## 2 Shell Dynamics  
### Energy Equation of the Shell  
Considering an MRCS with a length of \(L\), a thickness of \(h\), and a radius of \(R\), as shown in Figure 1, the \(u\), \(v\), and \(w\) represent the displacement functions of the three orthogonal directions in the MRCS, and their directions correspond to the coordinate axes \(x\), \(\theta\), and \(z\) of the orthogonal curve, respectively. The cylindrical shell is supported by a Pasternak elastic foundation, which is described by the following equation [26]:  
\[q=k_{i}w-k_{j}(\psi_{x}+\psi_{\theta}) \tag{1}\]  
where \(q\) represents the stress per unit area, \(k_{i}\) represents the stiffness of the Winkler foundation, and \(k_{j}\) represents the shear subgrade modulus. The lower corner label \(i=1\), \(j=2\) indicates the external support, while \(i=3\), \(j=4\) represents the internal support. To determine the material properties at any point along the thickness direction of the MRCS, the following expression can be used:  
\[P=(P_{a}-P_{b})\ V(z)+P_{b} \tag{2}\]  
where \(P_{a}\) and \(P_{b}\) are the maximum and minimum values of the material properties, respectively. \(V(z)\) represents the volume fraction, and its expression as a polynomial is as follows:  
\[V(z)=\sum\limits_{i=0}^{N_{b}}a_{i}x^{i} \tag{3}\]  
where \(a_{i}\) and \(N_{b}\) represent the coefficient and order of the polynomial, respectively. According to the elastic shell theory, the normal and shear strains of the MRCS can be expressed as:  
\[ \begin{array}{l}\varepsilon_{i}=\frac{\partial}{\partial\xi_{i}}\Big{(} \frac{s_{i}}{A_{i}}\Big{)}+\frac{1}{A_{i}}\sum\limits_{k=1}^{3}\frac{s_{k}}{A _{k}}\frac{\partial A_{i}}{\partial\xi_{k}}\\ \gamma_{ij}=\frac{1}{A_{i}A_{j}}\Big{[}A_{i}^{2}\frac{\partial}{\partial\xi_ {j}}\Big{(}\frac{s_{i}}{A_{i}}\Big{)}+A_{j}^{2}\frac{\partial}{\partial\xi_{ j}}\Big{(}\frac{s_{j}}{A_{i}}\Big{)}\Big{]}(i\
eq j)\end{array}  \tag{4}\]  
where the subscripts \(i\), \(j=1\), \(2\), \(3\) represent three different directions along the coordinate axis curves. For example, \(\xi_{1}=x\), \(\xi_{2}=\theta\), \(\xi_{3}=z\), represent \(x\), \(\theta\), \(z\) directions on the coordinate axis curve, respectively. Additionally, \(s_{1}=u\), \(s_{2}=v\), and \(s_{3}=w\) represent mutually orthogonal displacement functions. The Lame coefficient \(A_{i}\) (\(i=1\), \(2\), \(3\)) can be expressed as follows:  
\[A_{1}=a_{1}\Big{(}1+\frac{z}{R_{1}}\Big{)},A_{2}=a_{2}\Big{(}1+\frac{z}{R_{2} }\Big{)},A_{3}=1 \tag{5}\]  
Figure 1: The structure and cells of the MRCS: (**a**) Axial cross section; (**b**) Radial section; (**c**) Coordinate system. where the Lame coefficient is \(a_{1}=1\) and \(a_{2}=R_{2}\). Based on the FSDT, the displacement functions can be obtained as follows:  
\[ \begin{array}{l}u(x,\theta,z)=u_{0}(x,\theta,t)+z\psi_{x}(x,\theta)\\ v(x,\theta,z)=v_{0}(x,\theta,t)+z\psi_{\theta}(x,\theta)\\ w(x,\theta,z)=w_{0}(x,\theta,t)\end{array}  \tag{6}\]  
where \(u_{0}\), \(v_{0}\), and \(w_{0}\) represent the displacements in the middle plane along the \(x\), \(\theta\), and \(z\) directions, respectively. Moreover, the correlation between the strain and displacement at any point within MRCS can be calculated using the following expression:  
\[ \begin{array}{l}\varepsilon_{x}=\frac{R_{1}}{R_{1}+z}(\varepsilon_{x0}+z \kappa_{x}),\ \varepsilon_{\theta}=\frac{R_{2}}{R_{2}+z}(\varepsilon_{\theta 0}+z \kappa_{\theta})\\ \gamma_{x\theta}=\frac{R_{1}}{R_{1}+z}(\gamma_{x\theta 10}+z\kappa_{x\theta 1})+ \frac{R_{2}}{R_{2}+z}(\gamma_{x\theta 20}+z\kappa_{x\theta 2})\\ \gamma_{\theta z}=\frac{R_{2}\gamma_{x\theta 0}}{R_{2}+z},\ \gamma_{xz}=\frac{R_{1} \gamma_{x\theta 0}}{R_{1}+z}\end{array}  \tag{7}\]  
where  
\[