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Proceedings of the American Control ConferenceArlington. VA June 25-27, 2001

THE EIGENSYSTEM REALIZATION ALGORITHM FOR AMBIENT VIBRATIONMEASUREMENT USING LASER DOPPLER VIBROMETERS

Hong Vu-Manh, Masato Abe, Yozo Fujino, Kiyoyuk~KaitoDepartment of Civil Engineering, U niversity of Tokyo

Hongo 7-3-1, Bunkyo-ku, Tohyo 113-8656,Japan

Ab stra ct: Laser Doppler Vibrometers (LDVs) used inambient vibration measurement have eliminated theexpense of using sensors and exciters, and the need ofclosing service of the structure during the measurementperiod. Since there are only available two laser sources formeasuring responses at two points, data acquired all overthe structure to provide inform ation of its modal parameterswill not be synchronous, leading to the fact that manycurrent modal an alysis identification techniques will not beapplicable. Introduced in this paper is a synchronizationtechnique to synchronize response measurements ftom theLDV, then feeding them to the Eigensystem RealizationAlgorithm (EM)o obtain the structure’s modalparameters. Two experiments have been conducted on asteel plate to verify the technique. It has been found out thatthe modal parameters can be successfully identified and achange in these parameters is prominent when the systemprope lty is changed. Th is finding is significant in structuralhealth monitoring, in particular of damage detection, in thatit can provide a fast and accurate, relatively ch eap routine topinpoint the location of dainage.

1. INTRODUCTION

Modal testing of structures generally can provide valuableinformation to assess the integrity and service condition ofstructures. Investigating the identified modal parameters(natural frequencies, modal damping ratios, mode shapes)can characterize and monitor the performance of thestructure. However, inspecting and monitoring large-scalestructures, such as bridges, tunnels, dams, high-risebuildings, generally bear a high cost mostly due to thestructure’s scale. This expense will also be added more if

the structure needs to be closed of service during the periodof inspection.

Application of non-contact measurement devices such asLaser Doppler Vibrometer (LDV) in conjunction with

ambient excitation source in modal testing is now receiv ingconsiderable attention from modal testing community dueto its low in cost and less in personnel. However, there is alimitation of numbers of LDV to measure spatiallysimultaneous responses at all desired points. This unique

feature of using the LDV makes its modal identificationtechnique different from the well-known curre nt methods.

Most of modal identification methods for ambient vibrationuse a spectral analysis technique developed by Crawfordand Ward [4]. In this technique, a structure motion is

measured by a series of sensors. The structure’s naturalfrequencies are estimated from peaks in the power-spectraldensity function of the measurement records, while themodal damping ratios are estimated using the half-powerbandwidth method. The cross-power spectra between adesignated reference measurement and other measurementswill contain information of the structure’s mode shapes.Although the drawbacks of this technique have beendetailed in [2] and [4], t is still widely used in ambientvibration because of its simplicity. Recent development inambient vibration is applications of EigensystemRealization Algorithm (ERA) [3J , [ 5 ] , Auto RegressiveMoving Average Vector (ARMAV) [SI; AutoregressiveModeling (AR) [IO], covariance matrix [ll], EigenspaceAlgorithm [13]. These techniques are either applied for asiindtaneous data, i.e. data acquired at all points at thesame time, or not suitable for a personal comp uter due to itsintensive computation. For modal parameter identificationusing the LDV, there are limited numbers of responsemeasurements that can be recorded at the same timebecause of the availability of laser sources. Among above-mentioned techniques, the ER4 s the most general multi-input-multi-output (MIMO) technique that can bedeveloped and applied to the LDV measurement if a datasynchronization technique can be formulated.

Introduced in this paper is the modal analysis identificationtechnique using the ERA that is developed for the LDV inambient vibration measurement. A mathematical model ispresented in the first part and an experiment of a steel platewas carried out in a laboratory environment to venfy themathematical model. The steel plate was tested underambient excitation that might be resulted from groundwaves caused by traflic on a nearby street. Identification ofthe plate’s modal parameters is successful by the ERA . Theplate is then attached by a mass to venfy the change in itsmodal parameters as a result of the change in its mass

property. This finding will provide background for astructural health monitoring program using the LDV tomeasure ambient vibration and to use identified modalparam eters to monitor the performance of a structure.

