BROADBAND VIBRATION MEASUREMENTS USING A CONTINUOUSLY SCANNING LASER VIBROMETER

BROADBAND VIBRATION MEASUREMENTS USING A CONTINUOUSLY SCANNING LASER VIBROMETER Steve Vanlanduit* Patrick Guillaume** Johan Schoukens* *Department of Fundamental Electricity (ELEC) **Department of Mechanical Engineering (WERK) Vrije Universiteit Brussel Pleinlaan 2, 1040 Brussel, Belgium e-mail : Steve. vanlanduitqvub ac. be ABSTRACT In this article we will apply the continuously scanning vibrometer measurement principle - introduced in literature for stepped sine measurements- to broadband excitations. Special attention will be given to the errors which are introduced in the procedure. Also, a method will be proposed to estimate the complexity of the vibration patterns. Different scanning signals to cover the measured area of the device under test will be studied. 1 INTRODUCTION The experimental study of the vibration behaviour of plate-like structures takes an important place in a large number of application areas: automotive industry. aerospace structures. household products... The use of optical measurement techniques like the Scanning Laser Vibrometer in combination with adapted modal analysis procedures made the modeling of complex structures possible. An important disadvantage of the scanning laser vibrometer in contrast to wholefield optical measurement techniques, like for instance Electronic Speckle Pattern Interferometry (ESPI) is that the measurements are performed point by point. This means that the visualisation and the interpretation of the vibration patterns is only possible after the completion of the measurements. which can typically take several hours. In the early nineties. an adapted method to efficiently measure vibration patterns using a scanning laser vibrometer was introduced in literature. The technique. which is denoted Continuously Scanning Laser Vibrometer (CSLV). is based on the use of a scanning laser vibrometer. where the scanning mirrors are driven by a continuous signal instead of the traditional discrete D,C steps used in the classical SLV measurement procedure. The CSLV technique was firstly studied in Il. 2. 31 an,, later improved L4, 5,, 7l and applied to rotating disks 181, nondestructive testing. and the measurement of angular degrees of freedom 1. 1 To preserve the selfcontainedness of the text, a short introduction of the continuously scanning technique will be given is Section 2. Currently. the CSLV technique is only developed in combination with stepped sine excitation. Therefore, allthough it is quite fast when a limited number of frequencies are desired, it is still too time consuming to apply to complex structures with a large number of closely spaced modes, where a high frequency resolution FRF is necessary For this reason, contributions were made in this paper to the development of the technique for broadband measurements. This will be discussed in Section 3 2 THE CONTINUOUS SCAN PROCEDURE The vibration velocity of a certain geometrical point on a sinusoidally excited linear time-invariant structure can be written as: u(t) = AR cos(wt) + AI sin(wt) (1) with w the frequency of vibration and 4~ and A, the in-phase and quadrature amplitudes of vibration (with respect to some reference, e.g. a force signal). When studying the vibration of a curved area on a structure. the vibration amplitudes will become position dependent, while the frequency is independent of the position: u(s(t),t) = A~(s(t))cos(d) + A,(s(t))sin(ut) (2) where s is a parameter for the position along the curve. The out-of-plane velocity of the curve is therefore equal to a single frequency vibration. modulated with both in-phase and quadrature vibration amplitudes. A straightforward method to demodulate these vibration amplitudes is by multiplying the vibration velocities in Equation 2 with in-phase and quadrature signals. giving: and v(s(t),t)cos(wt) = ;An(s(t)) + ;A~(s(t))cos(Zwt) +~Ar(s(t))sin(Zwt) (3) v(s(t), t)sin(wt) = ia,(s(t)) + iar(s(t))sin(2wt) -;A,(s(t)) cos(2wt) (4) 476

