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

THE EIGENSYSTEM REALIZATION ALGORITHM FOR AMBIENT VIBRATION MEASUREMENT USING LASER DOPPLER VIBROMETERS

Hong Vu-Manh, Masato Abe, Yozo Fujino, Kiyoyuk~ Kaito Department of Civil Engineering, University of Tokyo

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

Abstract: Laser Doppler Vibrometers (LDVs) used in ambient vibration measurement have eliminated the expense of using sensors and exciters, and the need of closing service of the structure during the measurement period. Since there are only available two laser sources for measuring responses at two points, data acquired all over the structure to provide information of its modal parameters will not be synchronous, leading to the fact that many current modal analysis identification techniques will not be applicable. Introduced in this paper is a synchronization technique to synchronize response measurements ftom the LDV, then feeding them to the Eigensystem Realization Algorithm (EM) to obtain the structure’s modal parameters. Two experiments have been conducted on a steel plate to verify the technique. It has been found out that the modal parameters can be successfully identified and a change in these parameters is prominent when the system propelty is changed. This finding is significant in structural health monitoring, in particular of damage detection, in that it can provide a fast and accurate, relatively cheap routine to pinpoint the location of dainage.

1. INTRODUCTION

Modal testing of structures generally can provide valuable information to assess the integrity and service condition of structures. Investigating the identified modal parameters (natural frequencies, modal damping ratios, mode shapes) can characterize and monitor the performance of the structure. However, inspecting and monitoring large-scale structures, such as bridges, tunnels, dams, high-rise buildings, generally bear a high cost mostly due to the structure’s scale. This expense will also be added more if the structure needs to be closed of service during the period of inspection.

Application of non-contact measurement devices such as Laser Doppler Vibrometer (LDV) in conjunction with ambient excitation source in modal testing is now receiving considerable attention from modal testing community due to its low in cost and less in personnel. However, there is a limitation of numbers of LDV to measure spatially simultaneous responses at all desired points. This unique feature of using the LDV makes its modal identification technique different from the well-known current methods.

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

measured by a series of sensors. The structure’s natural frequencies are estimated from peaks in the power-spectral density function of the measurement records, while the modal damping ratios are estimated using the half-power bandwidth method. The cross-power spectra between a designated reference measurement and other measurements will contain information of the structure’s mode shapes. Although the drawbacks of this technique have been detailed in [2] and [4], it is still widely used in ambient vibration because of its simplicity. Recent development in ambient vibration is applications of Eigensystem Realization Algorithm (ERA) [3 J , [5] , Auto Regressive Moving Average Vector (ARMAV) [SI; Autoregressive Modeling (AR) [IO], covariance matrix [ll], Eigenspace Algorithm [13]. These techniques are either applied for a siindtaneous data, i.e. data acquired at all points at the same time, or not suitable for a personal computer due to its intensive computation. For modal parameter identification using the LDV, there are limited numbers of response measurements that can be recorded at the same time because of the availability of laser sources. Among above- mentioned techniques, the ER4 is the most general multi- input-multi-output (MIMO) technique that can be developed and applied to the LDV measurement if a data synchronization technique can be formulated.

Introduced in this paper is the modal analysis identification technique using the ERA that is developed for the LDV in ambient vibration measurement. A mathematical model is presented in the first part and an experiment of a steel plate was carried out in a laboratory environment to venfy the mathematical model. The steel plate was tested under ambient excitation that might be resulted from ground waves caused by traflic on a nearby street. Identification of the plate’s modal parameters is successful by the ERA. The plate is then attached by a mass to venfy the change in its modal parameters as a result of the change in its mass property. This finding will provide background for a structural health monitoring program using the LDV to measure ambient vibration and to use identified modal parameters to monitor the performance of a structure.

2.

