where j if is the frequency of the j th mode of the ith dataset, and j calc f is the calculated frequency from Eq. (1) of the jth mode. The set of parameters that minimizes the cost function given in Eq. (9) are the optimal parameters. The optimal values of the elastic parameters are given in Table 2. This estimate provides a point estimate of the elastic parameters Table 2 Least Squares Estimated Elastic Parameters Elastic Modulus Poisson’s Ratio Fit Error 110,410 psi 0.173 0.006 3.2. Three Mode Least Squares Fit The second optimization technique of estimating the values of the elastic parameters is similar to the first technique except that all three frequencies are used in the estimation process. The cost function used in the optimization is given by ¦ » » ¼ º « « ¬ ª ¸ ¸ ¹ · ¨ ¨ © § ¸ ¸ ¹ · ¨ ¨ © § ¸ ¸ ¹ · ¨ ¨ © § 9 1 2 16 16 16 2 9 9 9 2 7 7 7 i calc calc i calc calc i calc calc i f f f f f f f f f e (3) The optimal values of the elastic parameters are given in Table 3. This estimate also provides a single estimate of the elastic parameters. The fit error on when using three frequencies is about twice the error than using two frequencies (Table 2) from each of the samples. This could indicate that mode 16 is either not as consistent with an elastic model as the first two modes or that there is more uncertainty in the identification of mode 16. Table 3 Least Squares Estimated Elastic Parameters Elastic Modulus Poisson’s Ratio Fit Error 112,090 psi 0.199 0.013 3.3. Fitting Each Sample Individually The third optimization technique used to identify the elastic modulus and Poisson’s ratio is to perform an optimization to estimate the elastic parameters from modes 7 and 9 for each sample individually. The cost function for the optimization is similar to Eq. (2) and is given by 2 9 9 9 2 7 7 7 ¸ ¸ ¹ · ¨ ¨ © § ¸ ¸ ¹ · ¨ ¨ © § calc calc i calc calc i f f f f f f e (4) The density used was the actual density of the sample and therefore varied between samples. The optimal values of the elastic parameters are given in Table 4. The fit error is the frequency error between the frequency calculated using the optimized parameters for the sample and the experimentally measured frequency. Small errors in the prediction of the third mode indicates that the elastic model may in fact be appropriate. 222
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