## 5 Results and Discussion
Figure 6: Relationship between normalized natural frequency and length-to-thickness ratio for the carbon/epoxy [45, \(-\)45\({}_{2}\), 45]s square plate. is presented to identify the current material constants of laminated composite cantilever plates with different length-to-thickness ratios. First, consider the sensitivity analysis of natural frequencies subject to the variation of each material constant for a graphite/epoxy [0]s cantilever plate with a/h = 120. The percentage changes in the natural frequencies for a 20% reduction in each material constant are listed in Table 8.  
It is noted that each material constant may have effects of different extents on the natural frequencies. For instance, compared with other material constants, [PERSON]'s modulus \(E_{1}\) has the most prominent effects on the first natural frequency of the plate while [PERSON]'s ratio \(v_{12}\) has insignificant effects on the first five natural frequencies. The information will be used in the identification procedure to estimate the current material constants \(E_{1}\), \(E_{2}\), and \(G_{12}\). For illustrating the idea of the proposed identification procedure, rather than testing, the current \"measured\" natural frequencies are determined using the present method. The first five original and \"measured\" natural frequencies associated with, respectively, the original and degraded material constants are assumed to be as follows:  
* Natural frequency (Hz)
* Original: \(f_{1}=11.66\), \(f_{2}=15.01\), \(f_{3}=31.86\), \(f_{4}=69.96\), \(f_{5}=72.90\);
* Measured: \(f_{1}\)* \(10.43\), \(f_{2}\)* \(13.42\), \(f_{3}\)* \(28.49\), \(f_{4}\)* \(62.58\), \(f_{5}\)* \(65.20\). * and material constants
* Original: \(E_{1}=112.00\) GPa, \(E_{2}=11.00\) GPa, \(v_{12}=0.2\), \(G_{12}=4.48\) GPa;
* Degraded: \(E_{1}=89.60\) GPa, \(E_{2}=8.80\) GPa, \(v_{12}=0.2\), \(G_{12}=3.58\) GPa. For the material degradation assessment of thin plates, it is reasonable to assume \(v_{12}\) to be constant and \(G_{12}=G_{13}=G_{23}\). After reviewing the sensitivity analysis results listed in Table 9, it is possible that a chosen natural frequency can be used to estimate a particular material constant. In this case, the natural frequency \(f_{1}\), \(f_{2}\), or \(f_{4}\) will be used individually to estimate, respectively, material constant \(E_{1}\), \(G_{12}\), or \(E_{2}\) using a minimization technique. First, let the original material constants be X(1) = \(E_{1}\), X(2) = \(E_{2}\), and X(3) = \(G_{12}\). The updated material constant X(i)* is equal to \(q\)(i) X(i) in which the parameter q(i) satisfies the condition of (\(q_{1}<q<q_{u}\)) with \(q_{l}=0\) and \(q_{u}=1\). The updated material constants are used in the proposed method to predict the selected natural frequency f(q(i)) which depends on the chosen values of the parameters \(q\)(i). The minimization procedure for identifying the current \(E_{1}\) using the first natural frequency is given as follows:  
* Step 1. Calculate the upper bound frequency \(f_{u}\) using \(q(1)=q(2)=q(3)=1\) and lower bound frequency \(f_{l}\) using \(q(1)=0.1\) and \(q(2)=q(3)=1\). Set \(q_{u}=1\) and \(q_{l}=0.1\). * Step 2. Set \(q(1)=0.5\), \(q(2)=q(3)=1\). * Step 3. Let X(i)* = q(i)X(i); choose \(f^{*}=f_{1}\)*. * Step 4. Calculate the error function \(e(q(i))=\langle f(q(i))-f^{*}\rangle/f^{*}\). If e < 0.01, go to Step 8. * Step 5. If e (\(q(i))>0\), then let \(q_{u}=q(1)\) and calculate the upper bound frequency \(f_{u}\) for \(q_{u}\). Otherwise, if e (\(q(i))<0\), then let \(q_{l}=q(1)\) and calculate the lower bound frequency \(f_{l}\) for \(q_{l}\). * Step 6. Update \(q(1)=q_{l}+\langle f^{*}-f_{l}\rangle\) (\(q_{u}-q_{l}\))/\(\langle f_{u}-f_{l}\rangle\). * Step 7. Go to Step 4. * Step 8. Stop the iteration.