Generating Non-Gaussian Vibration for Testing Purposes

Transcription

1 Generating Vibration for Testing Purposes David O. Smallwood, Albuquerque, New Meico There is increasing interest in simulating vibration environments that are non-, particularl in the transportation arena. A method is presented here for generating time histories that are realizations of a zero mean, non- random process with a specified spectrum, skewness, and kurtosis. A zero-memor nonlinear (ZMNL) monotonic function = f() is generated to convert a zero mean realization with a specified autospectral densit () into a non- waveform. The transformation is generated using the densit method. The zero crossings are preserved in the transformation, which preserves most of the spectral information. The nonlinear transformation does introduce some harmonic distortion of the spectrum that can be reduced in an iterative process. The method does not preserve the phase structure between frequencies and is unable to match higherorder spectra. The resulting time histor can then be used in a simulation or reproduced on a shaker using commerciall available waveform reproduction software. The generation of non- noise has had a renewed interest in the defense industr for two reasons. The first is the realization that man surface transportation and wave environments are non-. The second is the development of shaker control sstems that could replicate long time histories that are non-. The original, and man current shaker control sstems, for generating random vibration tests generate onl random noise. However, waveform replication techniques now allow the reproduction of an waveform whose characteristics are within the bounds of a shaker. Man articles have been written on the subject of generating non- waveforms. This article uses the method of a zero-memor nonlinear (ZMNL) function. This method was proposed as far back as 967., At that time, analog circuits were used to implement the methods. The basic method relies on the equation: f f d ( ) = X( ) d where: f () = probabilit distribution of the random variable f () = probabilit distribution of the random variable X It is understood that in Equation, stands for = g (), where = g() and the densit function depends on, not. If = g() is not monotonicall increasing, the function must be broken into parts, and each part handled separatel. For the purposes of this article = g() will be restricted to a monotonicall increasing function. Therefore, d/d is alwas positive. For this article, the distribution of X will be assumed to be a zero mean, unit-variance, distribution. If the distribution f () is known (the distribution of f X () is assumed to be ), d/d can be found, and integration will ield = g (). Inversion will ield = g(). = g() is the ZMNL function. The ZMNL function will be scaled so that a unit variance X will ield a unit variance. In man real applications the complete distribution f () is not known, but estimates of the skewness and kurtosis are known. The idea is to find a relativel simple function = g() so that the distribution f () is zero mean unit variance, with a specified skewness and kurtosis. The conversion of a realization of to using a ZMNL func- Based on a paper presented at the 7th Shock & Vibration Smposium, Virginia Beach, VA, October. () Figure. ZMNL function, S =, K =. tion will alwas produce an autospectral densit for that is different than the autospectral densit of. The effect is to add harmonics to all the Fourier components of. This will make the transformed data appear whiter than the original data. However, most of the spectral information is contained in the zero crossings, which are preserved, and if the nonlinearit is not too great, the spectral change is usuall acceptable and the autospectral densit of will be near the autospectral densit of X. Implementation In this implementation, a realization of X, is generated with the desired autospectral densit shape and scaled to a unit variance. The realization is converted to a realization of using = g(). The realization is then rescaled to the desired variance and used in a computer simulation or reproduced on a shaker using waveform replication techniques if a vibration test is desired. Five constraints have to be met the area under the probabilit densit function must be unit, the mean is zero, the variance is unit, the skewness is S and the kurtosis is K. A = f ( ) d = S Ú- Ú- Ú- = Ú f - = Ú- m s = f ( ) d = = f ( ) d = ( d ) K f ( ) d The first criterion is automaticall met for a valid distribution. The third is a convenience. The realization can be rescaled later to the desired variance. In this implementation the integrals are approimated with sums. Waveform Generation Using a ZMNL Function The central idea is to find a function = g(), which will convert a waveform into a non- waveform with a specified skewness and kurtosis. The ideas have been presented before, but not coupled with an algorithm to generate the waveform with an arbitrar spectrum shape. () 8 SOUND AND VIBRATION/OCTOBER

