## 3 Chebyshev Polynomial Approximation Method for Dynamic Responses  
As previously stated, the presence of uncertainties plays a crucial role in the dynamic responses of the arches. Here, the interval analysis is employed, and the uncertain parameters are defined as interval parameters. In consideration of the uncertainties involved in the arches, Equation (16) ought be rewritten in the interval form  
\[\mathbf{M}\left(\mathbf{X}^{t}\right)\ddot{\mathbf{U}}\left(\mathbf{X}^{t},t\right)+\mathbf{C} \left(\mathbf{X}^{t}\right)\dot{\mathbf{U}}\left(\mathbf{X}^{t},t\right)+\mathbf{K}\left(\mathbf{ X}^{t}\right)\mathbf{U}\left(\mathbf{X}^{t},t\right)=\mathbf{F} \tag{18}\]  
where \(\mathbf{X}^{t}\) is an interval parameter vector encompassing all the uncertain parameters of an arch. The superscript \(l\) denotes an interval variable. \(\mathbf{X}^{l}\) can be expressed as  
\[\mathbf{X}^{l}=\left[\mathbf{X}^{L},\mathbf{X}^{l}\right]=\left\{\mathbf{X}:\mathbf{X}^{L}\leq\bm {X}\leq\mathbf{X}^{l}\right\}=\left[\mathbf{X}^{C}-\beta\mathbf{X}^{C},\mathbf{X}^{C}+\beta\mathbf{ X}^{C}\right] \tag{19}\]  
in which \(\mathbf{X}^{L}\), \(\mathbf{X}^{l}\) and \(\mathbf{X}^{C}\) are the lower bound, upper bound and nominal value of the interval vector \(\mathbf{X}^{l}\), respectively. \(\beta\) denotes the deviation vector, which signifies the variation coefficients corresponding to the uncertain parameters. The bounds of the uncertain transient dynamic responses essentially represent the extremum values of the deterministic arch structure system subject to the constraints expressed as Equation (18). A multitude of methodologies can be utilized to probe the solution of Equation (18). To guarantee the computational procedure's efficiency and the accuracy of the results, the Chebyshev inclusion function (CIF) is utilized in this study [11; 16; 17; 18; 19; 21; 22; 23; 24; 25; 26; 27; 28]. It is assumed that there are \(n\) uncertain parameters in the arch, and an \(n\)-dimensional surrogate model of transient dynamic responses at an arbitrary time up to order \(p\), which is established via the CIF, is given as \[U\Big{(}\mathbf{X}^{t}\Big{)}=\sum_{i_{1}=0}^{p}\cdots\sum_{i_{n}=0}^{p}\frac{1}{2^{ \lambda}}\Gamma_{i_{1},\cdots,i_{n}}T_{i_{1},\cdots,i_{n}}\Big{(}\mathbf{X}^{t}\Big{)} \tag{20}\]  
where \(\lambda\) represents the total number of zeros occurring in the subscripts \(i_{1},\cdots i_{n}\), \(\Gamma_{i_{1},\cdots,i_{n}}\) is the expansion coefficient to be estimated, and \(T_{i_{1},\cdots i_{n}}(\mathbf{X}^{t})\) is the \(n\)-dimensional Chebyshev polynomials, which can be expressed as  
\[T_{i_{1},\cdots i_{n}}\Big{(}\mathbf{X}^{t}\Big{)}=\cos(i_{1}\theta_{1})\cdots\cos (i_{n}\theta_{n})=T_{i_{1},\cdots i_{n}}\Big{(}\mathbf{\theta}^{t}\Big{)} \tag{21}\]  
with \(\theta_{k}=\arccos\frac{2\pi l_{k}^{t}-\big{(}\pi l_{k}^{t}+\pi l_{k}^{t}\big{)} }{\pi l_{k}^{t}-\pi l_{k}^{t}}\in[0,\pi]_{k}=1,2,\cdots n\). \(x_{k}^{t}\) is the \(k\)th member of the interval vector \(\mathbf{X}^{l}\). The expansion coefficient can be expressed as  
\[\Gamma_{i_{1},\cdots,i_{n}}=\left(\frac{2}{\pi}\right)^{n}\int_{0}^{\pi} \cdots\int_{0}^{\pi}U\Big{(}\mathbf{X}^{l}\Big{)}T_{i_{1},\cdots,i_{n}}\Big{(}\bm {X}^{t}\Big{)}d\theta_{1}\cdots d\theta_{n} \tag{22}\]  
It is difficult to directly derive the integral result of Equation (22), so a numerical integration formulation with an interpolation technique named Gauss-Chebyshev quadrature will be employed for the integral formulas. It can be expressed as  
\[\int_{-1}^{1}\rho(x)g(x)dx=\frac{\pi}{\hat{q}}\underset{i}{\overset{s}{\sum}}g (x_{i}) \tag{23}\]  
where \(\rho=1/\sqrt{1-x^{2}}\) is the weight function and \(\hat{q}\) is the interpolation number.