies of the analysed

system.

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2. ESTIMATION TECHNIQUE

2.1. :

The WT of a signal x(t) is an example of a time-scale decomposition obtained by dilating

and translating along the time axis a chosen analysing function (wavelet). The continuous

wavelet transform is defined as follows:

Wg(a, b) =1

za g+a

a

x(t)g*0t ba 1 dt (1)were b is the parameter localising the wavelet function in the time domain, a is the dilationparameter defining the analysing window stretching and g* is the complex conjugate ofthe basic wavelet function. The wavelet function in this paper is the well-known Morletwavelet [5, 6]:

g(t) = ejv0t et2/2 (2)

where v0 is the wavelet frequency.An alternative formulation of the continuous WT can be obtained by transforming both

the signal x(t) and the wavelet function g(t) in the frequency domain:

Wg (a, b) =za g+a

a

X(v)G*(av) ejvb dv (3)

where X(v) and G*(av) ejvb are the Fourier transform ofx(t) and g*(t b)/a respectively.This formulation of the WT can be expressed in a discrete form as:

W(m, n) = zmDasn

X(fn)G*(mDafn ) ej2pfnnDb (4)

where fn is the discrete frequency and Da and Db are discrete increments of dilation and

translation parameters. Equation (4) allows an easy implementation of the WT.As a signal decomposition the WT cannot be compared directly to any timefrequencyrepresentation. However, b represents a time parameter and the dilation parameter a isrelated strictly to frequency. If we consider the dilated Morlet wavelet in the frequencydomain is considered:

G(av) = e(avv0)2

(5)

this function reaches its maximum value when av=v0 that is when:

a =v02pf

(6)

So the WT gives a timefrequency representation of the signal performing a lineartransformation. Its main advantage over other linear timefrequency representations, such

as STFT, consists in assuring a multiresolution analysis in the timefrequency domain: theWT resolution is high at low frequencies and low at high frequencies [49].

2.2.

The discrete WT of a signal results in a complex-valued matrix, whose modulus and

phase can be presented easily. To understand better the modulus and phase of the WTrepresentation, we can consider a simple harmonic signal:

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x(t) = X0cos (vnt) (7)

whose spectrum is:

X(v) =X0

2p

[d(vvn)+d(v+vn)] (8)

where d is the Dirac delta function.The spectrum of the dilated and translated Morlet wavelet is given by:

Ga, b (v) = e(avv0)

2

ejvb (9)

Using the delta function property

g+a

a

d(t t0)f(t) dt =f(t0) (10)

and the frequency domain formulation of the WT expressed by equation (3), the WT of

the harmonic signal is:

W(a, b) = zaX0 e(avn v0)2 ejvnb (11)

For a fixed value of the dilation parameter a (a = ai), that is for a fixed value of frequency,the modulus and the phase of the WT are:

=W(ai, b)= = zaiX0 e(aivn v0)2

{[W(ai, b)] =vnb(12)

This equation highlights how the modulus is directly proportional to the amplitude of theharmonic signal, while the phase gives the signal phase variation vs time.If a more general signal is considered:

x(t) = k(t) cos (8(t)t) (13)where k(t) and 8(t) are time-varying envelope and phase functions, the WT of x(t) canbe expressed as:

W(a, b) = zak(t) e(a8(t)v0)2 ej8(t)b (14)

again for a = ai= constant we have:

=W(ai, b)= = zaik(t) e(ai8(t)v0)2

{[W(ai, b)] =8(t)b(15)

Equation (15) shows how a general time varying envelope k(t) or phase 8(t) of theanalysed signal can be determined effectively using the modulus and the phase of the WT

for a fixed value of frequency.

2.3.