2.

The Laser Doppler Vibrometer basically uses the Dopplerprinciple to measure velocity at a point where its coherentlaser beam is directed to. The reflected laser light is

LASER DOPP LER VIBROMETER AND DATA- ACQUISITION SYSTEM

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compared with the incident light in an interferometer togive the Doppler-shfied wavelength. This shiftedwavelength provides information of surface velocity in thedirection of the incident laser beam. For the LDV used inthis experiment, two deflecting mirrors are provided toenable the light beam to be directed at any desired point.

Providing non-contact measurement, the LDV is capable todetect a velocity of magnitude of 0.5pds at a distance ofmaximum 30m. Detailed characteristics of the LDV can befound in [91.

There are two laser sources to be used to measure thestructural vibration response in this research. One is tomeasure a fixed point on the measured surface to provideglobal information, i.e. to synchronize other measurements;the other is directed by the two deflecting mirrors tomeasure the respon se spatially. The first one will be calledreference laser and the second one the scanning laser

hereafter. A mesh of measured points should be definedfirst, and then under control of a computer, vibrationresponses of these pre-defined points will be measuredpoint by point.

Personal Computer Interface,---------------------------2 . I

-8

5

E2

eference laser m

Figure 1: Data Acq uisition System and the LDV

.Table 1: Specific ations of this data acquisition system

is automatic, requires the least command from an operator.Some specifications for this data acquisition system can beseen in Table 1.

3.

A viscously dam ped system with N DOF can be m odeled in

the following equation:

(1)

where [A4], C] , K ]are the N x N mass matrix, damping

matrix, stiffness matrix, respectively; { j i ( r ) } , { q r ) } , {q)}

are the N x 1 acceleration vector, velocity vector,displacement vector, respectively; [ F ] s a niatris of input

coefficients and {u( t ) }is the input vector at q locations.

The above system is equal to the continuous state-spacemodel:

A BRIEF REVIEW OF THE ERA

[h.iXWJ+[c]{w>+Kl{x(t)} = [FI {W}

bw}= [.4b(r)}+~ I I { ~ W ) (2){.(t>>= [RIbO))

where

By considering {x(r ) } as the free responses at p measured

locations, Juang and Pappa [6 ] have show n that equation '(2)can be written in a discrete form as follows:

fx(k)} = [RI * [AI"-' {B > (3 1

At

[B]= Je"' lr[B']dr0

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it also provides criteria on the calculated mode to judgebetween genuine and computational modes. In the aboveprocedure, the free response measurements at p locationsare needed, and they should be measured at the same time.Followings are a technique to synthesize the free responsesfrom ambient vibration measurement and to synchronizethose responses that will be fed into the ERA.

4. FREE RESPONSE SYNTHESIS FROMAMBIENT VIBRATION MEASUREMENT

Farrar and James [4 ] has pointed out that if the unknoivn

excitation is a white-noise random process, the cross-correlation function between two response measurementswill have the sam e form as the free response of the structure.

Denoting x R ( f ) the measured velocity at the reference

location, ~ , ( t )he measured velocity at the scanning

position i ( i = 1 ,2 , . . ). The cross-correlation fimction

between the scanning laser and the reference lasermeasurem ents is defined by Bendat and Piersol [2] as:

R , (z)= E{x, r + T ) X , r)} where E{*} denotes the

expectation operator.

As shown by Farrar and James [a], this cross-correlationfunction will have a form of:

R , z) = t F r ~os(w, r ) ]+G, [e -~~"~ 'in(o,r)] (4)

where the subscript r denotes values associated with the rth

mode, c,, a,, d, re the damping ratio, natural frequency

and the damped natural frequency, respectively, asso ciatedwith mode r , and N is the number of modes present in themeasurement. Parameters F, and G, contain modal mass atmode Y and the phas e and m agnitude difference at each time

the crosstorrelation is taken. Detailed formulation of thesetwo parameters can be found in [4]. Equation (4) shows thatthe above cross-correlation functions between tw o responsemeasurements that caused by an unknown white-noiserandom excitation have the same form of decayingsinusoids having the same characteristics as the structure'sfree response. If all p locations are measured at the sametime, the factor that represents the phase and magnitudedifference can be cancelled out in the ERA. However,measurement by the LDV does not possess this

characteristic. It, therefore, requires a synchronizationprocess before inp utting these free responses to the ERA.