2 6 d/d Figure. d/d, S =, K = Figure. Probabilit densit, S =, K = iteration of amp Frequenc, Hz Figure. Autospectral densities, S =, K =. In this article, the waveform is generated using the same methods used for generating waveforms for random shaker testing. The waveforms are then distorted using the ZMNL function to create non- waveforms. Three different methods are presented for generating the ZMNL function. Each has slight advantages and disadvantages. Each will cover a slightl different range of skewness and kurtosis. All Figure. Small portion of time histories, S =, K =. will give similar results. Method. We would like the function = g() and its derivative, d/d, to be continuous. We will actuall generate the inverse function = g - () and use a linear interpolation to generate the function values. The function chosen for this implementation is: A a Ê a ˆ ( )= Á + a - AB a a Ë a a = A + AB - b a () =-A Ê b ˆ Ê -- ( b ) ˆ Ë Á b - b + b Ë Á b + < -b d -a a- = Aa d > a = A - b a -b =-Ab b - - b ( ) <- There are si parameters in the model (A, α, a, b, β, B), more than needed, which ma lead to multiple solutions. However, the sith parameter, B, was found to be necessar to force the mean to zero in some cases. B is the onl parameter allowed to be negative. The function and its first derivative are continuous over the entire range of real values. The function is monotonicall increasing. The function is linear for small values of ( b a). The MATLAB function fminsearch was used to find the parameters for a given skewness and kurtosis. The error function: E = A- + m + s - + S - S + K - K () was minimized. In general, solutions for negative skewness are not necessar. A time histor with a positive skewness can alwas be inverted to achieve the same amplitude of negative skewness. Method, Cubic Function. Other suitable functions can be found. Winterstein, et al., used a cubic transformation and the inverse (which form depended on K): 6 = c + c + c + c K = c + c + c + c K < To assure the function is monotonic, the derivative must be positive at zero and should have no real roots. This results in the following constraints: c >, c >, c < cc (7) This requires use of a nonlinear optimization method with multiple inputs and constraints. The eample here uses fmincon from the MATLAB optimization toolbo. The cubic function does have the advantage that the second derivative is () (6) DNAMIC TESTING REFERENCE ISSUE 9

3 - iteration of amp Figure 6. ZMNL function, S =, K = Frequenc, Hz Figure 9. Autospectral densities, S =, K = 6. d/d Figure 7. d/d, S =, K = Figure 8. Probabilit densit, S =, K = 6. also continuous which results in a smoother distribution. Winterstein, et al., also considered using distributions other than for X. 6 Method, Hermite Polnomials. A closed-form solution for a cubic function is given b Winterstein 7 using Hermite polnomials for K >. In this formulation, is the non- waveform, and is the waveform. / / ( ) = È ( ) + c+ ( ) - È ( ) + c-( ) - a ÎÍ ÎÍ (8) Figure. Small portion of time histories, S =, K = 6. where: ( ) =. b Ê ˆ hˆ a+ a c b a Ë Á k -, a =, b, = ( - - ) hˆ = h ˆ ˆ S h ˆ =, h = + +. K - k = + h ˆ + 6h ˆ ( ) ( ) - +. K - 8 The positive roots are used. An improvement suggested b Gurle 8 uses the above solution as a starting point to solve the following pair of nonlinear equations: ( ) S= k 8hˆ + 8hh ˆ ˆ + 6hh ˆ ˆ + 6hˆ (9) K = k ( 6hˆ + 8hˆ + hˆ hˆ + 6hˆ + hˆ + 96 ˆ 76 ˆ ˆ ˆ () h + h h + h + ) These solutions do not guarantee a monotonic function for all S and K. Generall, a kurtosis greater than three is associated with a softening nonlinearit (widening distribution tails), and a kurtosis less than three is associated with a hardening nonlinearit (narrowing distribution tails). The range of the skewness is limited b: 7 S - + K The range of skewness and kurtosis for a unimodal probabilit densit function is much less 7 than that indicated above. A closed-form solution for a cubic function is given b SOUND AND VIBRATION/OCTOBER