The impulse response (IR) of a viscously damped single degree of freedom (sdof) systemis:

x(t) = A ezvnt cos (vdt +f0) (16)

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where A is the residue magnitude, vn is the undamped angular frequency, vd isthe damped angular frequency (vd=vnz1 z2), f0 is the phase lag and z is the dampingratio. The signal x(t) can also be represented as a rotating vector in the complex plane:

z(t) = A ezvnt +j(vdt +f0) (17)

being x(t) = Re[z(t)].A well-established technique for modal parameters estimation is based on theapproximation of the complex signal z(t) with the corresponding analytic signal za(t)

(z(t)3za(t)) defined as [13]:

za(t) = x(t)+jH[x(t)] (18)

where j=z1 and H[x(t)] is the HT ofx(t). This approximation is exact (i.e. z(t)=za (t))only ifx(t) is analytic. This condition is only achieved if the system is conservative, whilethe error that occurs in approximating z(t) with its corresponding analytic signal za(t)increases with the damping ratio z (Fig. 1) [2]. The HT, however, allows us to estimate

Figure 1. HT envelopes (left) and instantaneous frequencies (right) for (a) z= 0.1%; (b) z= 0.6%, and (c)z= 1.2%.

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the envelope and the phase functions of the signal that, in the case of the free responseof a sdof viscously damped system [equation (16)], are:

=za (t)=3A ezvnt

{[za (t)]3vdt +f0(19)

We now define the instantaneous frequency of the signal as [1, 2, 3, 9, 10]:

8(t)=1

2pd

dt({[za(t)]) (20)

In our case, the phase function is a straight line whose slope represents the damped angularfrequency:

8(t)3vd (21)

The envelope function on a logarithmic scale plot is again represented by a straight linewhose slope is the decay rate:

s= zvn (22)

Once the decay rate and the instantaneous frequency of the free response have beenestimated, it is possible to identify both the natural frequency and damping ratio of thesystem.

This identification technique has proved to be very simple and effective, still it bears

some limitations. Its basic principle consists of using the HT to obtain a complex signalfrom the real-valued signal [x(t) in equation (16)] which approximates the rotating vectorz(t) representing x(t) in the complex plane. This approximation is strictly exact only ifx(t)is analytic, that is only if the analysed system is undamped (it introduces errors increasingwith the system damping [2]). Furthermore, it is important to underline that this methodonly works for sdof systems: for real structures, where several modes are excited, it is firstnecessary to band-pass filter the system response in order to pick out each mode of interestand to reject all the others.

An improvement can be achieved if the WT is used instead of the HT. If we consider

the signal defined in equation (16), from equations (14) and (15) we have:

k(t)==W(ai, b)=

zai e(ai8(t)v0)2= A ezvnt

8(t)={[W(ai, b)=vdt +f0(23)

which means, from Section 2.2, that choosing a constant frequency line in the waveletrepresentation [equation (15)] makes the identification of the decay rate and of the phasevariation in the time domain possible. If a definition of the instantaneous frequency,similar to the one expressed by equation (20), is used:

8(t)=1

2pddt

({[W(ai, b)]) (24)

an identification procedure analogous to that using the HT can be followed, but with theconsistent difference that no assumption on the analytic nature of the signal has to bemade. In fact, the WT gives a complex signal directly from the original real-valued one[see equations (1) and (15)] without introducing any kind of approximation, i.e. without

introducing errors in the identification procedure.Figure 2 shows that using the WT, the entity of the damping ratio does not influence

the envelope and instantaneous frequencytime histories.

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Figure 2. WT envelopes and instantaneous frequencies for (a) z=0.1%; (b) z=0.6%; and (c) z= 1.2%.

2.4.

The wavelet representation in the timefrequency domain is very useful when theanalysed signal is the free response of a mdof system. In fact, the timefrequencymaps allow the decay of each mode of the structure to be followed separately from theothers, by selecting the right frequency value corresponding to the mode of interest. Theautomatic decoupling performed by the WT can be shown expressing the free responsein one point of the system as the superposition of the n most relevant modes of the

structure:

x(t)=sn

j= l

Aj ezjvnjt sin (vdjt) (25)

where Aj is the residue magnitude, zj is the damping ratio, vnj is the undamped angularfrequency and vdj is the damped angular frequency associated to the jth mode.

