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Reducing measurement time for a laser Doppler vibrometer using regressive techniques
ARTICLE IN PRESS
Optics and Lasers in Engineering 45 (2007) 49–56
Reducing measurement time for a laser Doppler vibrometer using regressive techniques Joris Vanherzeele, Steve Vanlanduit, Patrick Guillaume Vrije Universiteit Brussel (VUB), Acoustics and Vibration Research Group (AVRG), Department of Mechanical Engineering (MECH), Pleinlaan 2, B-1050 Brussel, Belgium Received 12 January 2006; received in revised form 13 March 2006; accepted 21 March 2006 Available online 12 May 2006
Abstract In the last decade the laser Doppler vibrometer (LDV) has become a widely spread instrument for measuring vibrations. It often offers accurate measurements with a high spatial resolution. However, the measurement time of the LDV and especially for the scanning LDV is long. Therefore, reducing the measurement time is an attractive objective. A way to achieve this is to use a single sine excitation (on a resonance frequency). However, this technique has two major drawbacks: the inability to provide information on the damping and a operational deflection shape that can differ from the true mode shape. In this article two methods will be introduced to reduce measurement time without these defaults. In the first method introduced in this article a narrow band multisine is used as excitation signal and the measured vibration signal in the time domain is represented by a model using sines and cosines with these fixed narrow band frequencies. The coefficients of those sines and cosines are then estimated on a global scale by means of a least-squares estimator. An important advantage of this particular technique is that one does not have to measure a full period of the signal, reducing time. The second method accelerates the measurement time for scanning LDV measurements. Using the time domain sequence from each previous scan point and a limited number of time samples from the current scan point, the full time domain sequence of the current scan point can be estimated. Both these methods are a key benefit for in-line quality control, which can have upwards of 1000 spatial measurement locations. The proposed techniques will be validated on both simulations and experiments of varying complexity. r 2006 Elsevier Ltd. All rights reserved. Keywords: Scanning laser Doppler vibrometer; Measurement time reduction; Regressive Fourier series; Spatial regressive technique
1. Introduction Modal analysis has for some time now been an important tool in structural analysis in many different fields of application [1]. The measurements themselves were mostly executed using contact-based measurement systems such as accelerometers. They have, however, some disadvantages such as the cumbersome task of applying the accelerometers to the structure (sometimes several times when working in different patches) and of course there is always the mass loading aspect (important for lightweight structures). Recently however, with the ever ongoing progress in optical techniques some interesting alternatives have been developed. However, this also led to entirely new measurement Corresponding author.
E-mail address: [email protected] (J. Vanherzeele). 0143-8166/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2006.03.004
systems themselves such as the laser vibrometer, or scanning laser Doppler vibrometer (LDV) [2]. The LDV made it possible to increase the number of degrees of freedom (DOF) measured on the surface by two to three orders of magnitude. Where in the past one was limited to a few hundred measurement points it was now feasible to measure tens of thousands of points. However, measuring such high spatial resolution data takes a relatively long time (e.g. a simple plate with 1000 points and a frequency resolution of 1 Hz, will roughly take about 20 min). In short, there are two possibilities to execute a scanning vibrometer measurement with the existing commercial LDV-systems. The first mode is to perform a so-called ‘full scan’ where all frequency lines are excited with which a frequency response function (FRF) over the entire frequency range is obtained. This mode takes a considerable amount of time inversely proportional to the frequency resolution. The second is a fast scan mode where
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only one frequency line is excited (normally on a resonance frequency). The measurement time of this mode is of course proportional to the inverse of the imposed frequency and is therefore faster. However, it is not possible to estimate damping, for which at least two frequencies are needed, nor will the estimated deflection shape be exact and even in this mode at least one period of the frequency has to be measured. The only way to reduce the actual measurement time is to simply measure less of the vibration time signal for each individual point on the structure. However, this is, as already stated, limited by the period of that signal. For one always has to measure at least one entire period (or an integer number of periods) of the vibration sequence in order to avoid leakage as is the case with a discrete Fourier transform (DFT). In the first method introduced in this paper, the signal is approximated by a series of sines and cosines with fixed frequency. The frequencies in the measured signal are known a priori for linear systems, because excitation is a multisine with a narrow fixed frequency band, which makes it possible to model an entire signal period using only a portion of the period. In this paper a regressive discrete Fourier series (RDFT) technique [3,4] was used to estimate the sine and cosine coefficients, yielding an approximation of the vibration signal. Using the proposed technique on a time sequence, it is clearly possible to not only generate an estimate for the entire sequence but maybe even more importantly to reduce the measurement time. Theoretically, one can estimate a single frequency using only two points, however in practice this strongly depends on several parameters in the data. The second method presented in this article is a spatially accelerated regressive technique used to reduce measurement time specifically for scanning LDV measurements. The technique is based on the fact that the time sequences measured in each sequential scan point are basically the same and only differ by their amplitude. Therefore, it is possible to approximate the vibration signal in a particular scan point by means of a previous scan point by simply estimating the amplitude. This can be done by using a small number of time samples of each adjacent scan point, hence reducing the measurement time. In the following section the methods will be unveiled. In Section 3 simulations on the regressive Fourier series technique will be shown on different examples showing the capabilities towards high spatial resolution data and multiple frequency signals. In Section 4 both methods will be put to practice on measurements on an aluminium beam and finally some conclusions will be drawn in the last section. 2. Measurement time reduction using regressive algorithms 2.1. Measurement time reduction using a regressive Fourier series Before going into full detail on the method, a short insight into the methodology of a measurement with the
presented technique will be given. Using the regressive Fourier series technique, the method boils down to the following steps: Perform a full frequency band measurement on a rough scan grid to determine the frequency band of interest. Single out this frequency band in the FRF, putting all other frequencies to zero and inverse Fourier transform this obtaining a band limited time sequence. Use this band limited time domain signal to determine the minimal number of time samples needed for the regressive Fourier series. Excite the structure with a multisine signal containing the chosen frequency band, only measuring the determined minimal number of time samples for each scan point. Perform the estimation of the vibration signal using the measured time samples and the input frequencies in the regressive Fourier series. The method is based on representing the vibration time signal by a series of sines and cosines. The frequencies of these functions are known a priori as they are imposed on the object during testing. These frequency lines are not chosen at random but somewhere in the vicinity of a resonance frequency. The resonance frequency can be traced by e.g. performing a full scan test with frequency resolution f res and sample frequency f sample in a limited number of object points. By using just a limited number of frequencies F concentrated around the resonance frequency with begin frequency F b it will be shown possible to speed up the measurement process significantly in comparison to a full scan where all frequency lines are excited. The excitation signal used to generate this limited band signal is following multisine in the time domain: xðnÞ ¼
F X
Am sinð2pF m nÞ þ Bm cosð2pF m nÞ,
(1)
m¼1
where Am and Bm represent the respective input amplitudes of the sines and cosines with frequency F m ¼ ½F b þ ðm 1Þf res =f sample ; n ¼ 0; . . . ; N 1. The time sequence of the measured vibration signal yðnÞ can be written in the same way as the input signal but with different amplitudes C m and Dm yðnÞ ¼
F X
C m sinð2pF m nÞ þ Dm cosð2pF m nÞ.
