. i SAND 972283C. SALVAGNG PYROTECHNC DATA WTH MNOR OVERLOADS AND OFFSETS David. Sallwood and Jeroe S. Cap Sandia National Laboratories P.O. BOX8 MS86 Albuquerque, NM 87 8 BOGRAPHY David Sallwood received his BSME degree fro New Mexico State University in 962 and his FASME degree fro New York University in 964. He has worked for Sandia National Laboratories since 967 and is currently a Distinguished Meber of the Technical Staff in the Environents Engineering Group of the Mechanical and Theral Environents Departent at Sandia. He is a fellow of the EST. Jeroe (Jerry) Cap received his BSME fro The Pennsylvania State University in 98, and his MSME fro the sae institution in 982. He has worked for Sandia National Laboratories since 983 and has worked in the area of shock and vibration analysis since 989. ABSTRACT We are soeties presented with data with serious flaws, like saturation, overrange, zero shifts, and ipulsive noise, including uch of the available pyrotechnic data. Obviously, these data should not be used if at all possible. However, we are soeties forced to use thcse data as the only data available. A ethod to salvage these data using wavelets is discussed. The results ust be accepted with the understanding that the answers are credible, not necessarily correct. None of the ethods will recover inforation lost due to saturation and overrange with the subsequent nonlinear behavior of the data acquisition syste. The results are illustrated using analytical exaples and flawed pyrotechnic data. KEYWORDS pyrotechnic, shock, wavelets, shock response spectru NTRODUCTON We are soeties presented with data with inor flaws3 caused by aliasing and ore serious flaws, like overloads, zero shifts, and ipulsive noise (spikes or dropouts), including uch of the available pyrotechnic data'.'. Obviously, the later data should not be used if at all possible. When presented with flawed data, the best procedure is to reject the data and collect a ore valid FFR 998 data set'. Other authors' have shown the risks involved if the data are accepted. We are soeties forced to use this data as the only data available and it is not possible to gather ore data. The data ay have been gathered as a one tie experient that can not be repeated, or additional experients would be prohibitively expensive. n this paper we will be discussing acceleration data, typically pyrotechnic data, but the ethods could be applied to other data sets. The shock response spectru (SRS) is a coon tool for the specification of pyrotechnic environents. f the flawed data are used, serious errors in the shock response spectru result. These errors are usually the prediction of uch higher responses at the low frequencies than the true environentjustifies. For this reason the flawed data ust be corrected if the data is used in a specification. The inor errors caused by aliasing can also lead to serious errors in the nuerically integrated velocity and displaceent, which can cause probles with wavefor reproduction progras on shaker systes. Earlier papers showed how soe of the data can be salvaged using a paraetric for for the corrections3". The purpose of this paper is to show that wavelets can also be used to salvage the data. Using these ethods requires judgent, and the results ust be accepted with the understanding that the answers are credible, not necessarily correct. None of the ethods will recover inforation lost due to overloads or nonlinearities of the data acquisition syste. The best that can be accoplished is the recovery of data after the data acquisition syste has recovered fro the overload. The ethods require assuptions on the characteristics of a credible data set and a odel of the corrections. The authors374 described a series of corrections which could be ade to correct data with inor flaws. n the investigation of soe data with ore serious flaws it was found that these tools did not work very well. This paper will discuss a new correction technique based in wavelets. CORRECTON WTH WAVELETS Wavelets are a new ethod (about years old) for analyzing data. Unfortunately space will perit only a Sdia is a uhiprogra laboratory operated by Sandia Corporation, a Lockheed Martin Copany, for the United States Departent of Energy under contract DEAC494ALS.