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The WT is a linear transform, so the WT of x(t) defined in equation (25) is, fromequation (14):

W(a, b)=za sn

j= l

Aj ezjvnjt e(avdjv0)2

ejvdjb (26)

The term e(avdjv0)2

in equation (26), derived from the Fourier transform of the dilatedMorlet wavelet [equation (5)], is a Gaussian window in the frequency domain whichoperates as a band-pass filter. For a fixed value of the dilation parameter (a = aj), onlythe mode whose frequency is [equation (6)]:

vdj=v0aj

e(ajvdjv0)23 l

(27)

gives a relevant contribution in equation (26). Therefore, we can easily say that, for a = aj:

W(aj, b)3zajAj ezjvnjt e(ajvdjv0)2 ejvdjb

W(aj, b)3zajAj ezjvnjt ejvdjb(28)

Using this wavelet property, it is possible to follow the envelope decay and the phase

variation in time of each mode and estimate the damping ratio and the frequencyassociated to the isolated mode. This technique requires a previous choice of the frequencyof interest from the signal spectrum to obtain, using equations (6) or (27), the value ofthe dilation parameter a corresponding to the analysed mode.

The validity of this decoupling procedure is strictly linked with the wavelet frequencyresolution, that is with the WT capability of separating close different harmoniccomponents in the time signal. As already stated, the WT resolution depends on thefrequency range of the analysis [48]: the application of this identification techniquebecomes crucial with close modal superposition at high frequencies. If the wavelet analysislacks in resolution then it means that the term e(avdjv0)

2is no longer able to separate two

close modes and equation (28) becomes:

W(aj, b)3zaj Aj ezjvnjt ejvdjb+zaj A( j+ 1) ez( j+ 1)vn (j+ 1)t ejvd( j+ 1)b (29)

In this case, the decoupling is not possible unless, in the measured point, AjwAj+ l or viceversa.

Figure 3. Sketch of the 4 dof system and its parameters.

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T 1

Four dof system simulation results

Theoretical Estimated Theoretical Estimatedfrequency frequency damping ratio damping ratio

Mode (Hz) (Hz) (%) (%)1 8.37 8.38 0.26 0.262 15.73 15.74 0.49 0.503 22.68 22.54 0.71 0.724 26.26 24.01 0.83 0.77

3. NUMERICAL EXAMPLE: A 4 DOF SYSTEM

To prove the effectiveness of the identification procedure based on the WT, a 4 dofsystem was analysed (Fig. 3).

The decay rate of the envelope for each mode was calculated from the slope of the linearinterpolation performed on the wavelet modulus decay, while the damped naturalfrequency was estimated as the mean value of the instantaneous frequency time history,calculated deriving the phase of the WT [equation (24)]. The estimation technique wasapplied to the free response of each of the four masses; the modal parameters estimatedin Table 1 are the averaged values of the results obtained from each response.

The results in Table 1 show the accuracy of the technique in estimating naturalfrequencies and damping ratios. Some problems occur with the fourth mode, due to thelow wavelet resolution at high frequencies and to the low residual magnitude of the fourthmode if compared to the third mode [equation (29)].

4. MODAL PARAMETERS IDENTIFICATION FROM AMBIENT DATA

The WT identification technique presented above was applied to the analysis ofacceleration responses of a civil engineering structure: the Queensborough bridge across

the Fraser river in Vancouver, Canada, whose dynamic behaviour has been studiedextensively [1116].The bridge was excited by ambient forces which, in principle, included wind, traffic and

low intensity ground motion, although during the measurements, the main load was due

to traffic. A full description of the test set-up, equipment disposition and bridge geometrycan be found in [15]. Measurements were collected at 23 positions along the bridge on bothsides. The data were recorded in eight segments of 4096 points at a sampling frequencyof 40 Hz.

4.1.

Figure 4 shows the spectrum of a typical recorded signal. The upper half frequency bandappears very noisy and there are no obvious vibration modes. For this reason, the datawere low-pass filtered with a cut-off frequency of 10 Hz. The 010-Hz frequency band

revealed the presence of a large number of modes. To avoid problems due to the WTlacking of resolution, the flexural vibrations were separated from the torsional ones bysimply adding and subtracting respectively the responses from two opposite positions ofthe bridge. This method requires the assumption of a symmetric behaviour of the bridge.

