Multipoint laser vibrometer for modal analysis

William N. MacPherson,1,* Mark Reeves,1,2 David P. Towers,1,3 Andrew J. Moore,1

Julian D. C. Jones,1 Martin Dale,4 and Craig Edwards4

1School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK2MRA Technology, Limited, Begbroke Science Park, Sandy Lane, Yarnton, Oxfordshire OX5 1PF, UK

3School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK4Land Rover Applied Optics Laboratory, Gaydon Test Centre, Building 102A, Banbury Road,

Warwick CV35 0RR, UK

*Corresponding author: w.n.macpherson@hw.ac.uk

Received 8 September 2006; revised 23 January 2007; accepted 27 January 2007;posted 29 January 2007 (Doc. ID 74859); published 15 May 2007

Experimental modal analysis of multifrequency vibration requires a measurement system with appro-priate temporal and spatial resolution to recover the mode shapes. To fully understand the vibration itis necessary to be able to measure not only the vibration amplitude but also the vibration phase. Wedescribe a multipoint laser vibrometer that is capable of high spatial and temporal resolution withsimultaneous measurement of 256 points along a line at up to 80 kHz. The multipoint vibrometer isdemonstrated by recovering modal vibration data from a simple test object subject to transient excitation.A practical application is presented in which the vibrometer is used to measure vibration on a squealingrotating disk brake. © 2007 Optical Society of America

OCIS codes: 120.3930, 120.7280, 120.6160.

1. Introduction

While theoretical and computational techniques aremuch advanced, there remains an important role forexperimental methods in the characterization of vibra-tion modes of complex structures [1]. An ideal vibrom-eter would enable measurements under conditions ofnatural excitation, whether transient or harmonic, andcomprising a range of frequencies. It would provide thetime history of the motion of every point on the testsurface with sufficient spatial resolution to map thehighest-order modes and enough temporal resolutionto adequately sample at the highest frequency present.In addition, the instrumentation would not perturb thesystem under test.

Single-point laser vibrometers are capable of high(megahertz) measurement bandwidths and being anoncontacting technique, it avoids loading the testobject. Such measurements are suitable for determin-ing component resonant frequencies and can assist inunderstanding the vibration modes [2,3]. Scanningthe measurement point across the test object can of-

fer an insight into the spatial distribution of the vi-bration mode. When it is possible to lock the vibrationmeasurement timing to a repeatable vibration actua-tor, then the amplitude and phase data can be deter-mined. However, for nonharmonic (e.g., transient)vibrations, the relative vibration phase at spatiallyseparate points is lost due to low-bandwidth scan-ning of the measurement point [4]. Higher-bandwidthscanning is afforded by using acousto-optic deflectors[5], however, such point-to-point vibration measure-ment remains unable to recover vibration phase inthe general case. Vibration amplitude and frequencyover an extended field can be measured by holo-graphic or electronic speckle pattern interferometry,but the phase of the motion is lost due to temporalaveraging imposed by the low bandwidth of videoframe rate detectors, and applications are again lim-ited to quasi-static or harmonic deformations [6,7].Recently, we recognized that, for many practicalproblems, valuable quantitative information could begained from one-dimensional spatial data, revealingthe complete temporal evolution of a complex motion[8]. The reduced spatial resolution allows a concom-itant increase in the temporal resolution. This com-bination of high (1D) spatial and temporal resolution

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permits the instrument to be viewed as a multipointvibrometer.

We illustrate the efficacy of the technique by report-ing optical modal analysis of a single ringdown test forthe first time. Traditional methods would require sev-eral repeated tests to obtain the same data: a single-point vibrometer would require each point to be testedindependently while time-averaged speckle pattern in-terferometry would require single-frequency excitationacross the frequency bandwidth of interest. The tech-nique can therefore reduce the measurement timefor harmonically excited objects compared with tradi-tional methods and enables multipoint modal testingfrom a single transient excitation, which is not possiblewith current methods. Finally, as a demonstration ofthis technique on a practical test object, measurementsof the surface vibration of a squealing disk brake underconditions of natural excitation are presented.

