# Uncertainty Representation of Natural Frequency for
## 2 Problem Description  
In this research, the laminated composite cylindrical shells of T300/QY8911 are studied. Due to the vibration characteristics of composite materials being different from that of isotropic structures, the natural frequency \(\omega\) is considered as the performance objective function to investigate the vibration characteristics of laminated composite cylindrical shells [44]. The uncertainty factors of laminated composite cylindrical shells mainly include elastic modulus, shell thickness, dimension parameters and strength coefficient. To simplify the problem, some elastic modulus parameters, including the modulus of longitudinal elasticity \(E_{1}\), the transverse modulus of elasticity \(E_{2}\), the main and secondary Poisson ratio \(\mu_{1}\) and \(\mu_{2}\), and in-plane shear modulus \(G\), are considered to be uncertain. In addition, other parameters, such as the shell thickness h, the radius of middle surface \(R\), shell density \(\rho\), shell length \(X\) and the number of layers \(L\) are assumed as determinate [8]. When the above parameters are determined, the solving method of natural frequency of laminated composite cylindrical shells becomes a difficulty [45]. Therefore, the solving method of natural frequency is discussed as follows. First, based on the Love shell theory, the free vibration equations of laminated composite cylindrical shells are acquired [46]. Considering the coupling effect, the vibration characteristics of laminated composite cylindrical shells with different laying were studied [47]. According to the Love shell theory, the middle surface micro-element at laminated cylindrical shells was taken as the research object. The acting forces and moments on the micro-element are shown in Figure 1. \(r\), \(\theta\) and \(x\) represent the radial, circumferential, and axial of laminated composite cylindrical shells in cylindrical coordinates. \(N_{x}\), \(N_{\theta}\) and \(N_{x\theta}\) represent internal force on a micro plane, \(M_{x}\), \(M_{\theta}\) and \(M_{x\theta}\) represent internal moment on a micro plane, \(u\), \(v\) and \(w\) represent the axial, circumferential and radial displacement of middle surface of cylindrical shells [48; 49]. Therefore, the differential equations of free vibration of the laminated composite cylindrical shells are expressed as  
\[\left\{ \begin{array}{l}\frac{\partial N_{x}}{\partial x}+\frac{\partial N_ {x\theta}}{\partial x\theta}=\rho h\frac{\partial^{2}u}{\partial x},\\ \frac{\partial N_{x\theta}}{\partial x}+\frac{\partial N_{x\theta}}{\partial x \theta}+\frac{\partial M_{x\theta}}{\partial x}+\frac{\partial M_{\theta}}{ \partial x\theta}=\rho h\frac{\partial^{2}u}{\partial x^{2}}g,\\ \frac{\partial^{2}M_{x}}{\partial x}+\frac{\partial^{2}M_{x\theta}}{\partial x \theta\partial x}+\frac{\partial^{2}M_{\theta}}{\partial x^{2}\theta^{2}}- \frac{N_{\theta}}{R}=\rho h\frac{\partial^{2}u}{\partial x^{2}},\end{array} \right. \tag{1}\]  
where the forces and moments of laminated composite cylindrical shells are described by  
\[\left\{ \begin{array}{l}(N_{x},N_{\theta},N_{x\theta})=\int_{-h/2}^{h/2}( \sigma_{x},\sigma_{\theta},\sigma_{x\theta})dz,\\ (M_{x},M_{\theta},M_{x\theta})=\int_{-h/2}^{h/2}(\sigma_{x},\sigma_{\theta}, \sigma_{x\theta})dz,\end{array} \right. \tag{2}\]  
where \(\sigma_{x}\), \(\sigma_{\theta}\) and \(\sigma_{x\theta}\) are the micro-element stress in different directions. According to the generalized [PERSON]'s law, the stress in Equation (2) can be expressed as  
\[\left\{ \begin{array}{l}\sigma_{x}\\ \sigma_{\theta}\\ \sigma_{x\theta}\end{array} \right\}=\left[ \begin{array}{ccc}Q_{11}&Q_{12}&0 \\ Q_{12}&Q_{22}&0\\ 0&0&Q_{66}\end{array} \right]\left\{ \begin{array}{l}e_{x}\\ e_{\theta}\\ e_{x\theta}\end{array} \right\}, \tag{3}\]  
The coefficient \(Q_{ij}\)(\(ij\) = 11, 12, 22, 66) is the reduction stiffness matrix coefficient of monolayer composite materials, where \(Q_{11}=E_{1}/(1-\mu_{1}\mu_{2})\), \(Q_{12}=E_{1}\mu_{2}/(1-\mu_{1}\mu_{2})\), \(Q_{22}=E_{2}/(1-\mu_{1}\mu_{2})\) and \(Q_{66}=G\).