## 3 Analysis of Stochastic Natural Frequency
### Laminated Plate
Twenty-seven design points were employed for the meso-scale uncertainty scenario and 79 design points were employed for the micro-scale uncertainty scenario. In comparison to other experimental design methods such as the Box-Behnken method, the central composite design method  
Figure 1: Flowchart of calculation of natural frequency statistics by different uncertainty consideration. could provide high quality predictions over the entire design space, and it is more flexible considering the number of input random variables [32]. For all laminate configurations shown in Table 3, \(1\times 10^{5}\) group samples of random variables were drawn to conduct stochastic analysis based on the surrogate model derived by the RSM, and corresponding relative error on estimation of the mean value of first order natural frequency was less than 0.01% (calculated by Equation (6)). For the scenario of 'correlated meso-scale uncertainty', Gibbs sampling was employed to draw samples of meso-scale random variables to account for their correlation [21], by the procedure as follows:  
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in Figure 2b, it is seen that there is notable difference in the tail region between the two probability density functions. The cumulative probability of the first order natural frequency is shown in Figure 3, which shows that the 'independent meso-scale uncertainty' provides a notable overestimation on the cumulative probability. This demonstrates a necessity to account for statistical correlation between meso-scale uncertainty to derive accurate stochastic natural frequency. To achieve better understanding on the contribution of different uncertainty to the stochastic variation of the natural frequency, linear correlation coefficients between the first order natural frequency of Laminate 1 and meso-scale random variables were derived, as shown in Figure 4. The linear correlation coefficient reflected a stochastic dependence or sensitivity of stochastic variation of the natural frequency over input random variables. It is seen that neglecting correlation between meso-scale uncertainty suppressed the stochastic dependence on \(E_{22}\) and \(G_{12}\), which was similar to observation in [1] where stochastic deflection of a laminate structure was investigated. An interesting observation is on the dependence over laminate density. If the statistical correlation between the meso-scale input random variables was ignored, the stochastic dependence of the first order natural  
Figure 3: Cumulative probability of first order frequency of Laminate 1, derived by considering correlated and independent meso-scale random variables. Figure 2: Probabilistic distribution of first order frequency of Laminate 1 by considering correlated and independent meso-scale random variables: (**a**) histogram; (**b**) fitted by normal distribution. frequency over ply density was negative. It seems to be under expectation as it is known that larger density results into smaller natural frequency if plate stiffness and boundary conditions remain unchanged. However, from a multi-scale viewpoint, the ply density is actually positively correlated with ply modulus, and a larger ply density also indicates a larger stiffness. Hence, for the case of 'correlated meso-scale uncertainty', a positive correlation was seen between the natural frequency and the ply density. This highlights the importance to consider statistical correlation between meso-scale mechanical properties to achieve comprehensive understanding on the stochastic free vibration of laminate plates. For Laminates 2, 3 and 4 listed in Table 3, similar characteristics of the standard deviation, cumulative probability and sensitivity of the first order natural frequency were obtained as compared to Laminate 1, considering the discrepancy between the three different uncertainty considerations.