## 5 Conclusions  
The present study provides statistical natural frequency of composites structures derived from micro-scale and meso-scale uncertainty, for the circumstance composite structures are composed of plies with unidirectional fibre reinforcement. Meso-scale uncertainty is obtained from micro-scale uncertainty by a combination of Monte-Carlo simulation and bridging micromechanical model. It is observed that significant statistical correlation may not only exist between ply elastic properties but also associated with ply density. The statistical correlation between meso-scale physical properties is commonly ignored in current study on stochastic free vibration of composite structures. Comparison is conducted on natural frequency statistics of composite plates and cylindrical shell considering'micro-scale uncertainty', 'correlated meso-scale uncertainty' and 'independent meso-scale uncertainty'. Several conclusions are drawn as follows:  
1. Natural frequency statistics of composite structures could be reasonably derived from either micro-scale random variables or meso-scale random variables but their statistical correlation needs to be well characterised;
2. Ignoring the statistical correlation between meso-scale uncertainty may result in large overestimation of standard deviation of structure natural frequency;
3. If the natural frequency statistics of composite structures are calculated by meso-scale uncertainty, it is especially important to consider the statistical correlation associated with ply density, which is somehow not noticed by previous studies. It also shows that dependence of the stochastic variation of natural frequency over ply density is positive rather than negative.  
It is very important to achieve accurate statistics of structure natural frequency in engineering design, so that potential structure resonance could be well avoided. Therefore, this work further  
\begin{table}
\begin{tabular}{c c c c} \hline  & **Meso-Scale Correlation** & **Meso-Scale Correlation** & **Meso-Scale** \\  & (\(E_{11}\), \(E_{22}\), \(G_{12}\), \(v_{12}\)) & (\(E_{11}\), \(E_{22}\), \(G_{12}\), \(v_{12}\)) & **Independent** \\ \hline \multirow{2}{*}{(0/90)\({}_{8}\) plate} & 5.10 Hz & 6.70 Hz & 5.61 Hz \\  & – & (31.4\%) \({}^{*}\) & (10.0\%) \({}^{*}\) \\ \hline \multirow{2}{*}{(90)\({}_{8}\) cylindrical shell} & 37.8 Hz & 56.0 Hz & 42.2 Hz \\  & – & (48.2\%) \({}^{*}\) & (11.6\%) \({}^{*}\) \\ \hline \end{tabular} \({}^{*}\) Difference from its counterpart value of ‘meso-scale correlation (\(E_{11}\), \(E_{22}\), \(G_{12}\), \(v_{12}\), \(\rho\))’.  
\end{table}
Table 5: Standard deviation of first order natural frequency considering different uncertainty.  
suggests establishing standard experimental approaches to obtain statistical correlation between ply elastic properties and ply density.  
Conceptualization, S.Z. and X.C.; methodology, S.Z.; software, S.Z.; formal analysis, S.Z.; investigation, S.Z. and X.C.; writing--original draft preparation, S.Z.; writing--review and editing, X.C.; funding acquisition, S.Z.  
This research was funded by National Natural Science Foundation of China, grant number 51405501.  
The authors declare no conflict of interest.