# Free Vibration Analysis of Arches with Interval-Uncertain Parameters
## 3 General Polynomial Surrogate Model for Interval Natural Frequency Analysis  
In this section, the uncertain natural frequency bounds of an arch will be analyzed with Chebyshev interval algorithms. It has been demonstrated that for a continuous function \(f(x)\) defined in [a, b], there always exists a polynomial \(g(x)\) that approximates it with an error no larger than an arbitrary small positive quantity \(\varepsilon\). On this basis, the Chebyshev polynomial surrogate (CPS) model is utilized to deal with uncertain dynamic problems; the frequencies are treated as the proposed function \(f(x)\) and can be approximated by orthogonal polynomials with acceptable errors in the following. The original dynamic questions can be modeled as simple mathematical functions by using a few interpolation points, which can save the computational cost of the traditional Monte Carlo simulation (MCS) without losing accuracy. As mentioned previously, the geometric dimensions of the cross-sectional area and physical parameters such as the Young's modulus may be uncertain. Due to the lack of necessary information, it is a complicated task to model the uncertainties probabilistically. However, only their bounds are needed in the interval approach, which can be achieved easily. Therefore, it is more reasonable to regard the uncertainties as interval variables. An uncertain parameter is regarded as an interval parameter \(q^{I}\), where superscript \(I\) denotes the interval variable, and can be denoted as:  
\[q^{I}=\left[q^{L},q^{U}\right]=\left\{q\in R\Big{|}q^{L}\leq q\leq q^{U}\right\} =\left[q^{C}-\beta q^{C}\leq q\leq q^{C}+\beta q^{C}\right] \tag{18}\]  
in which \(q^{L},q^{U}\), and \(q^{C}\) denote the lower bound, upper bound, and nominal value of the interval parameter \(q^{I}\), respectively. Deviation degree \(\beta\) denotes the deviation coefficients. \(R\) is the set of all real numbers. If there is more than one uncertain parameter, the expression above can be expressed in the form of multi-dimensional interval vectors, as:  
\[\mathbf{q}^{I}=\left[\mathbf{q}^{L},\mathbf{q}^{U}\right]=\left\{\mathbf{q}^{L}\leq\mathbf{q}\leq \mathbf{q}^{U}\right\}=\left\{q_{i}\in R\Big{|}q_{i}^{L}\leq q_{i}\leq q_{i}^{U} \right\} \tag{19}\]  
\(\mathbf{q}^{I}\) is the uncertain parameters vector, where \(i=1\), \(2_{r}\ldots r\). \(r\) is the total number of uncertain parameters, and the parameters are assumed to be mutually independent in the following. Considering the matrices in the free-vibration equations for the uncertain arches as the function of the uncertain parameter, Equation (17) can be rewritten in the interval form  
\[\left[\widetilde{K}\Big{(}\mathbf{q}^{I}\Big{)}\right]\left\{\delta_{d}\Big{(}\bm {q}^{I},\hat{\mathbf{\sigma}}\Big{)}\right\}+\omega\Big{(}\mathbf{q}^{I}\Big{)}^{2} \left[\widetilde{M}\Big{(}\mathbf{q}^{I}\Big{)}\right]\left\{\delta_{d}\Big{(}\bm {q}^{I},\hat{\mathbf{\sigma}}\Big{)}\right\}=0 \tag{20}\]  
The goal of this study is to determine the interval range of natural frequencies, and a surrogate model based on Chebyshev orthogonal polynomial approximation theory is introduced to achieve this. First, the one-dimensional uncertain model is analyzed. The first type of Chebyshev orthogonal polynomial is \(T_{n}(x)=\cos(n\arccos x)\), \(x\in[-1,1]\), where \(n\) is an arbitrary positive integer. \(T_{n}(x)\) is orthogonal on \([-1,1]\) with respect to the weight function \(\rho(x)=(1-x^{2})^{1/2}\), and its recurrence formulas are as follows:  
\[\left\{ \begin{aligned} T_{0}(x)=1\\ T_{1}(x)=x\\ T_{n+1}(x)=2 xT_{n}(x)-T_{n-1}(x)\end{aligned}