(2)
m¼1
By simply measuring these time signals as is, the minimum measurement time is always restricted to at least one period. However by approximating the signal by a series of sines and cosines with a certain number of fixed frequencies the measurement time can be brought down to less than one period. Eq. (2) can be written as a linear matrix equation: a ¼ Bc,
(3)
ARTICLE IN PRESS J. Vanherzeele et al. / Optics and Lasers in Engineering 45 (2007) 49–56
where a, B and c are given by the following matrices: 1 0 1 0 C1 yð0Þ BD C C B B 1C C B yð1Þ C B C B B . C a¼B C; c ¼ B .. C, .. C C B B . A @ BC C @ FA yðN 1Þ DF 0 B B B B¼B B @
time even further with a regressive approach. To explain the principle, a modal model in the frequency domain [5] will be used from which the time sequence yðnÞ which is actually measured will be derived. The FRF H in a certain scan point i for one single mode can be written as fi , io l
H i ðoÞ ¼
sinð2pF 1 0Þ
cosð2pF 1 0Þ
sinð2pF m 0Þ
cosð2pF m 0Þ
sinð2pF 1 1Þ .. .
cosð2pF 1 1Þ .. .
sinð2pF m 1Þ .. .
cosð2pF m 1Þ .. .
sinð2pF 1 ðN 1ÞÞ
cosð2pF 1 ðN 1ÞÞ
sinð2pF m ðN 1ÞÞ
cosð2pF m ðN 1ÞÞ
The unknown coefficients C m and Dm are estimated (^c) by means of a least-squares approach taking the pseudoinverse of the matrix B and multiplying it with a c^ ¼ Bþ a.
51
(4)
It is obvious that for each frequency two coefficients have to be estimated so this means that for each frequency at least two equations are necessary. So this means that theoretically for a sequence yðnÞ an estimate can be derived using only 2F time points. Of course in practice this depends on the time resolution of the signal. Indeed, the closer two sequential points lie together the harder it becomes to fit a sine through them. Nonetheless, this clearly illustrates the possibility to reduce the measurement time for each individual frequency. 2.2. Measurement time reduction using a spatial regressive technique Using the spatially accelerated regressive technique the method can be summed up as follows: Perform a full frequency band measurement on a rough scan grid to determine the frequency band of interest. Single out this frequency band in the FRF, putting all other frequencies to zero and inverse Fourier transform this obtaining a band limited time sequence. Use this band limited time domain signal to determine the minimal number of time samples needed between the first scan point and its adjacent neighbor to estimate their amplitude relation. Excite the structure with a multisine signal containing the chosen frequency band, only measuring the determined number of time samples for each scan point. Reconstruct the full time domain sequence of each scan point using each sequentially determined amplitude relation and the adjacent scan point’s time sequence. When performing scanning laser Doppler measurements with L scan points it is possible to reduce the measurement
(5) 1 C C C C. C A
where o represents the angular frequency, fi is the mode shape amplitude at that certain point i and l is the system pole containing natural frequency and damping. Now it is well known that for uncoupled normal modes, the amplitude fi1 of the mode shape in a scan point i 1 is in direct linear relation to fi in a point i by fi ¼ ki fi1 ,
(6)
ki
denotes the amplitude relation of the mode shape where in two adjacent scan points i and i 1. Now in the time domain, the measured vibration sequence yi ðnÞ in each respective scanning point i on the structure can be written in exactly the same fashion as Eq. (6), revealing Eq. (7). Indeed, the input frequencies are the same so only the amplitude changes (and in some cases phase, but this is not considered here). This means that it is possible to estimate this amplitude of each sequential yi ðnÞ on the structure using the result of the previous scan point yi1 ðnÞ. The main advantage is since the multiplier ki is a constant value, it can be estimated using a limited number of time samples N k of the measurement yi ðnÞ yi ðnÞ ¼ ki yi1 ðnÞ,
(7)
with n ¼ 0; . . . ; N 1; i ¼ 2; . . . ; L and the ki denotes the amplitude relation of two adjacent scan point times sequences. An estimate of this factor k^i in Eq. (7) is given by the following least-squares equation: k^i ¼ yi1 ðnÞþ yi ðnÞ,
(8)
with n ¼ 0; . . . ; N k 1; N k 5N; i ¼ 2; . . . ; L. In short this means that the measurement time can be reduced by N=N k times. The minimal number of points necessary N k to perform the estimate in each scan point can be determined when performing the full frequency band measurement on the coarse scan grid. The obtained band limited time sequences in two adjacent time points on the structure can then be used to estimate the minimal number of points N k necessary to estimate their amplitude relation.
