# Steady-State Response Analysis of an Uncertain Rotor Based on Chebyshev Orthogonal Polynomials
## 2 The Proposed Method
### Chebyshev Orthogonal Polynomial Approximation  
Suppose \(\xi=[\xi_{1},\xi_{2},\cdots,\xi_{n}]\), \(\xi_{j}\) (\(j=1,\)\(\cdots,\)\(n\)) is a standard random variable with a uniform distribution \(\mathcal{U}(-1,\)\(1)\). Using the tensor product approach, an \(n\)-dimensional, \(k\)-order Chebyshev polynomial \(T_{i_{1},\cdots,i_{n}}(\xi)\) can be defined as follows:  
\[T_{i_{1},\cdots,i_{n}}(\xi)=\prod_{i_{1}+\cdots+i_{n}=k}T_{i_{1}}(\xi_{1}) \cdots T_{i_{n}}(\xi_{n}) \tag{1}\]  
where \(i_{1}\), \(i_{2}\), \(\ldots\), \(i_{n}\) are non-negative integers. \(T_{i_{j}}\) denotes a Chebyshev polynomial with order \(i_{j}\). For example, if \(n=2\), then \(\xi=\{\xi_{1},\xi_{2}\}\), and we have  
\[\begin{array}{rl}k=0:&T_{0,0}(\xi)=T_{0}(\xi_{1})T_{0}(\xi_{2})=1\\ k=1:&T_{1,0}(\xi)=T_{1}(\xi_{1})T_{0}(\xi_{2})=\xi_{1},\ L_{0,1}(\xi)=T_{0}(\xi_{1})T_{1}( \xi_{2})=\xi_{2},\\ k=2:&T_{2,0}(\xi)=T_{2}(\xi_{1})T_{0}(\xi_{2})=2\xi_{2}^{2}-1,\ T_{1,1}(\xi)=T_{1}( \xi_{1})T_{1}(\xi_{2})=\xi_{1}\xi_{2},\\ &T_{0,2}(\xi)=T_{0}(\xi_{1})T_{2}(\xi_{2})=2\xi_{2}^{2}-1\\ \cdots\end{array} \tag{2}\]  
A Chebyshev orthogonal polynomial of order \(p\) is used to approximate the multi-dimensional continuous function \(f(\xi_{1},\cdots,\xi_{n}):\Re^{n}\rightarrow\Re\), which can be expressed as follows:  
\[\begin{array}{rl}f(\xi)\approx&\sum\limits_{\begin{subarray}{c}i_{1}+\cdots,i_{n}\leq p\\ i_{1}+\cdots+i_{n}=1\end{subarray}}c_{i_{1},\cdots,i_{n}}T_{i_{1},\cdots,i_{n}} (\xi)=\sum\limits_{\begin{subarray}{c}i_{1}+\cdots+i_{n}=0\\ i_{1}+\cdots+i_{n}=p\end{subarray}}c_{i_{1},\cdots,i_{n}}T_{i_{1},\cdots,i_{n}} (\xi)+\\ &\sum\limits_{\begin{subarray}{c}i_{1}+\cdots+i_{n}=1\end{subarray}}c_{i_{1},\cdots,i_{n}}T_{i_{1},\cdots,i_{n}}(\xi)+\cdots+\sum\limits_{i_{1}+\cdots+i_{ n}=p}c_{i_{1},\cdots,i_{n}}T_{i_{1},\cdots,i_{n}}(\xi)\end{array} \tag{3}\]  
where \(N_{c}\) is the total number of undetermined coefficients, which is given by the following equation:  
\[N_{c}=1+\frac{n!}{(n-1)!}+\frac{(n+1)!}{(n-1)!2!}+\frac{(n+2)!}{(n-1)!3!}+ \cdots+\frac{(n-1+p)!}{(n-1)!p!}=\frac{(n+p)!}{n!p!} \tag{4}\]  
This weight function serves to guarantee that Chebyshev polynomials of varying orders remain orthogonal over the specified interval. For the standard Chebyshev polynomials, the weight function is expressed as follows:  
\[\mathcal{W}(\xi)=\frac{1}{\sqrt{1-\xi_{1}^{2}}\cdots\sqrt{1-\xi_{n}^{2}}} \tag{5}\]  
Multiplying both sides of Equation (3) by \(T_{i_{1},\cdots,i_{n}}(\xi)\) and integrating the weight function \(\mathcal{W}(\xi)\), the following results can be obtained:  
\[\int_{-1}^{1}\cdots\int_{-1}^{1}T_{l}(\xi)T_{l}(\xi)\mathcal{W}(\xi)\mathrm{d }\xi_{1}\cdots\mathrm{d}\xi_{n}=\left\{\begin{array}{rl}\left(\frac{1}{2} \right)^{l}\pi^{n}&I=J\\ 0&I\
eq J\end{array}\right. \tag{6}\]  
where \(l\) is the number of non-zeroes in \(i_{1},\cdots,i_{n}\). The undetermined coefficients in Equation (3) can be formulated as  
\[c_{i_{1},\cdots,i_{n}}=\frac{2^{l}}{\pi^{n}}\int_{-1}^{1}\cdots\int_{-1}^{1}f (\xi)T_{i_{1},\cdots,i_{n}}(\xi)\mathcal{W}(\xi)\mathrm{d}\xi_{1}\cdots\mathrm{ d}\xi_{n} \tag{7}\]  
When the dimension \(n\) is small (\(n\leq 5\)), the coefficient in Equation (7) can be calculated by full factor numerical integration, and the calculation formula is as follows:  
\[c_{i_{1},\cdots,i_{n}}\approx\frac{2^{l}}{\pi^{n}}\sum_{j_{1}=1}^{q}\cdots \sum_{j_{n}=1}^{q}f\Big{(}\xi_{1}^{(j_{1})},\cdots,\xi_{n}^{(j_{n})}\Big{)} \Big{(}w_{1}^{(j_{1})}\cdots w_{n}^{(j_{n})}\Big{)} \tag{8}\]where \(\xi_{1}^{(i_{1})},\cdots,\xi_{n}^{(j_{n})}\) is integration node, and \(q\) is the number of integration nodes in each dimension.