r= l

5. DATA SYNC HRO NIZATION AND AVERAGING

Laser D oppler Vibrom eter is an optical device outputting its

measurement in a form of voltage. It is, therefore, subjectedto electrical noise resulted from magnetic fields surroundingthe w orking environment. It is also affected by some opticaleffects such as missing of reflecting light (so called specklenoise). Although the laser manufacturer has done its best toprotect its device from these effects, to some ex-ent these

noises have affected the measurement so that onemeasurement record at one point is usually not sufficient.Therefore, many measurement records at one point areneeded and then averaged out to cancel the random noiseeffect. It is practical to measure the velocities of thestructure at the reference point and scanning points longenough, then splitting the time history into small pieces thatcan represent the whole range of frequency of interest.Providing that these measurements are stationary andergodic, this technique is well applicable. However, thetime history is now no longer infinite leading to the fact thatthe energy flowing into the system at each timemeasurement will be different. And so will be the phase.The former is actually the magnitude difference while thelater is really a mattcr of where on the time axis themeasurement starts. To make the averaging processmeaningful, these measurements should be synchronized,i.e. beginning at the same time base and resulted from thesame exitation energy. This is the Data SynchronizationProcess that is introduced hereafter.

Denoting xR 0t o )

the base reference laser measurement,xs0 t o ) he simultaneous scanning laser measurement at a

specific scanning point. It is recommended that for a modeto be detected, it should be controllable and observable.Therefore the reference location shou ld be chosen so that itcontains all information about all possible modesandx,, (to>should be a record that is the least contaminated

by noise. The benefit of t h i s technique will be clear later inthe subsequent section. For an appropriate set-up in whichthe reference laser measurement will observe all possiblemodes and have maximum signal-to-noise ratio, xRo(t0)

can be the reference laser measurement at the starting timeof the measurement process.

Again denoting XR i (t i ) the reference laser measurement

while the scanning laser at point i starting time f , x, ( t , )

the scanning laser at point i starting time $.

Let a hypothetical frequency response function Hxflnoxm,f)

be defined, as in Bendat and Piersol [2], by:

where F [ o ] denotes finite Fo urier Transform of (a)

This hypothetical frequency response function containsinformation about the phase lag and magnitude differencebetween each point and each m easurement. If the excitationis a stationary and erg odic process, this relation will also be

applied for the scanning laser measurement, i.e.

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For simplicity, parameter t is dropped. Combiningequations (5 ) and (6) results in:

(7)

Multiplying equation (7) by the complex conjugate of its

left hand side, equation (7) now becomes:

Arranging the above equation leads to:

For a long enough record, the cross-spectrum and auto-spectrum between two records are defined by Bendat andPiersol [2] as:

G, (f) F' ( X ) F ( Y )

P,(f)

F' ( x ) F ( x )Therefore, equation (9) can be simply written as:

The meaning of equation (10) is that the cross-spectrum ofthe subsequent measurements will differ from the base

measurement by a factor of %L. Therefore, for a noise

reduction scheme by averaging in the frequency domain,any cross-spectrum that is not the base spectrum should bedivided by this factor to convert it to the base spectrum. Itcan be seen that this factor is calculated by the auto-spectrum only; it hence does not eliminate the effect ofcorrelated noise at the reference laser measurement. It is,therefore, necessary to set up the reference lasermeasu remen t that will have the larges t signal-to-noise ratio.

PX , ,

Scanning laser

Reference laser

Synchronizing8 Averaging

Modal Parameters:ERAmodel Natural frequency

Damping ratioModeshape

Figure 2: Determination of modal parameters from ambientvibration measurement

By inverse Fourier Transform of the averaged cross-

spectrum, the averaged crossconelation function can be

estimated and subsequently entered to the ERA. Details ofthis modal analysis identification process can besummarized in Figure 2. Since the averaging process iscarried out in the frequency domain, windowing should beapplied to minimize the effect of leakage. The Hanningwindow is chose n for this type of signal.