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This spatial regressive technique uses only N k points of the time sequence of each sequentially scanned point on the structure. In Section 4 it will be shown that this reduces the measurement time even further compared to using only the proposed regressive technique. 3. Computer simulations The simulations were carried out on a multisine signal x which made it possible to explore the effect of different parameters such as number of frequency components, sample frequency for different frequency bands. The following was investigated: How the bandwidth B (or equivalent the number of frequency components (F) in the multisine) influences the minimal number of time samples N k needed to estimate the sequence. How the sampling frequency f sample effects the number of data points needed to estimate the sequence. How noise effects these estimations. The frequency resolution f res ¼ 1 Hz for all simulations. Fig. 1 gives an example where the number of frequency components F was incremented with each iteration up to a maximum of 25 frequencies, starting from different begin frequencies F b . The multisine was sampled at f sample ¼ 500 Hz and the error value err (Eq. (9)) between estimate and true signal was set to 0.1%. The error value for the simulations with added noise was of course set above the respective noise levels. The measurement time was then determined by calculating the minimum number of time samples N k necessary to achieve this error. The time reduction is given by: time reduction ¼ 1 N k =N (in %) where N denotes the total number of time
samples. PN k 1 err ¼
n¼0
jðyest ðnÞ yðnÞÞj2 , jyðnÞj2
(9)
where yest represents the estimate of the signal y. The linear relation which can be derived from Fig. 1 for a small number of frequencies (that number increases with F b ), shows that only two points are needed for each frequency as is predicted theoretically. Above this value more data points are needed to obtain good conditioning. Also for the lowest begin frequency F b ¼ 1 Hz, a significantly lower time reduction is achieved in comparison with higher begin frequencies. For these higher begin frequencies F b , the time reduction in function of the number of input frequencies F is almost independent of F b . The sampling frequency is another important factor when trying to reduce the measurement time. By gradually increasing the sample frequency, it is clear that up to a certain point the measurement time is unhampered by the sample frequency for the case where F b ¼ 1 Hz (Fig. 2). After that the number of data points used must be increased to allow estimation. The relative number of points needed to perform the estimation stays constant for an increasing sampling frequency. Figs. 3 and 4 illustrate that for higher frequencies (F b ¼ 50 and 100 Hz, resp.) a low sampling frequency gives estimation problems due to aliasing, resulting in large spikes. Fig. 4 also shows that with a small number of frequency components (e.g. F ¼ 10) the theoretical minimum of two data points for every frequency is respected. It can be noted that the measurement time reduction for F ¼ 10 is far better than for a larger amount of frequency components at F ¼ 20. However in practice it is not necessary to use that many frequency lines to obtain a
100
90
F=20 F=15 F=10
95 90
80
time reduction (%)
time reduction (%)
100
Fb=400 Hz Fb=200 Hz Fb=100 Hz Fb=50 Hz Fb=25 Hz Fb=1 Hz
70
60
50
85 80 75 70 65 60 55
40 0
5
10
15
20
25
F Fig. 1. Time reduction (%) obtained for an increasing number of frequencies F in the multisine; frequency band with F b ¼ 1, 25, 50, 100, 200 and 400 Hz, respectively
50 50
100
150
200
250 300 fsample (Hz)
350
400
450
500
Fig. 2. Time reduction (%) obtained for sample frequencies between 40 and 500 Hz (multisine frequencies F ¼ 10, 15, 20, resp.; F b ¼ 1 Hz).
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100
100 F=20 F=15 F=10
90
-30 dB -20 dB -10 dB
80 time reduction (%)
95 time reduction (%)
53
90
85
70 60 50 40 30 20
80
10 50
100
150
200
250 300 fsample (Hz)
350
400
450
0
500
0
5
10
15
20
25
F
Fig. 3. Time reduction (%) obtained for sample frequencies between 40 and 500 Hz (multisine frequencies F ¼ 10, 15, 20, resp.; F b ¼ 50 Hz).
Fig. 5. Time reduction (%) obtained for multisine frequencies F incrementing up to 30 Hz; F b ¼ 1 Hz; noise level 10, 20 and 30 dB under signal level, respectively.
100 F=20 F=15 F=10
100 90
-30 dB -20 dB -10 dB
80 90
time reduction (%)
time reduction (%)
95
85
80
70 60 50 40 30 20
50
100
150
200
250 300 350 fsample (Hz)
400
450
500
Fig. 4. Time reduction (%) obtained sample frequencies between 40 and 500 Hz (multisine frequencies F ¼ 10, 15, 20, resp.; F b ¼ 100 Hz).
reliable prediction of the mode shapes, as will be shown in Section 4. Finally, the influence of noise on the examples above can be examined. When comparing the figures for the higher frequency band (Figs. 1 and 5) one can conclude that the overall appearance is quite similar, except that because of the added noise more data points are needed to perform the estimation. This was of course to be expected. Fig. 5 also shows that once noise is introduced to the signal, the actual level of the noise does not influence the estimates a lot. One can also note that as the number of data points used approaches the number of points in the true signal the curve becomes asymptotic. This is understandable because if one uses all points in the signal the estimation has to be exact.
10 50
100
150
200
250 300 fsample (Hz)
350
400
450
500
Fig. 6. Time reduction (%) obtained for increasing sample frequency between 40 and 500 Hz. F ¼ 10; F b ¼ 1 Hz; noise level 10, 20 and 30 dB under signal level, respectively.