After eliminating the last two terms in Equations 3 and 4 by applying a low-pass filter with a cutoff frequency below 2w, the vibration patterns are available. The succes of the described procedure is dependent on the curve s(t) that is used. Clearly, if s(t) contains high frequency components, because the scan through the curve was performed too fast or because the vibration pattern along the scan line contained high frequency components (e.g. discontinuities), then the applied low-pass filter will eliminate this information. More insight in the technique can be given by expanding the vibration amplitudes of a constant speed (s(t) - t) continuous scan as a spatial Fourier Series : n AR(~) = ARK + c AR* cm k(qt + &) k=l (5) n AI(~) = AI, + c AI, sin k(rt + &) (6) k=l where R is the scan frequency at which the curve is periodically scanned, and & are spatial phases. I SW 1000 1500 2000 Frequency (in HZ, Figure 1: Mean amplitude (in db) of Velocities at 128 discrete circular locations on an aluminium disk. Combining equations 1, 5 and 6, one can express[;;e velocity of the vibrating object along the scanned curve as v(t) = Azqo cos(wt)+ c n ARk ; AJk cos (cd - k (lx + fk)) k=-!-to +A~,sin(wt) n + c &I. -AI, 2 cm (d + t (at + Ek))!a--n *+o (7) From this equation it can be seen that velocities of a continuous, uniform rate scan will exhibit sidebands around the excitation frequency w with a spacing equal to the scan rate R. It is also evident that the complex amplitudes (in-phase AR and quadrature AI) can be reconstructed in the form of the spatial series in equations 5 and 6 from the measurement of the phases and amplitudes of the sidelobes. 3 PERFORMING BROADBAND FRF MEASUREMENTS The continuously scanning vibrometer procedure offers a means to quickly measure the vibration patterns at a single frequency. When modal parameters are required. the procedure should be repeated for a number of frequencies in a frequency band of interest. For structures with closely spaced modes, this procedure can be time-consuming. In this section the continuously scanned vibrometer will be combined with broadband excitation to enable a fast measurement of FRFs. The method will be illustrated on measurements at a circular curve on an aluminium disk. For comparison, discrete scan velocities (see Figure 1) where measured at 128 locations. The structure was excited upto 2 khz with a Schroeder Multisine signal and a frequency resolution of about 1 Hz. To obtain continuously scanned broadband FRFs the mirrors were driven with sine and cosine signals, with a period equal to a multiple n times the period of the excitation. Measuring n excitation periods and applying an FFT results in n - 1 empty frequency lines between the excitation lines, which can be used by the continuous scan modulation sidelobes. In Figure 2 the result using a 64 times slower scan is illustrated. From the zoom in Figure 2-lower, it is clear that only a small number of the 64 modulation lines are needed. The vibration patterns can be demodulated by computing the Inverse Fast Fourier Transform of the n = 64 sidelobes around each excited line. From figure 3 it can be seen that the error between the discrete and continuously measured vibration patterns are in the order of a few percentages of the vibration amplitudes for most frequencies. Moreover, for operating shapes at frequencies where the relative error is larger, the continuous scan shape looks better, since the pattern is represented by low-frequency Fourier series. Therefore, the continuous scan procedure implicitly contains a method to reduce high spatial frequency noise (see Figure 4.top) and dropouts (see Figure 4-bottom). When the noise level is low, the discrete scan shape is a sampled version of the continuously scanning measurements (see Figure 5). Another interesting property of the continuous scan measurement is the spatial decomposition that is inherently performed. This enables to view the spatial complexity in function of the excitation frequency. as could be done with a spatial Fourier analysis. Figure 6 shows that, for instance. the second spatial frequency component is dominantly present at lower frequencies, while the ninth spatial frequency component exists mainly at higher frequencies. as could be expected. From this same Figure 6 it is also clear that the spatial Fourier analysis from the discrete scan (dots) agrees very good with the modulated sidebands (full line). An important aspect of the use of a continuous scan is the speckle noise. For discrete scans this phenomenon does not pose 477

500 IO-30 1500 2 Frequency (in Hz, 00 Figure 3: RMS values of the discrete scan vibration patterns (full line) and error between discrete and continuous (demodulated) vibration patterns (dots) for excited frequencies. -2oL 1---.-~-L-- ---- L -L 4 982 984 985 988 990 992 Frequency (in Hz) Figure 2: Amplitude (in db) of Velocities using a continuous circular scan on an aluminium disk. The scan rate is 64 times slower than the frequency resolution. Top : full frequency band. Bottom : zoom. One possibility to reduce the speckle noise is to use a slower scan. The speckle noise level will decrease linearly with the speed, until a certain scan rate, where it remains at a constant value-which is in this case approximately two times larger than the discrete scan measurement noise (see Figure 9). Quantifying the speckle noise level is an important step to obtain an automatic procedure to measure continuously scanning broadband FRFs. The aim is to use enough modulation sidelobes to represent all vibration patterns in the frequency range of interest, without unnecessarely increasing the measurement time. The procedure includes the following steps: a problem, since measurements are made at a single speckle, at least when the in-plane motion is negligible. When using a (periodic) continuous scan. several speckles are visited which introduces random signal intensity fluctuations. The fact that this error is periodic in nature can be seen in Figure 7. where a continuous scan without exciting the structure is performed over 10 periods. Indeed. the energy at frequency lines 1OIi is about 10 db higher compared to the intermediate frequency lines. This implies that the standard deviation over several excitation periods is not a good quantifier for the speckle error. Instead a measurement without excitation is performed. Comparison of the estimated spleckle noise level (velocities from the nonexcited structure) and the excited velocity spectrum illustrate how to select which frequency components contain signal and which components are below the noise level (see Figure 8). From this same Figure 8 is can be seen that the speckle noise level is about 20 db above the noise level in the case of a discrete scan measurement. l Use an excitation sigal with a reduced number of frequencies (here 512 instead of 2048) but with a fixed number of timepoints. l Estimate the noise level using recorded velocities without applied excitation.. Compute for every excited frequency the number of sidelobes which are above the noise level.. Extract from these values (see Figure 10) the number of extra excitation frequencies which can be introduced without overlapping of neighbouring excitation frequency sidebands. This step is based on the assumption that the complexity of neighbouring vibration patterns does not change significantly. l Excite the structure with the properly filled-up excitation signal. From Figure 10 it can be seen that the number of sidelobes above the noise level increases from 5 at the lower frequencies 478

