## 2 Shell Dynamics
### The Motion Equation of the Shell  
The motion equation of the MRCS based on [PERSON]'s principle can be described by the following expression:  
\[\int_{t_{1}}^{t_{2}}\left(\delta U_{e}+\delta U_{P}-\delta K_{e}\right)dt=0 \tag{17}\]  
Substituting Equation (6) into Equation (13) and taking the first-order variation of the strain energy for the MRCS, the expression can be obtained as follows:  
\[ \begin{array}{rl}\delta U_{e}&=\frac{1}{2}\int_{0}^{L}\int_{0}^{2\pi}\int_{ -\frac{1}{2}}^{\frac{b}{2}}\widetilde{\mathbf{N}}\cdot\delta\epsilon A_{1}A_ {2}dzd\theta dx\\ &=\frac{a_{1}a_{2}}{2}\int_{0}^{L}\int_{0}^{2\pi}\int_{-\frac{1}{2}}^{\frac{b} {2}}\left(\begin{array}{c}\sigma_{x}\delta\varepsilon_{x}+\sigma_{\theta} \delta\varepsilon_{\theta}+\tau_{x\theta}\delta\gamma_{x\theta}\\ +\sigma_{\theta:}\sigma\delta\gamma_{\theta:\theta}+\tau_{x\theta}\delta\gamma_{ x\theta}\\ \end{array}\right)\Big{(}1+\frac{z}{R_{1}}\Big{)}\Big{(}1+\frac{z}{R_{2}}\Big{)}dzd \theta dx\\ &=\frac{a_{1}a_{2}}{2}\int_{0}^{L}\int_{0}^{2\pi}\left(\begin{array}{c} \mathbf{N}_{s}\delta\varepsilon_{x0}+M_{xx}\delta\kappa_{x}+N_{\theta}\delta \varepsilon_{x0}+M_{\theta}\delta\kappa_{\theta}\\ +N_{x\theta}\delta\gamma_{x\theta 10}+M_{x\theta}\delta\kappa_{x\theta 1}+N_{ \theta x}\delta\gamma_{x\theta 20}\\ +M_{\theta x}\delta\kappa_{x\theta 2}+Q_{\theta z}\delta\gamma_{x\theta 20}+Q_{ xx}\delta\gamma_{x\theta 20}\end{array}\right)dzd\theta dx\end{array}  \tag{18}\]  
Similarly, by substituting Equation (6) into Equation (15) and taking the first-order variation of the kinetic energy of MRCS, the expression can be obtained as follows:  
\[\delta T_{e}=\frac{a_{1}a_{2}}{2}\int_{0}^{L}\int_{0}^{2\pi}\int_{\frac{1}{2}}^ {\frac{b}{2}}\rho\Big{(}\dot{w}\,\dot{w}+\dot{u}\,\dot{\delta u}+\dot{v}\,\dot{ \delta v}\Big{)}\Big{(}1+\frac{z}{R_{1}}\Big{)}\Big{(}1+\frac{z}{R_{2}}\Big{)} dzd\theta dx \tag{19}\]  
Substituting Equations (16), (18), and (19) into Equation (17) yields the expression of  
\[\begin{array}{rl}\int_{t_{1}}^{t_{2}}\int_{0}^{L}\int_{0}^{2\pi}\int_{- \frac{1}{2}}^{\frac{b}{2}}\rho\Big{(}\dot{w}\,\dot{\delta w}+\dot{u}\,\dot{ \delta u}+\dot{v}\,\dot{\delta v}\Big{)}\Big{(}1+\frac{z}{R_{1}}\Big{)} \Big{(}1+\frac{z}{  
where  
\[\left\{ \begin{array}{l}\Gamma_{1}=\left[RN_{x}\delta u_{0}+RN_{x0}\delta v_{0}+RQ_{ xz}\delta w_{0}+RM_{x}\delta\psi_{x}+RM_{x0}\delta\psi_{\theta}\right]_{0}^{L}\\ \Gamma_{2}=\left[N_{\theta x}\delta u_{0}+N_{\theta}\delta v_{0}+Q_{\theta z} \delta w_{0}+M_{\theta x}\delta\psi_{x}+M_{\theta}\delta\psi_{\theta}\right]_{0 }^{2\pi}\end{array} \right. \tag{21}\]  
The paper calculates the natural frequency and the modal loss factor of the MRCS through the Rayleigh-Ritz method. The accuracy of the calculation results is closely linked to the choice of admissible functions. The Chebyshev polynomial, Legendre polynomial, and Gram-Schmidt orthogonal polynomial are the commonly used admissible functions [30; 31]. However, since the boundary conditions of the MRCS are determined by boundary functions at both ends, the Chebyshev and Legendre polynomials are not suitable. Thus, this paper adopts the Gram-Schmidt orthogonal polynomials to construct admissible functions. Then, the displacement field function is expressed as follows:  
\[\left\{ \begin{array}{l}u_{0}(x,\theta,z,t)=e^{-j\omega t}\cos(n\theta)U(x)\\ v_{0}(x,\theta,z,t)=e^{-j\omega t}\sin(n\theta)V(x)\\ w_{0}(x,\theta,z,t)=e^{-j\omega t}\cos(n\theta)W(x)\\ \psi_{x}(x,\theta,z,t)=e^{-j\omega t}\cos(n\theta)\Psi_{x}(x)\\ \psi_{\theta}(x,\theta,z,t)=e^{-j\omega t}\sin(n\theta)\Psi_{\theta}(x)\end{array} \right. \tag{22}\]  
where, \(U(x)\), \(V(x)\), \(W(x)\), \(\Psi_{x}(x)\), and \(\Psi_{\theta}(x)\) are expressed as vibration shape functions, and their expressions can be shown as follows:  
\[