The effect of the added noise to an increasing sampling frequency can be seen in Fig. 6. One can clearly see that the ‘flat line’ where the number of data points was constant for increasing sample frequency (Fig. 2) is no longer visible. A higher sampling frequency (f sample ) proves more sensitive to noise. 4. Experimental results In this section the regressive technique will be put to a practical test. A measurement was carried out on an aluminium beam, using the LDV, scanning along the center line in 80 scan points. The beam, suspended in
ARTICLE IN PRESS J. Vanherzeele et al. / Optics and Lasers in Engineering 45 (2007) 49–56
54
free–free conditions was excited by a shaker placed at one far end of the beam Fig. 7. The excitation signal was a broadband multisine with sampling frequency f sample ¼ 2048 Hz and frequency resolution f res ¼ 1 Hz. This broadband scan revealed two distinct modes (Fig. 8). The number of averages was limited to one. Taking the full frequency band measurement (or full scan) and singling out one of the two modes, five frequency lines localized around the resonance frequency were used as input signal for the shaker and hence for the regressive estimation. The regressive technique was tried out on the second mode, with input frequency lines 725–730 Hz. The number of averages was limited to one. The problem is determining the measurement time necessary for an error value err no higher than 3%, a priori. Deriving an analytical expression for the number of data points needed is a cumbersome task at best because of
the large number of variables in the problem. Therefore, the best method of operation is to measure a complete period of the time signal in a single point and calculate the number of points needed. This test was performed when doing the full scan on the rough grid. This calculation can then be used as a reference value for the following points. The measurement time for the full frequency band was 80 s, with f res ¼ 1 Hz and 80 scan points. The measurement time for this full scan and the technique with a fixed number of excitation lines is principally the same because the frequency resolution is the same. By using the proposed regressive Fourier series technique (RDFT) (F begin ¼ 725 Hz; F ¼ 5) the measurement time for the second mode was reduced by a factor 3 for each measurement scan point totalling 26.50 s ðerrp3%Þ. The modes were calculated from the FRF-matrices using a classic maximum likelihood approach [6], for which the stated error value obviously has little to no effect. Fig. 9 shows a comparison between the mode shape estimated with the full scan data and the RDFT with measurement time reduction. The calculated mode shape using the RDFT clearly fits the full scan mode very well. An estimate of the second mode shape can also be generated using the fast scan method. This is done by exciting just one frequency line on the resonance frequency. Of course, the choice of frequency line is touchy because of the deviations due to the estimation method, or a resonance which might have a tendency to shift through time. Also, trouble will be encountered when two modes are positioned closely together in the frequency domain (e.g. strongly coupled modes). A fast scan will not be an optimal measurement method in this case, which is reflected in the estimated operational deflection shape (ODS). On the other hand, the fast scan offers a very fast
Fig. 7. Test set-up.
1400 1200
60 1000
40
800
Ampl
50
Ampl (dB)
30
600
20 400
10 0
200
-10
0 10
-20 -30 0
200
400
600 800 Frequency (Hz)
1000
Fig. 8. Broadband scan of the aluminium beam.
1200
20
30
40 L
50
60
70
80
Fig. 9. Second mode of an aluminium beam using the entire frequency band (solid line) and the regressive Fourier series technique (F begin ¼ 725 Hz; F ¼ 5) with measurement time reduction (asterix line); absolute values.
ARTICLE IN PRESS J. Vanherzeele et al. / Optics and Lasers in Engineering 45 (2007) 49–56
1500
55
1400 1200 1000
Ampl
Ampl
1000
800 600
500 400 200 0 0
10
20
30
40 L
50
60
70
0
80
10
20
30
40
50
60
70
80
L
Fig. 10. Second mode of an aluminium beam using the entire frequency band (solid line) and the fast scan technique (F b ¼ 727 Hz; F ¼ 1) (asterix line); absolute values.
Fig. 11. Second mode of an aluminium beam using the entire frequency band (solid line) spatial regressive technique (F b ¼ 725 Hz; F ¼ 5) (asterix line); absolute values.
measurement sequence, using 727 Hz as input frequency line for the measurement. The measurement time for the fast scan was 0.11 s. In this case, the fast scan is obviously faster than the regressive technique but in this case the ODS will differ quite a bit from the mode shape obtained from a full scan. In this particular case, the estimated ODS differs significantly because the excited frequency line was chosen coincidental with the DFT-grid. Fig. 10 offers a comparison between the full scan mode and the deflection shape estimated with a fast scan. When comparing these measurement times, the fast scan proves to be the fastest technique but also with the lowest accuracy. Using the spatial regressive technique it is possible to reduce the measurement time with respect to the full scan for each scan point by about 200 times ðN k ¼ 10Þ. The measurement time for this mode was 0.39 s, making it almost exactly as fast as the fast scan technique, but with a far greater accuracy level. The mode shape estimated with the spatially accelerated regressive technique compared with the full scan is shown in Fig. 11. Table 1 shows a comparison of the measurement times and the relative error of the estimated mode shape for the second bending mode of the aluminium beam with 80 scan points with the different techniques.
Table 1 Comparison of the attained measurement time reduction, the total measurement time for the aluminium beam with 80 scan points and the relative error of the estimated mode shape
5. Conclusions In this paper a technique was introduced where a certain sequence with known fixed frequencies was represented by a model using a set of sinusoids with coefficients left to estimation. The method, a regressive Fourier transform technique revealed an estimation for the unknown coefficients by means of a least-squares approach thereby utilizing only a portion of the ‘time’ signal. This gave way to an attractive application: reduction of measurement
Full scan Fast scan RDFT technique Spatially regressive technique a
Reduction factor
Measurement time T (s)
Mode shape error (%)
1 727 3 200
80 0.11 26.50 0.39
0 487a 0.02 0.03
Large error due to excitation coincidental with DFT-line.
time for scanning laser vibrometer measurements. Furthermore, a technique was presented allowing further measurement time reduction for scanning laser vibrometer experiments, using a spatial regressive technique. Simulations were performed on multisines in different frequency bands, different number of frequencies and for different sampling frequencies. Experiments on an aluminium beam showed that a single mode could be measured accurately with the regressive Fourier series approach three times faster with five excited frequency lines. The technique proved faster than a full scan and more accurate than a fast scan. The spatial regressive technique showed nearly the same measurement time as a fast scan but with much more reliable results. 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
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authors also acknowledge the Flemish government (GOAOptimech) and the research council of the Vrije Universiteit Brussel (OZR) for their funding.
References [1] Heylen W, Lammens S, Sas P. Modal analysis theory and testing. Katholieke Universiteit Leuven, Departement Werktuigkunde; 1997. [2] Halliwell N. Laser Doppler measurement of vibrating surfaces: a portable instrument. J Sound Vib 1979;62:312–5.
[3] Arruda JRF. Surface smoothing and partial derivatives computation using a regressive discrete Fourier series. Mech Syst Signal Process 1992;6(1):41–50. [4] Arruda JRF, Rio SAV, Santos LASB. A space–frequency data compression method for spatially dense laser Doppler vibrometer measurements. J Shock Vib 1996;3(2):127–33. [5] Ewins D. Modal testing: theory and practice. Research Studies Press LTD; 1986. [6] Guillaume P, Verboven P, Vanlanduit S. Frequency-domain maximum likelihood identification of modal parameters with confidence intervals. In: Sas P, editor. Noise and vibration engineering, vol. 1, 1998. p. 359–76.
