On the Time Compression (Test Acceleration) of Broadband Random Vibration Tests

Transcription

1 PACKAGING TECHNOLOGY AND SCIENCE Packag. Technol. Sci. 2011; 24: Published online 17 September 2010 in Wiley Online Library (wileyonlinelibrary.com).915 On the Time Compression (Test Acceleration) of Broadband Random Vibration Tests By David Shires Pira International, Leatherhead, UK Many broadband random vibration tests are time compressed. This is done by increasing test intensity according to the Basquin model of cyclic fatigue. Conventionally, the test level is accelerated from the root mean acceleration and an assumed power constant (k = 2) is applied. Using conventional analysis the potential error in test severity can be very large if k is incorrect. The Miner Palmgren hypothesis of accumulated fatigue is used to re-assess the potential error in test severity accounting for the non-stationarity found in road distribution. This shows a substantially reduced sensitivity to the value of k depending on the distribution of actual vibration intensities around the time-compressed test intensity. Using an example of a leaf-sprung vehicle, the conventional level of time compression is shown to have low sensitivity to errors in k, whereas for an example of an air-ride vehicle a lower level of time compression is needed to reduce error sensitivity. Copyright 2010 John Wiley & Sons, Ltd. Received 12 February 2010; Revised 4 July 2010; Accepted 7 July 2010 KEY WORDS: random vibration; time compression; test acceleration; vibration testing; non-stationarity INTRODUCTION Broadband random vibration (BRV) testing is commonly used for the laboratory simulation of transport vibration: it is used to predict the durability of packaged items against vibration hazards that occur in distribution. A single level BRV test is characterized by defining a mean power density spectrum (PDS). The overall intensity of the test is simplified to the root mean square acceleration (G rms ) which corresponds to the square root of the area under the PDS. Products are commonly distributed across continents by road or rail and between continents by sea or air with correspondingly long journey times. From a testing perspective, it is convenient, economic and efficient to have a test duration substantially shorter than actual journey duration i.e. to have a time-compressed test. Time compression is based on the fatigue model developed by Basquin. 1 It is achieved by increasing the intensity of the test. The assumed (assumed because for packaging testing the applicability of fatigue models has not been proven) relationship between actual and test duration is: t t j t at = a j k (1) * Correspondence to: D. Shires, Chief Consultant, Pira International, Cleeve Road, Leatherhead, KT22 7RU, UK. david.shires@pira-international.com Copyright 2010 John Wiley & Sons, Ltd.

2 76 D. SHIRES where: t t = test duration t j = actual journey duration a t = test intensity (G rms ) a j = actual journey intensity (G rms ) k is a constant whose value depends on the material/product in question. For packaging testing, the quotient t j is typically limited to 5. 2,3 Time compression is tt commonly expressed as a multiple (for example, x5 ) and this form of expression is used throughout this paper. The value k = 2 is commonly used; it is understood that the value of k = 2 was set at one end of a range proposed by Curtis et al. (that giving the higher test levels). 1 For k = 2, and to establish the G rms of a test, Equation 1 is commonly expressed as: a a t j t = j t t 05. (2) Equation 1 is often (mistakenly) called Miner s rule or the Miner Palmgren rule see for example MIL-STD 810 and DEF_STAN ,5 The Miner Palmgren hypothesis is given in Equation 5. Time compression is widely established in the environmental testing field. Equation 1 is discussed in defence environmental test standards MIL-STD and DEF_STAN (both standards state typical values for k are between 5 and 8) but not in the international broadband vibration standard IEC Packaging test and performance standards differ in their approach to time compression. For example: ASTM D states Test levels are often increased over the actual field data to shorten test time. Any attempt to do so should be done with caution. Use of equivalence techniques of this type may assume linearity of specimen response to test input which is, in fact, not likely. ASTM D vibration schedules and International Safe Transit Association (ISTA) general simulation procedures 3A and 3F 8 appear from the duration and intensity of tests to use time compression but make no statement on this. Kipp suggests from analysis that the ASTM D4169 vibration schedules are time compressed at x5. 3 ISTA procedures 3E and 3H 8 give the relationship: t t d j = (3) 8 where: t t = test time (min) d j = journey distance (km). This changes the relationship with test time to journey distance (instead of journey time) but gives time compression of x5 when mean journey speed is 96 km/h. ISTA procedure 4AB 8 states that time compression is used. The final report of the European Union funded project SRETS 9 uses time compression but with a value for k (Equation 1) of 5. Over the last decade there has been an increase in the quality and quantity of recorded vibration data, in part because of improved instrumentation. This has led to an improved understanding of the vibration characteristics of different transport modes and regions and to an increase in the analysis of the statistical nature of transport vibrations, in particular for road distribution. Particular attention has been given recently to the non-stationarity of road vibration and to the occurrence of higher energy transients (shock events) during distribution. To better simulate the (non-stationary) distribution of vibration energies found in road distribution Young et al., and later Singh et al., introduced and used the split level BRV test 10,11 Two (or more) test intensities are used corresponding to a split in the population of recorded data according to ranked intensity.

