# Free Vibration Analysis of Arches with Interval-Uncertain Parameters
## 3 General Polynomial Surrogate Model for Interval Natural Frequency Analysis
\right.,n=1,2,\ldots \tag{21}\]  
The one-dimensional n-order Chebyshev polynomial approximation of function \(f(x)\) is formulated as:  
\[f(x)\approx g_{n}(x)=\frac{a_{0}}{2}+\sum_{i=1}^{n}a_{i}T_{i}(x),x\in[-1,1] \tag{22}\]  
in which \(a_{i}\) is the expansion coefficient and  
\[a_{i}=\frac{2}{\pi}\int_{-1}^{1}\frac{\widetilde{f}(x)T_{i}(x)}{\sqrt{1-x^{2} }}dx,\ i\ =\ 0,1,2,\ldots,n \tag{23}\]  
where \(\widetilde{f}(x)\) is the deterministic value of \(f(x)\) and \(n\) is the expansion order. In general, the direct integration of \(a_{i}\) poses significant complexities. However, when compared to this approach, employing the Gauss-Chebyshev integration formula proves to be a more efficient and expeditious method for obtaining \(a_{i}\). The Gauss-Chebyshev integration formula is a type of interpolation integral formula expressed as follows:  
\[\int_{a}^{b}\rho(x)p(x)dx=\sum_{k=1}^{m}A_{k}p(x_{k}) \tag{24}\]  
in which \(m\) is the number of integral points and should satisfy \(m\geq n+1\). In the Gauss-Chebyshev integration, the interpolation points involved in the integration process are the zeros of \(m\)-degree Chebyshev polynomials, which are given by:  
\[x_{k}=\cos\left(\frac{2k-1}{2m}\pi\right),k=1,2,\ldots,m \tag{25}\]  
For Equation (24), the weight function \((1-x^{2})^{1/2}\) of the Chebyshev orthogonal polynomial is chosen as \(\rho(x)\), and in this case, all the integral coefficients \(A_{k}=\pi/\text{m}\)[27]. Substituting this into Equation (24), the expansion coefficient \(a_{i}\) can be expressed as  
\[a_{i}=\frac{2}{\pi}\int_{-1}^{1}\frac{f(x)T_{i}(x)}{\sqrt{1-x^{2}}}dx\approx \frac{2}{\pi}\frac{\pi}{m}\sum_{k=1}^{m}f(x_{k})T(x_{k})=\frac{2}{m}\sum_{k=1 }^{m}f(x_{k})T(x_{k}) \tag{26}\]  
Substituting this into Equation (22), we obtain  
\[g_{n}(x)=\frac{1}{m}\sum_{k=1}^{m}f(x_{k})+\frac{2}{m}\sum_{i=1}^{n}\sum_{k= 1}^{m}f(x_{k})T_{i}(x_{k})T_{i}(x) \tag{27}\]  
Now the polynomial surrogate model function based on Chebyshev orthogonal polynomials can be constructed. It is evident that once the original frequency values at interpolation points, as given by Equation (25), are obtained from the deterministic model in Equation (17), the expansion coefficient \(a_{i}\) can be calculated immediately, thereby simultaneously obtaining the surrogate-model function. The upper and lower bounds of the function being approximated can be obtained by searching the maximum and minimum values of \(g_{n}(x)\) on standard interval \([-1,1]\), which can be easily achieved based on the maximum theorem of a continuous function on closed intervals. Moreover, linear transformation does not change the relative locations of the extreme values. Considering the arbitrary interval distribution of the uncertain parameter \(x\), and the fact that the Chebyshev polynomial surrogate model is applicable only for the standard interval [\(-\)1,1], it is necessary to transfer the parameter to a standard interval:  
\[x_{i}=\frac{2q_{i}-\left(q^{U}+q^{L}\right)}{q^{U}-q^{L}},\ q_{i}\in\left[q^{L},q ^{U}\right] \tag{28}\]  
The roots of the derivative equation are:  
\[\left\{x_{r}\middle|g_{n}^{\prime}(x)=0\right.\right\} \tag{29}\]  
The extremal value points \(x_{min}\) and \(x_{max}\), which correspond to the uncertain parameter \(x\), are determined by comparing the values of \(g_{n}(x_{r})\), \(g_{n}(-1)\), and \(g_{n}(1)\). This implies that the natural frequencies attain their maximum and minimum values at these points. The corresponding locations of \(x\) will be marked as \(x_{\min}\) and \(x_{\max}\). Supposing that multiple uncertain parameters are independent, for \(r\)-dimensional uncertain parameter vector \(\mathbf{q}^{I}\), the process of searching the extremum should be performed \(r\) times to obtain the maximum and minimum frequency value locations for each parameter. Two extremum vectors will be obtained:  
\[ \begin{cases}x_{\min}=\left[x_{1,\min},x_{2,\min},\cdots,x_{r,\min}\right]\\ x_{\max}=\left[x_{1,\max},x_{2,\max},\cdots,x_{r,\max}\right]\end{cases}