Optics and Lasers in Engineering 45 (2007) 49–56
Reducing measurement time for a laser Doppler vibrometer using regressive techniques Joris Vanherzeele, Steve Vanlanduit, Patrick Guillaume Vrije Universiteit Brussel (VUB), Acoustics and Vibration Research Group (AVRG), Department of Mechanical Engineering (MECH), Pleinlaan 2, B-1050 Brussel, Belgium Received 12 January 2006; received in revised form 13 March 2006; accepted 21 March 2006 Available online 12 May 2006
Abstract In the last decade the laser Doppler vibrometer (LDV) has become a widely spread instrument for measuring vibrations. It often offers accurate measurements with a high spatial resolution. However, the measurement time of the LDV and especially for the scanning LDV is long. Therefore, reducing the measurement time is an attractive objective. A way to achieve this is to use a single sine excitation (on a resonance frequency). However, this technique has two major drawbacks: the inability to provide information on the damping and a operational deflection shape that can differ from the true mode shape. In this article two methods will be introduced to reduce measurement time without these defaults. In the first method introduced in this article a narrow band multisine is used as excitation signal and the measured vibration signal in the time domain is represented by a model using sines and cosines with these fixed narrow band frequencies. The coefficients of those sines and cosines are then estimated on a global scale by means of a least-squares estimator. An important advantage of this particular technique is that one does not have to measure a full period of the signal, reducing time. The second method accelerates the measurement time for scanning LDV measurements. Using the time domain sequence from each previous scan point and a limited number of time samples from the current scan point, the full time domain sequence of the current scan point can be estimated. Both these methods are a key benefit for in-line quality control, which can have upwards of 1000 spatial measurement locations. The proposed techniques will be validated on both simulations and experiments of varying complexity. r 2006 Elsevier Ltd. All rights reserved. Keywords: Scanning laser Doppler vibrometer; Measurement time reduction; Regressive Fourier series; Spatial regressive technique
1. Introduction Modal analysis has for some time now been an important tool in structural analysis in many different fields of application [1]. The measurements themselves were mostly executed using contact-based measurement systems such as accelerometers. They have, however, some disadvantages such as the cumbersome task of applying the accelerometers to the structure (sometimes several times when working in different patches) and of course there is always the mass loading aspect (important for lightweight structures). Recently however, with the ever ongoing progress in optical techniques some interesting alternatives have been developed. However, this also led to entirely new measurement Corresponding author.
E-mail address: [email protected] (J. Vanherzeele). 0143-8166/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2006.03.004
systems themselves such as the laser vibrometer, or scanning laser Doppler vibrometer (LDV) [2]. The LDV made it possible to increase the number of degrees of freedom (DOF) measured on the surface by two to three orders of magnitude. Where in the past one was limited to a few hundred measurement points it was now feasible to measure tens of thousands of points. However, measuring such high spatial resolution data takes a relatively long time (e.g. a simple plate with 1000 points and a frequency resolution of 1 Hz, will roughly take about 20 min). In short, there are two possibilities to execute a scanning vibrometer measurement with the existing commercial LDV-systems. The first mode is to perform a so-called ‘full scan’ where all frequency lines are excited with which a frequency response function (FRF) over the entire frequency range is obtained. This mode takes a considerable amount of time inversely proportional to the frequency resolution. The second is a fast scan mode where
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only one frequency line is excited (normally on a resonance frequency). The measurement time of this mode is of course proportional to the inverse of the imposed frequency and is therefore faster. However, it is not possible to estimate damping, for which at least two frequencies are needed, nor will the estimated deflection shape be exact and even in this mode at least one period of the frequency has to be measured. The only way to reduce the actual measurement time is to simply measure less of the vibration time signal for each individual point on the structure. However, this is, as already stated, limited by the period of that signal. For one always has to measure at least one entire period (or an integer number of periods) of the vibration sequence in order to avoid leakage as is the case with a discrete Fourier transform (DFT). In the first method introduced in this paper, the signal is approximated by a series of sines and cosines with fixed frequency. The frequencies in the measured signal are known a priori for linear systems, because excitation is a multisine with a narrow fixed frequency band, which makes it possible to model an entire signal period using only a portion of the period. In this paper a regressive discrete Fourier series (RDFT) technique [3,4] was used to estimate the sine and cosine coefficients, yielding an approximation of the vibration signal. Using the proposed technique on a time sequence, it is clearly possible to not only generate an estimate for the entire sequence but maybe even more importantly to reduce the measurement time. Theoretically, one can estimate a single frequency using only two points, however in practice this strongly depends on several parameters in the data. The second method presented in this article is a spatially accelerated regressive technique used to reduce measurement time specifically for scanning LDV measurements. The technique is based on the fact that the time sequences measured in each sequential scan point are basically the same and only differ by their amplitude. Therefore, it is possible to approximate the vibration signal in a particular scan point by means of a previous scan point by simply estimating the amplitude. This can be done by using a small number of time samples of each adjacent scan point, hence reducing the measurement time. In the following section the methods will be unveiled. In Section 3 simulations on the regressive Fourier series technique will be shown on different examples showing the capabilities towards high spatial resolution data and multiple frequency signals. In Section 4 both methods will be put to practice on measurements on an aluminium beam and finally some conclusions will be drawn in the last section. 2. Measurement time reduction using regressive algorithms 2.1. Measurement time reduction using a regressive Fourier series Before going into full detail on the method, a short insight into the methodology of a measurement with the
presented technique will be given. Using the regressive Fourier series technique, the method boils down to the following steps: Perform a full frequency band measurement on a rough scan grid to determine the frequency band of interest. Single out this frequency band in the FRF, putting all other frequencies to zero and inverse Fourier transform this obtaining a band limited time sequence. Use this band limited time domain signal to determine the minimal number of time samples needed for the regressive Fourier series. Excite the structure with a multisine signal containing the chosen frequency band, only measuring the determined minimal number of time samples for each scan point. Perform the estimation of the vibration signal using the measured time samples and the input frequencies in the regressive Fourier series. The method is based on representing the vibration time signal by a series of sines and cosines. The frequencies of these functions are known a priori as they are imposed on the object during testing. These frequency lines are not chosen at random but somewhere in the vicinity of a resonance frequency. The resonance frequency can be traced by e.g. performing a full scan test with frequency resolution f res and sample frequency f sample in a limited number of object points. By using just a limited number of frequencies F concentrated around the resonance frequency with begin frequency F b it will be shown possible to speed up the measurement process significantly in comparison to a full scan where all frequency lines are excited. The excitation signal used to generate this limited band signal is following multisine in the time domain: xðnÞ ¼
F X
Am sinð2pF m nÞ þ Bm cosð2pF m nÞ,
(1)
m¼1
where Am and Bm represent the respective input amplitudes of the sines and cosines with frequency F m ¼ ½F b þ ðm 1Þf res =f sample ; n ¼ 0; . . . ; N 1. The time sequence of the measured vibration signal yðnÞ can be written in the same way as the input signal but with different amplitudes C m and Dm yðnÞ ¼
F X
C m sinð2pF m nÞ þ Dm cosð2pF m nÞ.
