NONLINEAR RANDOM DISPLACEMENT AND FATIGUE ESTIMATES USING PDF TRANSFORMATIONS

Transcription

1 NONLINEAR RANDOM DISPLACEMENT AND FATIGUE ESTIMATES USING PDF TRANSFORMATIONS K. A. Sweitzer and N. S. Ferguson ITT Industries Space Systems Division LLC, Rochester, NY, USA Institute of Sound and Vibration Research, University of Southampton, UK INTRODUCTION The estimation of displacement and fatigue life of nonlinear structures subjected to random excitation presents several analysis complications that will be discussed. Recent nonlinear random vibration experiments performed at Wright Patterson Air Force Base (WPAFB) [1] have generated data that has been analyzed in the time and frequency domains to model this complicated behavior. This paper will discuss how the probability density function transform and inverse distribution function methods have been applied. Probability density function (PDF) transformations have been shown to be useful for estimating nonlinear random fatigue damage, but preliminary applications were limited [2]. Analysis of the WPAFB data has employed additional normal PDF, peak PDF and nonlinear peak PDF transformations. These PDF transforms were used to estimate rates of zero crossings and more importantly, two dimensional rainflow matrices (RFM) of total stress. The inverse distribution function method has been shown to be very effective for estimating linear to nonlinear functional relationships. Several functional relationships of linear to nonlinear displacement and stress are presented, illustrating the method using response and peak distribution functions. Standard closed form PDF transformation methods require the linear to nonlinear functional relationships to be both differentiable and invertible. Numerical PDF transformation methods have been developed that require only differentiable functional relationships. Examples of the numerical methods will be presented for a two step serial application of the PDF transform. Analyses of the WPAFB data and subsequent ODE numerical experiments with Duffing equation systems have shown that total stress peak PDFs (or 2D RFMs) can be estimated using the PDF transform method. These estimated RFMs and the resultant time to failure will be compared to estimates using the raw WPAFB experimental results. PROBABILITY FUNCTIONS FOR NONLINEAR SYSTEMS The response PDFs of linear systems to Gaussian (normal) random inputs has been shown to be normal. If the system is dominated by one mode, the distribution of response peaks can be assumed (to a first approximation) to have a Rayleigh PDF [3]. A generalized peak PDF for the response of any linear system has also been developed [3, 4]. If the system is nonlinear, or the input distribution is not normal, the closed form equations for the response and peak PDFs are not valid. In these nonlinear cases, the PDF transform method can be used to

2 estimate the resultant PDFs. To review, the basic transform of a PDF p(x) of a know function to a PDF p(y) of another function is: p( x) p( y) = (1) dy dx The key to this equation is determining a one-to-one relationship between x and y, i.e.: y ( ) = g x (2) where x and y are the linear and nonlinear value respectively. Previous attempts to determine this functional relationship were based on a power law model using the RMS value of response at different input levels [2]. Research has uncovered the inverse distribution function (IDF) method to estimate linear to nonlinear functional relationships using data from the full range of distribution functions of x and y [5, 6]. Consider the distribution function of x (or F(x)) (also referred to as the cumulative distribution function (CDF)) which is defined as the probability of occurrence ρ up to a value x: ( ) Probability ( ) F x = X x = ρ (3) The distribution function monotonically increases from F ( ) = 0 to F ( ) = 1. The distribution function is related to the PDF by: x ( ) ( α ) 1 The inverse distribution function (IDF) or F ( ρ ) F x = p dα (4) determines the corresponding x value of a distribution function given a value of probability of occurrence: ( ρ ) ( ( )) F 1 F 1 F x x = = (5) The linear to nonlinear functions can be estimated discretely by solving the inverse problem, given distribution functions for both the linear x and nonlinear y. The IDF method produces pairs of x and y values that can be used numerically or curve fit to form an equation like (2). When working with nonlinear data, it is convenient to start with the nonlinear estimates of the distribution function determined from a histogram or PDF of the data: ( ) F y = ρ (6) NL i i At each value of ρ i find the point on the linear distribution function with equal probability, and then use the IDF to determine the corresponding linear function value: 1 ( ) ρ ( ); ( ( )) F y = = F x x = F F y (7) NL i i Lin i i Lin NL i Inverse distribution function of normal response displacement. The response distribution function for a linear system exposed to normal random excitation is normal, and can be expressed (for data with zero mean) as [3]: 1 x Fnorm( x) = 1 erf ρ 2 + = σ 2 where erf is the error function. This normal distribution function can be written as an IDF: (8)