The WT estimation technique presented in Section 3 operates on the free response ofthe analysed system. A well-established method to convert random responses of a structureto free decay responses is the random decrement technique (RDT) [1720]. The rigorous

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proof of the theory behind RDT is given in [20]; however, its basic concept is that a randomresponse of a linear structure is composed of a deterministic part (impulse or step response)and of a random part assumed to have a zero average. Hence, averaging enough samplesof the same random response removes the random part of the response, leaving itsdeterministic part. In the theoretical developments of RDT, it is assumed that the inputs

to the structure are white noise, zero mean, stationary Gaussian processes, but it is shownin [18] that this strict condition, hardly found in real-life applications, is not alwaysnecessary. For the purposes of modal parameters identification, the conditions required

are that the inputs are zero mean, stationary random processes. If this assumption is madeon ambient excitations, the free response x(t) of the system in one measurement point canbe estimated as follows:

x(t)=1N

sN

i= l

y(ti+ t) (30)

where y is the acceleration response, N is the number of trig points, that is the numberof samples averaged, ti is the time corresponding to the ith trig point and t is the freeresponse time length. Many kind of trig conditions can be used in RDT signaturesestimation [1719]. In the analysis performed here, a very simple trig condition was used:

ti= t0c 8y(t0)=00dydt1t = t0 q0 (33)in order to estimate the free decay positive IR [17]. In Fig. 5, the RDT signatures of atorsional and flexural vibration and their spectra are shown.

4.2.

Once the free responses of the structure were estimated, the WT identification procedurewas applied. In Fig. 6, the contour plot of the WT of the free decay of a torsional vibration

signature is presented. The natural frequencies were estimated as the mean values of theinstantaneous frequencytime histories. Damping ratios were identified estimating thedecay rate of the linear interpolation performed on the WT modulus. Figure 7 shows two

Figure 4. Typical spectrum of a recorded signal.

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(a)

(b)

Figure 5. RDT signatures and their spectra for (a) torsional and (b) flexural vibrations.

examples of instantaneous frequency and envelope time histories, together with theinterpolated decays. The estimated modal parameters are shown in Table 2. The dampingratios and natural frequencies were calculated by averaging the results obtained from theanalysis of the time histories from each sensor location.

The natural frequencies obtained using this estimation technique displayed a good matchwith those found using other techniques, operating in the time or frequency domain[1116]. Damping ratios estimation is always crucial when lightly damped structures areconsidered; however, some damping coefficients estimated using this technique are closeto the parameters obtained in [14].

5. CONCLUSIONS AND FUTURE WORK

The application of the wavelet analysis to the free response of the system represents agood improvement of the technique based on the HT [13].

Figure 6. WT of RDT torsional signature.

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Figure 7. Envelope and instantaneous frequencies for (a) flexural (frequency = 2.266 Hz) and (b) torsionalvibrations (frequency = 1.875 Hz); ***, WT decay; , interpolated decay.

The results obtained on the numerical simulation underline the accuracy of theprocedure in estimating both natural frequencies and damping ratios, whose identificationis always critical.

When the identification technique is performed on ambient data, some time should bespent on data preprocessing. The estimated natural frequencies here are in good agreementwith those found in all the papers dealing with Queensborough Bridge dynamic behaviour.Some differences can be found with the damping ratios estimated in [11], while there isa good match with some of the values found in [14].

The authors believe that this identification technique could allow also the estimation ofthe mode shapes of the analysed structure, and currently we are working in this directionto complete the estimation procedure.

T 2

Estimated modal parameters

Frequency DampingFrequency variance Damping variance

Mode (Hz) (103) (%) (102)

1F 1.09 0.00 0.91 5.212F 1.88 0.00 1.23 1.703T 2.27 0.00 0.52 5.004F 2.42 0.00 1.15 5.26

5T 3.20 0.00 0.75 8.016T 3.44 0.00 0.80 5.067F 3.75 0.00 0.87 8.438T 5.16 0.00 0.33 1.049F 5.73 0.23 0.30 1.79

10T 7.17 0.40 0.67 1.6711T 7.52 1.32 0.34 1.9712T 8.56 1.46 0.15 0.10

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ACKNOWLEDGEMENTS

Authors wish to thank Dr Felber from EDI Ltd. for his collaboration and for supplyingQueensborough Bridge data.

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