frequency. The error in predicting the third frequency represents the error due to noise, experimental errors, and errors due to the elastic model assumption. When three frequencies are used to estimate the parameters, these same errors exist, but the errors are spread over all three modes due to the least squares approach used here, producing a smaller error in the prediction of all three frequencies. The elastic parameters estimated using three frequencies will provide a more representative elastic model than will the parameters estimated using only two frequencies. Table 6 Comparison of Optimal and Means of Elastic Parameters using two modes Fit Method Elastic Modulus Poisson’s Ratio All Samples Least Squares (Table 2) 110,410 psi 0.173 Individual Samples Least Squares (Table 4) 110,410 psi 0.173 Bayesian 110,430 psi 0.173 Table 7 Comparison of Optimal and Means of Elastic Parameters using three modes Fit Method Elastic Modulus Poisson’s Ratio All Samples Least Squares (Table 3) 112,100 psi 0.199 Individual Samples Least Squares (Table 5) 112,100 psi 0.199 Bayesian 112,120 psi 0.199 Probability density functions can be estimated using the posterior samples of the parameters obtained from the Bayesian calibration and from the optimizations where the elastic parameters were fit for each individual sample. KDEs were used to estimate the probability density functions of the parameters obtained with both the optimization and the Bayesian calibration. These KDEs are compared in Figure 5 and Figure 6 For the Young’s modulus, the KDE representing the posterior distribution obtained from Bayesian calibration has a smaller variance than the optimization distribution. For Poisson’s ratio, the trend is reversed with the PDF from the optimization results having the smaller variance. Figure 5 KDE comparisons for Young’s modulus and Poisson’s ratio using two frequencies for estimation. 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 Optimization Bayes 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 20 40 60 80 100 120 140 160 Poisson's ratio PDF Poisson's ratio Optimization Bayes 226
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