## 5 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multicolumn{5}{c}{**Natural Frequency Reduction**} \\ \cline{2-6}  & **1** & **2** & **3** & **4** & **5** \\ \hline \(E_{1}\) & \(-10.50\%\) & \(-6.40\%\) & \(-1.66\%\) & \(-6.70\%\) & \(-6.09\%\) \\ \hline \(E_{2}\) & \(-0.05\%\) & \(-0.15\%\) & \(-5.54\%\) & \(-8.53\%\) & \(-0.05\%\) \\ \hline \(G_{12}\) & \(-0.01\%\) & \(-3.83\%\) & \(-3.12\%\) & \(-1.27\%\) & \(-0.06\%\) \\ \hline \(v_{12}\) & \(-0.09\%\) & \(-0.06\%\) & \(-0.04\%\) & \(0.03\%\) & \(-0.09\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Percentage changes in natural frequency induced by reductions in the material constants of a graphite/epoxy plate (\(a/h=120\)). After performing the above minimization procedure, the best estimate of the current \(E_{1}\) has been attained with \(q(1)=0.7993\). Once the best estimate of \(E_{1}\) has been attained, the above minimization procedure can be used to find the best estimate of the current \(G_{12}\) by initiating \(q(1)=0.7993\), \(q(2)=1\), \(q(3)=0.5\), and \(f^{\star}=f_{2}{}^{\star}\) in the minimization process. Similarly, once the best estimates of the current \(E_{1}\) and \(G_{12}\) are available, i.e., \(q(1)=0.7993\) and \(q(3)=0.7957\), the above minimization procedure can be used to find the best estimate of the current \(E_{2}\) by initiating \(q(1)=0.7993\), \(q(2)=0.5\), \(q(3)=0.7957\), and \(f^{\star}=f_{4}{}^{\star}\) in the minimization process. It is noted that the proposed minimization procedure requires fewer than 10 iterations to identify each current material constant. The identified current material constants are listed in Table 9 in comparison with the actual ones. It is noted that the proposed material constant identification method can produce good results for a thin plate with a maximum error of less than 0.6%. The proposed minimization procedure is then used to identify the material constants of the same [0\({}^{\circ}\)]\({}_{8}\) cantilever plate but with \(a/h=20\). Again, consider the sensitivity analysis of natural frequencies subject to the variation of each material constant for the plate. The percentage changes in the natural frequencies for a 20% reduction in each material constant are listed in Table 10.  
It is noted that each material constant may have effects of different extents on the natural frequencies. For instance, with respect to the first natural frequency, Young's modulus \(E_{1}\) has the most prominent effects on the first natural frequency of the plate while the effects of the other material constants are insignificant. The sensitivity information will be used in the identification procedure to estimate the current material constants \(E_{1}\), \(E_{2}\), \(G_{12}\), and \(G_{23}\). For illustration purposes, the current \"measured\" natural frequencies are again determined using the present method instead of testing. The first five original and \"measured\" natural frequencies associated with, respectively, the original and degraded material constants are assumed to be as given below. The original and degraded material constants used in this case are the same as those of \(a/h=120\). Natural frequency (Hz):  
Original: \(f_{1}=69.10,f_{2}=89.53,f_{3}=189.14,f_{4}=404.19,f_{5}=410.12\);  
Measured: \(f_{1}{}^{\star}=61.77,f_{2}{}^{\star}=80.05,f_{3}{}^{\star}=169.14,f_{4}{}^{ \star}=361.37,f_{5}{}^{\star}=366.89\). Herein, following the same idea of the proposed current material constant identification procedure, the first \"measured\" natural frequency is used to identify the current \(E_{1}\), the second to identify \(G_{12}\), the third to identify \(E_{2}\), and the fourth to identify \(G_{23}\). The best estimates of the current material constants are listed in Table 11 in comparison with