3 F ( ) 1 ( ) 2erf norm ρ σ ρ = = x (9) 1 where erf is the inverse error function. Estimates of nonlinear displacement response PDFs from the WPAFB data are shown in Figure 1-a overlaid with a normal distribution function, with the results being normalized by the standard deviation of the corresponding data set. An example distribution function (CDF) determined from the 8 g PDF is shown in Figure 1-b overlaid with a normal CDF. The inverse method can be traced on Figure 2, which shows the positive response portion of the linear and nonlinear CDFs. Given a nonlinear normalized displacement of 0.8 σ, what linear displacement will have the same CDF? By inspection, the corresponding linear displacement is approximately 0.7 σ. Another example is 1.8 σ nonlinear mapping to 2.0 σ linear. Figure 1. Normalized displacement response PDFs. Normalized nonlinear displacement response distribution function (CDF) and linear response CDF Linear 0.80 Nonlinear 1.81 Nonlinear 2.00 Linear Figure 2. Linear and nonlinear displacement cumulative distribution functions for 8 g Input.

4 The IDF method can be repeated at every nonlinear response CDF value. The resulting x and y pairs are shown in Figure 3-a for the WPAFB data. The positive pairs are shown again in Figure 3-b on log-log scale. Note that the results from each input level form a set of unique curves. Curve fits of these functions can be used to estimate σ, kurtosis and rates of zero crossings. Figure 3. Linear to nonlinear displacement response functions. Log-log plot of linear to nonlinear displacement response functions for WPAFB data. Curve fitting the linear to nonlinear functions. In previous attempts [2] a piecewise power-law was used to curve fit the RMS stress response functions. Research with similar nonlinear functions [5] suggests that a polynomial function of the nonlinear response is the preferred method to curve fit stress stiffening data: ( ) h y = x 2 n cy 1 + cy 2 + L + cy n = x where y and x are the nonlinear and linear displacement levels respectively. Note that the polynomial is now on the left hand side of the equation. A third order polynomial proved to be a good fit of the displacement data. Nonlinear peak displacement PDF. The IDF method can also be applied to estimate linear peak to nonlinear peak functions. In this case, one needs to first estimate nonlinear peak PDFs and corresponding linear peak PDFs. Peak PDF estimates for the WPAFB data were generated using Rain Flow Matrices determined using MATLAB functions developed by the WAFO group [7]. A typical normalized displacement RFM [8] is shown in Figure 4. Positive peak (or maxima) histograms can be calculated by summing the bin values along the rows of the RFM; histograms of minima can be calculated by summing the columns. Peak PDFs can then be estimated from the histograms of peaks. (10)

5 Figure 4. Normalized peak displacement RFM for 8 g input WPAFB data, σ = 1.09 mm. Normalized peak displacement PDFs are shown in Figure 5-a for three of the input levels, with a linear peak PDF, determined from equation (12), for comparison. The data shows strong nonlinear behavior, with increasing amounts of negative-valued maxima. Further analysis of peak PDFs requires knowledge of the rates of zero crossings and peaks. Response rates. The standard method of determining response rates based on spectral moments [3, 4] cannot be used with nonlinear data [9]. Instead, the rates must be estimated directly from the time domain data. The WPAFB results files were processed using the WAFO MATLAB functions to determine the rates of zero crossings. The rates of positive slope zero crossings Φ 0 + and the rates of displacement peaks Φ P + are found by dividing the total number of positive slope zero level crossings and the number of rainflow cycles (from a RFM) for each simulation level by the analysis time period. Table 1 gives the rates of positive slope zero displacement crossings as well as the rates of displacement peaks for the WPAFB data. Also tabulated is the ratio of displacement zero to peak rates ϖ x : Φ ϖ x = (11) Φ + 0 x + P x Table 1. Rates of displacement zero crossing and peaks for WPAFB experiments. Input + Φ 0 x Hz Φ P + x Hz ϖ x 0.5 g g g

