Table 4 Estimated Elastic Parameters for Each Sample Sample Number Elastic Modulus (psi) Poisson’s Ratio Fit Error for Mode 3 1 119,280 0.174 0.06% 2 111,120 0.176 0.11% 3 114,370 0.173 0.11% 4 101,230 0.170 0.16% 5 116,280 0.175 0.09% 6 108,030 0.172 0.13% 7 106,130 0.174 0.09% 8 107,860 0.175 0.13% 9 109,410 0.170 0.12% Mean 110,410 0.173 0.11% 3.4. Fitting Each Sample Individually using Least Squares The final optimization strategy identifies parameters for each sample individually, but uses all three frequencies for the fit. The cost function for this optimization is given as 2 16 16 16 2 9 9 9 2 7 7 7 ¸¸ ¹ · ¨¨ © § ¸¸ ¹ · ¨¨ © § ¸¸ ¹ · ¨¨ © § calc calc i calc calc i calc calc i f f f f f f f f f e (5) The results are given in Table 5. The estimated Poisson’s ratio for all the samples is about 10% higher when all three frequencies are used than when only modes 7 and 9 are used to estimate the parameters. The fit error in the mode 16, however is much lower when all three frequencies are used rather than only estimating the parameters using modes 7 and 9. When all three frequencies are used, parameters that best match all three frequencies are estimated so the resulting error in the third mode would be expected to be lower. Table 5 Estimated Elastic Parameters for Each Sample Sample Number Elastic Modulus (psi) Poisson’s Ratio Fit Error for Third Mode 1 120,600 0.192 0.01% 2 112,780 0.201 0.02% 3 116,090 0.199 0.02% 4 103,080 0.201 0.03% 5 117,940 0.198 0.02% 6 109,850 0.200 0.02% 7 107,630 0.197 0.02% 8 109,710 0.204 0.02% 9 111,190 0.197 0.02% Mean 112,100 0.199 0.02% 223
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