## 2 Shell Dynamics
### The Motion Equation of the Shell
\begin{array}{l}U(x)=\sum\limits_{m=1}^{N}a_{m}\varphi_{m}^{u}(x),V(x)=\sum \limits_{m=1}^{N}b_{m}\varphi_{m}^{v}(x),W(x)=\sum\limits_{m=1}^{N}c_{m} \varphi_{m}^{uv}(x)\\ \Psi_{x}(x)=\sum\limits_{m=1}^{N}d_{m}\varphi_{m}^{\varphi x}(x),\Psi_{\theta} (x)=\sum\limits_{m=1}^{N}e_{m}\varphi_{m}^{\varphi\theta}(x)\end{array}  \tag{23}\]  
where \(a_{m},b_{m},c_{m},d_{m}\), and \(e_{m}\) are the coefficients of the vibration shape functions. In addition, \(\varphi_{m}^{i}(x)\) (\(i=u,v,w,\varphi_{x},\varphi_{\theta}\)) are the \(m\)-th order Gram-Schmidt orthogonal polynomials. The expression for the Gram-Schmidt orthogonal polynomials can be obtained using the following recursive formula:  
\[ \begin{array}{l}\eta_{2}^{p}(x)=\left(\frac{x}{L}-B_{2}\right)\eta_{1}^{p} \left(\frac{x}{L}\right)\\ \eta_{m}^{p}(x)=\left(\frac{x}{L}-B_{m}\right)\eta_{m-1}^{p}\left(\frac{x}{L} \right)-C_{m}\eta_{m-2}^{p}\left(\frac{x}{L}\right),m\geq 2\end{array}  \tag{24}\]  
where  
\[B_{m}=L\frac{\int_{0}^{1}x\left(\eta_{m-1}^{p}\right)^{2}\!dx}{\int_{0}^{1} \left(\eta_{m-1}^{p}\right)^{2}\!dx},C_{m}=L\frac{\int_{0}^{1}x\eta_{m-1}^{p} \eta_{m-2}^{p}dx}{\int_{0}^{1}\left(\eta_{m-2}^{p}\right)^{2}\!dx} \tag{25}\]  
In addition, the calculation parameter \(\eta_{k}\varphi(x)\) can be obtained by the following expression:  
\[\left\|\eta_{k}^{p}\left(\frac{x}{L}\right)\right\|_{2}=\sqrt{\int_{0}^{1} \left[\eta_{k}^{p}\left(\frac{x}{L}\right)\right]^{2}\!dx} \tag{26}\]  
where the upper corners \(p=u\), \(v\), \(w\), \(\varphi_{x}\), and \(\varphi_{\theta}\) represent orthogonal polynomials for different vibration shape functions. Then, the normalized vibration shape functions can be obtained as follows:  
\[\varphi_{k}^{p}\left(\frac{x}{L}\right)=\frac{\eta_{k}^{p}\left(\frac{x}{L} \right)}{\left\|\eta_{k}^{p}\left(\frac{x}{L}\right)\right\|}(p=u,v,w) \tag{27}\]  
The vibration shape function has the additional following properties:  
\[\int_{0}^{1}\varphi_{k}^{p}\left(\frac{x}{L}\right)\varphi_{l}^{p}\left(\frac {x}{L}\right)dx=\frac{\delta_{kl}}{L},(p=u,v,w,\varphi_{x},\varphi_{\theta}) \tag{28}\]This paper considers three types of boundary conditions: clamped boundary conditions, simply-diaphragm boundary conditions, and free boundary conditions. The first term of the Gram-Schmidt orthogonal polynomials is determined by the boundary function. The boundary functions expressing the different types of boundary conditions are as follows [22; 32; 33]:  
For the clamped edge:  
\[U=V=W=\Psi_{x}=\Psi_{\theta}=0:\ \varphi_{0}^{i}=\left(\frac{x}{L}\right)^{2}- \left(\frac{x}{L}\right),\ (i=u,v,w,\varphi_{x},\varphi_{\theta}) \tag{29}\]  
For the simply-diaphragm edge:  
\[V=W=0:\ \left\{ \begin{array}{l}\varphi_{0}^{i}=\left(\frac{x}{L}\right)^{2}- \left(\frac{x}{L}\right),\ (i=v,w)\\ \varphi_{0}^{i}=1,\ (i=u,\varphi_{x},\varphi_{\theta})\end{array} \right. \tag{30}\]  
For the free edge:  
\[\text{No constraints}:\varphi_{0}^{i}=1,\ (i=u,v,w,\varphi_{x},\varphi_{ \theta}) \tag{31}\]  
Equation (23) can be written in vector product form, and its expression can be expressed as follows:  
\[ \begin{array}{l}U(x)=\mathbf{A}\cdot\mathbf{\varphi}_{u},\ V(x)=\mathbf{B}\cdot\mathbf{ \varphi}_{v},\ W(x)=\mathbf{C}\cdot\mathbf{\varphi}_{w}\\ \Psi_{x}(x)=\mathbf{D}\cdot\mathbf{\varphi}_{\varphi_{x}},\ \Psi_{\theta}(x)=\mathbf{E}\cdot\mathbf{ \varphi}_{\varphi_{\theta}}\end{array}  \tag{32}\]  
where \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{C}\), \(\mathbf{D}\), and \(\mathbf{E}\) as well as \(\mathbf{\varphi}_{u},\mathbf{\varphi}_{v},\mathbf{\varphi}_{w},\mathbf{\varphi}_{\theta\theta}\), and \(\mathbf{\varphi}_{\theta x}\) represent the coefficient vector and function vector, respectively. Their expressions can be expressed as follows:  
\[