CORRELATION ANALYSIS OF AUTOMOBILE CRASH RESPONSES USING WAVELETS

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1 CORRELATON ANALYSS OF AUTOMOBLE CRASH RESPONSES USNG WAVELETS Zhiqing Cheng, Walter D. Pilkey, Kurosh Darvish, William T. Holowel, and Jeff R. Crandall Department of Mechanical and Aerospace Engineering University of Virginia Charlottesville, VA 22904, USA *National Highway Traffic Safety Administration 400 7th St. SW, Washington, DC 20590, USA ABSTRACT Wavelets are used to analyze automobile crash responses. Crash signals are decomposed into a wavelet basis, which provides an intuitive vision of impact behavior of the vehicle structure. The decomposed signals are further divided into segments that represent vibrations occurring in certain time spans. A correlation analysis is then performed on the decomposed and segmented signals, in order to determine the relationship between the dynamic behaviors of different structural components, particularly the relationship between the short-time vibrations in different parts. The structural responses in a full frontal impact test are analyzed. The information obtained is useful for the refinement and validation of a finite element automobile crash model. 1. NTRODUCTON A finite element model of a four-door 1997 Honda Accord DX Sedan was created at the Automobile Safety Laboratory of the University of Virginia [1,2]. The model was developed from data obtained from the disassemby and digitization of an actual automobile using a reverse engineering technique. This was a part of a series of efforts to develop detailed, dynamic finite element (FE) models of vehicles representing today s highway fleet. The model was to be validated with test results of actual vehicles. This requires that the impact dynamic responses of the model be compared with test signals recorded from prescribed locations on the actual test vehicle structure. During the model refinement and validation, it is necessary to know how critical structural components behave and what is the relationship between the dynamic responses at different locations of the vehicle. Conventionally, the relationship between two signals can be determined using a correlation analysis [3]. However, automobile crashes, such as car-to-car impacts and car-tobarrier impacts, occur in very short time durations. Due to stress wave propagation effects, the automobile crash responses are transient and strongly localized in the time domain. n physical automobile impact tests, the measurements of crash responses are often contaminated with noise. As a consequence, the conventional correlation analysis used for stationary signals may not be appropriate or efficient for the analysis of automobile crash signals. nstead, as a new tool for signal analysis, wavelets are introduced. Wavelets are localized in both frequency and time domains, which matches with major characteristics of automobile crash signals. 2. DECOMPOSTON AND CORRELATON ANALYSS Wavelet analysis consists of decomposing a signal into a hierarchical set of approximations and details. Since the original signal s(t) is obtained through data acquisition with a sampling rate At, it can be considered as the approximation at level 0, denoted as 4, with the time scale At,, = At, (1) The hierarchical decomposition of the signal s(t)in a wavelet basis can be expressed as [4,5] S=A,+D, =A,+D,+D, =... = A, +zjgdj (2) (3) 1166

2 and Dj(t)=zd,yj(t-k). k Here, Aj is the approximation the detail of S at the level j the constants (4) of S at level j, and D, is Given by wavelet transforms, cjk and d,, are referred to as scaling function coefficients and wavelet coefficients, respectively. The functions tij (t) and,!/, (t) are the scaling function and the wavelet at level j for reconstruction, respectively. f orthogonal wavelets are used, the hierarchical decomposition of Eq. (2) is orthogonal. That is,. AJ is orthogonal to D,, DJm,, D,-,,...,. Dj is orthogonal to D, for j # k Suppose two signals x(t) and y (t)are from stationary and ergodic random processes. The correlation function coefficient may be defined by P,(4= R LrJ R&-wy ~[R_(O)-~51[R,,.(o)-~:l pu, and,ll~ are the mean values of X(Z) and y(t), R,, and Rw are the auto- and cross-correlation functions of x(t) and y(t), respectively. (5) and equations, the total length of time is divided into several sub-intervals in which a decomposed signal can be considered as stationary. That is,, without the loss of generalty, A,m(t)=~S,4(-m) nl=, The correlation analysis can be performed between the segments of the two signals at each level. That is, superscript j, denotes the m-th segment at level j (9) (10) The coefficient pi; describes the correlative relationship between the segments of the two decomposed signals. n general, for orthogonal wavelets, relations: there are the following T, is the observation time or the length of a record. f the approximations and details of two originally nonstationary signals become stationary after decomposition at a certain level, the correlation analysis between these two signals can be performed between the details or approximations of the two signals at each level using the above definitions and equations. That is, Pgd = ky Cd- P1Pj ~[R~(0)-(~~)21[R~~(0)-(~~)21 the superscript j denotes the j-th level. The coefficient pcy represents the correlative relationship between the two signals at different time scales or frequency ranges. This information is frequently required in engineering practice. However, it is possible that the decomposed signals (details or approximations) at each level are still not stationary. For instance, as shown in Figs. 1 to 8, neither the approximations nor the details of the automobile crash responses in a frontal impact are stationary. n order to use the above definitions (6) (7) which can be used in the calculation of the coefficients of p*i, and p$ 3. ANALYSS OF AUTOMOBLE CRASH RESPONSES The structural crash responses in a full frontal impact test of a two-door 1997 Honda Accord (Number 2475 in the vehicle crash test database of the National Highway Traffic Safety Administration) will be investigated. n this test, the accelerometers were placed at the locations of the top and bottom of the engine, the left and right brake calipers, the center of the dash panel, and the right and left positions of the rear floor. When a crash signal is decomposed in a wavelet basis, it is important to choose an appropriate type of wavelet and right level of decomposition. n this paper, the fifth order of Daubechies wavelet, db5, is selected [6], since it is orthogonal, which is essential to the orthogonal decomposition, and compactly supported with the length appropriate for the crash signals being analyzed. The maximum decomposition level is determined such that at that level, the approximation of an acceleration response basically represents the gross motion of the corresponding 1167