(2)
m¼1
By simply measuring these time signals as is, the minimum measurement time is always restricted to at least one period. However by approximating the signal by a series of sines and cosines with a certain number of fixed frequencies the measurement time can be brought down to less than one period. Eq. (2) can be written as a linear matrix equation: a ¼ Bc,
(3)
ARTICLE IN PRESS J. Vanherzeele et al. / Optics and Lasers in Engineering 45 (2007) 49–56
where a, B and c are given by the following matrices: 1 0 1 0 C1 yð0Þ BD C C B B 1C C B yð1Þ C B C B B . C a¼B C; c ¼ B .. C, .. C C B B . A @ BC C @ FA yðN 1Þ DF 0 B B B B¼B B @
time even further with a regressive approach. To explain the principle, a modal model in the frequency domain [5] will be used from which the time sequence yðnÞ which is actually measured will be derived. The FRF H in a certain scan point i for one single mode can be written as fi , io l
H i ðoÞ ¼
sinð2pF 1 0Þ
cosð2pF 1 0Þ
sinð2pF m 0Þ
cosð2pF m 0Þ
sinð2pF 1 1Þ .. .
cosð2pF 1 1Þ .. .
sinð2pF m 1Þ .. .
cosð2pF m 1Þ .. .
sinð2pF 1 ðN 1ÞÞ
cosð2pF 1 ðN 1ÞÞ
sinð2pF m ðN 1ÞÞ
cosð2pF m ðN 1ÞÞ
The unknown coefficients C m and Dm are estimated (^c) by means of a least-squares approach taking the pseudoinverse of the matrix B and multiplying it with a c^ ¼ Bþ a.
51
(4)
It is obvious that for each frequency two coefficients have to be estimated so this means that for each frequency at least two equations are necessary. So this means that theoretically for a sequence yðnÞ an estimate can be derived using only 2F time points. Of course in practice this depends on the time resolution of the signal. Indeed, the closer two sequential points lie together the harder it becomes to fit a sine through them. Nonetheless, this clearly illustrates the possibility to reduce the measurement time for each individual frequency. 2.2. Measurement time reduction using a spatial regressive technique Using the spatially accelerated regressive technique the method can be summed up as follows: Perform a full frequency band measurement on a rough scan grid to determine the frequency band of interest. Single out this frequency band in the FRF, putting all other frequencies to zero and inverse Fourier transform this obtaining a band limited time sequence. Use this band limited time domain signal to determine the minimal number of time samples needed between the first scan point and its adjacent neighbor to estimate their amplitude relation. Excite the structure with a multisine signal containing the chosen frequency band, only measuring the determined number of time samples for each scan point. Reconstruct the full time domain sequence of each scan point using each sequentially determined amplitude relation and the adjacent scan point’s time sequence. When performing scanning laser Doppler measurements with L scan points it is possible to reduce the measurement
(5) 1 C C C C. C A
where o represents the angular frequency, fi is the mode shape amplitude at that certain point i and l is the system pole containing natural frequency and damping. Now it is well known that for uncoupled normal modes, the amplitude fi1 of the mode shape in a scan point i 1 is in direct linear relation to fi in a point i by fi ¼ ki fi1 ,
(6)
ki
denotes the amplitude relation of the mode shape where in two adjacent scan points i and i 1. Now in the time domain, the measured vibration sequence yi ðnÞ in each respective scanning point i on the structure can be written in exactly the same fashion as Eq. (6), revealing Eq. (7). Indeed, the input frequencies are the same so only the amplitude changes (and in some cases phase, but this is not considered here). This means that it is possible to estimate this amplitude of each sequential yi ðnÞ on the structure using the result of the previous scan point yi1 ðnÞ. The main advantage is since the multiplier ki is a constant value, it can be estimated using a limited number of time samples N k of the measurement yi ðnÞ yi ðnÞ ¼ ki yi1 ðnÞ,
(7)
with n ¼ 0; . . . ; N 1; i ¼ 2; . . . ; L and the ki denotes the amplitude relation of two adjacent scan point times sequences. An estimate of this factor k^i in Eq. (7) is given by the following least-squares equation: k^i ¼ yi1 ðnÞþ yi ðnÞ,
(8)
with n ¼ 0; . . . ; N k 1; N k 5N; i ¼ 2; . . . ; L. In short this means that the measurement time can be reduced by N=N k times. The minimal number of points necessary N k to perform the estimate in each scan point can be determined when performing the full frequency band measurement on the coarse scan grid. The obtained band limited time sequences in two adjacent time points on the structure can then be used to estimate the minimal number of points N k necessary to estimate their amplitude relation.
