## 2 Theory and Formulation
### Hamilton's Principle
\begin{array}{c}\bigg{\{}-I_{0}\frac{ \partial^{2}u}{\partial t^{2}}-I_{1}\frac{\partial^{2}\phi}{\partial t^{2}}+I _{3}c_{1}\bigg{(}\frac{\partial^{2}\phi}{\partial t^{2}}+\frac{\partial^{3}w }{\partial R\partial^{2}}\bigg{)}\bigg{)}\delta u\\ +\bigg{\{}-I_{1}\frac{\partial^{2}u}{\partial t^{2}}-I_{2}\frac{\partial^{2} \phi}{\partial t^{2}}+I_{4}c_{1}\bigg{(}\frac{\partial^{2}\phi}{\partial t^{2 }}+\frac{\partial^{3}w}{\partial R\partial^{2}}\bigg{)}\bigg{)}\delta\phi\\ +\bigg{\{}c_{1}I_{3}\frac{\partial^{2}u}{\partial t^{2}}+c_{1}I_{4}\frac{ \partial^{2}\phi}{\partial t^{2}}-I_{6}c_{1}^{2}\bigg{(}\frac{\partial^{3} \phi}{\partial R\partial^{2}}+\frac{\partial^{3}w}{\partial R\partial^{2}} \bigg{)}\bigg{)}\delta\phi\\ +\bigg{\{}-c_{1}I_{3}\frac{\partial^{3}u}{\partial R\partial^{2}}-c_{1}I_{4} \frac{\partial^{3}\phi}{\partial R\partial^{2}}+I_{6}c_{1}^{2}\bigg{(}\frac{ \partial^{3}\phi}{\partial R\partial^{2}}+\frac{\partial^{4}w}{\partial R \partial^{2}}\bigg{)}\bigg{)}\delta w\\ +\bigg{\{}-I_{0}\frac{\partial^{2}w}{\partial t^{2}}\delta w\end{array} \right] \tag{34}\]  
where \(\{I_{0},I_{1},I_{2},I_{3},I_{4},I_{5},I_{6}\}=\int\limits_{-\frac{h}{2}}^{h} \rho^{NCM}\Big{\{}1,z,z^{2},z^{3},z^{4},z^{5},z^{6}\Big{\}}dZ\) represent the mass inertias. Substituting Equations (22), (26), (31), and (34), into Equation (21) yields the Euler-Lagrange equations of MHCAP  
\[\delta u:\frac{\partial N_{RR}}{\partial R}-\frac{N_{\theta\theta}}{R}=I_{0} \frac{\partial^{2}u}{\partial t^{2}}+I_{1}\frac{\partial^{2}\phi}{\partial t ^{2}}-c_{1}I_{3}\bigg{(}\frac{\partial^{2}\phi}{\partial t^{2}}+\frac{ \partial^{3}w}{\partial R\partial t^{2}}\bigg{)} \tag{35}\]  
\[\delta w:c_{1}\frac{\partial^{2}P_{RR}}{\partial R^{2}}-\frac{c_{1}}{R}\frac{ \partial P_{0\theta}}{\partial R}+\frac{\partial Q_{R2}}{\partial R}-3c_{1} \frac{\partial S_{R2}}{\partial R}-K_{W}w \tag{36}\]  
\[+K_{p}\frac{\partial^{2}w}{\partial R^{2}}-N^{T}\frac{\partial^{2}\phi}{ \partial R^{2}}=c_{1}I_{3}\frac{\partial^{3}u}{\partial R\partial^{2}}+c_{1}I _{4}\frac{\partial^{3}\phi}{\partial R\partial^{2}}\] \[-c_{1}I_{6}\bigg{(}\frac{\partial^{3}\phi}{\partial R\partial^{2}}+ \frac{\partial^{4}w}{\partial R^{2}\partial^{2}}\bigg{)}+I_{0}\frac{\partial^ {2}w}{\partial t^{2}}\]  
\[\delta\phi:\frac{\partial M_{RR}}{\partial R}-c_{1}\frac{\partial P_{RR}}{ \partial R}-\frac{M_{\theta\theta}}{R}+\frac{c_{1}}{R}P_{\theta\theta}-Q_{R2} +3c_{1}S_{R2}=I_{1}\frac{\partial^{2}u}{\partial t^{2}}+I_{2}\frac{\partial^ {2}\phi}{\partial t^{2}} \tag{37}\]  
The boundary conditions are given as  
\[\delta u=0\ \ \text{or}\ \ \text{N}_{RR}n_{R}=0 \tag{38}\]  
\[\delta w=0\ \ \text{or}\ \ \bigg{[}c_{1}\frac{\partial P_{RR}}{\partial R}-c_{1} \frac{P_{\theta\theta}}{R}+Q_{RZ}-3c_{1}S_{RZ}-K_{P}\frac{\partial w}{\partial R }+N^{T}\frac{\partial w}{\partial R}\bigg{]}n_{R}=0 \tag{39}\]  
\[\delta\phi=0\ \ \text{or}\ \ \bigg{[}M_{RR}-c_{1}P_{RR}\bigg{]}n_{R}=0\]  
differential quadrature method (DQM), which was proposed by [PERSON] et al. [78,79] in the 1970s. In DQM, the results depend on the numbers of grid points. As in any computational method, choosing the optimal numbers of grid points is of high importance. [PERSON] [80] proposed an efficient approach to determine the weighting coefficients for an infinite number of grid points which is known as generalized differential quadrature method (GDQ). [PERSON] and [PERSON] [81] applied a domain decomposition method which can be implemented in multidomain problems. In GDQ, the approximate \(r\)th derivative of \(f(x)\) is determined as  
\[\left.\frac{\partial^{r}f(x)}{\partial R^{r}}\right|_{x=x_{p}}=\sum\limits_{j= 1}^{n}g_{ij}^{