# Uncertainty Representation of Natural Frequency for
## 2 Problem Description
\(e_{x}\), \(e_{\theta}\) and \(e_{x\theta}\) are the axial, circumferential and shear strains at the distance from the reference Z-place. The relationship between the main and secondary Poisson ratios is \(E_{1}\mu_{1}=E_{2}\mu_{2}\). Based on the Love shell model, the strains are expressed as \(e_{x}=e_{1}+zk_{1}\), \(e_{\theta}=e_{2}+zk_{2}\) and \(e_{x\theta}=\gamma+2z\tau\), where \(e_{1}\), \(e_{2}\) and \(\gamma\) are  
Figure 1: The force and moment distribution of laminated composite cylindrical shells. reference surface strain, \(k_{1}\), \(k_{2}\) and \(\tau\) are reference surface strain curvature. The reference surface strain and curvature are expressed as  
\[\left\{ \begin{array}{l}(e_{1},e_{2},\gamma)=\left(\frac{\partial u}{\partial x },\frac{1}{R}\left(\frac{\partial v}{\partial\theta}+w\right),\frac{\partial v }{\partial x}+\frac{\partial u}{R\partial\theta}\right),\\ (k_{1},k_{2},\tau)=\left(-\frac{\partial^{2}w}{\partial x^{2}},\frac{1}{R^{2 }}\left(-\frac{\partial^{2}w}{\partial\theta^{2}}+\frac{\partial v}{\partial \theta}\right),\frac{1}{R}\left(-\frac{\partial^{2}w}{\partial\theta\alpha}+ \frac{\partial v}{\partial\lambda}\right)\right),\end{array} \right. \tag{4}\]  
By substituting Equation (4) into Equation (2), the internal force of the cylindrical shells can finally be expressed as the following matrix form:  
\[\left\{ \begin{array}{l}N_{x}\\ N_{\theta}\\ N_{x\theta}\\ M_{\theta}\\ M_{x\theta}\\ M_{x\theta}\end{array} \right\}=\left\{ \begin{array}{l}A_{11}\quad A_{12} \quad A_{16}\quad B_{11}\quad B_{12}\quad B_{16}\\ A_{12}\quad A_{22}\quad A_{26}\quad B_{12}\quad B_{22}\quad B_{26}\\ A_{16}\quad A_{26}\quad A_{66}\quad B_{16}\quad B_{26}\quad B_{66}\\ B_{11}\quad B_{12}\quad B_{16}\quad D_{11}\quad D_{12}\quad D_{16}\\ B_{12}\quad B_{22}\quad B_{26}\quad D_{12}\quad D_{22}\quad D_{26}\\ B_{16}\quad B_{26}\quad B_{66}\quad D_{16}\quad D_{26}\quad B_{66}\end{array}  \right\}\left\{ \begin{array}{l}e_{1}\\ e_{2}\\ \gamma\\ k_{1}\\ 2\tau\end{array} \right\}, \tag{5}\]  
where \(A_{ij}\), \(B_{ij}\) and \(D_{ij}\) are the tensile stiffness, coupling stiffness and bending stiffness, which are the key factors in representing the natural frequency of different ply angles of laminated composite cylindrical shells. The tensile stiffness, coupling stiffness and bending stiffness are expressed as  
\[\left\{ \begin{array}{l}A_{ij}=\int_{-h/2}^{h/2}\overline{Q}_{ij}^{k}dz=\sum \limits_{k=1}^{n}\overline{Q}_{ij}^{k}(z_{k}-z_{k-1}),\\ B_{ij}=\int_{-h/2}^{h/2}\overline{Q}_{ij}^{k}dz=\sum\limits_{k=1}^{n} \overline{Q}_{ij}^{k}\Big{(}z_{k}^{2}-z_{k-1}^{2}\Big{)},\\ D_{ij}=\int_{-h/2}^{h/2}\overline{Q}_{ij}^{k}z^{2}dz=\sum\limits_{k=1}^{n} \overline{Q}_{ij}^{k}\Big{(}z_{k}^{3}-z_{k-1}^{3}\Big{)},\end{array} \right. \tag{6}\]  
Due to the research object being the symmetrical layering of cylindrical shells, \(B_{ij}\) is generally calculated as 0. Assuming that \(V_{1}=\cos 2\theta_{i}\), \(V_{2}=\cos 4\theta_{i}\), \(V_{3}=\sin 2\theta_{i}\) and \(V_{4}=\sin 4\theta_{i}\), \(\theta_{i}\) represents the ply angle of laminated composite cylindrical shells. The partial axis stiffness coefficient \(\overline{Q}_{ij}^{k}\) of the \(k\)th layer is described by  
\[\left\{ \begin{array}{l}\overline{Q}_{11}^{k}\\ \overline{Q}_{12}^{k}\\ \overline{Q}_{22}^{k}\\ \overline{Q}_{66}^{k}\end{array} \right\}=\left[ \begin{array}{ccc}S_{1}&V_{ 1}&V_{2}\\ S_{4}&0&-V_{2}\\ S_{1}&-V_{1}&V_{2}\\ S_{5}&0&-V_{2}\end{array} \right]\left\{\begin{array}  
\[\left\{ \begin{array}{l}u=\sum\limits_{n=0}^{\infty}\sum\limits_{s=1}^{\infty}U_{ ns}\cos(n\theta)\exp[i(wt-k_{ns}x)],\\ v=\sum\limits_{n=0}^{\infty}\sum\limits_{s=1}^{\infty}V_{ns}\sin(n\theta)\exp[ i(wt-k_{ns}x)],\\ w=\sum\limits_{n=0}^{\infty}\sum\limits_{s=1}^{\infty}W_{ns}\cos(n\theta)\exp[ i(wt-k_{ns}x)],\end{array} \right.