ARTICLE IN PRESS J. Vanherzeele et al. / Optics and Lasers in Engineering 45 (2007) 49–56
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This spatial regressive technique uses only N k points of the time sequence of each sequentially scanned point on the structure. In Section 4 it will be shown that this reduces the measurement time even further compared to using only the proposed regressive technique. 3. Computer simulations The simulations were carried out on a multisine signal x which made it possible to explore the effect of different parameters such as number of frequency components, sample frequency for different frequency bands. The following was investigated: How the bandwidth B (or equivalent the number of frequency components (F) in the multisine) influences the minimal number of time samples N k needed to estimate the sequence. How the sampling frequency f sample effects the number of data points needed to estimate the sequence. How noise effects these estimations. The frequency resolution f res ¼ 1 Hz for all simulations. Fig. 1 gives an example where the number of frequency components F was incremented with each iteration up to a maximum of 25 frequencies, starting from different begin frequencies F b . The multisine was sampled at f sample ¼ 500 Hz and the error value err (Eq. (9)) between estimate and true signal was set to 0.1%. The error value for the simulations with added noise was of course set above the respective noise levels. The measurement time was then determined by calculating the minimum number of time samples N k necessary to achieve this error. The time reduction is given by: time reduction ¼ 1 N k =N (in %) where N denotes the total number of time
samples. PN k 1 err ¼
n¼0
jðyest ðnÞ yðnÞÞj2 , jyðnÞj2
(9)
where yest represents the estimate of the signal y. The linear relation which can be derived from Fig. 1 for a small number of frequencies (that number increases with F b ), shows that only two points are needed for each frequency as is predicted theoretically. Above this value more data points are needed to obtain good conditioning. Also for the lowest begin frequency F b ¼ 1 Hz, a significantly lower time reduction is achieved in comparison with higher begin frequencies. For these higher begin frequencies F b , the time reduction in function of the number of input frequencies F is almost independent of F b . The sampling frequency is another important factor when trying to reduce the measurement time. By gradually increasing the sample frequency, it is clear that up to a certain point the measurement time is unhampered by the sample frequency for the case where F b ¼ 1 Hz (Fig. 2). After that the number of data points used must be increased to allow estimation. The relative number of points needed to perform the estimation stays constant for an increasing sampling frequency. Figs. 3 and 4 illustrate that for higher frequencies (F b ¼ 50 and 100 Hz, resp.) a low sampling frequency gives estimation problems due to aliasing, resulting in large spikes. Fig. 4 also shows that with a small number of frequency components (e.g. F ¼ 10) the theoretical minimum of two data points for every frequency is respected. It can be noted that the measurement time reduction for F ¼ 10 is far better than for a larger amount of frequency components at F ¼ 20. However in practice it is not necessary to use that many frequency lines to obtain a
100
90
F=20 F=15 F=10
95 90
80
time reduction (%)
time reduction (%)
100
Fb=400 Hz Fb=200 Hz Fb=100 Hz Fb=50 Hz Fb=25 Hz Fb=1 Hz
70
60
50
85 80 75 70 65 60 55
40 0
5
10
15
20
25
F Fig. 1. Time reduction (%) obtained for an increasing number of frequencies F in the multisine; frequency band with F b ¼ 1, 25, 50, 100, 200 and 400 Hz, respectively
50 50
100
150
200
250 300 fsample (Hz)
350
400
450
500
Fig. 2. Time reduction (%) obtained for sample frequencies between 40 and 500 Hz (multisine frequencies F ¼ 10, 15, 20, resp.; F b ¼ 1 Hz).
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100
100 F=20 F=15 F=10
90
-30 dB -20 dB -10 dB
80 time reduction (%)
95 time reduction (%)
53
90
85
70 60 50 40 30 20
80
10 50
100
150
200
250 300 fsample (Hz)
350
400
450
0
500
0
5
10
15
20
25
F
Fig. 3. Time reduction (%) obtained for sample frequencies between 40 and 500 Hz (multisine frequencies F ¼ 10, 15, 20, resp.; F b ¼ 50 Hz).
Fig. 5. Time reduction (%) obtained for multisine frequencies F incrementing up to 30 Hz; F b ¼ 1 Hz; noise level 10, 20 and 30 dB under signal level, respectively.
100 F=20 F=15 F=10
100 90
-30 dB -20 dB -10 dB
80 90
time reduction (%)
time reduction (%)
95
85
80
70 60 50 40 30 20
50
100
150
200
250 300 350 fsample (Hz)
400
450
500
Fig. 4. Time reduction (%) obtained sample frequencies between 40 and 500 Hz (multisine frequencies F ¼ 10, 15, 20, resp.; F b ¼ 100 Hz).
reliable prediction of the mode shapes, as will be shown in Section 4. Finally, the influence of noise on the examples above can be examined. When comparing the figures for the higher frequency band (Figs. 1 and 5) one can conclude that the overall appearance is quite similar, except that because of the added noise more data points are needed to perform the estimation. This was of course to be expected. Fig. 5 also shows that once noise is introduced to the signal, the actual level of the noise does not influence the estimates a lot. One can also note that as the number of data points used approaches the number of points in the true signal the curve becomes asymptotic. This is understandable because if one uses all points in the signal the estimation has to be exact.
10 50
100
150
200
250 300 fsample (Hz)
350
400
450
500
Fig. 6. Time reduction (%) obtained for increasing sample frequency between 40 and 500 Hz. F ¼ 10; F b ¼ 1 Hz; noise level 10, 20 and 30 dB under signal level, respectively.
