# Free Vibration Analysis of Arches with Interval-Uncertain Parameters
## 2 Model of the Arch and Procedure of Deterministic Free-Vibration Equations  
### Establishment of Free-Vibration Equations  
The model of the arch is shown in Figure 1. Only the elastic deformation and the in-plane vibration are considered in the model. The term \(\vartheta\) denotes the central angle formed between the radius of any cross-section and the \(Y\) axis, which is positively oriented in the clockwise direction; \(\vartheta\)\(\in\)\([-\vartheta_{0},\vartheta_{0}]\), where \(\vartheta_{0}\) is half of the central angle of the whole arch. \(R\) is the radius and \(u(\vartheta,t)\), \(v(\vartheta,t)\), and \(\varphi(\vartheta,t)\) are the tangential displacement, radial displacement, and bending slope of the cross-section at a given central angle \(\vartheta\) and arbitrary time \(t\), respectively. The kinematic equation can be written as follows [6]:  
\[\left\{ \begin{array}{l}\frac{1}{R}\frac{\partial N(\vartheta,t)}{\partial \vartheta}-\frac{Q(\vartheta,t)}{R}=\rho A\frac{\partial^{2}u(\vartheta,t)}{ \partial t^{2}}\\ \frac{1}{R}\frac{\partial Q(\vartheta,t)}{\partial\vartheta}+\frac{N( \vartheta,t)}{R}=\rho A\frac{\partial^{2}\varphi(\vartheta,t)}{\partial t^{ 2}}\\ \frac{1}{R}\frac{\partial M(\vartheta,t)}{\partial\vartheta}-Q(\vartheta,t)= \rho I\frac{\partial^{2}\varphi(\vartheta,t)}{\partial t^{2}}\end{array} \right. \tag{1}\]  
where \(\rho\) is the mass density for unit volume; \(A\) and \(I\) are the area and moment of inertia of the cross-section, respectively. \(N(\vartheta,t)\), \(Q(\vartheta,t)\), and \(M(\vartheta,t)\) represent the axial force, shear force, and bending moment, respectively, and these internal forces can also be expressed by the constitutive relation of displacement as:  
\[\left\{ \begin{array}{c}N(\vartheta,t)=\frac{EA}{R}\left(\frac{\partial u( \vartheta,t)}{\partial\vartheta}-v(\vartheta,t)\right)\\ Q(\vartheta,t)=\frac{GA}{Rx_{0}}\left(u(\vartheta,t)+\frac{\partial v( \vartheta,t)}{\partial\vartheta}+R\varphi(\vartheta,t)\right)\\ M(\vartheta,t)=\frac{EI}{R}\frac{\partial\varphi(\vartheta,t)}{\partial \vartheta}\end{array} \right. \tag{2}\]  
in which \(E\) is [PERSON]'s modulus and \(G\) is shear modulus; \(\kappa_{0}\) is the shear factor. Substituting the relations in (2) into Equation (1), and with \(u(\vartheta,t)\), \(v(\vartheta,t)\), and \(\varphi(\vartheta,t)\) indicated as \(u\), \(v\), and \(\varphi\), the equations can be expressed as follows:  
\[\left\{ \begin{array}{l}\frac{EA}{R^{2}}\frac{\partial^{2}u}{\partial\vartheta ^{2}}-\frac{GA}{Rx_{0}}u-\left(\frac{EA}{R^{2}}+\frac{GA}{Rx_{0}}\right)\frac {\partial v}{\partial\vartheta}-\frac{GA}{Rx_{0}}\varphi=\rho A\frac{ \partial^{2}u}{\partial t^{2}}\\ \left(\frac{EA}{R^{2}}+\frac{GA}{Rx_{0}}\right)\frac{\partial u}{\partial \vartheta}+\frac{GA}{Rx_{0}}\frac{\partial^{2}v}{\partial\vartheta^{2}}- \frac{EA}{R^{2}}\
u+\frac{GA}{Rx_{0}}\frac{\partial\vartheta}{\partial t^{ 2}}\\ -\frac{GA}{Rx_{0}}u-\frac{GA}{Rx_{0}}\frac{\partial\varphi}{\partial\vartheta} +\frac{EI}{R}\frac{\partial^{2}\varphi}{\partial\vartheta^{2}}-\frac{GA}{Rx _{0}}\varphi=\rho I\frac{\partial^{2}\varphi}{\partial t^{2}}\end{array} \right. \tag{3}\]  
Figure 1: Model of the arch. Equation (3) comprises the in-plan free-vibration equations. In general, the separation of variables can be employed to seek solutions that exhibit temporal harmony with frequency \(\omega\), so the displacement can be expressed as:  
\[\left\{ \begin{aligned} & u(\vartheta,t)=U(\vartheta)\sin(\omega t)\\ & v(\vartheta,t)=V(\vartheta)\sin(\omega t)\\ &\varphi(\vartheta,t)=\Phi(\vartheta)\sin(\omega t)\\ \end{aligned} \right. \tag{4}\]  
\(U(\vartheta)\), \(V(\vartheta)\), and \(\Phi(\vartheta)\) are indicated as \(U\), \(V\), and \(\Phi\), which represent the tangential, radial, and rotational vibration amplitude values of the structure, respectively. Substituting the relations in (4) into Equation (3), we obtain:  
\[\left\{