## 2 Generally Shaped Arch Model and Equations of Motion  
Figure 1 illustrates a generally shaped arch with the span \(L\) and mid-span rise \(H\) (vertical height from the abutments to the top), referred to an orthogonal local system o'x'y', and a global polar system defined using a central angle \(\theta\) regarding the vertical Y axis and a radius of curvature \(R=R(\theta)\) (\(R(\theta)\) is constant for the circular arch and can be expressed as \(R\)), and \(\theta\) is positive if it is clockwise, \(\theta\in[-\theta_{0},\theta_{0}]\). It is assumed that the arch is linear elastic and symmetric, with its transverse section remaining constant along the principal axis. The arch is composed of elastic, homogeneous, and uniform isotropic materials, \(u(\theta,t)\), \(v(\theta,t)\) and \(\varphi(\theta,t)\) are the tangential displacement, radial displacement, and bending slope of the centroid of transverse section corresponding the angle \(\theta\) at an arbitrary time \(t\), respectively. These displacements consistently align with the positive direction of the corresponding local system. The strain-displacement relations of the arch are determined through the utilization of the definitions for the strain displacements in the polar system and are presented as follows  
\[\left\{ \begin{matrix}\varepsilon(\theta,t)=\frac{1}{R(\theta)}\left(\frac{ \partial u(\theta,t)}{\partial\theta}-v(\theta,t)\right)\\ \gamma(\theta,t)=\frac{1}{R(\theta)}\left(u(\theta,t)+\frac{\partial\varepsilon (\theta,t)}{\partial\theta}\right)+\varphi(\theta,t)\\ \chi(\theta,t)=\frac{1}{R(\theta)}\frac{\partial\varphi(\theta,t)}{\partial \theta}\end{matrix} \right. \tag{1}\]  
in which \(\varepsilon(\theta,t)\), \(\gamma(\theta,t)\) and \(\chi(\theta,t)\) are the normal strain, the shear strain and the curvature components, respectively. Generally, the equations of motion of a generally shaped arch can be deduced by the Hamilton principle. The principle provides the actual boundary conditions of the problem and stipulates that among the set of all the admissible configurations of a system, its actual motion renders the potential \(\Pi\) stationary once the configuration of the system is defined at the limits \(t=t_{1}\) and \(t=t_{2}\), which implies:  
\[\delta\mathbf{\Pi}=\int_{t_{1}}^{t_{2}}(\delta\mathbf{\Omega}-\delta\mathbf{T}-\delta\mathbf{ W})dt=0 \tag{2}\]  
where \(\delta\mathbf{\Omega}\), \(\delta\mathbf{T}\), and \(\delta\mathbf{W}\) refer to the variation of the strain energy, kinetic energy and work of the external excitation forces, respectively. The strain energy can be expressed as  
\[\mathbf{\Omega} =\frac{1}{2}\int_{-\theta_{0}}^{\theta_{0}}\left(EA\varepsilon^{ 2}(\theta,t)+GA\gamma^{2}(\theta,t)/\kappa_{0}+EI\chi^{2}(\theta,t)\right)R( \theta)d\theta \tag{3}\] \[=\frac{1}{2}\int_{-\theta_{0}}^{\theta_{0}}\varepsilon^{T}\left[ \begin{array}{ccc}EA&0&0\\ 0&GA/\kappa_{0}&0\\ 0&0&EI\end{array} \right]\varepsilon R(\theta)d\theta\]where \(E\) is [PERSON]'s modulus, \(G\) is the shear modulus, and \(G=\frac{E}{2(1+\
u)}\), in which \(\
u\) represents the Poisson ratio. \(A\) is the area of the transverse section, \(I\) is the moment of inertia, \(\kappa_{0}\) is the shear factor, and \(\epsilon=[\epsilon(\theta,t)\), \(\gamma(\theta,t)\), \(\chi(\theta,t)]^{\mathrm{T}}\) is the generalized strain vector. Accordingly, the kinetic energy can be expressed as  
\[ \begin{array}{ll}\mathbf{T}&=\frac{1}{2}f_{-\theta_{0}}^{\theta_{0}}\left(\rho A \left(\frac{\partial u(\theta,t)}{\partial t}\right)^{2}+\rho A\left(\frac{ \partial v(\theta,t)}{\partial t}\right)^{2}+\rho I\left(\frac{\partial\varphi (\theta,t)}{\partial t}\right)^{2}\right)R(\theta)d\theta\\ &=\frac{1}{2}f_{-\theta_{0}}^{\theta_{0}}\frac{\mathbf{u}^{T}}{\partial t} \left[\begin{array}{ccc}\rho A&0&0\\ 0&\rho A&0\\ 0&0&\rho I\end{array}\right]\frac{\partial\mathbf{u}}{\partial t}R(\theta)d\theta \end{array}