## 2 Generally Shaped Arch Model and Equations of Motion
 \tag{4}\]  
where \(\rho\) is the mass density for the unit volume, \(\mathbf{u}=[u(\theta,t),v(\theta,t),\)\(\varphi(\theta,t)]^{\mathrm{T}}\) is the generalized displacement vector, and \(\frac{\partial\mathbf{u}}{\partial t}=\left[\frac{\partial u(\theta,t)}{\partial t }\quad\frac{\partial v(\theta,t)}{\partial t}\quad\frac{\partial\varphi( \theta,t)}{\partial t}\right]^{T}\) is the generalized velocity vector. The work of the external force can be expressed as  
\[ \begin{array}{ll}W&=\int_{-\theta_{0}}^{\theta_{0}}\left(p(\theta,t)u( \theta,t)+q(\theta,t)v(\theta,t)+m(\theta,t)\varphi(\theta,t)\right)R(\theta)d \theta\\ &=\int_{-\theta_{0}}^{\theta_{0}}\mathbf{u}^{T}\mathbf{q}R(\theta)d\theta\end{array}  \tag{5}\]  
where \(\mathbf{q}=[p(\theta,t),\)\(q(\theta,t),\)\(m(\theta,t)]\) is the vector of the external excitation force, with the components \(p(\theta,t),\)\(q(\theta,t)\) in the tangential and normal direction, and \(m(\theta,t)\) the flexural moments per unit of length. Substituting Equations (3)-(5) into Equation (2), the equations of motion can be expressed as follows [5]:  
\[ \begin{array}{ll}\left(\frac{EA}{R^{2}(\theta)}\frac{\partial^{2}u(\theta,t )}{\partial\theta^{2}}-\frac{EA}{R^{3}(\theta)}\frac{\partial R(\theta)}{ \partial\theta}\frac{\partial u(\theta,t)}{\partial\theta}-\frac{GA}{R^{2}( \theta)\kappa_{0}}u(\theta,t)-\left(\frac{EA}{R^{2}(\theta)}+\frac{GA}{R^{2}( \theta)\kappa_{0}}\right)\frac{\partial v(\theta,t)}{\partial\theta}\\ +\frac{EA}{R^{3}(\theta)}\frac{\partial R(\theta)}{\partial\theta}v(\theta,t) -\frac{GA}{R(\theta)\kappa_{0}}\varphi(\theta,t)+p(\theta,t)=\rho A\frac{ \partial^{2}u(\theta,t)}{\partial t^{2}}\\ \left(\frac{EA}{R^{2}(\theta)}+\frac{GA}{R^{2}(\theta)}\frac{\partial u(\theta,t)}{\partial\theta}\frac{\partial u(\theta,t)}{\partial\theta}-\frac{GA}{R^{ 3}(\theta)\kappa_{0}}\frac{\partial R(\theta)}{\partial\theta}u(\theta,t)+ \frac{GA}{R^{2}(\theta)\kappa_{0}}\frac{\partial^{2}v(\theta,t)}{\partial \theta^{2}}\\ -\frac{GA}{R^{3}(\theta)\kappa_{0}}\frac{\partial R(\theta)}{\partial\theta} \frac{\partial v(\theta,t)}{\partial\theta}-\frac{EA}{R^{2}(\theta)}v(\theta, t)+\frac{GA}{R(\theta)\kappa_{0}}\frac{\partial\varphi(\theta,t)}{\partial \theta}+q(\theta,t)=\rho A\frac{\partial^{2}v(\theta,t)}{\partial t^{2}}\\ -\frac{GA}{R(\theta)\kappa_{0}}u(\theta,t)-\frac{GA}{R(\theta)\kappa_{0}}\frac{ \partial v(\theta,t)}{\partial\theta}+\frac{EI}{R^{2}(\theta)}\frac{\partial ^{2}\varphi(\theta,t)}{\partial\theta^{2}}\\ -\frac{EI}{R^{3}(\theta)}\frac{\partial R(\theta)}{\partial\theta}\frac{ \partial\varphi(\theta,t)}{\partial\theta}-\frac{GA}{\kappa_{0}}\theta( \theta,t)+m(\theta,t)=\rho I\frac{\partial^{2}\varphi(\theta,t)}{\partial t^{2}} \end{array} \right.\end{array} \tag{6}\]  
The boundaries at both ends of the arch studied in this paper are fixed, which can be described as \(u(\theta,t)=0,v(\theta,t)=0,\)\(\varphi(\theta,t)=0\), at \(\theta=\pm\theta_{0}\). The differential quadrature (DQ) method is introduced to eliminate the differential terms concerning \(\theta\) as follows [3]. According to the DQ method, the derivative of the smooth function concerning the independent variable can be substituted by a weighted summation of the function values at multiple sample points, which is obtained by dividing the domain of the independent variable according to specific rules. This means that the \(n\)th ordered derivative of a smooth function \(f(\theta)\) with respect to \(\theta\) at the \(i\)th sample point \(\theta_{i}\) can be replaced as follows  
\[\frac{\partial^{n}f(\theta)}{\partial\theta^{n}}\left|{}_{\theta=\theta_{i}} \right.=\sum_{j=1}^{N}n_{ij}^{(n)}f\big{(}\theta_{j}\big{)},\ i=1,2,\cdots N \tag{7}\]  
where \(\eta_{ij}^{(n)}\) represents the \(n\)th order derivative weighting coefficient of \(\theta\) at the \(i\)th sample point calculated for the \(j\)th sample point in the domain, \(f(\theta_{j})\) is the value of the function \(f(\theta)\) at the \(j\)th sample point, and \(N\) is the total number of sample points.