3 TIME COMPRESSION OF BRV TESTS 77 Rouillard and Sek showed that the non-stationarity in time sampled and averaged road vibration data equates to the kurtosis from a continuous acceleration record. 12 They also showed that a nonstationary time record of vehicular vibration can be broken into a series of short periods of vibration each satisfying the characteristics of stationary, Gaussian random vibration. 13 From this, a multi-level BRV test was developed with a controller which modulates the test intensity to reproduce the nonstationary characteristics of road vibration. 14 Various researchers have studied the higher energy transients that are recorded during vibration studies. Studies have focussed on understanding the characteristics of shocks that occur during road and rail distribution and proposing the development of methods to simulate such characteristics in BRV tests (shock on random testing or real-time high level on time-compressed BRV) These newer developments have demonstrated that conventional BRV tests are not true simulations of the distribution vibration environment and have raised (unanswered) questions over the validity of the single level BRV test. However, the single level test remains the method defined by most packaging performance standards. Additionally, the single level test might be viewed as the preferred method where time compression is required. The (time compressed) test level generally falls within the distribution of on road intensity and so does not expose the test items to extreme levels of vibration. It is emphasized that neither Young, Singh nor Rouillard and Sek have proposed time compression of their split level/modulated tests. The objective of this research is to understand the implications of the value of the constant k in Equation 1; in particular to use current understanding of non-stationarity in vibration to re-assess the errors associated with an incorrect value for k. This research does not validate or prove the applicability of the Basquin model (though there is merit in such research); it explores the assumption of the value of k given the model is widely used. The fact that time-compressed tests have been widely used over many years with only few documented reports of poor correlation suggests that time-compressed test levels are reasonably correct. However non-stationarity impacts on the relationship between time compression and k. FATIGUE MODELS The Basquin and Miner Palmgren models were developed to describe fatigue in metal structures. Fatigue is the progression of structural weakening resulting from the development and propagation of micro-cracks within a structure as a response to applied stress cycles. 1 5,19 Fatigue life (N f ) is defined as the number of stress cycles of a specified character that a specimen sustains before failure or a response of a specified nature occurs. 19 For vibration which is constant in nature, fatigue life can also be expressed as the duration of vibration resulting in a response of a specified nature (T f ). Basquin related the number of stress cycles to cause fatigue failure to the magnitude of the stress cycles. For stress cycles within the elastic region of a material, the Basquin relationship can be simplified to: k z = N f a (4) where: a = magnitude of the stress cycle N f = number of stress cycles to cause fatigue failure (i.e. the fatigue life) k and z are constants (k is the same as in equation 1). Miner promoted the application of a hypothesis previously developed by Palmgren that fatigue is linearly accumulative. For example, if an item is subject to two sets of stress cycles each equal to 10% of fatigue life, the item will be fatigued to 20% of its fatigue life. The Miner Palmgren hypothesis can be used to evaluate the cumulative effect of a history of stress cycles at x different stress levels. This is described by the equation: x ni = 1 (5) N i= 1 fi

