# Free Vibration Analysis of Arches with Interval-Uncertain Parameters
## 2 Model of the Arch and Procedure of Deterministic Free-Vibration Equations
\begin{aligned} &\frac{EA}{R^{2}}\frac{d^{2}U}{d^{2}}-\frac{ GA}{R^{2}\kappa_{0}}U-\left(\frac{EA}{R^{2}}+\frac{GA}{R^{2}\kappa_{0}}\right)\frac{dV}{d \vartheta}-\frac{GA}{R_{0}}\Phi=-\omega^{2}\rho AU\\ &\left(\frac{EA}{R^{2}}+\frac{GA}{R^{2}\kappa_{0}}\right)\frac{dU }{d\vartheta}+\frac{GA}{R^{2}\kappa_{0}}\frac{dV}{d\vartheta^{2}}-\frac{EA}{ R^{2}}V+\frac{GA}{R_{0}}\frac{d\Phi}{d\vartheta}=-\omega^{2}\rho AV\\ &-\frac{GA}{R\kappa_{0}}U-\frac{GA}{R\kappa_{0}}\frac{dV}{d \vartheta}+\frac{EI}{R^{2}}\frac{d^{2}\Phi}{d\vartheta^{2}}-\frac{GA}{\kappa_ {0}}\Phi=-\omega^{2}\rho I\Phi\\ \end{aligned} \right. \tag{5}\]  
The equations describing the boundary conditions are written as follows:  
\[\text{Clamped edge boundary }U=0,\,V=0,\,\Phi=0\text{ at }\vartheta=\pm\vartheta_{0} \tag{6}\]  
### Establishment of Free-Vibration Equations  
In this subsection, our primary objective is to derive a solution for the system of ordinary differential equations. Directly integrating the equations incurs significant complexity, thus necessitating the discovery of a concise and efficient approach. The differential quadrature (DQ) method is a highly effective approach for solving partial and ordinary differential equations. Its fundamental principle involves partitioning the domain of independent variables for smooth functions into discrete sample points according to specific rules and approximating the derivative of the function with respect to these independent variables as a weighted sum of their corresponding function values [11; 12; 13]. Therefore, the \(n\)th order derivative of function \(f(\vartheta)\) with respect to \(\vartheta\) at the \(i\)th point \(\vartheta_{i}\) is approximated as follows:  
\[\left.\frac{\partial^{n}f(\vartheta)}{\partial\vartheta^{n}}\right|_{\vartheta =\vartheta_{i}}=\sum\limits_{j=1}^{H}\hat{\xi}_{ij}^{(n)}f\big{(}\vartheta_{ j}\big{)},i=1,2,\ldots,H \tag{7}\]  
where \(\hat{\xi}_{ij}^{(n)}\) denotes the \(n\)th-order derivative weighting coefficients of \(\vartheta\) at the \(i\)th sample point calculated for the \(j\)th sample points in the domain, \(H\) is the total number of sample points, and \(f(\vartheta_{i})\) is the function value at point \(\vartheta_{i}\). The weighting coefficients can be determined by Lagrange polynomial functions, and the first-order derivative weighting coefficients are:  
\[\hat{\xi}_{ij}^{(1)}=\frac{L^{(1)}(\vartheta_{i})}{\big{(}\theta_{i}-\vartheta _{j}\big{)}L^{(1)}\big{(}\vartheta_{j}\big{)}},i,j=1,2,\ldots,H,i\
eq j\]  
where \(L^{(1)}\big{(}\vartheta_{j}\big{)}=\prod\limits_{i=1,j\
eq j}^{H}\big{(} \vartheta_{j}-\vartheta_{i}\big{)}\). \[\hat{\xi}_{ii}^{(1)}=-\sum\limits_{j=1,j\
eq i}^{H}\hat{\xi}_{ij}^{(1)},i,j=1, 2,\ldots,H \tag{8}\]The second- and higher-order derivative weighting coefficients are:  
\[\xi_{ij}^{(n)}=n\left(\xi_{ii}^{(n-1)}\xi_{ij}^{(1)}-\frac{\xi_{ij}^{(n-1)}}{\theta _{i}-\theta_{j}}\right),i,j=1,2,\ldots,H,i\
eq j,n=2,3,\ldots,H-1 \tag{9}\]  
Each order of the derivative weighting coefficients can be obtained according to the above formulas. It is evident that the computation of derivative weight coefficients beyond the second order necessitates prior acquisition of first-order weight coefficients. The selection of appropriate sample points serves as a fundamental basis for ensuring both accuracy and efficiency in DQ method calculations. The Chebyshev-Gauss-Lobatto (C-G-L) grid distribution [11] is employed in this paper. According to the C-G-L rule, the sample point \(r_{i}\) on the standard interval [0,1] is expressed as:  
\[r_{i}=\frac{1-\cos\left(\frac{i-1}{H-1}\pi\right)}{2},i=1,2,\ldots,H \tag{10}\]  
and its values can be established by mapping the relationship with \(\theta_{i}\) as follows:  
\[\theta_{i}=-\theta_{0}+r_{i}\cdot 2\theta_{0},i=1,2,\ldots,H \tag{11}\]  
The required sample points for this method are now available. After selecting a specific number of sample points based on the calculation requirements, the derivative weighting coefficient can be obtained through relations (8) and (9).