2. Multipoint Laser Vibrometer

The basic optical arrangement is shown in Fig. 1 [8].The object beam is derived from the output from an830 nm 80 mW laser diode that is collimated andreflected by a 50�50 nonpolarizing beam splitter(BS1) to pass through a cylindrical lens (L1). Thehorizontal line generated by L1 illuminates the testobject via a small redirecting mirror (M) and is im-aged by a photographic zoom lens (L2) passingthrough a second beam splitter (BS2) onto the 256-element linear array detector (CCD1). Mirror M issmall in comparison with the lens aperture, thereforeshadowing effects only slightly reduce the light in-tensity reaching the detector. The beam transmittedby BS1 forms the reference beam, which is coupledinto a fiber-pigtailed waveguide optical phase modu-lator (WGM). Fiber lengths were chosen to give ap-proximately equal path lengths in the object andreference arms, thus minimizing the effects of laserfrequency noise. The fiber pigtails of the WGM use asingle-mode polarization maintaining fiber to providea stable output polarization state and excellent spa-tial coherence thus optimizing fringe visibility. Thefiber output was transmitted by a second cylindricallens (L3) to a second beam splitter (BS2), the reflectedcomponent of which provided an on-axis referencebeam superimposed over the object beam on the 256detector array. A second camera (CCD2) is used inreal time for alignment purposes to ensure that theobject and reference beams are accurately superim-posed and aligned to the detector array. It can also beused to record traditional time-averaged speckle in-terferometry results when the cylindrical lenses L1and L3 are removed.

The motion of a point on the test surface is relatedto the optical path difference between light scattered

from the point x, y on the surface and the referencebeam, measured at the detector, I�x, y, t�, where t istime. In our case, we restrict the intensity measure-ment to a single line, for example, along the x axis.The light intensity I measured at the detector corre-sponding to point x on the target at time t arises fromtwo-beam interference [9], which can be rewritten as

I � I0�1 � V cos�� � ���, (1)

where I0 is the bias intensity, V is the visibility ofspeckle modulation, � is the interferometer phase,and � is the random speckle phase.

Determining � from I is ambiguous, modulo 2�.Furthermore this relationship is susceptible to in-tensity fluctuations (source power fluctuations orchanges in light intensity scattered back from thetest surface) that would be interpreted as a change inoptical phase. This familiar problem in interferome-try is amenable to a solution by a number of tech-niques. The one adopted here is based on phasestepping, i.e., introducing a known phase step be-tween successive measurements (camera frames).The reference beam optical fiber incorporates anintegrated-optic lithium niobate phase modulator(WGM in Fig. 1), generating a phase shift in theguided beam in response to applied voltage, with amodulation bandwidth of �1 MHz and a maximummodulation amplitude of �2� rad. A four-step stair-case function, synchronized with the camera framerate, produces phase steps of ��2 rad betweenframes. We choose ��2 rad phase steps because itmaximizes the measurement dynamic range [9]. Theoptical phase is obtained by applying the Carré algo-rithm [10] to the intensity values from any four con-secutive frames (n, n � 1, n � 2, and n � 3),

�n � � � tan�1���3�In�1 � In�2� � �In � In�3����In � In�3� � �In�1 � In�2��1�2

�In�1 � In�2� � �In � In�3� . (2)

Fig. 1. Optical configuration for a line scan vibrometer. T, testobject; SG, signal generator. The vibrometer consists of laser diodesource (LD), beam splitters (BS1 and BS2), cylindrical lenses L1and L3, and imaging lens L2. One-dimensional and full-field im-ages are captured using CCD1 and CCD2, respectively. Thestepped signal generater, SWG, modulated the phase modulator(WGM) and is synchronized to CCD1 frame capture.