DO 20 I 0-20 -2s -30- F.1. *I -3s 20 40 60 so 100 120 LDC~tiO -40 I 40; A-2 0 500 1000 1 so0 2000 Frequency (in HZ, Figure 4: Operating shape at 698 Hz (top) and 12 Hz Figure 6: Second (top) and nineth (bottom) Spatial (bottom) using discrete (dots) and continuous scan (full Fourier coefficients of discrete (dots) and continuous scan line). (full line). 500, I to 11 at the higher frequencies. Therefore, the fully excited signal could be used with e.g. 16 modulation sidelobes. ::::. -1-500 20 40 60 80 100 120 Location An incorrect chaise of the number of sidelobes can have drastic effects on the measurement results. Firstly the vibration patterns are smoothed, because higher spatial frequency components are lost. But more important. the sidelobes from different frequencies will overlap, giving rise to bad FRFs. Indeed. from Figure 11 it is clear that at least 16 modulation lines are needed. From this figure it is also visible that the number of necessary modulation lines is dependent on the frequency. Around 9OHz (see Figure ll-top) four modulation lines are enough. while 16 lines should be taken to perform a correct measurement at 1kHz (see Figure 11-bottom). Figure 5: Op&ating shape at 188 Hz using discrete (dots) and continuous scan (full line). 479

I Not Excited 0 Not Scanned 10-J I 10* 10-z 10. 10 10 scan Frequency (i HZ, Figure 7: Amplitude spectrum of ten periods of a continuous scan wihout applying excitation, zoom in at a small frequency band. Figure 9: Uncertainty (sample standard deviation) of nonexcited velocity responses in function of scan speed (squares) and standard deviation of nanscanning velocity responses (circles). 11t *I I- 1 d 5 10~ g 9-. 9 a 8. D7 i $6 S g 4 : ;1113 200 400 so0 800 loo0 1200 14061600 1800 2004 Frequency (in HZ, Figure 8: Amplitude spectrum of continuous scan, excited with Schroeder Multisine (full line), not excited (squares) and both not excited and not scanned (circles). Figure 10: Number of sidebands above noise level (filtered using length three median filter). 480

fq%ijii;g;j 1 B8 so 92 hquency,in HZ, 1 Figure 12: Top : coverage of the scan area with a six component optimal coverage multisine (left) and two sines with periods 6 and 128 seconds respectively (right). Bottom : velocity spectra for optimal coverage multisine scan (left) and two sine scan (right). -j&p;%] 530 540 sso Frequency (In HZ, / 4 CONTINUOUSLY MEASURING TWO-DIMENSIONAL VIBRATION SHAPES When instead of a simple one-dimensional curve. the vibration of a two-dimensional plate has to be measured, then the coverage of the surface by the laser beam is essential. Two possibilities to perform a coverage of a rectangular aluminium plate were studied. The first one is to drive the laser mirrors with multisines with a small number of components (six in the experiment), with amplitudes and phases computed to optimally cover the surface (see Figure 12. top-left). Because the curve is again scanned at a uniform rate, the velocity spectra exhibit symmetrical sideband spectra (Figure 12, bottom-left), which can be demodulated using the an FFT. To obtain values at a rectangular grid an interpolation procedure is necessary. 1ow 100s 10101015 1020 102s 1030 Frequency,I HZ, Figure 11: FRFs zoom at different frequencies, measured using a continuous scan with 2.4,8,16 and 64 modulation lines respectively. When selecting two sines with different frequencies RI and Rz as driving signals, the velocity spectra (Figure 12, bottom-right) are more complex. Sidebands with frequency spacing RI are positioned at multiples &I from the excitation frequencies. These sidebands can be converted by a linear matrix transform to coefficients of a two-dimensional Chebycheff polynomial ~61 The measurement results, which are shown in Figure 13. are reasonable for both discussed techniques. The multisine scan resulted in slightly larger errors near the corners of the surface (due to the interpolation scheme). 481