3 structural component. For instance, as illustrated by Fig. 1, if the acceleration response of the left rear floorpan is decomposed into 6 levels, the approximation at level 6 is not zero initially, which does not match the gross motion of the rear floorpan that should be at rest initially during the impact. The maximum level of 5 is found to be appropriate for the decomposition of the signals from this test. As the original signals are sampled with the sampling rate of 0.1 ms, the time scale of level 5 is 3.2 ms, roughly corresponding to the frequency range from 0 to 120 Hz, if the cut-off frequency f,, is calculated by f,,,, = l/(2.56. At). (12) n a conventional analysis, automobile structural acceleration responses are filtered with SAE-60 filter for which the cut-off frequency f,,, =loo Hz [2]. Therefore, the approximation at level 5 basically matches with the filtered signal with SAE-60 filter in terms of frequency range. Using db5, the crash responses are decomposed at five levels S=A,+D,+D,+D,+D,+D,, (13) as shown in Figs. 2 to 8. From these figures, it can be seen that the approximation of a signal at level 5,!&, is not cyclic and thus describes while the gross motion acceleration of a part, D, to D,, the details at different levels, are cyclic and thus can be considered as vibrations in different frequency ranges, These vibrations are superimposed on the gross motion. Note that lower case letters are used for the labeling in the figures with Wavelet Toolbox in MATLAB. While the gross motions (approximations) of structural components may be related to each other, our interest is in the relationship between high frequency components. Therefore, the correlative relationship between the details at each level is investigated. Several trial correlation analyses are performed between the details for the entire length of record. The absolute values of these correlation function coefficients pi, are very small, which indicates that in terms of the entire length of record, decomposed signals are not closely related. n fact, Figs. 2 to 8 show that the details at level 1 to 5 are still localized in the time domain. However, decomposed signals are probably related to each other in certain time spans. Therefore, the details at each level are further divided into segments that are defined in subintervals of time. For ease in automatic computation, the length of each subinterval is identical. A logical selection of this length is to use the support length of wavelets at each level. Figure 9 illustrates the support lengths of db5 at levels 1 to 5. As such, the subinterval length is given by T, = (2N, - l)atj, (14) N, is the order of db wavelets, and Atj is the time scale of level j Ati =2 At. (15) The segments, denoted as D,, can be overlapped, jointed, or separated from each other. n this paper, the segments at level j are overlapped with the shift of Atj That is, Djm (t) = Dj(t), t E [(m - l)atj,(m 0, otherwise - l)atj + Tj] (16) m = 1,2,..., Mj, and 9, the number of segments at level j, is given by M,=qmTj / 7+1=2- Ns-2(Nw-1), (17) i N, is the number of points of an original signal When the correlation function coefficient P%(Z) is calculated, the time shift z is allowed to vary in a small range with the stepsize Atj, with the consideration time required for a signal (stress wave) the structure. A quantity of the to transmit through p$ = mm, fd; (r)}, z E [.... -Atj,O,Atj,... 1, (18) is used to measure the the correlative relationship between a segment (a short signal) of the signal x(t)and its counter part of the signal y(t)(with the same length and at about the same time). The correlation function coefficients of the automobile crash responses are calculated, with the absolute values being plotted in Figs. 10 to DSCUSSON The decomposition of a signal in a wavelet basis clearly illustrates the composition of the signal at different time scales, including the amplitude and time location of a component. The energy distribution of a signal over different time scales can be determined easily. For the automobile crash responses being analyzed, the approximation at level 5 basically represents the gross motion of a structural part during impact, while the details at level 1 to 5 describe vibrations of the part that occur in certain time spans and frequencies. Figures 10 to 15 display correlation function coefficients that are calculated over the entire record length. However, in practice, whether a major impact pulse is transmitted through the structure from one part to another is concerned. Thus, only those vibrations with large amplitudes are important. From the decomposition of a crash response (Figs. 2 to 8), significant vibrations and their time spans can be identified. From the correlation analysis of this response with other responses (Figs. 10 to 15), the absolute values of 1168