4 - iteration of amp Figure. ZMNL function, S =., K =.. -6 Frequenc, Hz. Figure. Autospectral densities, S =., K =... d/d Figure. d/d, S =., K = Figure. Probabilit densit, S =., K =.. Winterstein 7 using Hermite polnomials for K <. S K - ( )= - ( -)- ( - ) 6 () As above, this formulation will work onl for a limited range of S and K. Eamples This section illustrates several eamples. We will start with Method and compute realizations for skewness of zero with Figure. Small portion of time histories, S =., K =.. two values of kurtosis, one less than three, and one greater than three. Remember a skewness of zero and a kurtosis of three is a waveform. We will then increase the skewness and compute an eample. Method is then used and compared to Method with zero skewness and a kurtosis less than three. Method is then compared to Method for a nonzero skewness and a kurtosis greater than three. In each case, the ZMNL function, d/d, the (probabilit densit function), the autospectral densit, the time histor and the non- time histor will be plotted. S =, K =, Method. The parameters needed to achieve a skewness of and a kurtosis of two are: A =.767, a =.87, a =.68, b =.68, b =.87, B = Note that the eponents are greater than one, which will generall be the case for a small skewness and a kurtosis less than three. The resulting ZMNL function is shown in Figure. In all the eamples, the ZMNL has been normalized for a standard deviation of one. The derivative d/d and the resulting are shown in Figures and. A is also shown for reference. Note that a trimodal is generated b the requirement to suppress large values of to achieve a small kurtosis. (The kurtosis of a waveform is three.) A realization of, points was created using the target spectrum shown in Figure. The frequenc-domain method discussed b Smallwood and Paez 9 was used to generate the time histories. This is basicall the same method used to generate time histories for random vibration control sstems. A DNAMIC TESTING REFERENCE ISSUE

5 - iteration of amp Figure 6. ZMNL function, S =, K =, using the cubic function. -6 Frequenc, Hz 6 Figure 9. Autospectral densities, S =, K =, using a cubic function. d/d Figure 7. d/d, S =, K =, using a cubic function normal Figure 8. Probabilit densit, S =, K =, using a cubic function. Hanning window with 7% overlap was used. The sample rate was,, and the block size was 8. This method is basicall the inverse of the Welch method for estimating an autospectral densit. After the autospectral densit () of was estimated, a single iteration of the spectrum of was performed to correct the of in an attempt to improve the of. The resulting is plotted on Figure. Also shown in Figure are the estimated of the waveform,, and the non- waveform,. A small portion of the time histories is shown in Figure. Note how the large values Figure. Small portion of time histories, S =, K =, using a cubic function. of have been suppressed in. The whitening effect is seen mainl in the notch and the roll-off at high frequencies. S =, K = 6, Method. For this eample, the skewness was kept at zero, and the kurtosis was increased to si. The same target in the previous eample was used. A realization of, points was generated as in the previous eample. The parameters for this eample are: A =.89, α =., a =.779, b =.779, β =., B = In this case, the eponents are less than one. Figures 6 through show the same information as the previous eample. Notice that now the peaks in are eaggerated versions of the peaks in the. A slight ecess in the of is seen near Hz. This is the third harmonic of the peak in the at Hz. As in the previous eample, the largest deviations from the reference spectrum are at the minima. S =., K =., Method. This is an eample of a nonsmmetric waveform. The parameters are: A =.8, α =., a =.8, b =.98, β =.766, B =. The same in the previous eamples is used. As before, a realization of, points was generated. Note that α is smaller than β, which suggests a positive skewness. The results are shown in Figures through. S =, K =, Method, Using a Cubic Function. The same spectrum was used in the first eample. In this case, the cubic function was: SOUND AND VIBRATION/OCTOBER