6. EXPERIMENTAL VERIFICATION

A cantilevered steel plate was used as a test piece toexperimentally demonstrate the modal analysisidentification technique using the LDV. The dimension ofthe plate is shown in Figure 3(a), and a mesh of scanningpoints is shown in Figure 3(b).

t3 0mm 4 Scanningpt ion 1

9

0

Ref.E pointE

3

(a) Can tilevered plate (t=2mm) (b) Scanning mesh ( l o x 10)

Figure 3: Experimental steel plate

The LDV w as set at a distance of 6m from the plate surface.Main excitation is believed to be from ground wavesgenerated by traffic on a nearby street. It is assumed thatthis excitation source is a white-noise random process.Testing time was allowed for about 7 hours in which mostof time, the acquisition process was unattended. This test isdesignated as Test A. Following this test, a small magnet,

weighed about 150grams, was attached to the backside ofthe plate (as shown in Figure 3 (b)). This test is designatedTest B in order to show that the technique is capable todetect the ch ange in modal parameters as a result of changein structural mass distribution.

6.1 Data Acquisition Set-up

The LDV was set to sample at the rate of 1OkHz ince thereis only available an anti-alias filter with minimum cut-offfrequency of 5kHz. Sampling number was selected as 215

(32768) to utilize the Fast Fourier Transform algorithm. Ateach point, time history was allowed for 163.84 secondsand then split to 50 records for averaging. Each channel wasapplied a gain of 10 to reduce effects of noise and AD

converter's quantisation.

A typical vibration response measured at the tip of the platein Test A and its simultaneous measurement at the referencelocation are shown in Figure 4.

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1V-0.1 m m l s4 ,

2 :::I0

2

s....

. . . . I I

0 0 5 1 1 5 2 2 .5 3 3. 5Scanni ng l aser meaLuremenl

4

2 ........................................ : ............:............. ............ ............

-2

4 I

............:............. :........... ; .........:............. L ............ ...........

8 i

0 0. 5 1 1.5 2 2 .5 3 3.5Reference laser measuremen1 Tim e (secs)

Figure 4: Typical response at the tip of the plate and itssimultaneous reference response measurem ent

Figure 5 shows the averaged cross-spec- betweenmeasurem ent record at this po int and the -measurem entrecord at the reference location after synchronization andaveraging and its cross-correlation function after InverseFourier Transform.

I I 1

x 10

100 2 0 0 300 40 0 50 0

Averaged crowspectrum F w Hz )5

5 1 ! 1

. . . . . . . .5 1 ‘

0 0. 2 0 . 4 0.6 0.8 1 1 . 2 1. 4 1.6

Averaged crosscorrelation function Time (secs)

Figure 5: Averaged cross-spectrum and crosscorrelation

function6.2 Identified Modal Parameters

There are 7 identified modes for Test A (Figure 6) while 10modes can be detected for Test B (Figure 7). The dampingratios estimated from the ambient test are rather low.However, many researchers have believed that dampingratio i s dependent on excitation level. In this experiment,ambient excitation source and damping of the steel platemay just@ the fact. It also shows that the addition of amass to one side of the plate has:

1. Caused anti-symmetric modes to be excited(modes 2,4 ,7 , 9 in Test B);

2. Lowered natural frequency and modal dampingratio of the structure. It is so true since the naturalfrequency and m odal dam ping ratio carry -themodal mass in its denominator.

These findings are sigrufcant for a structural healthmonitoring program developed in future research. In that,upon the change in the system’s modal parameters,

system’s dynamic properties can be monitored. A damagethat may be represented by a decrease in stiffness or mass,

i.e. an addition of a negative stiffness or mass, can bedetected by using m easured modal parameters.

7. CONCLUSIONS

It has been demonstrated that the LDV used in conjunctionwith ambient vibration is capable to i d e n w modalparameters of a system. The ERA has been utilized for themodal parameter identification after a technique of freeresponse synthesis and synchronization is developed. Thetechnique successfully identified modal parameters of asteel plate excited by unknown ambient source. It is alsoshown that the technique can detect a change in these modalparameters after a change in system’s dynamic properties.This finding will be used in a structuralhealth monitoringdeveloped later in the lin e of this research .

