9th National Congress on Theoretical and Applied Mechanics, Brussels, 9-10 May 2012

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1 Random Vibration Testing Using a Pseudo-Random Method with Crest-Factor Limiting: An experimental comparison with the classical method J. MARTINO, ir. 1 1, K. HARRI, dr. ir Royal Military Academy, Department of Mechanics Avenue de la Renaissance 30, 1000 Bruxelles jonathan.martino@rma.ac.be, kristof.harri@rma.ac.be Abstract A classical, broadly implemented testing method, often considered as a reference standardized testing method (e.g. MIL-STD ) is the well-known random method. Although it commonly offers fast testing capability with relative good reliability, it presents two major drawbacks. Firstly, a basic control system generates an amplitude time signal following a normal (Gaussian) distribution with a (little) probability of having extremely high values exceeding the maximum admissible value for the amplifier which reduce the RMS level of the signal and the dynamic range of the testing facility. Secondly, the only way to reduce the crest factor and to avoid these extreme values consists of a sigma clipping truncating the signal and so increasing the PSD noise floor and reducing the dynamic range. Alternatively, the crest factor can be limited by using a multisine approach. A multisine composition can cover the same PSD requirements as the classical random by only dimensioning the amplitude of the sines. On the other hand, the phases can be chosen freely and, if good chosen, can lead to a reduced crest-factor (e.g. Schroeder phase coding). This article presents a comparison between the experimental results of both classical and pseudo-random methods. The analysis is done for a certain acceleration amplitude passing through the amplifier and introduced into the device under test. Obtained FRFs will also be compared. Keywords Multisine, Kurtosis, Random control, Pseudorandom, experimental comparison, NCTAM I. INTRODUCTION WE can not ignore the importance of vibration tests for both the industry and the research. The industry has long understood the need to design its products and to perform environmental tests to guarantee a certain resistance to the future load that will be undergone by the material during its life cycle. This led to the establishment of standards (e.g. Mil-Std 810). These have the advantage of standardizing environmental testing for the entire industry, providing the same characteristics, types of signals, energy content, levels,... However, these standards have a downside: the loads imposed do sometimes not accurately (or even sometimes not at all) reflect the stresses incurred by the equipment during use. Remind that the purpose of environmental testing is to predict the behavior of the material in its future operating conditions. Note also that opportunities for fixing material on the test facilities are limited and very often interfaces must be used to approach the real-life conditions or more simply to allow the fixture e.g. head-expander. These interfaces have their own behavior which can lead to control perturbations of the load entering the device under test (DUT). In some sectors such as aerospace, there are sometimes very specific requirements in terms of signals, contents and other. This often goes out of the standards and has to be tailored according to the application. It is therefore interesting to study less conventional signals to offer a closer approach of the reality and to better customize the tests. Apart shocks, the most commonly encountered loads are the sine sweep and random, the first allows an independent determination of modes of vibration but need more time while the latter is faster but requires many averages making this gain relative. The sine sweep also has the advantage of being completely deterministic and therefore the possibility to avoid leakage but this type of load can often not be used for environmental testing. This is why preference is given to the random method to approximate reality. As mentioned above, they are certainly faster to obtain a frequency response but it contains a heavy random component. This results in two needs: a first need is to average the signals and a second one is to use windows to avoid leakage. Moreover, the generation of random signal in the controller is mostly exclusively Gaussian. This means that the drive can contain any value under the distribution curve, theoretically from to +. Clearly, a limit exists beyond which it can damage the amplifier. To avoid this, the generally used method, the sigma clipping, truncate the drive with or without a polynomial remodeling [1], resulting in a higher noise floor and thus a lower dynamic range. It is also worth noting that most basic controllers only allow generation of Gaussian noise while in many cases the actual loads are not (named as non-gaussian loads) [2]. The use of advanced control software, often expensive, is then required to control the kurtosis. This parameter will be introduced in the next section and describes non-gaussian noise. It is possible to profit of the benefits of the sine sweep while maintaining a 1

(pseudo-)random behavior without the constraints of classical random. This alternative involves the time replication of pseudo-random multisine.