6 ZMNL Interpolated values Figure. ZMNL function, S =., K = Frequenc, Hz. Figure. Autospectral densities, S =., K =...8 d/d Figure. d/d, S =., K = pdf Figure. Small portion of time histories, S =., K =.. S =., K =., Method. This is the same as one of the previous eamples, ecept Method was used. The parameters are: c =.87, c =.8, c =.66, c =.878 This is an eample of a nonsmmetric waveform. The same from the previous eamples is used. As before, a realization of, points was generated. The results are shown in Figures through. The results are ver similar to the previous eample. Figure shows that the iteration energ is removed from the near the minima to correct the spectrum of with some success. Method results in essentiall identical results for this eample Figure. Probabilit densit, S =., K =.. with the parameters: = c + c + c + c c =, c =.799, c =, c =.7 The results are shown as Figures 6 through. A comparison to Figures through shows that the two ZMNL functions give ver similar results. This eample does illustrate that the solution is not unique. Man functions can be found that will result in a transformation with the same skewness and kurtosis. Conclusions Three simple but effective methods for generating realizations of a stationar random process with specified skewness and kurtosis using a zero-memor nonlinear (ZMNL) function are presented. The all give similar, but not identical results. Each covers similar but slightl different ranges of skewness and kurtosis. Each will result in slightl different distributions. All the methods are relativel eas to implement using modern computational tools such as MATLAB. MATLAB is available from The MathWorks, Acknowledgment Sandia National Laboratories supported this work. Sandia is a multiprogram laborator operated b Sandia Corporation, a Lockheed Martin Compan, for the United States Department of Energ under contract DE-AC-9-AL8. DNAMIC TESTING REFERENCE ISSUE

7 s. Gujar, U. G., Generation of Random Signals With Specified Probabilit Densit Functions and Power Spectral Densit Spectra, M.Sc.E. Thesis, Dept. of Elec. Engrg., Universit of New Brunswick, Fredericton, NB, Canada, Gujar, U. G., and Kavanagh, R. J., Generation of Random Signals with Specified Probabilit Densit Functions and Power Densit Spectra, IEEE Transactions on Automatic Control, pp , December Wirsching, Paez, and Ortiz, Random Vibrations, Theor and Practice, ISBN , Wile, 99., Wise, Gar L., Traganitis, Apostolos, P., and Thomas, John B., The Effect of a Memorless Nonlinearit on the Spectrum of a Random Process, IEEE Transactions on Information Theor, Vol. IT-, No., Januar Bendat and Piersol, Random Data, Analsis and Measurement Procedures, Third Edition, Sec..., ISBN -7-7-, Wile,. 6. Winterstein, S. R., Lange, C.H, and Kumar, S. Fitting: A Subroutine to Fit Four-Moment Probabilit Distributions to Data, SAND9-9, Sandia National Laboratories, Albuquerque, NM, Januar Winterstein, S. R., 988, Nonlinear Vibration Models for Etremes and Fatigue, J. of Engr. Mech., Vol., No., pp , Oct Gurle, K., and Kareem, A., 997, Analsis Interpretation Modeling and Simulation of Unstead Wind and Pressure Data, J. of Wind Engineering and Industrial Aerodnamics, 69-7 pp , Smallwood, D. O. and Paez, T. L., A Frequenc Domain Method for the Generation of Partiall Coherent Stationar Time Domain Signals, Shock and Vibration, Vol., No., pp., 99.. Tebbs, J. D., and Hunter, N. F., Digitall Controlled Random Vibration Tests on a Sigma V Computer, 97 Proceedings of the Institute of Environmental Sciences, pp. 6-, 97. MATLAB scripts for generating non- vibration test signals using the methods presented in this article are available from the author: dsmallwood@comcast.net. SOUND AND VIBRATION/OCTOBER