Applications of DSP to Explicit Dynamic FEA simulations of … · 2019. 8. 1. · 168 T. Diehl et al. / Applications of DSP to Explicit Dynamic FEA simulations of elastically-dominated

167

Applications of DSP to Explicit DynamicFEA simulations of elastically-dominatedimpact problems

Ted Diehla, Doug Carrollb and Ben NagarajcaMechanical Technology Center, PCS, Motorola, 8000W. Sunrise Blvd., Fort Lauderdale, FL 33322, USATel.: +1 954 723 8024; Fax: +1 954 723 5584;E-mail: [email protected] Technology, Smart and ConnectedProducts, Motorola, 1500 Gateway Blvd., BoyntonBeach, FL 33426, USATel.: +1 561 739 3818; Fax: +1 561 739 3486;E-mail: [email protected] Technology Center, PCS, Motorola, 8000W. Sunrise Blvd., Fort Lauderdale, FL 33322, USATel.: +1 954 723 3098; Fax: +1 954 723 5584;E-mail: [email protected]

Received 16 August 1999

Revised 6 April 2000

Explicit Dynamic Finite Element techniques are increasing-ly used for simulating impact events of personal electron-ic devices such as portable phones and laptop computers.Unfortunately, the elastically-dominated impact behavior ofthese devices greatly increases the tendency of Explicit Dy-namic methods to calculate noisy solutions containing high-frequency ringing, especially for acceleration and contact-force data. For numerous reasons, transient FEA results areoften improperly recorded by the analyst, causing corruptionby aliasing. If aliasing is avoided, other sources of distortioncan still occur. For example, filtering or decimating ExplicitDynamic data typically requires extremely small normalizedcutoff frequencies that can cause significant numerical prob-lems for common DSP programs such as MATLAB. Thispaper presents techniques to combat the unique DSP-relatedchallenges of Explicit Dynamic data and then demonstratesthem on a very challenging transient problem of a steel ballimpacting a plastic LCD display in a portable phone, corre-lating simulation and experimental results.

1. Introduction

Impact and drop analysis of personal electronic de-vices, such as portable phones and laptop computers,differs dramatically from the more established analy-sis of car crashes. Whereas a car crash is dominatedby plasticity, the impact behavior of these electronicdevices is dominated by elasticity. When a portablephone is dropped to the floor, its housing usually doesnot dent or crush like when a car smashes into a rigidbarrier. Without sufficient inelastic deformation to dis-sipate energy, the resulting solution variables for anelastically-dominated impact problem can contain sig-nificant amounts of high-frequency energy, often lead-ing to extremely noisy results when solved using Ex-plicit Dynamic FEA codes. Acceleration and contactforce have been the most susceptible to noise, and to alesser extent, velocity, strain, and stress.

Attempts to validate predictions from Explicit Dy-namic models of elastically-dominated structures us-ing physically measured accelerometer data usually re-sult in very poor correlation. There are several reasonsfor this, both on the experimental and simulation side.Concentrating on the simulation error sources, one ofthe most likely and frequent problems is corruption ofthe FEA data by aliasing. While antialiasing filters arecommon in most (but not all) modern digital data ac-quisition systems, it is very important for everyone tounderstand that there is no such safety feature in com-mercial Explicit Dynamic FEA programs! Whereasexperimental systems deal with 10 or maybe hundredsof transient data channels, the FEA codes typically dealwith 104 to 107 (or more) “data channels”. The natureof these codes currently makes it infeasible to have an-tialias filtering internal to the program. Many of theFEA vendors that supply these Explicit codes do notprovide proper tools and training to correctly deal withhighly transient data. Hence, it is up to the FEA us-er to properly store and process their transient data –

Shock and Vibration 7 (2000) 167–177ISSN 1070-9622 / $8.00 2000, IOS Press. All rights reserved

168 T. Diehl et al. / Applications of DSP to Explicit Dynamic FEA simulations of elastically-dominated impact problems

something which is not a trivial task. It is important tonote that it has been the observation of these authorsthat many engineers doing such impact analyses withExplicit Dynamic codes have had little, if any, trainingin the processing of highly transient signals. DigitalSignal Processing (DSP) is not commonly a part of aMechanical Engineer’s core classes.