4 78 D. SHIRES where: n i = number of stress cycles at stress level i N fi = fatigue life at stress level i x = number of stress levels in a stress cycle history ultimately resulting in failure. Note that the summation of Equation 5 has been shown to vary from 1 in practice. Relating Equation 4 at stress levels 1 and 2 gives: k x = N a = N a k f 1 1 f 2 2 (6) Equation 6 simplifies to Equation 1 and the constant k is the same in both equations. The end of a journey of interest may not result in failure, but it can be argued to be the accumulation of fatigue to the point of interest. The Basquin and Miner Palmgren equations are imperfect in describing the fatigue behaviour of many metals: this point is not elaborated here as it falls outside the scope of this research. However, their applicability to the responses of packaged products to vibration merits question: Many packaging materials are non-metallic. Many failure modes seen in packaged product distribution are not the result of fatigue. The transmissivity of vibration through an assembly of packages may be non-linear with input amplitude (especially near structural resonances). Some failure mechanisms (for example, surface scuffing or closure back-off) may have an endurance limit i.e. there will be a magnitude of stress cycle below which failure will not occur regardless of the number of stress cycles endured. There is, however, an argument that failure modes seen in packaged product distribution are accumulative in nature. If this was not so, failure would be seen after a few cycles of test and the duration of a test would have no bearing on the result. Intuitively and empirically, it also is expected that test severity has a bearing on the rate of development of vibration responses. There are therefore reasons to doubt the applicability of the fatigue equations to package testing and reasons to argue they may offer a reasonable basis for time compression of single level BRV tests. To the author s knowledge, there has been no research to measure the S/N (stress vs. number of cycles) behaviour of packaged items. This is not surprising considering the subjective nature of the assessment of many packaged product responses; the range of possible responses; the vast range of combinations of primary, secondary and tertiary packs; and the large number of samples that would be required per combination studied. It remains that the Basquin and Miner Palmgren equations are used as the basis for time-compressed BRV testing and that time-compressed testing is undertaken widely. In this context, this research explores the implications of the choice of value of the constant k in Equation 1. The research examines the implications of non-stationarity in the original vibration record on time compression and examines how the significance of the higher energy events in non-stationary vibration depends on assumptions in the Miner Palmgren model. ESTABLISHED INTERPRETATION In fatigue development in metals, k varies with material and sample shape and typically has a value between 5 and 8. 5,6,19 As stated earlier, a value for k of 2 is typically used for the time compression of BRV tests, though researchers in the SRETS project used a value of 5. 9 Accepting the Basquin equation is applied to describe the S/N response of packaged items, the difference between the value of k used for packaging testing (2) and that measured for fatigue processes in metals (5 8) can be explained in part by the assumed degree of damping which affects the value of k. Thus, the range of possible values for k will be lower than for metals. 1 As a wide range of packaging materials, packaging formats and failure responses are seen in packaging distribution, it seems reasonable to expect that there is a range of applicable k values (not a single value).

5 TIME COMPRESSION OF BRV TESTS 79 Figure 1. Test level (x5 time compressed from G rms ) for k = 1,2,3 and 5. Table 1. Data capture set-up: leaf-sprung journey record. Instrument Lansmont 3X90 (Lansmont Inc., Monterey, CA, USA) Measurement Location and Axis Vertical axis on chassis over rear axle Sampling frequency 1000/s Sample duration s Sampling interval 10 s Anti-aliasing filter 500 Hz; subsequent analysis truncated spectrum at 100 Hz Active journey duration 15 h, 21 min Data analysis Instrument supplier s SaverXware software and export to spreadsheet As Equation 1 uses k as a power factor, the errors resulting from an incorrect value of k are large. Equations 1 and 2 are applied using the G rms of the entire journey record. Actual journeys vary in intensity throughout the journey (they are statistically non-stationary). 12,13 Typically for packaging research, journeys are recorded in short time samples and the G rms of each time sample calculated. The statistical distribution of values of sample G rms can be summarized in a histogram. Figure 1 shows an example of such a histogram the data were collected on a fully laden 2-axle leaf sprung semi-trailer on mixed quality roads in China in 2009 (details of the journey record are summarized in Table 1). Superimposed on Figure 1 are test levels calculated using a t = k 1 for k = 1,2,3,5 (Equation 2 for a x5 time-compressed BRV). This shows how widely the test level varies if k has a value other than the assumed value of 2. If k is incorrectly assumed to be 2, an error in time compression results. Figure 2 shows the magnitude of error in time compression as a function of k when a test intensity is set assuming k = 2. Figure 2 shows the possibility of significant errors in test severity if k differs from the assumed value of 2. A few workers have reported on, or commented on, the risk of poor correlation between time-compressed BRV tests and field performance The author knows of no study of the correlation of BRV tests with field data across a range of laboratories and samples but reports of poor correlation are relatively few implying general satisfaction. IMPLICATIONS OF NON-STATIONARITY The terms fatigue and fatigue life are used in this section for consistency in the interpretation and application of the Basquin and Miner Palmgren equations.