6 Linear peak probability functions. The normalized (σ = 1) linear peak PDF is a function of the ratio ϖ [3]: 2 2 k1 x x x ppeak ( x) = exp ϖ xexp 1 Q π 2k1 2 k2 The additional terms k n and Q n are also functions of the ratio ϖ: 1 x 2 2 Qn ( x) = erfc k 1 ϖ k k 2 = = kn where erfc is the complimentary error function. The normalized linear peak distribution function (or CDF) is [3]: 2 x FPeak ( x) = 1 Q1 x exp 1 Q2x 2 ( ) ϖ [ ] The linear peak inverse distribution function can be determined from equation (14). Inverse distribution function method for peak displacement. One major observation from the WPAFB data is that the ratio ϖ was not constant. This change in the ratio ϖ with the input level (and degree of non-linearity) requires a decision with regard to how to proceed with the IDF method. The logical choices were to use one linear peak CDF based on the linear ratio ϖ, or to allow the ratio ϖ to change and therefore determine a corresponding linear peak CDF for each input level. ϖ (12) (13) (14) Figure 5. Normalized nonlinear peak displacement PDFs and linear peak PDF Normalized nonlinear peak displacement CDFs (experiment) and linear peak CDFs. Figure 5-b shows the normalized linear and nonlinear CDFs for the data. Notice that the CDF for the nonlinear data at the highest input of 8 g crosses the zero displacement level at a cumulative probability of approximately 0.8 σ, while the CDF for the lowest level input of 0.5 g crosses the zero displacement level of approximately 0.2 σ. The linear peak CDFs based on variable ratios ϖ cross at about the same level as the nonlinear data.

7 A more compelling argument for a variable ratio ϖ comes when one inspects the linear to nonlinear functions determined using the IDF method. Figure 6-a shows the linear to nonlinear function based on a fixed linear peak CDF, while Figure 6-b shows the function based on a variable linear peak CDF. Log-Log plots of these functions are shown in Figure 7-a and Figure 7-b. Notice that in both cases, the data does not all fall along one common curve. The data for the variable linear peak CDF functions does tend to cross the zero-zero point, which is considered to be desirable when curve fitting the functions. A curve fit of the linear to nonlinear function through data values above approximately 1 σ was used. It is reassuring to note that polynomial curve fits of the data points determined from the fixed ϖ and variable ϖ linear peak CDF yield similar coefficients as given in Table 2. Figure 6. Linear to nonlinear peak displacement functions for WPAFB data: Fixed ratio linear peak CDF and variable ratio linear peak CDF. Figure 7. Log-log plot of linear to nonlinear peak displacement functions for WPAFB data: Fixed ratio linear peak CDF and Variable ratio linear peak CDF.

8 Table 2. Polynomial coefficients for linear to nonlinear curve fits of WPAFB peak displacement data. c 1 mm/mm c 2 mm/mm 2 c 3 mm/mm 3 Fixed ϖ CDF Variable ϖ CDF PDF TRANSFORMS FOR PEAK RESPONSE PDF transform for peak displacement. Nonlinear peak PDFs were estimated using the coefficients in Table 2 using the PDF transform method. The PDF transfer function for the linear to nonlinear peak displacement left handed polynomial (10) is: NLpeak ( ) ( ) p y = dx dy p x (15) Taking the derivative of the polynomial in (10) and substituting into (15) yields: NLpeak Peak 2 ( ) 2 3 ( ) p y = c + c y+ c y p x (16) ˆ Peak Instead of trying to solve this equation in closed form, the numerical result can be easily determined by first estimating the linear peak bin center values using: ˆx = cy+ cy + cy (17) where the y values are be based on the nonlinear histogram or PDF bin centers. The linear peak PDF is given in equation (12). PDF transform for nonlinear total stress. In this section, the PDF transform method is used to estimate the nonlinear peak total stress PDF from the nonlinear peak displacement PDF. The data is from nonlinear SDOF ODE simulations of the WPAFB experiments. The nonlinear peak displacement to peak total stress function was determined based on a nonlinear fit [10] of filtered WPAFB data: ( ) z = g y 2 3 ( ) z = E d y+ d y + d y where z and y are nonlinear peak total stress and nonlinear peak displacement respectively. The material elastic modulus E = 204 GPa was used to convert the strain to stress. The coefficients d i are given in Table 3. Equation (18) is the standard form of the nonlinear function with the polynomial on the right hand side of the equation. Table 3. Nonlinear displacement to bottom strain parameters for WPAFB data Run d 1 µε/mm d 2 µε/mm 2 d 3 µε/mm g g g (18)