To provide a base of understanding for the DSP con-cepts used here, the reader is directed to [3,4] for anapplied overview. Additional details on DSP theoryare found in [1,6,7,9,11–14]. Lastly, all the DSP re-lated calculations presented here are performed usingDiehl’s DSP Extensions [2], an easy-to-use extensionpack that works with Mathcad, a mathematical and en-gineering program for the MS Windows operating sys-tem. The extension pack is specially designed for tran-sient impact analysis and can be downloaded for freeat http://mathcad.adeptscience.co.uk/dsp/.

2. Unique challenges in storing and processingExplicit Dynamic FEA data

A typical portable-phone drop analysis might simu-late 5 milliseconds of physical time using an averagetime increment of 0.1 microseconds (due to solutionstability requirements). Each variable in the simula-tion is a digital signal containing approximately 50,000data points (approximately 0.2 Megabytes, single pre-cision). Considering that a typical model may easilycontain over 106 variables (acceleration, velocity, dis-placement, stress, strain, etc.), that implies a total of5 × 1010 data points are computed for the model (ap-proximately 186 Gigabytes, single precision). In gen-eral, this amount of data is infeasible to store in its en-tirety. Out of necessity, a common approach used in theFEA community is to simply request results output atsome time interval that is much greater than the actualsolution’s time increment, reducing the data size downto say a couple hundred points per variable. For noisyvariables such as acceleration or contact forces, thisapproach frequently results in corruption by aliasing.

Besides the huge amount of data that is generated,there are several additional challenges to working withExplicit Dynamic FEA data. The stability requirementof the central difference integration scheme typicallyrequires the time increment for these models to be twoto four orders of magnitude greater than the desired fre-quency content of interest for the structure. ApplyingDSP filters to this type of heavily oversampled data cancause significant numerical distortions. Worst yet, the

time increment throughout the solution is not constant;it is always changing to achieve as efficient a solutionas possible. Thus, the data must be regularized to atime vector that has a constant time increment (constantsampling rate) before any DSP can be applied.

2.1. Potential problems with digital filtering

To properly deal with this transient data, DSP pro-grams must be used. The most significant DSP methodthat is employed is lowpass filtering. This sectiondemonstrates several undesirable filter distortions thatvarious commercial programs create when filtering Ex-plicit Dynamic data. To demonstrate the problems,simple test signals are passed through lowpass filters invarious programs. While the signals tested here seemtrivial, they will clearly point out many issues. Morecomplicated, real-world signals are evaluated in the ballimpact problem at the end of this paper.

For the filtering analyses, the following definitionsare used. Normalized frequency Ω and normalizedcutoff frequency Ωc are defined as

Ω =ω

ωsΩc =

ωc

ωs(1)

where ω is frequency, ωs is the sampling rate, andωc is the cutoff frequency. (Note that these normal-ized frequency definitions are different than those usedby MATLAB.) The generic z-domain transfer functionH(z) for a Infinite Impulse Response (IIR) filter is de-fined as

H(z) =(2)

B0 + B1z−1 + B2z

−2 + . . . + BLz−L

1 + A1z−1 + A2z−2 + . . . + ALz−L

where L is the filter order, vectors A and B are filtercoefficients, and z is a complex variable related to fre-quency. If the A coefficients are all zero, a Finite Im-pulse Response (FIR) filter is realized. A direct methodto compute the filter’s gain over a range of frequenciesΩ is simply

Gain = H(z = e2πjΩ

)(3)

where j =√−1 and an ejωt time dependence is as-

sumed. Similar to the use of Laplace transforms withcontinuous systems, we use z-transforms with digi-tal signals to map between the frequency and time-domains. Through the z-transform, we can implementdigital filters entirely in the time-domain. Given a dis-crete time domain data sequence xi (digital signal) andfilter coefficient vectors A and B, the filtered time do-