6 80 D. SHIRES Figure 2. Error in time compression as a function of k (test level k set at an assumed k value of 2 with x5 time compression). We can apply the Basquin equation to a statistical distribution of field data to relate each bin level in turn (rather than the overall journey G rms ) to a test level of interest. Equation 5 allows us to add the fatigue contribution of each bin level. The duration T f1 of stationary vibration at a single RMS (Root Mean Square) acceleration a 1 giving the same fatigue as that accumulated by the total actual journey is: T f1 k n ai = ti a (7) i= 1 where t i and a i are the duration and RMS acceleration of each of n bin records representing the total journey. If we accept the end of a total journey as representing a point of interest, then T f1 can be interpreted as the fatigue life at acceleration a 1. Using Equation 5, the fatigue, f i, contributed by the vibration of bin i to that of the total journey is: 1 ti fi = (8) T where t i is total duration of all vibration in the range of G rms of bin i and T fi is the fatigue life for vibration at the level of bin i. Applying Equation 5: fi n f i = i= 1 1 (9) Table 2 shows the calculation by spreadsheet of f i for the leaf-sprung journey summarized in Figure 1 (using k = 2). For clarity, a number of rows and columns associated with the central bins of the distribution are hidden. Figure 3 shows the fatigue contribution (% total journey fatigue) by bin for k = 2, 4 and 6. The distribution of G rms is given for reference. As expected, the calculated fatigue caused by the higher energy bands increases as k increases. At k = 2, the majority of fatigue is caused by mid-intensity acceleration; at k = 4, fatigue is distributed across all but the lowest intensity acceleration; and at k = 6, most of the fatigue is caused by higher intensity acceleration. The significance of the highest energy bins is shown in Table 3 which gives the combined contribution to fatigue of the top 4 bins (0.088% of the journey record) as a function of k.

7 TIME COMPRESSION OF BRV TESTS 81 Table 2. Spreadsheet calculation of f i (fatigue contribution by bin). Figure 3. Distribution of G rms and fatigue contribution (%) by G rms histogram bin. Table 3. Fatigue contribution (% total journey fatigue) from top four acceleration bins as a function of k. k Fatigue contribution (%) Whether we need be concerned with the occasional high-energy events in a journey depends on the actual (rather than presumed) value of k. If k is approximately 2, the higher energy events contribute little to the total response to vibration. Figure 2 shows the calculated error in time compression as a function of k when applying the Basquin equation to the G rms of the journey record. Implicit in this application of the Basquin equation

8 82 D. SHIRES is an assumption of stationarity, or that the fatigue caused by a journey is centred and concentrated around the G rms value: Figures 1 and 3 show that this is not the case. Equation 7 allows time compression to be calculated for a stationary test intensity against the distribution (by bin) of non-stationary vibration. The test time equivalent to each acceleration bin is calculated and all are summed to give total test time. Table 4 shows the spreadsheet used to calculate time compression from a non-stationary journey analysis (some columns are hidden for clarity). Figure 4 shows test time compression (calculated using Equation 7 for the leaf-sprung journey record shown in Figure 1) as a function of k for a number of test intensities. There is a range of test intensities where the sensitivity of time compression to the assumed value of k is relatively small. The time-compressed test level normally calculated for the journey in Figure 1 is G rms which Table 4. Spreadsheet calculation of time compression from non-stationary journey record. Figure 4. Time compression as a function of k (non-stationary analysis): leaf-sprung truck.