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

LASER DOPPLER VIBROMETER AND DATA - ACQUISITION SYSTEM

0-7803-6495-3/01/$10.00 0 2001 AACC 435

compared with the incident light in an interferometer to give the Doppler-shfied wavelength. This shifted wavelength provides information of surface velocity in the direction of the incident laser beam. For the LDV used in this experiment, two deflecting mirrors are provided to enable the light beam to be directed at any desired point. Providing non-contact measurement, the LDV is capable to detect a velocity of magnitude of 0.5pds at a distance of maximum 30m. Detailed characteristics of the LDV can be found in [ 91.

There are two laser sources to be used to measure the structural vibration response in this research. One is to measure a fixed point on the measured surface to provide global information, i.e. to synchronize other measurements; the other is directed by the two deflecting mirrors to measure the response spatially. The first one will be called reference laser and the second one the scanning laser hereafter. A mesh of measured points should be defined first, and then under control of a computer, vibration responses of these pre-defined points will be measured point by point.

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

- 8 5

E

2 Reference laser m

Figure 1: Data Acquisition System and the LDV

.Table 1: Specifications of this data acquisition system

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

3.

A viscously damped system with N DOF can be modeled 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] is a niatris of input coefficients and {u(t)} is the input vector at q locations. The above system is equal to the continuous state-space model:

A BRIEF REVIEW OF THE ERA

[h.iXWJ+ [c]{w>+ [Kl{x(t)} = [ F I { 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 shown that equation'(2) can be written in a discrete form as follows:

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

At

[B] = Je"'lr[B']dr 0

436

it also provides criteria on the calculated mode to judge between genuine and computational modes. In the above procedure, the free response measurements at p locations are needed, and they should be measured at the same time. Followings are a technique to synthesize the free responses from ambient vibration measurement and to synchronize those responses that will be fed into the ERA.

4. FREE RESPONSE SYNTHESIS FROM AMBIENT 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 measurements will have the same form as the free response of the structure.

Denoting x R ( f ) the measured velocity at the reference location, ~ , ( t ) the measured velocity at the scanning position i ( i = 1,2,. .. ). The cross-correlation fimction between the scanning laser and the reference laser measurements 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-correlation function will have a form of:

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

where the subscript r denotes values associated with the rth mode, c,, a,, ad, are the damping ratio, natural frequency and the damped natural frequency, respectively, associated with mode r, and N is the number of modes present in the measurement. Parameters F, and G, contain modal mass at mode Y and the phase and magnitude difference at each time the crosstorrelation is taken. Detailed formulation of these two parameters can be found in [4]. Equation (4) shows that the above cross-correlation functions between two response measurements that caused by an unknown white-noise random excitation have the same form of decaying sinusoids having the same characteristics as the structure's free response. If all p locations are measured at the same time, the factor that represents the phase and magnitude difference can be cancelled out in the ERA. However, measurement by the LDV does not possess this characteristic. It, therefore, requires a synchronization process before inputting these free responses to the ERA.

r=l

5. DATA SYNCHRONIZATION AND AVERAGING

Laser Doppler Vibrometer is an optical device outputting its measurement in a form of voltage. It is, therefore, subjected to electrical noise resulted from magnetic fields surrounding the working environment. It is also affected by some optical effects such as missing of reflecting light (so called speckle noise). Although the laser manufacturer has done its best to protect its device from these effects, to some ex-ent these

noises have affected the measurement so that one measurement record at one point is usually not sufficient. Therefore, many measurement records at one point are needed and then averaged out to cancel the random noise effect. It is practical to measure the velocities of the structure at the reference point and scanning points long enough, then splitting the time history into small pieces that can represent the whole range of frequency of interest. Providing that these measurements are stationary and ergodic, this technique is well applicable. However, the time history is now no longer infinite leading to the fact that the energy flowing into the system at each time measurement will be different. And so will be the phase. The former is actually the magnitude difference while the later is really a mattcr of where on the time axis the measurement starts. To make the averaging process meaningful, these measurements should be synchronized, i.e. beginning at the same time base and resulted from the same exitation energy. This is the Data Synchronization Process that is introduced hereafter.