T. Diehl et al. / Applications of DSP to Explicit Dynamic FEA simulations of elastically-dominated impact problems 169

main response yi is simply computed as (Ifeachor [7,p. 143])

yi =L∑

j=0

Bjxi−j −L∑

j=1

Ajyi−j (4)

To improve numerical accuracy, especially for cas-es of very small normalized cutoff frequencies (sayΩc < 0.01), the actual numerical implementation ofan Lth-order filter should be done as a cascade of 2nd-order filters (see [14, p. 100], for more details). Whilethis distinction may seem trivial, it is important to real-ize that many DSP programs are written with the expec-tation that the data is not “enormously oversampled”.The idea that you may sample at 10 MHz and be in-terested in data at or below 5 kHz will seem foolish tomany DSP programmers. Hence, it is likely that thiscondition will not be tested for in their codes. How-ever, this is the exact condition that the FEA analystsfinds themselves in when modeling these elastically-dominated structures.

For time domain analyses, undesirable filter-induceddistortions can be created by filter start-up and endingeffects. These are caused by the common assumptionthat the signal to be filtered has zero amplitude for alltime before and after the signal. While this commonassumption is correct for most DSP applications, it isnot correct for many FEA analyses. For example, stressand strain components for a preloaded structure havenonzero values at the beginning of the analysis. In adrop analysis that begins just prior to impact, the initialvelocity will be nonzero. In almost all FEA simula-tions, the computation will be stopped prior to mostvariables settling to a steady state or zero value. Filter-ing any of these signals that violate the “zero value” as-sumption can cause transient distortions because of theinherent non-smooth transition between the assumedzero amplitude region before the signal’s beginning andthe actual signal content itself. Similar distortions canoccur at the end of the signal. This type of error canbe minimized by artificially projecting the original sig-nal back in time by a finite amount. MATLAB’s filt-filt function uses a reflected mirror algorithm to mini-mize start-up distortions. The filter algorithms used inDiehl’s DSP Extensions offer several methods to min-imize filter-induced distortions, including the follow-ing assumptions: zero, constant, reflected mirror, anda prediction algorithm based on Mathcad’s “Predict”function [10]. In all cases for this section of the paper,the prediction algorithm is used by Diehl.

Figure 1 evaluates MATLAB’s Signal ProcessingToolbox, V5.1 [12] relative to results computed by

Diehl’s DSP Extensions [2]. The evaluation consistsof defining an 8th-order Butterworth lowpass IIR filter,computing the magnitude of it’s gain, and then filteringa sloped line and a 1.0 kHz sine wave (with a 70 de-gree phase shift). The sample rate for all the signals is6.0 MHz. Two cutoff frequencies are studied, 30 kHz(Ωc = 0.005) and 18 kHz (Ωc = 0.003). All parame-ters are chosen to be representative of values seen in atypical Explicit Dynamic simulation.

Figure 1(a) shows the magnitude of the filter gainscomputed by Diehl’s DSP Extensions (denoted asDiehl) and by MATLAB (using two different MAT-LAB algorithms). The results from Diehl are quite ac-curate and not sensitive to the normalized cutoff fre-quency. Diehl simply evaluated the filter gain usinga modified version of Equation (3), coded for a cas-caded filtering approach as defined in [14]. The plotsshow that MATLAB can yield very poor results andthat their algorithms’ numerical stability are sensitiveto normalized cutoff frequency. Neither of the MAT-LAB algorithms utilize a cascade filtering approach.Additionally important to note is that MATLAB pro-vided no warning to the user that the gain computationwould potentially have numerical problems. While asavvy DSP expert might have suspected potential prob-lems with such small cutoff frequencies, a FEA analystwould likely not.

