## 3 Analysis of Stochastic Natural Frequency  
### Laminated Plate  
In this work, stochastic natural frequency of several different configurations of laminated plates subjected to different boundary conditions as listed in Table 3 was investigated, including cross-ply laminate, angle-ply laminate, quasi-isotropic laminate and laminate of random ply orientations. For each laminate configuration, statistics of first order natural frequency were derived in three scenarios: (a) considering micro-scale uncertainty of constituent material properties as listed in Table 1, termed in short as'micro-scale uncertainty'; (b) considering meso-scale uncertainty as listed in Table 2, and their statistical correlation was also considered, termed in short as 'correlated meso-scale uncertainty'; (c) considering meso-scale uncertainty of ply properties as listed in Table 2, but the statistical correlation between ply mechanical properties and also the density was neglected, termed in short as 'independent meso-scale uncertainty'. The flowchart of calculation process of stochastic natural frequency by considering the three different scenarios of uncertainty is shown in Figure 1. It is important to notice that most current studies (as introduced in Section 1) were based on 'independent meso-scale uncertainty', and it is necessary to discuss whether 'independent meso-scale uncertainty' provides reasonable estimation on the statistics of resonance frequency of composite structures. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Random**} & \multicolumn{4}{c}{**Linear Correlation Coefficient**} & \multirow{2}{*}{**Mean**} & \multirow{2}{*}{**CoV**} & \multirow{2}{*}{**Experiment**} \\  & \(E_{1}\) & & & & & & [30] \\ \hline \(E_{1}\) (GPa) & 1 & & & & 44.5 & 0.083 & 45.6 \\ \(E_{2}\) (GPa) & 0.65 & 1 & & & 13.2 & 0.077 & 16.2 \\ \(G_{12}\) (GPa) & 0.63 & 0.98 & 1 & & 4.93 & 0.078 & 5.83 \\ \(v_{12}\) & \(-\)0.06 & \(-\)0.10 & \(-\)0.20 & 1 & & 0.29 & 0.107 & 0.28 \\ \(\rho\) (kg/m\({}^{3}\)) & 0.43 & 0.72 & 0.71 & \(-\)0.07 & 1 & 1964 & 0.029 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The statistics of UD E-glass/Epoxy composite. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**No.** & **Ply Thickness** & **Ply Orientation** & **Size** & **Boundary Condition** \\ \hline Laminate 1 & 0.45 mm & (0/90)\({}_{4}\) & 0.6 \(\times\) 0.2 m & Simply supported \\ Laminate 2 & 0.45 mm & [(\(\pm\)60)\({}_{2}\)]s & 0.6 \(\times\) 0.2 m & Simply supported \\ Laminate 3 & 0.45 mm & [0/\(\pm\)45/90]s & 0.6 \(\times\) 0.2 m & Clamped \\ Laminate 4 & 0.45 mm & [15/45/60/90/30/0/10/20] & 0.6 \(\times\) 0.2 m & Clamped \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Laminate plate configurations. The statistics of first order natural frequency were derived by a combination of Monte-Carlo simulation and FE modeling. Four-node linear-elastic shell element (Shell181 in Ansys 14.5) is used to model the laminate. A mesh convergence study was conducted, which showed that a mesh size of \(0.01\times 0.01\) m provided a convergent solution on the first order natural frequency. Considering heavy computation cost of direct Monte-Carlo simulation on FE modeling, response surface method (RSM) with quadratic expression was employed to derive an approximate surrogate model to map random variables to the first order natural frequency, as shown in Equation (8):  
\[g(\mathbf{X})=a+b_{i}\sum_{i=1}^{n}x_{i}+c_{ij}\sum_{i=1}^{n}x_{i}\sum_{i=1}^{n }x_{j} \tag{8}\]  
where \(g\) represents the first order natural frequency, \(x_{i}\) represents micro-scale or meso-scale random variable shown in Table 1 or Table 2, and \(a\), \(b\) and \(c\) are regression terms. In this work, the regression terms were determined by central composite design method [31]. The central composite design was composed of one central point, two axis points per input variable and factorial points at the corners of the hypercube.