## 2 Shell Dynamics
### Energy Equation of the Shell
\begin{array}{l}\varepsilon_{x0}=\frac{1}{a_{1}}\bigg{(}\partial_{x}u_{0}+ \frac{\partial_{\theta}a_{1}}{a_{2}}v_{0}+\frac{a_{1}}{R_{1}}w_{0}\bigg{)}, \ \varepsilon_{\theta 0}=\frac{1}{a_{2}}\bigg{(}\partial_{\theta}v_{0}+ \frac{\partial_{x}a_{2}}{a_{1}}u_{0}+\frac{a_{2}}{R_{2}}w_{0}\bigg{)}\\ \gamma_{\theta z0}=\frac{1}{a_{2}}\bigg{(}\partial_{\theta}w_{0}+a_{2}\psi_{ \theta}-\frac{a_{2}}{R_{2}}v_{0}\bigg{)},\ \gamma_{x\geq 0}=\frac{1}{a_{1}} \bigg{(}\partial_{x}w_{0}+a_{1}\psi_{x}-\frac{a_{1}}{R_{1}}u_{0}\bigg{)}\\ \gamma_{x\theta 10}=\frac{1}{a_{1}}\bigg{(}\partial_{x}\psi_{0}- \frac{\partial_{\theta}a_{1}}{a_{2}}u_{0}\bigg{)},\ \gamma_{x\theta 20}=\frac{1}{a_{2}} \bigg{(}\partial_{\theta}u_{0}-\frac{\partial_{x}a_{2}}{a_{1}}v_{0}\bigg{)}\\ \kappa_{x}=\frac{1}{a_{1}}\bigg{(}\partial_{x}\psi_{x}+ \frac{\partial_{\theta}a_{1}}{a_{2}}\psi_{\theta}\bigg{)},\ \kappa_{\theta}=\frac{1}{a_{2}} \bigg{(}\partial_{\theta}\psi_{\theta}+\frac{\partial_{x}a_{2}}{a_{2}}\psi_{ x}\bigg{)}\\ \kappa_{x\theta 1}=\frac{1}{a_{1}}\bigg{(}\partial_{x}\psi_{\theta}- \frac{\partial_{\theta}a_{1}}{a_{2}}\psi_{x}\bigg{)},\ \kappa_{x\theta 2}=\frac{1}{a_{2}} \bigg{(}\partial_{\theta}\psi_{x}-\frac{\partial_{x}a_{2}}{a_{1}}\psi_{ \theta}\bigg{)}\end{array}  \tag{8}\]  
According to [PERSON]'s law, the stress-strain relationship of the MRCS can be expressed as follows:  
\[ \begin{array}{l}\begin{array}{l}\sigma_{xx}\\ \sigma_{\theta\theta}\\ \tau_{x\theta}\\ \tau_{\theta z}\\ \tau_{xz}\\ \end{array}\end{array}  \tag{9}\]  
where  
\[ \begin{array}{l}Q_{11}=Q_{22}=\frac{E\ (1+i\eta_{1})}{1-\mu^{2}},\ Q_{12}=Q_{21}= \mu Q_{11}\\ Q_{44}=Q_{55}=Q_{66}=G(1+i\eta_{2})\end{array}  \tag{10}\]  
Equation (10) introduces the complex Young's modulus \(E\ (1+i\eta_{1})\) and the complex shear modulus \(G\ (1+i\eta_{2})\), following the MSE [27; 28] and MSKE [29] methods. Here, \(\eta_{1}\) and \(\eta_{2}\) represent the loss factors of MRCS in normal and shear strains, respectively. In addition, the resultant force and resultant moment in the MRCS can be defined as follows:  
\[ \begin{array}{l}\mathbf{N}_{s}=\int_{-\frac{h}{2}}^{\frac{h}{2}}\begin{cases} \sigma_{x}(1+z/R_{2})\\ \sigma_{\theta}(1+z/R_{1})\\ \tau_{x\theta}(1+z/R_{2})\\ \tau_{x\theta}(1+z/R_{1})\\ \end{cases}dz,\ \mathbf{M}_{s}=\int_{-\frac{h}{2}}^{\frac{h}{2}}\begin{cases} \sigma_{x}(1+z/R_{2})\\ \sigma_{\theta}(1+z/R_{1})\\ \tau_{x\theta}(1+z/R_{2})\\ \tau_{x\theta}(1+z/R_{1})\\ \end{cases}zdz\\ \mathbf{Q}_{s}=K_{s}\int_{-\frac{h}{2}}^{\frac{h}{2}}\begin{cases}\sigma_{xz}(1+z/ R_{2})\\ \sigma_{\theta z}(1+z/R_{1})\\ \end{cases}dz\end{cases}\end{array}  \tag{11}\]  
where \(K_{s}=5/6\). In addition, the normal strain and shear strain in the midface of MRCS can be expressed as follows:\[\epsilon_{0}=\left\{ \begin{array}{c}\varepsilon_{x0}\\ \varepsilon_{\theta 0}\\ \gamma_{x\theta 10}\\ \gamma_{x\theta 20}\end{array} \right\}\kappa=\left\{ \begin{array}{c}\kappa_{x} \\ \kappa_{\theta}\\ \kappa_{x\theta 1}\\ \kappa_{x\theta 2}\end{array} \right\}\gamma_{0}=\left\{ \begin{array}{c} \gamma_{\theta:0}\\ \gamma_{x0}\end{array} \right\} \tag{12}\]  
Based on Equations (8) and (9), the strain energy of the MRCS can be obtained as follows:  
\[U_{e}=\frac{1}{2}\int_{0}^{L}\int_{0}^{2\pi}\widetilde{\mathbf{N}}\cdot \epsilon d\theta dx \tag{13}\]  
where  
\[\widetilde{\mathbf{N}}=\left\{ \begin{array}{c}\mathbf{N}_{s}\\ \mathbf{M}_{s}\\ \mathbf{Q}_{s}\end{array} \right\},\epsilon=\left\{ \begin{array}{c}\epsilon _{0}\\ \kappa\\ \gamma_{0}\end{array} \right\} \tag{14}\]  
Furthermore, the kinetic energy of the MRCS and the potential energy of the Pasternak elastic foundation can be expressed as:  
\[T_{e}=\frac{1}{2}\int_{0}^{L}\int_{0}^{2\pi}\int_{-\frac{1}{2}}^{\frac{b}{2}} \rho(z)\Big{(}\dot{u}^{2}+\dot{v}^{2}+\dot{w}^{2}\Big{)}A_{1}A_{2}dzd\theta dx \tag{15}\]  
\[U_{P}=\frac{1}{2}\sum_{i=1}^{2}\int_{0}^{2\pi}\int_{0}^{L}\Big{[}k_{2i-1}w^{2} +k_{2i}\Big{(}\psi_{x}^{2}+\psi_{\theta}^{2}\Big{)}\Big{]}\,A_{1}A_{2}dxd\theta \tag{16}\]