Figures 1(b) and (c) compare results from passingthe test signals through the lowpass filters. Both the fil-ter from Diehl and the MATLAB filter (using the filtfiltfunction) used a double-pass, zero-phase filtering ap-proach (see [2,12] for details). Also, both filters use preand post data projection algorithms to minimize end-effects (more to be said shortly). Since both signalshave no frequency content above either cutoff frequen-cy, the Ideal results after filtering should be the orig-inal signals. The results show that Diehl matches thebenchmark while MATLAB has significant problemsas the normalized cutoff frequency is made smaller. Asbefore, Diehl used a cascaded filtering approach andMATLAB did not. This problem may seem trivial, butit is not because when filtering Explicit Dynamic data,values of normalized frequencies similar to these willbe encountered. Furthermore, it is reasonable to expectlow frequency content (structural deformation modes)to be contained in the signal and we would expect thisfrequency content to be undisturbed by a lowpass filter.

Figure 2 evaluates the filtering capabilities ofABAQUS/Post V5.8 [5]. The figure shows the time-domain results of some simple signals before and af-ter lowpass filtering. The test signals depicted in (a)


Fig. 1. Comparing MATLAB lowpass filtering and Diehl’s DSP Extensions using some digital test signals. Sample rate of all signals is 6 MHz.

are a sloped line and three sine waves with frequenciesof 1.3 kHz, 6.0 kHz, and 29.0 kHz (note that the sinewaves are defined such that they begin with non-zeroamplitude). The curves depicted in (b)–(e) show the

ideal solution, results from ABAQUS/Post, and resultsfrom Diehl’s DSP Extensions. ABAQUS/Post uses asingle-pass sine-Butterworth IIR filter (a slight varia-tion on the more common Butterworth filter) with no


Fig. 2. Comparing ABAQUS/Post lowpass filtering and Diehl’s DSP Extensions using some digital test signals. Sample rate of all signals is240 kHz and cut-off frequency is 3.0 kHz.

filter distortion compensation. The filter from Diehlis a Butterworth double-pass zero-phase filter with fil-ter start-up minimization. (Similar results are easilyachieved with other common filters such as Cheby I ora sinc-based FIR.) Both ABAQUS and Diehl’s filterswere 6th-order. Figures 2(a–d) demonstrate the defi-ciencies of the ABAQUS filter implementation; end-distortions and time delay are clearly evident. Thesame figures show the relatively distortion-free resultsfrom Diehl.

Figure 2(e) shows a serious deficiency with the

ABAQUS lowpass filter implementation, it producedan aliased result! ABAQUS/Post has a default resam-pling algorithm that will resample (decimate or inter-polate) the original signal to a sample rate that is 10times the specified cut-off frequency of the filter (TheABAQUS/Post manual [5] incorrectly states the resam-pling to be 5 times the cut-off freq). This resamplingis done prior to applying the lowpass filter. In our testcase, the 29.0 kHz sine wave, which originally had asample rate of 240 kHz, was decimated to a sample rateof 30 kHz (10 times 3 kHz). The 30 kHz sample rate


caused the 29.0 kHz signal to be aliased to a 1.0 kHzsignal. This resampled signal then passed through the3 kHz filter “untouched”, except for the filter distor-tions already discussed. These simple cases show thatthe ABAQUS filter implementation not only createsundesirable distortions but that it is very susceptible toinducing aliasing errors.

Other commercial programs have also been testedby the authors and many show deficiencies similar tothose demonstrated in Figs 1 and 2. For example, LS-Taurus [8] (the post processor for LS-Dyna) aliasesoutput when the user requests the program to outputdata to an ascii file.

While these issues may seem trivial, or are the re-sult of special “trick” conditions, they are not. Theyare the result of the Explicit Dynamic FEA world andthe DSP world coming together in an imperfect fash-ion. Until greater awareness of the these DSP issuesare achieved in both communities, errors like the onesshown will continue to happen. The lesson learnedfrom these studies is that one should check out any DSPfunctionality of a potential program with simple func-tions. Then you can fully understand what the programis (and is not) doing to your data.

2.2. Summary of proper sampling and decimation ofExplicit Dynamic FEA data

The fundamental problem when dealing with tran-sient FEA data is that we generally do not know themaximum frequency content of the various solutionvariables ahead of time. If we did, then we wouldknow a safe sampling rate. Lacking knowledge of themaximum frequency content, the following templatewill help to insure aliased-free data. See [3] for morespecifics relative to the use of ABAQUS.