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It is assumed that the frame rate is sufficiently highthat the test object surface velocity is constant duringany four consecutive frames (to a first-order approx-imation). This velocity component is equivalent to anerror in the phase step between consecutive frames,however, Carré’s algorithm is relatively insensitive tosuch errors. It is also assumed that the velocity doesnot exceed the velocity measurement limits of thevibrometer [8], in this case �10 mm s�1.

The random speckle phase, �, can be removed if themeasurement includes some known or stationary ref-erence position. For the general case of continuousnonharmonic motion with unknown initial deforma-tion, the speckle phase can be removed by calculatingthe phase changes between frames, i.e., time-resolveddeformation or surface velocity, so the instrumentcan be considered to be a multipoint vibrometer. Al-ternatively, assuming that there is no gross objectposition change, it can then be useful to removespeckle phase by assuming a zero mean object posi-tion; this gives no information about the initial shapeof the test object but offers the amplitude, phase, andfrequency of the vibration about a mean zero position.In both cases, the dynamic component of the objectmotion is recovered, provided that the total phasechange (phase step and deformation) between framesat any point is less than � rad, leading to the velocitylimit that depends on the camera frame rate andlaser wavelength used [8].

3. Harmonic Vibration Measurement

As an initial demonstration, a simple well-understoodtest object with single-frequency excitation was used.A 170 mm diameter, 1 mm thick aluminum disk sup-ported in the center was chosen because single-frequency excitation results in known vibrationmodes [11]. Using continuous single-frequency exci-tation of the circular plate via a sinusoidally drivenpiezoelectric actuator attached to the disk, we dem-onstrated the simultaneous measurement of vibra-tion phase and amplitude. This test configuration isan example in which a scanning single-point vibrom-eter measurement suitably locked to the actuationsignal could recover the vibration amplitude andphase over an extended area; however we use thissetup to demonstrate the capability of the multipointvibrometer.

A typical optical intensity measurement is shownin Fig. 2 in which the vertical axis represents time,and the horizontal axis represents spatial positionon the test object. In these data, the four sequentialphase steps down the time axis can be clearly seen(as shown in the inset image). These data are pro-cessed using Eq. (2) to obtain the optical interfer-ence phase at each pixel as a function of time, whichwas then unwrapped along the time axis as shownin Fig. 3. A calculation of phase is possible for eachframe using this approach by using the currentframe and three previous frame intensities rear-ranged appropriately to give I�n, n�1, n�2, and n�3� for usein Eq. (2). In this case, we eliminated speckle phaseby assuming a mean zero displacement. Whereas tra-ditional time-averaged speckle interferometry wouldonly reveal the vibration amplitude, this result illus-trates the measurement of a vibration mode wherethe two outer antinodes (centered at pixels �60 and�250) are out of phase with the inner antinode(centered at pixel �150). The residual between themeasuredharmonicdisplacementandthelinear least-squares fit sinusoid for each measurement point was21 nm rms.

4. Transient Vibration Measurement

The same test object was used to demonstrate vibra-tion modal analysis directly from a transient vibrationexcitation. In this case, the disk was struck once andallowed to “ringdown.” The resulting vibration con-tains multiple frequencies, excited simultaneouslywith arbitrary phases between them, and amplitudethat reduces due to damping. This is an example of ameasurement that could not be made using conven-tional optical vibrometry: time-averaged techniquesare not applicable to the transient vibration case, whilea traditional single-point laser vibrometer would re-quire an exact repeat of the experiment for each ofthe measurement points, alternatively multiple single-point vibrometers could be used but in general that

Fig. 2. Intensity measurement of disk vibrations, with four se-quential phase steps illustrated in the inset.

Fig. 3. Vibration measurement on the circular plate excited usinga piezoelectric strip attached to the reverse of the disk, with 987 Hzexcitation frequency (camera running at 80 kHz). The vibrationdisplacement is given in nanometers.

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would prove to be prohibitively expensive and difficultto arrange experimentally.