Denoting xR0 ( t o ) the base reference laser measurement, xs0 ( t o ) the simultaneous scanning laser measurement at a specific scanning point. It is recommended that for a mode to be detected, it should be controllable and observable. Therefore the reference location should be chosen so that it contains all information about all possible modes andx,, (to> should be a record that is the least contaminated by noise. The benefit of this technique will be clear later in the subsequent section. For an appropriate set-up in which the reference laser measurement will observe all possible modes and have maximum signal-to-noise ratio, xRo( t0)

can be the reference laser measurement at the starting time of the measurement process.

Again denoting XRi (ti) 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 Fourier Transform of (a)

This hypothetical frequency response function contains information about the phase lag and magnitude difference between each point and each measurement. If the excitation is a stationary and ergodic process, this relation will also be applied for the scanning laser measurement, i.e.

437

For simplicity, parameter t is dropped. Combining equations (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 and Piersol [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 of the 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 be divided by this factor to convert it to the base spectrum. It can be seen that this factor is calculated by the auto- spectrum only; it hence does not eliminate the effect of correlated noise at the reference laser measurement. It is, therefore, necessary to set up the reference laser measurement that will have the largest signal-to-noise ratio.

P X , ,

Scanning laser

Reference laser

Synchronizing 8 Averaging

Modal Parameters: ERA model Natural frequency

Damping ratio Modeshape

Figure 2: Determination of modal parameters from ambient vibration measurement

By inverse Fourier Transform of the averaged cross- spectrum, the averaged crossconelation function can be

estimated and subsequently entered to the ERA. Details of this modal analysis identification process can be summarized in Figure 2. Since the averaging process is carried out in the frequency domain, windowing should be applied to minimize the effect of leakage. The Hanning window is chosen for this type of signal.

6. EXPERIMENTAL VERIFICATION

A cantilevered steel plate was used as a test piece to experimentally demonstrate the modal analysis identification technique using the LDV. The dimension of the plate is shown in Figure 3(a), and a mesh of scanning points is shown in Figure 3(b).

t 30mm 4 Scanningption 1 9

0 Ref.

E point E 3

(a) Cantilevered plate (t=2mm) (b) Scanning mesh ( l o x 10) Figure 3: Experimental steel plate

The LDV was set at a distance of 6m from the plate surface. Main excitation is believed to be from ground waves generated by traffic on a nearby street. It is assumed that this excitation source is a white-noise random process. Testing time was allowed for about 7 hours in which most of time, the acquisition process was unattended. This test is designated as Test A. Following this test, a small magnet, weighed about 150grams, was attached to the backside of the plate (as shown in Figure 3 (b)). This test is designated Test B in order to show that the technique is capable to detect the change in modal parameters as a result of change in structural mass distribution. 6.1 Data Acquisition Set-up

The LDV was set to sample at the rate of 1OkHz since there is only available an anti-alias filter with minimum cut-off frequency of 5kHz. Sampling number was selected as 215 (32768) to utilize the Fast Fourier Transform algorithm. At each point, time history was allowed for 163.84 seconds and then split to 50 records for averaging. Each channel was applied a gain of 10 to reduce effects of noise and AD converter's quantisation.

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

438

1 V-0.1 mmls 4 ,

2 :::I - 0

2 s

.... . . . .

I I

0 0 5 1 1 5 2 2.5 3 3.5 Scanning laser meaLuremenl

4

2 ........................................ : ............. : .............. : ............. 1 ............

-2

4 I

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

8 i 0 0.5 1 1.5 2 2 . 5 3 3.5

Reference laser measuremen1 Time (secs)

Figure 4: Typical response at the tip of the plate and its simultaneous reference response measurement

Figure 5 shows the averaged cross-spec- between measurement record at this point and the -measurement record at the reference location after synchronization and averaging and its cross-correlation function after Inverse Fourier Transform.