1. For every variable of interest, output the result ateverytime increment. Since the resulting outputfile might get quite large, you will need to beselective on how many variables you output, say10 or so. Additionally, the actual value of thechange in time at each of the time incrementsshould be requested directly form the solver andstored. Simply attempting to calculate the timeincrements from the stored ASCII output data ofthe total time vector might have significant errorsif the output is only stored with six or less digits ofaccuracy, which is common for many programs.

2. Because the time increment in an Explicit Dy-namic analysis changes throughout the solution,regularize the data to a constant time increment(DSP algorithms require this). The original da-ta must be interpolated onto a new constant-increment time vector. Two reasonable choicesfor the time increment of the new time vector arethe solution’s average time increment or it’s min-imum time increment. The user should evaluatea plot of the time increment as a function of timeto determine which one to select. When selectingthis choice, two error sources must be considered:A) aliasing caused by selecting the average timeincrement if this value is much greater than theminimum time increment and B) interpolation er-ror caused by selecting the minimum time incre-ment if it is much smaller than the average incre-ment and the minimum time increment is local-ized in time (occurs only a few times throughoutthe solution).

3. Reduce the regularized data sets down to 10 timesthe highest frequency of interest. To decimatethe data safely, it must be first lowpass filteredto sufficiently attenuate all the frequency contentthat is above the Nyquist frequency of the desired(reduced) sample rate. Note, the cutoff frequencyof the antialias filter must be sufficiently less thanthe ideal cutoff frequency (Nyquist frequency ofdesired sample rate) to account for the transitionband of the filter. Once this frequency content is“removed”, the data sets can be safely decimatedto the new sample rate. To avoid filter distortions,special precautions discussed previously must beutilized.

Any program with appropriate DSP capabilities maybe used, but they must be able to properly handle verysmall normalized cutoff frequencies (values on the or-der of Ωc = 0.01 or less). The functions in Diehl’s DSPExtensions are capable of handling the filtering plus itoffers functions to easily regularize and decimate thedata.

Some additional practical things to remember. Re-sponses such as acceleration and contact force will bevery noisy, especially in solid elements. This type ofdata often has large amplitude, high frequency noisecomponents. All data of this type that is to be evalu-ated should be post-processed in the manner describedabove. On the other end of the spectrum is displacementdata. By its nature, high frequency displacement com-ponents have very low amplitude and therefore pose a


relatively low risk of alias corruption. Stress, strain,and velocity are in between these two cases.

Unfortunately, application of the process outlinedabove is not generally feasible for animated contours,as too much initial data must be stored. A reasonableapproach around this limitation is to output data at a fewrepresentative nodes (or elements) within the contourby the method outlined above. Using this selected data-set, a safe sampling rate can be determined which willavoid aliasing. Then the entire data set can be stored(output) from the solution using this safe sampling rate.Remember, you must be sure that there is no significantfrequency content in the solution variables of interestgreater than one half of your output frequency. If thereis, aliasing will occur which you will not be able todetect nor correct.

3. Ball bearing impact on a portable phone lens

Figure 3 depicts a very challenging ball bearing im-pact problem. A plastic housing with plastic displaylens is supported at 4 bosses and subjected to the im-pact of a 130 g, 31.75 mm diameter steel ball which isdropped from 500 mm. The quantities of interest arethe len’s transient responses of acceleration and dis-placement under the point of impact (back side of thelens). The objective of this example is to correlate bothexperimental and FEA results.

