Figure 3 Probability density functions for Young’s modulus, Poisson’s Ratio, and standard error as calculated using Bayes theorem using two modes. Finally, the same Bayesian technique was used but with all three frequencies for each sample included in the likelihood. The likelihood then becomes ¸ ¸ ¹ · ¨ ¨ © § ¸ ¸ ¹ · ¨ ¨ © § ¸¸ ¹ · ¨¨ © § ¸¸ ¹ · ¨¨ © § ¸¸ ¹ · ¨¨ © § 9 1 2 16 16 16 2 9 9 9 2 7 7 7 2 2 1 exp , i calc calc i calc calc i calc calc i f f f f f f f f f p f V H T (10) with the same priors as described in Eq. (9). The results of the Bayesian analysis using Eq. (10) as a likelihood function are given in Figure 4. As was seen with the optimization results, the three frequency method results in PDFs that are slightly shifted, most notably in the distribution for the Poisson’s ratio, when compared to the results with the two frequency likelihood function. Figure 4 Prior and Posterior PDFs for Young’s modulus, Poisson’s ratio, and standard error from Bayesian updating using three modes. 5. COMPARISON OF RESULTS The different techniques can be grouped into techniques that used two frequencies to estimate the elastic parameters and those techniques that used three frequencies for the estimation. Table 6 summarizes the three techniques of fitting elastic parameters used in this paper when two modes are used in the estimation. Table 7 summarizes the techniques for when three frequencies are used in the estimation. In both cases, the differences in the estimated parameters between the number of modes used in the fitting process is much greater than the differences in fitting techniques. As discussed above, when only two frequencies are used to fit the two elastic parameters, the estimated parameters will exactly reproduce the fitting data, and produce an error in the third 0.8 0.9 1 1.1 1.2 1.3 1.4 x 10 5 0 1 2 3 x 10-4 Elastic Modulus PDF Elastic Modulus Posterior Prior -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 Poisson's ratio PDF Poisson's ratio Posterior Prior -6 -4 -2 0 2 4 6 0 1 2 3 4 5 6 Standard Error, (log(x)) PDF Standard Error Posterior Prior 0.8 0.9 1 1.1 1.2 1.3 1.4 x 10 5 0 1 2 3 x 10-4 Elastic Modulus PDF Elastic Modulus Posterior Prior -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 5 10 15 20 25 30 35 Poisson's ratio PDF Poisson's ratio Posterior Prior -6 -4 -2 0 2 4 6 0 1 2 3 4 5 6 7 Standard Error, (log(x)) PDF Standard Error Posterior Prior 225
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