9 The PDF transfer function for this nonlinear peak displacement to peak stress function is: p NLpeak ( z) p ( y) NLpeak = (19) dz dy where p NLpeak (y) is the nonlinear PDF of peak displacement. The derivative has two roots: 2 d2 ± d2 3d2d3 y = (20) d The range of results was such that only one root (the minimum) was observed. The next step in the process of estimating the total stress peak PDF is to assume that each positive displacement peak is paired with a negative minimum with the same magnitude but opposite sign. In other words, each point on a positive peak displacement PDF is assumed to map to a complete displacement cycle with amplitude equal to the positive peak value. This is considered fair based on the standard assumption that a narrow band response has one positive peak for each positive zero crossing. The displacement cycle assumption now lets one calculate the PDF for min and max stresses based on the total stress equation (18). It is also helpful to observe how the displacement maps to total stress with an example. Figure 8-a shows six cycles of an increasing displacement function, and Figure 8-b shows the corresponding total stress based on equation (18). The first three cycles behave normally, but the final three cycles show that the stress function now contains two cycles for one displacement cycle. 1 Figure 8. Six cycles of an increasing displacement function. Total stress determined from displacement. First, define a primary cycle that has a stress maxima mapping to the positive displacement maxima, and a stress minima mapping to the displacement minima. This one to one mapping works for displacement cycles up to the value determined by equation (20). Note that the standard PDF transfer function (19) is asymptotic at this point. The primary cycles have a minimum at roots (equation (20)) of the total stress function. Also, define a secondary cycle that has a minimum stress at this same minimum and a maximum stress at the point mapping to the negative minimum of the displacement cycle.

10 The total stress PDF can be determined numerically as follows. First, define a range of nonlinear peak displacement values at a convenient spacing (e.g. dy = 0.01σ y ) corresponding to the positive peak displacement range observed in the nonlinear experiment (e.g. 0.1 σ y y 4 σ y ). Determine the PDF of nonlinear positive displacement peaks pnlpeak ( y ) based on equation (15). Also, determine a second nonlinear displacement PDF of negative minima using the same PDF, but now assigned at the negative values of y. Figure 9. Nonlinear positive maxima and negative minima displacement PDFs for 8 g ODE simulation, ζ = Figure 9 is an example of this process for data from and ODE simulation of the 8 g experiment. Now, given these PDFs, numerically evaluate equation (19). At the lower levels of input, the mapping of displacement to stress is one to one as shown in Figure 10-a. At higher input levels, the secondary cycles start to occur resulting in two possibilities for generating stress maxima and minima. Figure 10-b gives an example of this result. Note that the total stress PDF has an asymptote at 20 MPa, and that now Figure 10-b shows PDFs of primary maxima and secondary maxima. These functions are be plotted in Figure 11-a in a form like the WAFO RFM of ODE stress shown in Figure 11-b. A visual comparison of the data is very encouraging. These two functions for the higher inputs are not strictly PDFs because the two to one mapping of stress results in an integrated area greater than one. The correct peak total stress PDF has to be normalized by the sum of the primary and secondary contributions. The normalization can be determined as discussed below. One complication of this numerical PDF transform method is that the total stress functions determined with equation (18) have uneven spacing. The resulting nonlinear unevenly spaced PDFs of primary and secondary cycles have different bin centers and widths. They can be interpolated at evenly spaced points and then normalized by the sum of the two functions. Another method is to generate histograms of the primary and secondary cycles and then simply sum the two to get an overall histogram of total cycles. This second method was used to generate histograms for comparison to those determined from the ODE simulations of total stress; there was no need to calculate normalized PDFs.

11 Figure 10. Peak total stress PDF determined from transformed positive and negative displacement PDFs for 0.5 g simulation and 2 g simulation, ζ = Figure 11. Nonlinear total stress primary and secondary ranges from displacement PDFs. Total stress rain flow matrix for nonlinear ODE simulation: Input = 2 g, ζ = Peak stress histograms of minima and maxima are shown in Figure 12. Note, the histograms are shown with log number of counts in order to allow for better observation of the results near the tails of the histograms. These figures show that the histogram results determined by PDF transform compare very well with the rain flow histograms determined from the direct time domain data of total stress.