9 TIME COMPRESSION OF BRV TESTS 83 falls in the range where sensitivity to k is small. A slightly higher test value of 0.2 G rms would give time compression nearest x5 across a range of k values. Thus, the use of k = 2 appears to work fairly well for the time compression of the steel-sprung journey record, but does not necessarily indicate that k actually is 2. Rather, it sets a test level which is relatively insensitive to the (unknown) value of k. This is because as k increases, the expected increase in time compression is offset by the increasing significance of the higher energy events. The higher energy events are time expanded because they are higher than the test level so a counterbalancing effect occurs. This counterbalancing effect will work less for journeys with narrower distributions of vibration intensity, for example, journeys on air-ride vehicles. This is because the conventional x5 time-compressed test level will be at a higher percentile of the journey record so there will be fewer higher energy events that are decelerated. Figure 5 shows the intensity by bin of a journey recorded on a fully laden 3-axis air-ride semi-trailer on mixed quality roads. The journey was recorded in Northern Ireland in 2008 and details of data capture are given in Table 5. The journey G rms and time-compressed test level (k = 2, time compression of x5 against G rms ) are indicated. The time-compressed test level is above the majority of the distribution. Figure 6 gives time compression as a function of k for a number of test intensities for the air-ride journey. Testing at the conventional x5 time compression level gives a large sensitivity to the value of k. This is because there are few higher intensity events to counterbalance the increasing time compression as k increases. If k is actually 2 (test G rms 0.2), then the time compression is reasonably valid; but if k is larger than 2, a large error in time compression results. The counterbalance is restored if a lower test level is used but at the cost of lower time compression. Figure 5. Distribution of journey intensity by bin: air-ride semi-trailer on mixed roads. Table 5. Data capture set-up: air-ride journey record. Instrument Lansmont 3X90 (Lansmont Inc., Monterey, CA, USA) Measurement Location and axis Vertical axis on vehicle bed over rear axle Sampling frequency 500/s Sample duration 1.0 s Sampling interval 1.0 s Anti-aliasing filter 250 Hz Active journey duration 1 h, 15 min Data analysis Instrument supplier s SaverXware software and export to spreadsheet

10 84 D. SHIRES Figure 6. Time compression as a function of k (non-stationary analysis): air-ride semi-trailer. DISCUSSION Time-compressed test levels are established on the assumption that the response of the test item to vibration can be described by the Basquin model of fatigue. This same assumption has been used in this analysis; this is discussed in the main text. To achieve increased confidence in the correlation between time-compressed testing and actual performance in distribution, this assumption needs to be researched; there is a need to understand the relationship between packaged product responses and vibration intensity. The model presented explains why severe over-testing does not result from the use of a conservative value for the constant k in the Basquin formula. In the examples shown, the lowest error in time compression is circa x5 for a steel-sprung journey record, but at circa x2 for an air-ride journey record. The method used can be applied to data records used for focussed simulation testing to determine the degree of time compression with lowest potential error. The method can also be used to explore different scenarios: two examples are given here. Example 1 considers the implications of stationery non-time-compressed testing (testing at journey G rms ). This approach is recommended by ASTM D and is known to be favoured by some laboratories 20 as it is thought to avoid errors in correlation arising from time compression. Figure 7 shows that testing at G rms does not reproduce the accumulated fatigue of a non-stationary journey and generally will be an under-test. This finding supports the use of split level or modulated tests for non-time-compressed testing; for a given journey record, the method of analysis presented here can be used to determine an appropriate number of test levels. Note however that time compressing split level tests will result in a greater potential error as a function of k than testing at a single level (because the distributions are narrowed by splitting the test and so the self-correcting mechanism will not apply). Example 2 considers the implications of a stressing mechanism having an endurance limit (i.e. a magnitude of stress cycle below which failure will not occur regardless of the number of stress cycles endured). Endurance limits seem likely in at least some packaged goods responses; for example, static friction will prevent relative surface movement at low stress levels so that phenomena such as scuffing and displacement will not occur. Figure 8 presents time compression as a function of k and test intensity for the steel-sprung journey assuming an endurance limit of 0.08 G rms (approximating to the overall journey G rms of 0.081). The results are directly compared with the same analysis without an endurance limit as originally presented in Figure 4. The fatigue contributions of the lower four bins in Figure 1 were set to zero but their contributions to journey time were unchanged; these four bins equate to 54% of the total journey. The change in time compression is generally small (and greatest where k is low).

11 TIME COMPRESSION OF BRV TESTS 85 Figure 7. Time Compression as a function of k testing at journey G rms (steel-sprung and air-ride journeys as illustrated in Figures 1 and 5). Figure 8. Time compression as a function of k calculated from the entire journey record (solid lines) and from the journey record with an applied endurance limit (EL) of 0.08 G rms (dashed lines). For laboratories undertaking focussed vibration testing, there is a choice between wishing to undertake a non-time-compressed test (so the test is most realistic) or a time-compressed test for efficiency and convenience. For non-time-compressed testing, this research suggests that testing at a single (G rms ) level is likely to result in an under-test and that split level or modulated testing will give a better correlation with actual life. For time-compressed testing, this research suggests that single-level testing is preferable to split level testing. The test level should be set to minimize sensitivity to the value of k, rather than using a time compression of x5 and an assumed value of k. CONCLUSIONS 1. It can be argued that packaging responses to vibration are accumulative in nature and develop more rapidly under more intense vibration; however, it is not known how well these responses are described by the Basquin and Miner Palmgren equations.