4 the correlation function coefficients can be obtained. High values (say, above 0.7) indicate that the vibrations occurring in these parts during these time spans are likely related. Based on the decomposition and the correlation analysis, several observations can be made on the dynamic hehavior of the vehicle structure in this particular crash. n terms of the gross motion, structural responses are fairly symmetrical. This can be seen from the approximations at level 5 for the left and right brake calipers (Figs. 4 and 5) and for the left and right rear floorpan (Figs. 7 and 8). For the rear floorpan, the vibrations that occurred around 20 ms on the left and right positions at levels 2 to 5 seem related. However, the vibrations that occurred at about 40 ms on the left at levels 1 to 3 are not related to those on the right (Figs. 7, 8, and 11). Figures 2, 7, and 13 indicate that the major vibrations that occurred during 25 to 45 ms at the bottom of the engine may have not transmitted to the right rear floorpan. The vibrations that occurred during 20 to 30 ms on the engine top seem to correlate with the vibrations of the dash panel at the same time (Figs. 3, 6 and 14). Vibrations of the dash panel and the left rear floorpan that occurred around 20 and 60 ms are correlated at certain levels (Figs. 6, 8, and 15). will be used as a crash partner for corresponding new generation vehicles developed by automobile manufacturers. REFERENCES [l] Thacker, J.G., Reagan, SW., Pellettiere, J.A., Pilkey, W.D., Crandall, J.R., and Sieveka, E.M., Experiences During Development of a Dynamic Crash Response Automobile Model, Finite Element in Analysis and Design, 30 (1998), [2] Cheng, Z.Q., Thacker, J.G., Sieveka, E.M., Reagan, S.W, Pilkey, W.D., and Crandall, J.R., Experience of Modification and Validation of a Finite Element Automobile Crash Model, Finite Element in Analysis and Design, to appear. [3] Bendat, J.S. and Piersol, A. G., Engineering Applications of Correlation and Spectral Analysis, John Wiley & Sons, New York,1993. [4] Strang G. and Nguyen T., Wavelets and Filter Banks, Wellesley-Cambridge Press, Massachusetts, [5] Misiti, M. and Misiti, Y., Wavelets Toolbox User s Guide, The MathWorks, nc., Massachusetts, 1997 [6] Daubechies,., Ten Lectures on Wavelets, The Society for ndustrial and Applied Mathematics, Pennsylvania, A comprehensive analysis is limited by the availability of the test data. To determine the structural dynamic bahavior in more detail, acceleration responses at more locations, especially in major loading transmission paths, need to be acquired. 5. CONCLUDNG REMARKS A wavelet analysis can provide valuable insight into automobile crash responses. The decomposition of crash signals in a wavelet basis produce intuitive vision of the dynamic responses of the vehicle structure in a crash event. Through the correlation analysis performed on the decomposed and segmented signals, the relationship between the dynamic responses of critical structural components can be determined. The information obtained from the decomposition and the correlation analysis is important and useful for the refinement and validation of a FE automobile crash model. t is recognized that there are limitations on the applicability of the correlation analysis, especially for automobile crashes nonlinearities, mulitple imputs, and multiple paths are involved in the structural responses. ACKNOWLEDGEMENTS The modeling work of a four-door 1997 Honda Accord DX Sedan was sponsored by the US National Highway Traffic Safety Administration (NHTSA). The finite element model Y 1 d : 2 d -lo rm m sm Em 7m BM 9M mo Figure 1 Acceleration response of the left floorpan decomposed at level

5 v u, ma m x m em?m Km 9(10 Km Figure 2 Acceleration response at the bottom of the engine Figure 4 Acceleration response of the right brake caliper.-. Km zm a0 m 5m Em 7m am om raa Figure 3 Acceleration response on the top of the engine Figure 5 Acceleration response of the left brake caliper 1170

6 M am 3m 400 Em MO 7m Km 900 rm Figure 6 Acceleration response at the dash panel center -2 i M m a0 400 soa uw x rmo Figure 8 Acceleration response at the left rear floorpan 2p 5 O 15 al Figure 9 The support length of dh5 at each level Figure 7 Acceleration response at the right rear floorpan 1171

7 Correlation analysis at each level Figure 10 Correlations between the bottom and top of the engine Figure 13 Correlations between the engine bottom and the right rear floor d%, 1 0 Correlation analysis al each level Correlation analysis at each level no0 600 Figure 11 Correlations between the right and left rear floor Figure 14 Correlations between the engine top and the dash panel center Correlation analysis at each level 1. -.L Figure 12 Correlation between the right and left calipers Figure 15 Correlations between the dash panel and the left rear floorpan 1172