12 Figure 12. Nonlinear peak stress histograms of minima and maxima from nonlinear ODE simulation and transformed displacement PDFs: input=0.5 g, input=2 g, ζ= DAMAGE ESTIMATES USING THE PDF TRANSFORM METHOD Having estimated the PDFs and histograms of nonlinear peak total stress, the next step in the analysis is to determine time to failure for the transformed displacement results. Gordon, et al. [1] only mentions that the beam material used in the WPAFB experiments was high carbon steel. The material SAE 1015 (a low carbon steel) was used for the damage analysis because fatigue properties for different fatigue models were available [11]. The baseline alternating stress model assumed no mean stress effects, whilst the Morrow (with true fracture strength) and Walker models included mean stress effects [8, 11]. In this case, damage estimates were determined directly from the primary and secondary histograms of peak total stress minima and maxima (the RFM were not calculated). Damage calculations were also made using only the primary histograms of peak total stress minima and maxima. The calculations based on only the primary cycles were equal to the combined damage of primary and secondary cycles up to the 5 th significant digit or more. The primary cycle results are given in Table 4. These compare favorably with the time to failure results from estimates determined from the raw WPAFB Strain data. Table 4. Time-to-failure estimates for transformed nonlinear displacement PDF primary cycle data from ODE simulations compared to estimates from WPAFB data. Run WPAFB Data Baseline (s) Morrow (s) Walker (s) ODE Sim WPAFB Data ODE Sim WPAFB Data ODE Sim 0.5 g 1.6e14 3.7e14 1.5e14 3.6e14 8.9e13 1.9e14 2 g 3.8e10 4.1e10 3.4e10 3.4e10 1.7e10 1.3e10 8 g 1.9e7 7.0e6 1.3e7 3.1e6 7.2e6 1.5e6

13 CONCLUSIONS PDF transforms have been shown to be effective for estimating nonlinear PDFs of response and peak displacement and total stress. The inverse distribution function method has been used to estimate polynomial linear to nonlinear functions. A two step process to transform linear to nonlinear peak displacement to total nonlinear stress PDFs compare well with measured stress rainflow matrices and subsequent fatigue life estimates. Acknowledgment. The authors thank the WPAFB Structural Mechanics Branch for providing the experimental data and ITT Industries for their educational support at the ISVR. REFERENCES 1. R.W. Gordon, J.J. Hollkamp, and S.M. Spottswood, Nonlinear Response of a Clamped- Clamped Beam to Random Base Excitation, VIII Int. Conf. on Recent Advances in Structural Dynamics, Southampton, UK, July K.A. Sweitzer, M. Veltri, S.C. Kerr, N.W.M. Bishop, A Preliminary Application of the PDF Transfer Function to Fatigue Calculations for Nonlinear Systems, VIII Int. Conf. on Recent Advances in Structural Dynamics, Southampton, UK, July J.S. Bendat, A.G. Piersol, Random Data: Analysis and Measurement Procedures (3 rd edition), John Wiley and Sons, New York, USA, K.A. Sweitzer, N.W.M. Bishop, V.L. Genberg, Efficient Computation of Spectral Moments for Determination of Random Response Statistics, ISMA2004 Int. Conf. on Noise and Vibration Engineering, Leuven, Belgium, September S.R. Winterstein, C.H. Lange, S. Kumar, Fitting: A Subroutine to Fit Four-Moment Probability Distributions to Data, Sandia National Laboratories Technical Report, SAND , Alburquerque, NM, USA 6. A. Papoulis, S.U. Pillai, Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York, (4 th edition) WAFO Toolbox - Wave Analysis for Fatigue and Oceanography, Version 2.1.0, 2004, Lund Institute of Technology, Lund University, Sweden K.A. Sweitzer, N.S. Ferguson, Mean Stress Effects on Random Fatigue of Nonlinear Structures, XII Int. Congress on Sound and Vibration, Lisbon, Portugal, July K.A. Sweitzer, N.S. Ferguson, Estimating Nonlinear Random Response Statistics, IX Int. Conf. on Recent Advances in Structural Dynamics, Southampton, UK, July J.S. Bendat, Nonlinear Systems Techniques and Applications, John Wiley and Sons, New York, USA, N.E. Dowling, Mean Stress Effects in Stress-Life and Strain-Life Fatigue, SAE Paper , Proc. of Fatigue 2004: Second SAE Brasil Int. Conf. on Fatigue, São Paulo, Brasil, 2004.