The resulting dynamic measurement using themultipoint laser vibrometer is illustrated in Fig. 4and shows displacement as a function of time (verti-cal axis) and spatial position (horizontal axis). Thedata shown correspond to a 50 ms section of a totalringdown period of �1000 ms. The multifrequencycontent is clearly evident, especially when Fig. 4 iscompared with the vibration due to single-frequencyexcitation measured at the same location on the diskshown earlier in Fig. 3.

The displacement measurements in Fig. 4 wereanalyzed to determine frequency and spatial contentof the multifrequency vibration. The 1D fast Fouriertransform (FFT) was calculated for the displacementdata recorded at each pixel, i.e., down columns of Fig.4. The FFT amplitude is plotted in Fig. 5 and indi-cates vibration modes at 117, 246, 566, and 987 Hz.This measurement allows us to determine the spatialdistribution of the vibration amplitude, for example,in Fig. 5 the position of two vibration nodes at 566 Hzis evident at pixel positions 110 and 225.

In the general case, the multipoint vibrometer pro-vides the frequency, amplitude, phase, and spatialcontent of the vibration. However, for the specific caseof a vibrating center-clamped circular plate the vi-

bration modes can be modeled [10] and comparedwith the experimental measurements. The plate de-flection, p�r, ��, for one of the two degenerate naturalfrequencies is given by

p�r, �� � A sin�nt�Jn�kr�cos�n��, (3)

where r and � represent position on the plate surfacein polar coordinates, A is the amplitude of vibration,and n is the mode frequency. n is an integer thatdescribes the number of diametral nodes, and wehave assumed that the fundamental circumferentialmode dominates. J is the Bessel function of order n ofthe first kind that describes the circumferential vi-bration component, where k2 � 5, 5.253, 12.23, 21.6for n � 1, 2, 3, 4. The term cos�n�� describes thediametral vibration component. This model was ex-tended to include a spatial phase and temporal phaseterms, thus,

p�r, �� � A sin�nt � �t�Jn�knr�cos�n� � �s�, (4)

where �t is the temporal phase between the vibrationmodes and �s is the spatial phase that describes theposition of the diametral nodes. In this experiment,we have sufficient information—the vibration ampli-tude and vibration phase along a known measure-ment line—to obtain the coefficients n, �t, �s, and A,thus fully describing each vibration mode both spa-tially and temporally.

Plots of the vibration amplitude and phase foreach of the four natural frequencies identified fromthe Fourier analysis of experimental data (Fig. 5)are shown in Fig. 6, columns 2 and 3. A linearleast-squares fit of the experimental data to Eq. (4)was used to determine coefficients n, �t, �s, and A foreach mode. It is then possible, for this particularobject, to reconstruct the mode shape for the en-tire plate using Eq. (4), reconstructed from the ex-perimentally determined coefficients from a line ofdata, Fig. 6, column 4. Discrepancies between themodel predictions and the measurements arise fromimperfections in the plate and its mounting. To val-idate the vibration modes recovered, the interferom-eter was switched to full-field time-averaged mode byusing CCD2 and removing cylindrical lenses L1 andL3. Single-frequency excitation was applied to thecircular plate, as in Section 3, at each of the naturalfrequencies identified. The time-averaged electronicspeckle pattern interferometry (ESPI) fringes recordedare shown in Fig. 6, column 5, and show good agree-ment in spatial position of the diametral nodes.

5. Application of Vibrometer to Brake SquealMeasurements

A typical test application, in which the vibrationmodes are unknown, was analyzed using the multi-point vibrometer. The test rig was composed of astandard automotive disk brake system. The drive-shaft was driven by an electric motor at �30 rpm,and the disk brake applied via a standard hydraulic

Fig. 4. Displacement measurement from transient excitation ofthe circular plate supported in the center. Time is the vertical axiswith an 80 kHz frame rate, with spatial measurement on thehorizontal axis.

Fig. 5. Frequency content of line scan data proportional to FFTamplitude.

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mechanism. To improve the optical signal returnedfrom the target, areas of interest were covered inretroreflective tape.