12 86 D. SHIRES 2. The calculation of test time compression using the Basquin equation changes substantially if non-stationarity is incorporated by using the Miner Palmgren equation. 3. How often the Basquin constant k is at or near 2 is not known but in the example of a journey record for a leaf-sprung truck, a conventional time compression sets a test level which is relatively insensitive to the actual value of k. 4. If k is higher than 2, then infrequent high intensity vibration events contribute to accumulated fatigue, increasing substantially as k increases. 5. For narrower distributions of vibration intensity, for example air-ride vehicles, the conventional level of time compression leaves the test susceptible to error but this susceptibility is reduced if the test level (and corresponding time compression) is reduced. 6. To achieve least sensitivity to errors in k, the time-compressed test level and corresponding degree of time compression need to be analysed for each specific journey record. It may be that analysis of a greater number of records will give rules of thumb that can be applied with reasonable confidence. 7. Testing at overall journey G rms (for non-time-compressed testing) risks under-testing, and split level or modulated testing will give a better correlation with the actual journey: conversely, split level and modulated testing appear to be more prone to time compression error than single-level testing when time compression is used. ACKNOWLEDGEMENTS The author wishes to thank: Pira International for providing the time and encouragement to undertake this work; The International Safe Transit Association for permission to publish this paper (originally prepared for oral presentation at their 2010 Transport Packaging Forum); and Tetra Pak Packaging Solutions for allowing use of the leaf-sprung truck data recorded on their behalf in China. REFERENCES 1. Curtis AJ, Tinling NG, Abstein HT Jr. Selection and performance of vibration tests shock & vibration monograph 8. The Shock and Vibration Information Center, United States Department of Defence. 1971; Chapter cgi-bin/gettrdoc?ad=ad737830&location=u2&doc=gettrdoc.pdf [accessed 5 February 2010]. 2. Young DE. Focused Simulation. Paper presented at ISTACON. International Safe Transit Association: East Lansing, MI, Kipp WI. Vibration Testing Equivalence: How Many Hours of Testing Equals How Many Miles of Transport? International Safe Transit Association: East Lansing, MI. Paper presented at ISTACON updated December 2008; org/forms/vibration_testing_equivalence-kipp_2000.pdf [accessed 5 February 2010]. 4. Test Method Standard MIL-STD-810G. United States Department of Defence, Method Annex A: Maryland, USA, Standard Environmental Handbook for Defence Materiel Part 5 Issue 4 Induced Mechanical Environments. United Kingdom Ministry of Defence, Chapter 6-01 Annex A: Glasgow, UK, Standard Test Method for Random Vibration Testing of Shipping Containers ASTM D ASTM International: Philadelphia, USA, Standard Practice for Performance Testing of Shipping Containers and Systems ASTM D ASTM International: Philadelphia, USA, Resource Book International Safe Transit Association: E. Lansing, MI, Braunmiller U. Source Reduction by European Testing Schedules (SRETS) Final Report. Fraunhofer ICT: Pfinztal, Germany. 1999; [accessed 5 February 2010]. 10. Young D, Gordon R, Cook B. Quantifying the Vibration Environment of a Small Parcel System. Proceedings of TransPack 97. IoPP: Herndon, VA, 1998, Singh J, Singh PS, Joneson E. Measurement and analysis of US truck vibration for leaf spring and air ride suspensions, and development of tests to simulate these conditions. Packaging Technology and Science 2006; 19(6): , doi: /pts Rouillard V, Sek MA. Monitoring and simulating non-stationary vibrations for package optimization. Packaging Technology and Science 2000; 13(4): , doi: / (200007). 13. Rouillard V, Sek MA. Statistical modelling of predicted non-stationary vehicle vibrations. Packaging Technology and Science 2002; 15(2): , doi: /pts Rouillard V, Sek MA, Synthesizing Nonstationary, Non-Gaussian Random Vibrations. Packaging Technology and Science; In Press.

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