Figure 2 Variation of natural frequencies with Poisson’s ratio. Once the curves (Figure 2) were generated (and consequently, the form of gi(U) was established), Eq. (1) was used to evaluate the frequency given a value of Young’s Modulus, density, and Poisson’s ratio. This simplification produced a model that could be evaluated very quickly. A fast running model is extremely useful in uncertainty quantification where the model may have to be evaluated thousands of times. Many times, more general meta models or surrogate models are developed to represent the response measure of interest, ([3]) to expedite calculations. 3. OPTIMIZATION Four optimization-based techniques for fitting the parameters of the model to the data were explored. The first technique was a least squares technique using only the first two frequencies in the cost function. The second was a least squares fit using all of the frequencies in a single optimization. The third technique is a least squares fit of the parameters individually for each sample. This optimization only uses two of the three available frequencies for the fitting. Statistics are derived from the collection of parameters from all of the fits. The final technique fits the parameters on each sample individually but uses all three frequencies for the fit. The statistics are derived from the collection of parameters from all of the fits similar to the second technique. 3.1. Two Mode Least Squares Fit The first optimization technique to identify the elastic modulus and Poisson’s ratio is to perform a least squares fit of all of the data using Eq. (1) to model the relationship between modes 7 and 9 and the elastic parameters. Two frequencies are the minimum number necessary since there are two parameters to estimate. The first two frequencies were chosen for the optimization because from Figure 2, it can be seen that the modes 9 and 16 are correlated with respect to Poisson’s ratio and therefore would give poor estimates. The density used was an average of all of the densities given in Table 1. The variation of the density is small between samples. The cost function, e, used in the optimization is given by ¦ » » ¼ º « « ¬ ª ¸ ¸ ¹ · ¨ ¨ © § ¸ ¸ ¹ · ¨ ¨ © § 9 1 2 9 9 9 2 7 7 7 i calc calc i calc calc i f f f f f f e (2) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Poisson's Ratio Frequency, (Hz) Frequency vs Poisson's Ratio (E = 1, density = 1) Mode 7 Mode 9 Mode 16 221
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