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SAND 972283C 2 /27/98 brief description of wavelets. A good introduction is contained in an article by Strang' and a thorough discussion is contained in a book by Strang and Nguyen6. Wavelets are a powerful new,way to look at transient data. Basically a function is described as a cobi,nationof basis functions (wavelets) An inverse exists: if you know the coefficierits bjk, you can reconstruct the original wavefor. A typical wavelet is copressed in tie and shifted wjk( t ) = 2j'2~ ( 2t ' k ) Noralized wavelet on [kat, ( k Figure Daubechies 3 wavelet function, psi +N)At] The wavelet, w, often has a quite coplicated shape, but is copletely described by a pair of filter!;: a high pass filter and a low pass filter. A nuber of useful wavelets have been defined, and one of the probles of a user is to pick the ost appropriate wavelet for a particular application. The wavelets typically have copact support. Siply, this eans that the wavelets are nonzero over a short finite period of tie. n practice the continuous for of the wavelet, Eq. (l),is seldo used, we alost always are dealing with finite wavefors sapled at discrete ties. nsz At = l,a = 2 ', b = k2', j E N, k E Z (3) The advantage of wavelets is that they provide a description of a wavefor that is localized in both tie and frequency. Although, users of wavelets don't like to talk about frequency, they use the ter level instead. This should prove useful for the task of this paper, since the flaws in the data are typically localized in tie and frequency. Soe of the wavelets are ebers of a faily of wavelets, identified by the order nuber. n general, the higher order wavelets provide better frequency localization at the expense of tie localization. The wavelet chosen for this paper was the Daube chies wavelet of order 3, abbreviated as the db3 wavelet in the MATLAB Wavelet Toolbox. The basic wavelet is plotted as Figure. The procedure used for this paper is alost the inverse of denoising of signals as described in the Wavelet Toolbox User's Guide7. n this case the "noise" is the shock we want to keep. The "signal" is the data correction we want to reove. For this paper, we used a level correction. We decoposed the signal using levels of decoposition. Each level of decoposition decoposes the signal into two bands, with a highpass filter and a low pass filter. The output of the high pass filter is called the detail, and the output of the low pass filter is called the approxiation, The approxiation and the detail are deciated by two and the process is repeated for the approxiation for the next level. The detail is not processed again. Rearkably, the desired wavelet coefficients are the deciated detail. At first glance it would appear the deciation of the detail would cause serious aliasing probles, but this is not the case. The two filters are carefully constructed such that the inforation lost in the deciation of the detail is retained in the approxiation. Exact reconstruction is possible fro the approxiation and details. Since each level of approxiation has about half the bandwidth of the previous level, the upper frequency liit of the nth level of approxiation is about f, / 2"+l, where f, is the sapling frequency. Thus the level chosen provides frequency localization. n this case the approxiation contains the low frequencies and the detail contains the high frequencies. We also need localization in tie, since the locations in tie of the errors in the original wavefor are not known. The wavelet transfor inherently provides the localization in tie. To suarize, the corrected acceleration was reconstructed fro the n levels of details (crudely the high frequency coponents). The correction was reconstructed fro the last level of approxiation (roughly the lowest
4 SAND 972283C 3 frequencies). The su of the correction (the approxiation) and the corrected wavefor (the details) will be the original wavefor. RESULTS The first two exaples will look at flaws associated with the nuerical integration of wavefors with inor aliasing errors. The first exaple is the su of a Hz and a 4 Hz exponentially decaying sinusoid saples at 2, saples/second fro Sallwood3. The analytical acceleration, velocity, and displaceent arc presented as Fig. 2. Figure 3 shows the wavefor integrated with a rectangular rule. The velocity wavefor which shows an offset and the displaceent clearly are in error, which is caused by alaising3. The intent of the correction is to change the acceleration in such a anner that the nuerically integrated velocity and displaceent are near the analytical wavefors. The correction and the corrected wavefors are shown as Figs. 4 and. As can be seen the corrections to the acceleration are sall, about parts per thousand, and isolated to the first hundredth of a second. The resulting corrected velocity and displaceent are nearly correct. The second exaple is also fro Sallwood3. The wavefor is the su of three exponentidly decaying sinusoids,, 3, and 4 Hz, sap;ed at 2, saples/second. A delay between the 3 and 4 Hz sinusoids akes this a ore severe test of h e correction ethods. The analytical wavefors (the correct result), the nuerically integrated wavefor (showing the nuerical integration errors), the correction using wavelets, and the corrected wavefors are shown as Figs. 69. For this exaple the corrected acceleration is very siilar to the analytical wavefor. The corrections to the acceleration are about 3 parts per ten thousand. The corrected velocity and displaceent show inor errors. The third and fourth exaples are fro Srnallwood and Cap4. These wavefors are two pyrotechnic: shocks with ore serious flaws. The saple rate was 2, saples/second. For these wavefors a credible result will be acceleration, velocity and