2 (pseudo-)random behavior without the constraints of classical random. This alternative involves the time replication of pseudo-random multisine. This offers the possibility of sizing a frequency content as controlling the kurtosis. The comparison of the use of these signals with conventional ones is the subject of this article. The aim is to see the advantages of using a low kurtosis signal in the control of a random test. The approach will be conducted on the control of the shaker equipped with its head expander because this configuration is generally the basis of a test and because the stress entering the DUT can be disrupted by internal resonances of the system shaker head expander. Two case studies were developped. The first one is a standard test according to the Mil-Std 810G Method Category 4 - Composite wheeled vehicle. The stress profile is relatively simple and is located in an area of relatively low frequencies ( Hz) [3]. The second case study will be conducted over a frequency range including resonances and pushed to a level approaching the limits of the test facility in terms of forces and accelerations, potentially making the control difficult. a probability distribution with thicker tails and a negative kurtosis a probability distribution more centered around the mean. Figure 2 shows three time histories which are characterized by the three cases of kurtosis: negative, zero and positive. Fig. 1. Probability density functions for three different kurtosis values: zero, positive and negative A. Multisine signals II. THEORETICAL ASPECTS A multisine signal is a linear combination of sinusoids of different amplitudes and phase shifts. x(t) = N k=1 A k sin(2π f k + ϕ k ) (1) There is a link between the desired PSD profile and amplitudes of sinusoids. A k = 2 f S( f k ) (2) With f the frequency resolution, S( f k ) the RMS level of the PSD profile in the considered frequency and ϕ k the phase shift. B. Kurtosis and Crest Factor Kurtosis is defined [4] as the fourth statistical moment of a stochastic phenomenon X γ 2 = E [ (X µ X ) 4] σ 4 X 3 (3) where µ X is the mean and σ X the standard deviation of this stochastic phenomenon. The term 3 is introduced to make γ 2 = 0 for a Gaussian distribution. Kurtosis is a parameter characterizing the concentration of random values around the mean and the thickness of the tails. As shown in figure 1, a zero kurtosis characterizes a Gaussian distribution while a positive kurtosis represents Fig. 2. Time histories corresponding to the probability density functions depicted in figure 1 Even if the kurtosis is an indication of the height of the peaks potentially contained in the time history, the maximum amplitude actually present in the time domain signal is subjected to stochastic process. The kurtosis can thus not fully characterize all the time signals of a same probability distribution, we have therefore to use another parameter: the Crest Factor. 2

3 The crest factor (CF) is defined as the ratio between the maximum value present in the signal and its RMS value. CF = max(x(t)) min(x(t)) 2 σ X (4) or can be expressed in function of the L 2p -norm [5]: 1 T x(t) 2p = 2p [x(t)] 2p dt (5) T 0 With: CF = M+ M 2 E e f f (6) E e f f = N k=1 It is clear that either M + = f, or M = f. Using signals with N =, it is obvious that a square wave has a CF = 1, which is the best that can be obtained. The crest factor of a pure single sine wave equals 2. The crest factor must be larger than one due to its definition and the equality between E e f f and L 2, as follows from Parceval s relationship [6]. As mentioned above, the maximum allowed value for the amplifier limits the ability of the test facility. For a same maximum value max( x(t) ) of the amplifier, a low crest factor signal will yield an RMS level larger than a high crest factor signal. The reduction of crest factor is therefore an objective in itself and different methods can be used for this purpose. For a given probability distribution and thus for the same kurtosis (e.g. Gaussian, γ 2 = 0) it is possible to obtain several different crest factors as shown in following figure. In a basic control software that commonly only implements Gaussian noise generation, this property is used to optimize the crest factor. Only the most advanced software that supports control of kurtosis can further reduce the crest factor by reducing the kurtosis and thus more reduce the probability distribution tails. The different methods will be explained later. A 2 k 2 (7) Fig. 3. Three different time histories with their corresponding CF for a same probability density functions Gaussian in this case III. MULTISINE VS. RANDOM Although the purpose of sizing a multisine and a random signal is identical i.e. to reproduce some frequency content and by extension a certain PSD profile, some subtleties make the use of tailored multisine signals more attractive. The overall approach of a random or a multisine is initially the same, starting with a PSD profile that is discretized in spectral lines and for which an inverse Fourier transform is performed to obtain a time signal. In regard to the theory of the Fourier transform, the time signal is a linear composition of sine waves. While generating a random signal requires further computation to obtain a pure random signal [7], the multisine signal generation stops here, that s why we talk about pseudo-random signals. The amplitudes of the sine waves are determined using the formula 2 and injected into the formula 1. We note however that a parameter is undefined: the phase shift. The difference between random and multisine lies in the phase shift control. A. Control of the peakiness beyond control of the phase shift An advanced control of the peakiness involves control of kurtosis and is made possible by a control of the phases of the harmonics present in the target PSD profile. Regarding the classical random signals and although basic controllers do not support the control of kurtosis, it is still possible to reduce the crest factor. A random distribution of the phases between 0 and 2π allows to generate a time signal with a Gaussian distribution. The typical order of magnitude of the crest factor for such a signal is about 3.5. To reduce this value, based on the principle illustrated in Figure 3, we generate a large amount of Gaussian time 3