Measurements on the brake calipers demon-strated 1 �m amplitude vibrations with a resonantfrequency of �3000 Hz. It also proved possible tomonitor the vibration of the brake disk as it rotates.This measurement is complicated due to the con-stantly changing mean disk position (due to a slightwobble in the rotating disk) and because test timeswere limited due to disk heating during braking melt-ing the retroreflective tape. The short time scaleavailable for these tests highlights the benefit of the

multipoint vibrometer that was capable of makingthe required measurement in a fraction of a second,thus minimizing effects due to heating and disk move-ment. Typical data recorded during brake squeal con-dition (equivalent to 3 ms of measurement time) areshown in Fig. 7. During the measurement time, thedisk rotation (and hence average in-plane displace-ment) at the measurement radius is �3.5 �m betweenframes corresponding to �1�200 pixel in the imageplane and 1050 �m (�1.5 pixels in image plane) overthe time scale of the experiment shown below.

Analysis of these data is of particular interest forvalidation of a proposed model of brake disk vibra-

Fig. 6. Vibrational modes of the circular disk. Column 2 plots the experimental vibration amplitude (in micrometers) obtained from themultipoint vibrometer in comparison with the modeled amplitude for the same portion of the disk. Column 3 plots the phase (in radians)measured by the multipoint vibrometer in comparison with modeled phase data. Column 4 shows the modeled mode shape. Column 5illustrates experimental measurement of the modes using full-field time averaged ESPI with single-frequency excitation.

Fig. 7. Brake disk vibration measurement.

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tions [12] based on two degenerate modes given byEq. (5):

S��, t� � A sin�t � �t1�sin�n� � �s1�

� A sin�t � �t2�cos�n� � �s2�, (5)

where A is the peak amplitude, n is the diametralmode order, t is time, � is the angular position aroundthe disk, � is the vibration frequency, �s is the spatialphase, and �t is the temporal phase. Once again, wecalculated the amplitude and phase data by applyinga FFT to the measured vibration in the time domain.Subsequently, a multiparameter linear least-squaresfit to the amplitude and phase data was performed(Fig. 8). Our measurement adds quantitative data tosupport the model of this brake squeal mode as twodegenerate modes at slightly different frequenciescombining to produce a mode that travels around thedisk circumference. Other optical techniques thathave been applied to measure brake squeal [13,14]have also provided support for this dynamic behavior.

6. Discussion

The vibrometer reported here is capable of resolving256 measurement points along a line projected ontothe test surface, with out-of-plane measurement ac-curacy of �21 nm rms when using an 830 nm lasersource and frame rates up to 80 kHz. Unambiguoussurface velocities measurement up to �10 mm s�1 ispredicted theoretically, and values approaching thiswere obtained in experimental tests ��4.2 mm s�1 inFig. 4). At present, the available laser power enablesa measurement line of �200 mm in length.

The work reported here used a 256-element detec-tor array. Recent advances in detector technologyhave resulted in linear arrays with 1024 elementarrays capable of operating at frame rates approach-ing 100 kHz. While there is no doubt that advances indetector technology will further increase this capabil-ity, there are always likely to be issues with datatransfer and processing due to the large amount ofdata that high bandwidth and high-resolution detec-tors generate. Based on PC data acquisition with dig-ital cameras, the data throughput limit is currently of

the order of 200 Mbits�s. Higher surface velocitiescan be measured with a corresponding reduction inthe number of measurement points within this limit.Should higher velocities be required, photodiode ar-rays with dedicated digital processing could be em-ployed, raising the effective data throughput limit togigabits per second.

The main advantage of the multipoint vibrometeris its ability to measure nonrepeating or transientvibrations when vibration phase over extended areasis required. In such cases, it is either not possibleto use scanning single-point vibrometers or time-averaged 2D techniques, or the multipoint vibrom-eter offers a significant time saving over traditionaltechniques. If it is desirable to have a 2D measure-ment array of the vibration, the spatial configurationof the measurement points could be altered usingadditional optical elements.