REFERENCES:

1.Abe M. (1998), “Structural Monitoring of CivilStructures using Vibration Measurem ent-Current Practice

and Future”, Structural Engineering Applications of

Artijkial Intelligence, Lecture Notes in ComputerScience, Sprin ger Verlag;

2. Bendat J.S., Piersol A.G. (1993), EngineeringApplications of Correlation and Spectral Analysis, 2nd d.,John W iley & Sons, Inc., USA;

3.Dyke S.J., Jolmson E.A. (2000), “Monitoring of aBenchmark Structure for Damage Identification”,Proceedings of the Engineering Mechanical SpecialtyConference, May 21-24, A ustin, Texas, USA;

4.Fanar C.R, James 111, G.H. (1997), “SystemIdentification From Ambient Vibration Measurement onA Bridge”, Journal of Sound and Vibration, Vo1.205,

5. arrar C.R., Comw ell P. J., D oebling S.W., Prime M.B.(ZOOO), Structural Health Monitoring Studies of theAlamosa C anyon and I 4 0 Bridges, Los Alamos N ationalLaboratory, USA;

6.Juang J.N., Pappa R.S. (1985), “An EigensystemRealization Algorithm For Modal ParameterId en ~ ca ti o n And Model Reduction”, Journal ofGuidance, Control, and Dynamics, Vol. 8, NO. 5., S ept.-Oct., pp620-627;

7.Juang J.N., Pappa, RS. (1986), “Effects of Noise OnModal Parameters Identified by the EigensystemRealization Algorithm”, Journal of Guidance, Control,and Dynamics, Vol. 9, No. 3, May-June, pp 294-303;

8.Garibaldi L., Giorcelli E., Piombo B.A.D. (1998),“ARMAV Techniques for T-c Excited B ridges”,Journal of Vibration and Acoustics, Transactions of the

9. Kaito K, Abe M., Fujono Y., Yoda H. (2000), “Detectionof Structural Damage by Ambient VibrationMeasurement using Laser Doppler Vibrometer”, Non-

N0.1, ppl-18;

ASME, Vol. 120, July 1998, ~ ~ 71 3 -7 1 8 ;

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Destructive Testing in Civil Engineering 2000, Uomoto

(ed.), Tokyo, Japan;

10. Kadakal U., Yuzugullu 0. (1996), “A comparativestudy on the identification methods for the autoregressivemodeling from the ambient vibration records”, Soil

Dynaniics and Earthquake Engineering, Vol. 15, pp45-49;

11. Lardies J., (1997), “Modal Parameter IdenhficationFrom Output-Only Measurements”, Mechanics ResearchConriiiunications, Vo1.24, No.5, pp521-528;

12. Maia N.M.M, Silva J.M.M (ed) (1997), Theoreticaland Experimental A4odal Analysis, Research StudiesPress Ltd, England;

13. Quek S.T, Wang W, Koh C.G. (1999), “SystemIdentification of Line ar MDOF Structures Under AmbientExcitation”, Earthquake Engineering and Structural

Dynamics, Vo1.28, pp61-77;

14. Stanbridge A.B., Ewins D.J. (1996), “Measurement oftranslational and angular vibration using a scanning laser

Doppler vibrometer”, Shock and Vibration, Vo1.3, N0.2,

15 . Vu Manh H. (2001), Application of Laser Doppler

Vibrometer in Structural Health A4onitoring, MasterThesis, Department of Civil Engineering, The Universityof Tokyo, Japan (to be appeared)

~ ~ 1 4 1 - 1 5 2 ;

0=10.744H?

6 0.147%

o 233.3 Hz

Mode 5

w = 10.235Hz

Mode 1

o 124.42Hz

C=0.013%

Mode 5

o 3 19.04Hz

Mode 9

o = 67.588Hz

Mode 2

w = 371.72Hz

Mode 6

o = 137.39Hz I o = 192.87Hz

Mode3 I Mode 4

Mode7 1Figure6: dentfied modal parameters for Test A

o 28.710Hz

C= 0.10396

Mode 2

w = 190.07Hz

C= 0.01 %

-Mode 6

o = 376.74Hz

C = 0.023% A

Mode 10

1) = 66.104Hz I 0=110.83Hz

Mode3 1 Mode 4

i)= 228.79Hz o 232.87Hr

= 0.03 1% r = 0.012% A

Mode 7

Figure 7: dentified modal parameters for Test B

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