4 signals and we retain only those where the crest fgactor is sufficiently reduced. Acceptable values after 100 generations are of the order of 2 [8] but this is time constraining and only allows a replication of the selected optimal signal. Regarding the multisine, different optimization methods have been developed and can reach crest factor values up to 1.65 for the Schroeder method [9], 1.5 for the Van den Bos [5] and genetic algorithm [8] methods or even 1.4 for an optimization of the L 2P -norm. Some of these methods (e.g. Schroeder method) also have the advantage of relative speed and can therefore be implemented to permit an adaptative control of the signal. B. High dynamic range despite a low crest factor Such levels of the crest factor are only achievable by optimizing the phases of a multisine. The only alternative in terms of random is a sigma clipping but truncation of peaks induces a distortion in the signal what leads to noise floor enhance and implies a significant loss in dynamic range [1]. This is not the case when using a tailored multisine, the height of the peaks is naturally reduced, therefore no distortion is present and the frequency content remains intact. time block. The first aim is to see how a single shot multisine control behaves; there will only be one replay of the time history and therefore no adaptation of the ITF. The comparison of these two methods will be done using the measurements taken by the control accelerometer to qualify the signals actually entering into the DUT. A second comparison will be based on the frequency response functions (FRF) measured on a simple structure. A preliminary sine sweep study has resulted in the FRF of this structure for a large frequency range from 10 to 2000 Hz. This reference FRF is given in figure 4 and will be used to compare the FRF obtained from each of the methods presented here. Case Study 1 The first case study is a standardized test according to standard MIL-STD-810G Method Cat 4 Composite wheeled vehicle [3] with an RMS level of 2.24 g Case Study 2 The second case study presents a flat profile with an overalls RMS level of 12.8 g on a frequency domain covering internal resonances in the shaker. C. Fully deterministic signals avoiding leakage problems Due to the further computations applied to a multisine based random signal, it has a problem of leakage and windowing has to be applied resulting in a dillution of the frequency content. Unlike the random, the multisine signals are fully deterministic and therefore do not have this problem. IV. EXPERIMENTS As mentioned earlier, tests were conducted on the shaker equipped with its head expander and two case studies were defined. The control of the random signal is performed by the Test.Lab Random Control software from LMS. This control is an adaptive control of the drive through the inverse transfer function (ITF) of the vibration facility. On the other hand, similar tests showing identical characteristics in terms of PSD will be executed using multisine signals generated off-line and then reproduced by time replication. This replication is done using the module SAWR (Single Axis Waveform Replication) of the LMS Test.Lab software. For both tests, the frequency resolution was defined to be 1 Hz. The duration of each measurement is then 1 second. While the random control is set to 60 seconds long with an adaptation of the ITF every 5 averages what gives 12 chances to optimize the control PSD, the multisine duration is only 1 second (1 block) without any adaptation of the ITF. Note that the software still offers the possibility to adapt the ITF between two replays of the Fig. 4. The Reference FRF of the DUT used to compare the two methods: Random and Multisine 4

5 A. Composite wheeled vehicle vibration test The PSD profile is given in figure 5. Fig. 7. Case study 1: PSD profile measured from the control accelerometer and the target PSD profile for the random method Fig. 5. Target PSD profile for the composite wheeled vehicle vibration test Taking a look at the ITF (figure 6) obtained for the Composite Vehicle Test, no major troubles are to be found there, we may expect a good control. Note only that an electrical disturbance at 100 Hz is present which tends to be compensated by the ITF. If we look at the FRF of the DUT, we can see that the two other little peaks are due to resonance frequencies. Fig. 8. Case study 1: PSD profile from the control accelerometer and the target PSD profile for the multisine method Fig. 6. Inverse Transfer Function for the composite wheeled vehicle vibration test The two figures below show the control PSD as the target PSD respectively for the random control and for the multisine control. Looking at Figure 8, we see directly the component at 100 Hz in the PSD of the multisine. This is due to the fact that the time block is being replicated only once without any adaptation of the ITF. However, the PSD profile for the multisine control is much closer to the target profile. In order to visualize this difference, figure 9 represents a zoom on Hz of the random, multisine and target PSD profiles. 5