7. Conclusions

The multipoint vibrometer presented here has beendemonstrated to be capable of measuring vibrationamplitude and phase of complex modes on the simplecase of a centrally supported circular plate. From asingle ringdown measurement, the vibration naturalfrequencies, mode amplitude, and their spatial andtemporal phase were determined. A practical appli-cation was also described in which the vibrometerwas used to measure brake squeal vibrations on astandard design of automotive brake calipers anddisk. In the case of the brake squeal test, informationfrom these measurements helps to support a previ-ously proposed model of such vibrations.

This work was funded by the UK Engineering andPhysical Science Research Council (EPSRC) and theUK Department of Trade and Industry LINK Scheme.W. N. MacPherson and A. J. Moore acknowledgeEPSRC for provision of funding via the Advanced Fel-lowship Programme. The authors thank C. Buckberryfor helpful discussions.

References1. S. Welaratna, “Vibration testing in the automotive industry,”

EE-Eval. Eng. 39, 132–140 (2000).2. D. Moreno, B. Barrientos, C. Perez-Lopez, and F. Mendoza

Santoyo, “Modal vibration analysis of a metal plate by using a

Fig. 8. Modeled data (solid curve) fitted to experimental data (�) for counterpropagating modes for a brake disk. Plots show the vibrationamplitude in micrometers (left) and vibration phase (right) for the brake disk.

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laser vibrometer and the POD method,” J. Opt. A 7, S356–S363 (2005).

3. C. Huang, “Transverse vibration analysis and measurementfor the piezoceramic annular plate with different boundaryconditions,” J. Sound Vib. 283, 665–683 (2005).

4. A. B. Stanbridge and D. J. Ewins, “Modal testing using ascanning laser Doppler vibrometer,” Mech. Syst. Signal Pro-cess. 13, 255–270 (1999).

5. B. K. A. Ngoi, K. Venkatakrishnan, B. Tan, N. Noel, Z. W.Shen, and C. S. Chin, “Two-axis-scanning laser Doppler vi-brometer for microstructure,” Opt. Commun. 182, 175–185(2000).

6. A. F. Doval, “A systematic approach to TV holography (reviewarticle),” Meas. Sci. Technol. 11, R1–R36 (2000).

7. J. M. Huntley, G. H. Kaufmann, and D. Kerr, “Phase-shifteddynamic speckle pattern interferometry at 1 kHz,” Appl. Opt.38, 6556–6563 (1999).

8. J. M. Kilpatrick, A. J. Moore, J. S. Barton, J. D. C. Jones, M.Reeves, and C. Buckberry, “Measurement of complex surfacedeformation by high-speed dynamic phase-stepped digital

speckle pattern interferometry,” Opt. Lett. 25, 1068–1070(2000).

9. P. Hariharan, Basics of Interferometry (Academic, 1992).10. P. Carré, “Installation et utilisation du comparateur photoélec-

trique et interférentiel du Bureau International des Poids etMesures,” Metrologia 2, 13–23 (1966).

11. A. W. Leissa, Vibration of Plates, NASA SP-160 (U.S. NationalTechnical Information Service, 1969).

12. M. Reeves, N. Taylor, C. Edwards, D. Williams, and C. H.Buckberry, “A study of brake disc modal behaviour duringsqueal generation using high-speed electronic speckle patterninterferometry and near-field sound pressure measurements,”Proc. Inst. Mech. Eng., Part D (J. Automob. Eng.) 214, 285–296(2000).

13. F. Chen, G. M. Brown, M. M. Marchi, and M. Dale, “Recentadvances in brake noise and vibration engineering using lasermetrology,” Opt. Eng. 42, 1359–1369 (2003).

14. R. Krupka, T. Walz, and A. Ettemeyer, “Fast and full-fieldmeasurement of brake squeal using pulsed ESPI technique,”Opt. Eng. 42, 1354–1358 (2003).

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