displaceent wavefors which are oscillatory. The initial and final values of the acceleration, velocity and displaceent should all be near zero. The previous paper gave credible corrections for these wavefors using a paraetric filter. The easured acceleration and the nuerically integrated velocity and displaceent are shown as Figs. and 3. The velocity of x shows erroneous behavior at large ties, which /27/98 causes large displaceents as tie increases. The velocity of h shows an alost step change in velocity. The wavefor corrections are shown as Figs. and 4. The corrected wavefors are shown as Figs. 2 and. The bandwidth of the correction is approxiately 2,/26 or 3 khz. The corrected wavefors are credible for the acceleration, velocity, and displaceent. The agnitudes of the corrections (a significant part of the agnitude of the original wavefor) raise significant questions about the validity of the corrections. These corrections are larger than the corrections fro the previous paper. However in defense of the wavelet technique, the peak agnitude of the acceleration is not changed very uch. The effects on the shock response spectru (SRS) are shown in Fig. 6. The effects on the SRS are very siilar to the effects on the SRS fro the previous paper. As can be seen the SRS is essentially unchanged except at the low frequencies. The SRS of both x and h are, now siilar at low frequencies, a credible result. To suarize, the agnitude of the corrections are larger and have ore high frequency content than the corrections fro the previous paper, but the results on the SRS are very uch the sae. The fifth and sixth exaples are a different pyrotechnic shock test sapled at, saples/second. These two wavefors also have serious flaws. These were separation shocks of a payload fro a bus. Unfortunately, as is often the case, no credible easureents are available for coparison. The easureents will be called B3 and B3. B3 was located near one of the explosive bolts in the separation syste. B3 was a few inches away. Other useful easureents were ade further fro the explosive bolt, but are not useful for this discussion. The original data together with the nuerically integrated velocity and displaceent are shown Figs. 7 and 2. B3 shows a sall zero shift in the acceleration and other possible errors which result in erroneous velocity and displaceent wavefors. B3 shows an even larger zero shift in the acceleration. The correction ethods fro the previous work4 did not give pleasing results. An attept was then ade to correct the data with wavelets. B3 was corrected with the wavelet transfor as described the previous section. The correction and the corrected wavefor are shown as Figs. 8 and 9. The bandwidth of the correction is about.6 khz. The correction has a peak of a little over 3 g (about?4of the peak aplitude of the original wavefor). The correction is confined priarily to about sec. The agnitude of
SAND 972283C 4 the correction raises serious quesb.s about correction. The corrected acceleration and velocity look credible. The displaceent still appears to have a little too uch low frequency oscillation. The correction and the corrected wavefor!; for B3 are shown as Figs. 2 and 22. The agnitude of the acceleration correction is about!4 of the peak aplitude of the original wavefor. A zero shift is clearly present in the original data and is also present in the correction. The effects on the SRS are shown in Figs. 23 and 24. The SRS was calculated for the original wavefor, and the wavelet corrected wavefor. The effwts of the corrections on the Fourier spectru are shown as Figs. 2 and 26. As can be seen the corrections priarily affected the low frequencies. CONCLUSONS Wavelets are a robust ethod to correct flawed acceleration wavefors. The corrections :ire easier to apply with fewer subjective choices than the ethods of the previous paper. The ethod produced credible data sets for a larger set of exaples than the previous paper. The data sets used in the exaples of this paper cover the range fro inor corrections to corrections which should probably not be used. f used with care, the corrections discussed can produce credible data sets fro data with inor flaws. Since the corrections can also produce credible data sets for seriously flawed data, it is iportant to preserve the original wavefor, the correction wavefor, as well as the corrected wavefor. The agnitude, duration and frequency content of the correction can serve as a guide to the validity of the correction. f the agnitude of the correction is a substantial fraction of the agnitude of the original wavefor the corrections should be viewed with great caution. The duration and frequency content of the correction should be consistent with a. reasonable correction. The corrected data sets ay underestiate the environent at high frequencies, but shoilld result in reasonable estiates in the id frequencies ( khz,for an SRS with an upper usable frequency of khz). The corrections prevent the gross overestiates of the SRS at the low frequencies (below khz) which are coon in flawed pyrotechnic data. The intent is not to hide the data flaws. The corrected data sets should never be placed in a data bank without references to the original data set, and a clear explanation of the correction ethod used. /27/98 and 4 Hz eqnentially decaying sinusiod, SR=2 < c $....2?.2....2.2.3.3.4.2.3.4.3. f t alv,....2.2.3.3.4 Figure 2 Wavefor coposed of the s u of a and 4 Hz exponentially decaying sinusoids " S and 4 Hz eonentialhr decavina sinusiod, SR=2 $ 8 '.2.2' n4..........2.2.3.3.2.2.3.3.4.2.3.3.4.2.4 Figure 3 Wavefor of Fig. 2 integrated with rectangular rule " S Correction using wavelet db3 x " $ 8....2.2.'...... d&&"oo.3.3.4.4.2.2.3.3.2.2.3.3.4 Figure 4 The wavelet correction for the wavefor of Fig. 3.