I I 1

x 10 100 2 0 0 300 400 500

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 T ime (secs)

Figure 5: Averaged cross-spectrum and crosscorrelation function

6.2 Identified Modal Parameters

There are 7 identified modes for Test A (Figure 6) while 10 modes can be detected for Test B (Figure 7). The damping ratios estimated from the ambient test are rather low. However, many researchers have believed that damping ratio i s dependent on excitation level. In this experiment, ambient excitation source and damping of the steel plate may just@ the fact. It also shows that the addition of a mass 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 damping ratio of the structure. It is so true since the natural frequency and modal damping ratio carry -the modal mass in its denominator.

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

system’s dynamic properties can be monitored. A damage that may be represented by a decrease in stiffness or mass, i.e. an addition of a negative stiffness or mass, can be detected by using measured modal parameters.

7. CONCLUSIONS

It has been demonstrated that the LDV used in conjunction with ambient vibration is capable to i d e n w modal parameters of a system. The ERA has been utilized for the modal parameter identification after a technique of free response synthesis and synchronization is developed. The technique successfully identified modal parameters of a steel plate excited by unknown ambient source. It is also shown that the technique can detect a change in these modal parameters after a change in system’s dynamic properties. This finding will be used in a structural health monitoring developed later in the line of this research.

REFERENCES: 1.Abe M. (1998), “Structural Monitoring of Civil

Structures using Vibration Measurement-Current Practice and Future”, Structural Engineering Applications of Artijkial Intelligence, Lecture Notes in Computer Science, Springer Verlag;

2. Bendat J.S., Piersol A.G. (1993), Engineering Applications of Correlation and Spectral Analysis, 2nd ed., John Wiley & Sons, Inc., USA;

3.Dyke S.J., Jolmson E.A. (2000), “Monitoring of a Benchmark Structure for Damage Identification”, Proceedings of the Engineering Mechanical Specialty Conference, May 21-24, Austin, Texas, USA;

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

5. Farrar C.R., Comwell P. J., Doebling S. W., Prime M.B. (ZOOO), Structural Health Monitoring Studies of the Alamosa Canyon and I 4 0 Bridges, Los Alamos National Laboratory, USA;

6.Juang J.N., Pappa R.S. (1985), “An Eigensystem Realization Algorithm For Modal Parameter Iden~cat ion And Model Reduction”, Journal of Guidance, Control, and Dynamics, Vol. 8, NO. 5., Sept.- Oct., pp620-627;

7.Juang J.N., Pappa, RS. (1986), “Effects of Noise On Modal Parameters Identified by the Eigensystem Realization 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 Bridges”, Journal of Vibration and Acoustics, Transactions of the

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

N0.1, ppl-18;

ASME, Vol. 120, July 1998, ~~713-718;

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Destructive Testing in Civil Engineering 2000, Uomoto (ed.), Tokyo, Japan;

10. Kadakal U., Yuzugullu 0. (1996), “A comparative study on the identification methods for the autoregressive modeling from the ambient vibration records”, Soil Dynaniics and Earthquake Engineering, Vol. 15, pp45-49;

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

12. Maia N.M.M, Silva J.M.M (ed) (1997), Theoretical and Experimental A4odal Analysis, Research Studies Press Ltd, England;

13. Quek S.T, Wang W, Koh C.G. (1999), “System Identification of Linear MDOF Structures Under Ambient Excitation”, Earthquake Engineering and Structural Dynamics, Vo1.28, pp61-77;

14. Stanbridge A.B., Ewins D.J. (1996), “Measurement of translational 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, Master Thesis, Department of Civil Engineering, The University of Tokyo, Japan (to be appeared)

~~141-152 ;

0=10.744H? 6 = 0.147%

o = 233.3 lHz

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 1 Figure 6: Identfied modal parameters for Test A

o = 28.710Hz C= 0.10396

Mode 2

w = 190.07Hz C = 0.01 5%

- 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: Identified modal parameters for Test B

440