The Explicit Dynamic FEA model is composed ofshell elements for the housing and solid elements forthe variable thickness lens (three elements through thethickness). The plastic material is modeled using on-ly Hooke’s law (no plasticity or viscous effects). Themodel is solved using ABAQUS/Explicit. The exper-imental measurement of the acceleration utilized anEndevco 2255B-01 Isotron accelerometer connectedto an Endevco model 133 Signal Conditioner with allthe Conditioner’s HP & LP filters turned off. Thelightweight accelerometer has a resonance of 300 kHzand an internal 2-pole Butterworth lowpass analog filterwith an approximate cutoff frequency of 28 kHz. Theexperimental acceleration was captured, alias free, at asample rate of 250 kHz and is displayed in Fig. 3(c).The ABAQUS/Explicit prediction of acceleration froma single node on the bottom side of the lens, directlyunder the point of impact, is displayed in Fig. 3(d).This data has already been regularized per the methodsoutlined in Section 2.2. (The regularized and raw FEAdata are very similar and would be indistinguishable onthis plot.) Figures 3(e–f) show these results plotted in

the frequency domain (The frequency spectrum of thetime-domain signal was computed with Diehl’s DSPExtensions). The question we need to answer is “Doesthe simulation and experiment correlate”? Based onthe data presented in Fig. 3, we would have to say “no”.Let’s try to further analyze the data to see if thingsimprove.

A common approach to improve correlation is to ap-ply a lowpass filter to remove noise that is likely presentin both the experiment and the simulation. A commoncutoff frequency used for this type of structure mightbe between 1 kHz and 10 kHz. For our case, we willchoose 5 kHz. Using the rule of thumb of 10x, wedecide that our desired sampling frequency should be50 kHz. Figure 4 presents what happens if we justsample the raw FEA data at 50 kHz without protectingagainst aliasing. The experimental acceleration (sam-pled at 250 kHz) and the FEA acceleration (sampledat 50 kHz) displayed in Fig. 4(a) definitely do not cor-relate. Figure 4(b) presents the results after lowpassfiltering with a 5 kHz Butterworth filter. To match filterresponses for the two data sets, the experimental datautilized an 8th-order Butterworth and the simulationdata used a 7th-order Butterworth. To match filter re-sponses as close as possible, different filter orders areused because the two data sets had different samplingrates. (Remember, filter responses are a function of thenormalized cutoff frequency.) Even after filtering, thetwo acceleration data sets look completely unrelated.Lastly, we compare displacement data (Fig. 4(c)). Dis-placements for the experiment are computed by doubleintegrating the experimental acceleration signal. TheFEA code computes displacements directly. Interest-ingly enough, the raw FEA displacement curve and theintegrated experimental displacement curve correlatequite well. However, acceleration and displacement aredirectly related and we observed that the accelerationswere completely different! Also plotted in Fig. 4(c)are “integrated FEA displacements” computed by dou-ble integration of the FEA acceleration curves fromFig. 4(a–b). Both of these integrated results are verydifferent from the raw displacements computed direct-ly by ABAQUS. How could that be? The answer isthat all the FEA acceleration data in Figure 4 is aliased!It is aliased because we sampled the acceleration da-ta without first removing the high frequency contentabove 25 kHz (half of 50 kHz). Figure 3(f) (regularizedbased on sampling every solution increment) showsthat the original acceleration signal contains significantfrequency content up to approximately 1 MHz. Thisextremely high frequency content is caused by, amongother things, the individual element vibrational modes.


Fig. 3. Ball bearing impact example, experiment and /Explicit FEA model.

Figure 5 presents data that is properly decimatedto 50 kHz by the process described in Section 2.2.In addition, the experimental data has also been dec-imated down to a sample rate of 50 kHz. The re-sults in Fig. 5(a), before the 5 kHz LP filter, lookvery promising. After filtering, the correlation betweenthe simulation and experimental accelerations is excel-lent (Fig. 5(b)). Now, the integrated displacements forthe simulation match the raw FEA displacement data(Fig. 5(c)). An important thing to note is that the suc-cess of obtaining correlation was dependent on propersampling technique, not filter form (IIR or FIR). Equal-

ly good results are obtained with a Cheby I IIR or sinc-based FIR filter provided that the “proposed filteringmethod” described previously is used and that the filterparameters are defined such that the magnitude of thefilter responses are similar.