S A N D 972283C /27/98, Corrected using wavelet db3 :..2,,.2; $ e Q '.b.2....2.2 o d.3 *...3.4.3.4.3. zk '. *..3.3 :.2.3.4..2 3.4 d 2..2.3.4. i d O..2.3. B..p d.3..4..4. Z Figure 8 The wavelet correction of wavefor of Fig. 7 Corrected usina wavelet db3...2.3.4..;..2.3.4. dr2..2.3.4 W. Figure 9 The wavelet corrected wavefor of Fig. 7 X.4 d&.&...2.2.b3 "._....2.2.b3....2.2.3 "". 7 3$..2.3.4 "" O2_i. n i.4 ntegrated wlth the rectangular tule lo.l.3.2.. Figure 6 A wavefor coposed of the su of, 3, and 4 Hz exponentially decaying sinusoids.2.4 :. d4 O.2.2. 2..A d 3 8 4 Hz exponentially decaying sinusiod, SFk2 ; Figure The corrected Fig. 3 wavefor ; :P 'E2 B Correction using wavelet db3 x O..2.3. Figure 7 Wavefor of Fig. 6 integrated with rectangular rule. i=. no.2 Lo s. Figure The wavefor called x, with the nuerically integrated velocity and displaceent
6 SAND 972283C wavelet x correction 2.&...b q O.dO...2 e. wavelet h correction, 28d $$OB /27/98.2.3 8p ' '....2.2.3 " L.2?8%.3 4....2.2.3.W...2.2.3 //....2.2..3?: Figure The wavelet correction to x Figure 4 The wavelet correction to h wavelet corrected x wavelet corrected h t! P Ooo $ow t 8.... R 6.b.2..3....2.2.b3 q.2.2.3 :8&. k...2.2.3..2.2.3 do... 2.2!3 O.dO. SRS of x and h daping = % '. Figure The wavelet corrected h h. Figure 2 The wavelet corrected x va~ %$~.2 d.2.3 e io3. Y c $lo2. 28p.6...2 L.2.3 Y e n ',: /.:,.< :. 2: ' Figure 3 The wavefor called h with the nuerically integrated velocity and displaceent...., ' o3 ' frequency (Hz) Figure 6 The SRS of the original and corrected x and h
/27/98 7 t 8 EE!!!E 83. H o l go 2 2? 2 f o ; l $ Y) 4? tie (s).figure 7 B3 with nuerically integrated velocity and displaceent tie (s) Figure 2 B3 with the nuerically integrated velocity and displaceent correction to 83, using db3, level correction corredion to 83, using db3, level correction feeef3 g oil/?.""" 2 2 s. 2 2 $ u ' tie (s) 2 corrected 83, using db3, level correction corrected 83. usina db3. level correctior i 8 o, c ; tie (s) Figure 9 The wavelet corrected B3 2 2 2 :3 :: a 6 tie (s) Figure 2 The wavelet correction to B3 Figure 8 The wavelet correction to B3? 2 B. 2 \i 8 L g :: s % 2 7 tie (s) Figure 22 The wavelet corrected B3 2
4 SAND 972283C 8 /27/98 o6 daping = % '...._ 3 ' z C ) &lo' oo ' ' o3 Frequency (Hz) ' ' \. Figure 26 The FFT of the original and corrected B3.~ ' REFERENCES daping = %. Galef, A., (983, "ZeroShifted Acceleroeter Outputs," 6" Shock and Vibration Syposiu, p 7779. l lo3? 2. Handbookfor Dynaic Data Acquisition and Analysis, E S T Design, Test, and Evaluation Division Recoended Practice 2., ESRPDTEO2., nstitute of EnvironentalSciences, Mount Prospect, L. / h W :lo2: W Y a '. ' oo ' / / ' o3 Natural frequency (Hz)..J " Figure 24 The SRS of the original and corrected B3 3. Sallwood,D. O., "Correcting Nuerical Errors Caused by Sall Aliasing Errors," 68" Shock and Vibration Syposiu, 998, SAVAC, Arlington VA. 4. Sallwood,D. O., and Cap, J. S., "Salvaging Transient Data with Overloads and Zero Offsets," 68" Shock and Vibration Syposiu, 998, SAVAC, Arlington VA.. Strang, G., "Wavelets,"Aerican Scientist 82 (April 994) 22. 6. Strang, G., and Nguyen, T., Wavelets and Filter Banks," WeleseyCabridge Press, Wellesley MA, 996. 7. Misiti, M., Misiti, Y., Oppenhei, G., and Poggi, J., Wavelet Toolbox User's Guide, The Mathworks nc. Natick MA, 996. Figure 2 The FFT of the original and corrected B3 Sandia is a ultiprogra laboratory operated by Sandia Corporation, a Lockheed Martin Copany, for the United States Departent of Energy under Contract DEACO494AL8.