The effect of the added noise to an increasing sampling frequency can be seen in Fig. 6. One can clearly see that the ‘flat line’ where the number of data points was constant for increasing sample frequency (Fig. 2) is no longer visible. A higher sampling frequency (f sample ) proves more sensitive to noise. 4. Experimental results In this section the regressive technique will be put to a practical test. A measurement was carried out on an aluminium beam, using the LDV, scanning along the center line in 80 scan points. The beam, suspended in
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free–free conditions was excited by a shaker placed at one far end of the beam Fig. 7. The excitation signal was a broadband multisine with sampling frequency f sample ¼ 2048 Hz and frequency resolution f res ¼ 1 Hz. This broadband scan revealed two distinct modes (Fig. 8). The number of averages was limited to one. Taking the full frequency band measurement (or full scan) and singling out one of the two modes, five frequency lines localized around the resonance frequency were used as input signal for the shaker and hence for the regressive estimation. The regressive technique was tried out on the second mode, with input frequency lines 725–730 Hz. The number of averages was limited to one. The problem is determining the measurement time necessary for an error value err no higher than 3%, a priori. Deriving an analytical expression for the number of data points needed is a cumbersome task at best because of
the large number of variables in the problem. Therefore, the best method of operation is to measure a complete period of the time signal in a single point and calculate the number of points needed. This test was performed when doing the full scan on the rough grid. This calculation can then be used as a reference value for the following points. The measurement time for the full frequency band was 80 s, with f res ¼ 1 Hz and 80 scan points. The measurement time for this full scan and the technique with a fixed number of excitation lines is principally the same because the frequency resolution is the same. By using the proposed regressive Fourier series technique (RDFT) (F begin ¼ 725 Hz; F ¼ 5) the measurement time for the second mode was reduced by a factor 3 for each measurement scan point totalling 26.50 s ðerrp3%Þ. The modes were calculated from the FRF-matrices using a classic maximum likelihood approach [6], for which the stated error value obviously has little to no effect. Fig. 9 shows a comparison between the mode shape estimated with the full scan data and the RDFT with measurement time reduction. The calculated mode shape using the RDFT clearly fits the full scan mode very well. An estimate of the second mode shape can also be generated using the fast scan method. This is done by exciting just one frequency line on the resonance frequency. Of course, the choice of frequency line is touchy because of the deviations due to the estimation method, or a resonance which might have a tendency to shift through time. Also, trouble will be encountered when two modes are positioned closely together in the frequency domain (e.g. strongly coupled modes). A fast scan will not be an optimal measurement method in this case, which is reflected in the estimated operational deflection shape (ODS). On the other hand, the fast scan offers a very fast
Fig. 7. Test set-up.
1400 1200
60 1000
40
800
Ampl
50
Ampl (dB)
30
600
20 400
10 0
200
-10
0 10
-20 -30 0
200
400
600 800 Frequency (Hz)
1000
Fig. 8. Broadband scan of the aluminium beam.
1200
20
30
40 L
50
60
70
80
Fig. 9. Second mode of an aluminium beam using the entire frequency band (solid line) and the regressive Fourier series technique (F begin ¼ 725 Hz; F ¼ 5) with measurement time reduction (asterix line); absolute values.
ARTICLE IN PRESS J. Vanherzeele et al. / Optics and Lasers in Engineering 45 (2007) 49–56
1500
55
1400 1200 1000
Ampl
Ampl
1000
800 600
500 400 200 0 0
10
20
30
40 L
50
60
70
0
80
10
20
30
40
50
60
70
80
L
Fig. 10. Second mode of an aluminium beam using the entire frequency band (solid line) and the fast scan technique (F b ¼ 727 Hz; F ¼ 1) (asterix line); absolute values.
Fig. 11. Second mode of an aluminium beam using the entire frequency band (solid line) spatial regressive technique (F b ¼ 725 Hz; F ¼ 5) (asterix line); absolute values.
measurement sequence, using 727 Hz as input frequency line for the measurement. The measurement time for the fast scan was 0.11 s. In this case, the fast scan is obviously faster than the regressive technique but in this case the ODS will differ quite a bit from the mode shape obtained from a full scan. In this particular case, the estimated ODS differs significantly because the excited frequency line was chosen coincidental with the DFT-grid. Fig. 10 offers a comparison between the full scan mode and the deflection shape estimated with a fast scan. When comparing these measurement times, the fast scan proves to be the fastest technique but also with the lowest accuracy. Using the spatial regressive technique it is possible to reduce the measurement time with respect to the full scan for each scan point by about 200 times ðN k ¼ 10Þ. The measurement time for this mode was 0.39 s, making it almost exactly as fast as the fast scan technique, but with a far greater accuracy level. The mode shape estimated with the spatially accelerated regressive technique compared with the full scan is shown in Fig. 11. Table 1 shows a comparison of the measurement times and the relative error of the estimated mode shape for the second bending mode of the aluminium beam with 80 scan points with the different techniques.
Table 1 Comparison of the attained measurement time reduction, the total measurement time for the aluminium beam with 80 scan points and the relative error of the estimated mode shape
5. Conclusions In this paper a technique was introduced where a certain sequence with known fixed frequencies was represented by a model using a set of sinusoids with coefficients left to estimation. The method, a regressive Fourier transform technique revealed an estimation for the unknown coefficients by means of a least-squares approach thereby utilizing only a portion of the ‘time’ signal. This gave way to an attractive application: reduction of measurement
Full scan Fast scan RDFT technique Spatially regressive technique a
Reduction factor
Measurement time T (s)
Mode shape error (%)
1 727 3 200
80 0.11 26.50 0.39
0 487a 0.02 0.03
Large error due to excitation coincidental with DFT-line.
time for scanning laser vibrometer measurements. Furthermore, a technique was presented allowing further measurement time reduction for scanning laser vibrometer experiments, using a spatial regressive technique. Simulations were performed on multisines in different frequency bands, different number of frequencies and for different sampling frequencies. Experiments on an aluminium beam showed that a single mode could be measured accurately with the regressive Fourier series approach three times faster with five excited frequency lines. The technique proved faster than a full scan and more accurate than a fast scan. The spatial regressive technique showed nearly the same measurement time as a fast scan but with much more reliable results. 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
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authors also acknowledge the Flemish government (GOAOptimech) and the research council of the Vrije Universiteit Brussel (OZR) for their funding.
References [1] Heylen W, Lammens S, Sas P. Modal analysis theory and testing. Katholieke Universiteit Leuven, Departement Werktuigkunde; 1997. [2] Halliwell N. Laser Doppler measurement of vibrating surfaces: a portable instrument. J Sound Vib 1979;62:312–5.
[3] Arruda JRF. Surface smoothing and partial derivatives computation using a regressive discrete Fourier series. Mech Syst Signal Process 1992;6(1):41–50. [4] Arruda JRF, Rio SAV, Santos LASB. A space–frequency data compression method for spatially dense laser Doppler vibrometer measurements. J Shock Vib 1996;3(2):127–33. [5] Ewins D. Modal testing: theory and practice. Research Studies Press LTD; 1986. [6] Guillaume P, Verboven P, Vanlanduit S. Frequency-domain maximum likelihood identification of modal parameters with confidence intervals. In: Sas P, editor. Noise and vibration engineering, vol. 1, 1998. p. 359–76.