4. Conclusions

It is imperative that highly transient Explicit Dynam-ic FEA solution data is properly handled, especially forelastically-dominated impact problems. Proper DSP


Fig. 4. Processing data without protecting against aliasing. FEA data is simply requested from solution at a 50 kHz sample rate.

processing can make the difference between realisticand nonsensical conclusions. The key DSP issues re-lated to this application are:

1. The proper method to process transient data andavoid aliasing errors is to output the desired FEAsolution variables at every solution increment,regularize this data, and then decimate the da-ta (incorporating an antialias lowpass filter) to amanageable sample rate. Applying this approachto large data sets, such as animations of contours,is presently not feasible. For large data sets, this

approach should be applied to a small selected setto determine a sufficient sample rate that wouldavoid aliasing for the entire set.

2. Quantities such as acceleration and contact forceare the most susceptible to alias errors, and toa lesser extent, velocity, strain, and stress. Dis-placements are the least susceptible to alias errors.In general, it is impossible to determine if a giv-en sampled signal has aliased frequency contentwithout detailed knowledge of the original signalprior to sampling. Even smooth, low-frequency


Fig. 5. Processing data with antialias filtering. FEA data is properly decimated down to a 50 kHz sample rate.

signals can be corrupted by aliasing.

3. Users need to be aware of the many problems

that can occur when processing and filtering Ex-

plicit Dynamic FEA data. The normalized cut-

off frequencies can be quite extreme. As a re-

sult, many commercial programs that offer DSP

features may not properly process Explicit Dy-

namic FEA data. Vendors of Explicit Dynamic

FEA programs need to significantly improve their

ability to process highly transient data.

4. When proper DSP methodology is utilized, ex-cellent correlation between experimental and ex-plicit dynamic FEA data can be achieved, evenat very challenging locations such as the point ofimpact.

Acknowledgments

The authors wish to sincerely thank colleagues Dr.Dave Yeager and Dr. Jason McIntosh for their numer-


ous discussions of and insight into the world of DSP.Additional appreciation is owed to Dr. McIntosh for hisefficient C++ coding of many of the algorithms usedin this work.

References

[1] Brillouin, L., Wave Propagation and Group Velocity,Academ-ic Press, 1960.

[2] Diehl, T., Diehl’s DSP Extensions,Available for download athttp://mathcad.adeptscience.co.uk/dsp/, 1999.

[3] Diehl, T., Nagaraj, B. and Carroll, D., Using Digital SignalProcessing (DSP) to Significantly Improve the Interpretationof Drop/Impact Simulation Results for Personal ElectronicDevices: Part I – Theory, 70th Shock & Vibration Symposium,November 15–19,Albuquerque, NM, SAVIAC.

[4] Diehl, T., Nagaraj, B. and Carroll, D., Using Digital SignalProcessing (DSP) to Significantly Improve the Interpretationof Drop/Impact Simulation Results for Personal ElectronicDevices: Part II – Applications, 70th Shock & Vibration Sym-

posium, November 15–19,Albuquerque, NM, SAVIAC.[5] Hibbitt, Karlsson and Sorensen, ABAQUS /Post,V5.8.[6] IES, Handbook for Dynamic Data Acquisition and Analysis,

Institute of Environmental Sciences, ISBN 1-877862-47-9, nocopyright date listed in document.

[7] Ifeachor, E. and Jervis, B., Digital Signal Processing: A Prac-tical Approach,Addison-Wesley, 1993.

[8] Livermore Software Technology Corp., LS-Taurus User’sManual,1997.

[9] Madisetti, V. and Williams, D., The Digital Signal ProcessingHandbook,CRC Press, 1998.

[10] Mathsoft, Mathcad 7 User’s Guide,Mathsoft Inc., 1997.[11] Mathsoft, Mathcad Signal Processing Function Pack,Math-

soft Inc., 1998.[12] Math Works, Signal Processing Toolbox For Use with MAT-

LAB: UserŠs Guide,The Math Works Inc., 1998.[13] Oppenheim, A. and Schafer, R., Digital Signal Processing,

Prentice-Hall, 1975.[14] Stearns, S. and David, R., Signal Processing Algorithms,

Prentice-Hall, 1988.[15] Ziemer, R., Tranter, W. and Fannin, D., Signals and Systems:

Continuous and Discrete,Macmillan Publishing, 1983.

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