6 expander can disrupt the stress entering the DUT due to a lack of control, it may be more subtle that the DUT itself can induce perturbation to its own load. The ITF has to be looked in regard with the FRF of the DUT because this latter influences the behavior of the shaker and makes its control more difficult. We see a lot of disturbances due to the DUT but also a large fall of the tension per g at 1785 Hz meaning that it is a internal resonance frequency of the shaker itself. Fig. 9. Case study 1: Zoom on the PSD profiles from the control accelerometer for the random and multisine methods Regarding the FRFs now, there is a big difference. The quality of the FRF is much better for the random control method. As the averages are used to smooth the frequency profiles, it is quite normal to obtain a better result. However the obtained FRF from the multisine covers the same major resonance frequencies (84 Hz and 430 Hz). Fig. 11. Inverse transfer function of the shaker facility between 900Hz and 2000Hz Fig. 10. FRFs obtained from the random control method and the multisine control method B. High level beyond resonance frequencies vibration test The aproach here is different, the aim is clearly to try to control the shaker into a field where it is difficult to control it to see how each method behaves. The used PSD profile is a flat profile at high level located around the resonance frequency of the test facility. The ITF, shown in figure 11, can give us an idea about the frequencies where the control could be problematic. If it is obvious that an internal resonance frequency of the system shaker head The next figure shows the PSD profiles of the control accelerometer for both methods random and multisine and the target PSD profile. We can see that the random control presents some difficulties to keep the signal as close as possible to the target profile. Along the test duration, the controller tries to minimize the exceeding peaks without observable results leading to a global reducing of the PSD level to compensate the uncontrollable peaks. The multisine signal also presents a control difficulty at 1034 Hz. Beyond this, the multisine signal appears to be truncated from 1750 Hz. The global shape of the PSD from the control accelerometer, compared to the PSD of the initially generated signal, suggests that a low-pass filter is applied and that we are in the roll-off zone of the filter. In order to play the time history by the shaker, the SAWR module performs a resampling and internal signal processing. This roll-off can be a limitation of the SAWR software but do not make the results before 1750 Hz irrelevant. 6

7 Fig. 12. Case study 2: PSD profiles from the control accelerometer for both methods and the target PSD profile If now we compare the FRFs obtained from random and multisine controls for the second case study, we see very clearly that the difficulties of random control and incessant adaptations of the ITF make the this FRF totally unusable because buried in noise. The FRF obtained by the multisine control is much smoother. (1 second compared to 60 seconds for the random control). The multisine control was implemented in the two settings without major problems while the random control was more laborious in the case of a high level load at high frequencies. The random control has only been possible by greatly widing the safety margins within the control software. For a normal test a common value of ±6 db is used to define the safety margins, saying that the test has to be aborted if at least one frequency of the control PSD deviates from the target PSD by four times the expected level, denoting a lack of control. This value was here set to ± 12 db. In all the cases, the multisine control is globally cleaner and closer to the target profile. About the measured FRFs for the first case study with an easy control of the random control, the latter does better but benefits of adapting its ITF as applying averages and thus smoothing the curves. The obtained FRF of the multisine control is somewhat buried in the noise, problem that could be solved by applying the same averaging methodology as for a random control. Regarding the second case study, the trend is reversed. The random control is difficult and can not be stabilized resulting in an unusable FRF. Multisine signals on the contrary are successful. As expected, multisine signals are faster while maintaining overall better control. Raw exploitation of the measurements is possible without requiring a particular processing (smoothing, averaging, windowing,...). The obtained FRF are representative with respect to the reference FRF. Further improvements may be implemented to improve results as the use of a adaptive ITF control method for the multisine. REFERENCES Fig. 13. FRFs obtained from the random control method and the multisine control method V. CONCLUSIONS This study has shown the advantages of the use of multisine signals in comparison with conventional random signals. This study was conducted through two very different settings, covering a wide frequency range and variety of levels. The outlined theoretical considerations showed the benefits of multisine signals i.e. naturally low crest factor and thus low introduced levels without loss of signal quality in the frequency domain, avoidable leakage and speed [1] Alexander Steinwolf, Random Vibration Testing With Crest Factor Limiting By Kurtosis Manipulation, Experimental Techniques, September/October [2] John Van Baren and Philip Van Baren, The Third Dimension Of Random Vibration Control, Vibration Research Corporation, MI [3] MIL-STD-810G, Department of Defense Test Method Standard for Environmental Engineering Considerations and Laboratory Tests, USA, 31 October 2008 [4] Kihong Shin and Joseph K. Hammond Fundamentals of Signal Processing for Sound and Vibration Engineers, John Wiley & Sons Ltd [5] Edwin Van Der Ouderaa, Johan Schoukens and Jean Renneboog Peak Factor Minimization of Input and Output Signals of Linear Systems, IEEE Transactions on Instrumentation and Measurement, Vol. 37. No 2. June 1988 [6] Steven. W. Smith, The Scientist and Engineer s Guide to Digital Signal Processing, California Technical Publishing, 1997 [7] David O. Smallwood, Vibration with Non-Gaussian Noise, Sandia National Laboratories [8] Andrew Hornera and James Beauchamp A Genetic Algorithm- Based Method for Synthesis of Low Peak Amplitude Signals, Acoustical Society of America, January

8 [9] M. R. Schroeder, Synthesis of Low-Peak-Factor Signals and Binary Sequences with Low Autocorrelation, IEEE Trans. Information Theory (Corresp.), vol. IT-16, pp , Jan [10] Alexander Steinwolf, Random Vibration Testing Beyond PSD Limitation, Sound and Vibration, September

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