0 50 100 150 200 250 300 Time (in Seconds) 10 r 5 b -f O -directlo X-direction so4 100 50 r: b 3 0.c -50 5-100 r r -5 8 k-10-150 -15 40 20 I 500,000 1500 20 Location 1 Y-dkeCtiO X-diVWi0 Figure 14: Time record (top) and demodulated vibration pattern (bottom) for optimal coverage multisine scan. Parts of the scanned path are outside the object. 10 r 5 c j 0 rz -5 8 f -10-15 40 Figure 13: Operating shapes around 370 Hz for discrete scan (top), optimal cover multisine scan (middle) and two sine scan (bottom). 20 The problem of continuously measuring the vibration behaviour of structures with a complex geometry is much more difficult. Likely the coverage will be poor and the laser beam will encounter holes or other discontinuities. When the laser beam falls off the object, a signal drop occurs and dropouts will be present because of laser beam defocus (see Figure 14-top). Also, jumps in the vibration pattern will result in a Gibbs phenomenon (see Figure 14-bottom). But more important, spatial modelling. like for instance polynomial models, cannot be used anymore when discontinuities are present. Therefore, as a conclusion. the continuous scan method is only practically applicable to study the vibration behavior of simple shape objects (no holes or discontinuities). 5 CONCLUSIONS By carefully selecting the laser mirror signals, it is possible to use the continuously scanning vibrometer in combination with broadband excitation. From measurement noise information it

was seen that the continuous measurements are of lower quality than the discrete measurements. Because of the low-pass filtering effect of the spatial Fourier series, however, the continuous scan procedure is more resistant to high frequency noise, like dropouts. The noise information was also used to automatically detect the spatial complexity of the vibration patterns. Therefore a measurement time reduction is achieved in comparison with the classical discrete scan techniques. which often use spatial oversampling. The continuous measurement of two-dimensional vibration patterns worked well for simple shapes, but unfortunately it can not efficiently handle holes and jumps in a structure. [8] Stanbridge, A. B. and Ewins, D. J., Modal testing of rotating discs using a scanning L/W, Proceedings of the ASME Design Engineering Technical Conference, Vol. 3. 1995. [9] Stanbridge, A. B. and Ewins, D. J.. Measurement of translational and angular vibration using a scanning laser Doppler Vibrometer, Proceeding of the SPIE, No. 2358, pp. 37-47, 1994. [lo] Stanbridge, A. B. and Ewins, D. J., Measurement of total vibration at a point using a conical-scanning LDV, Proceeding of the SPIE, No. 2868, pp. 126-136, 1996. ACKNOWLEDGEMENTS This research has been sponsored by the Flemish Institute for the Improvement of the Scientific and Technological Research in Industry (IWT). the Fund for Scientific Research - Flanders (FWO) Belgium, the Flemish government (GOA-IMMI) and the Belgian government as a part of the Belgizin program on Interuniversity Poles of attraction (IUAP50) initiated by the Belgian State, Prime Ministers Office Science Policy Programming. REFERENCES (11 Spiram. P., Hanagud, S., Craigh, J. and Komerath. N. M.. Scanning laser Doppler technique for velocity profile sensing on a moving surface. Applied Optics, Vol. 29, pp. 2409-2417. 1990. [2] Spiram, P., Hanagud, S. and Craigh. J.. Mode shape measurement using a scanning laser Doppler vibrometer, Journal of Analytical and Experimental Modal Analysis. Vol. 7. pp 169-178. 1992. [3] Barker, A. J., Rapid Full Field Vibration Pattern Imaging using a Laser Doppler vibrometer. Proceedings of the 10th international Modal Analysis Conference, pp. 650-655, 1992. [4] Stanbridge. A. B. and Ewins. D. J.. Structural modal analysis using a scannig laser Doppler vibrometer, Proceedings of the Royal Aeroelastics Society international Forum on Aeroelasiticty and structural dynamics. No. 85.1-85.7, 1995. [5] Stanbridge. A. B. and Ewins. D. J.. Using a continuously scannfng laser Doppler vibrometer for modal testing, Proceedings of the international Modal Analysis Conference, No. 14, pp. 816-822, 1996. [6] Stanbridge and Ewins, D. J.. Modal Jesting using a scanning laser Doppler Vibrometer, Mechanical Systems and Signal Processing, Vol. 13. No. 2, pp. 255-270, 1999. [7] Stanbridge. A. 6.. Martelli, M. and Ewins. D. J., Measuring area mode shapes with a continuous-scan LDV. Proceedings of the SPIE, pp. 176-183. 2000. 483