# Dynamic Response of Dual-Disk Rotor System with Uncertainties Based on Chebyshev Convex Method
## 3 The Chebyshev Convex Method for Dynamic Analysis
### Chebyshev Method for Convex Analysis of the Rotor Dynamic Response  
In this section, a Chebyshev interval expansion function based on the Chebyshev series expansion is proposed. Considering functions in the ellipsoid as follows:  
\[F=f(\mathbf{x}),\ \ \mathbf{x}=\left[x_{1},x_{2},\ldots,x_{n}\right]^{\mathrm{T}} \in E_{\mathbf{x}^{l},W_{e}}=\Big{\{}\mathbf{x}\Big{|}(\mathbf{x}-\mathbf{x}^ {c})^{\mathrm{T}}W_{x}(\mathbf{x}-\mathbf{x}^{c})\leq 1\Big{\}} \tag{13}\]  
Function \(f\left(x\right)\) can be approximated by Chebyshev polynomials. \[f(\mathbf{x})\approx p_{p}(\mathbf{x})=\frac{1}{2}f_{0}+\sum\limits_{i=1}^{k }f_{i}C_{i}(\mathbf{x}),\ \ \mathbf{x}\in E_{\mathbf{x}^{l},W_{e}}=\Big{\{}\mathbf{x}\Big{|}(\mathbf{x}- \mathbf{x}^{c})^{\mathrm{T}}W_{e}(\mathbf{x}-\mathbf{x}^{c})\leq 1\Big{\}} \tag{14}\]  
where, \(P\) is the order of Chebyshev polynomials, \(f_{i}\) is the constant coefficient of Chebyshev polynomials \(C_{i}\). For an \(n\)-dimensional problem, the Chebyshev polynomials are defined as  
\[C_{k_{1},k_{2},\cdots,k_{n}}(x_{1},x_{2},\cdots,x_{n})=\cos(k_{1}\theta_{1}) \cos(k_{2}\theta_{2})\cdots\cos(k_{n}\theta_{n}) \tag{15}\]  
where, \(k_{i}\) (\(I=1,2,\)... \(n\)) is the nonnegative integer, \(\theta_{i}\) (\(I=1,2,\)... \(n\)) is a function of the convex variable. \[\theta_{i}=\arccos\left(\frac{2x_{i}-\left(x_{i}^{\mathrm{R}}+x_{i}^{L} \right)}{x_{i}^{R}-x_{i}^{L}}\right)\in[0,\pi],\ \ i=1,2,\ldots,n \tag{16}\]  
The truncated Chebyshev polynomial of function \(f(x)\) can be rewritten as  
\[ \begin{array}{l}f(x)\approx p_{p}(x)=\frac{1}{2}f_{0}+\sum\limits_{k=0}^{k }\cdots\sum\limits_{k_{n}=0}^{k}f_{k_{1},\ldots,k_{n}}C_{k_{1},\ldots,k_{n}}( \mathbf{x})\\ =\frac{1}{2}f_{0}+\sum\limits_{k_{i}=0}^{k}\cdots\sum\limits_{k_{n}=0}^{k}f_{ k_{1},\ldots,k_{n}}\cos i[\theta_{i}],\ \theta\in[0,\pi]\end{array}  \tag{17}\]  
According to the Mehler integral method, \(f_{i}\) can be calculated by Equation (18). \[f_{i}=\frac{2}{\pi}\int_{-1}^{1}\frac{1}{\sqrt{1-x^{2}}}f(x)C_{i}(x)dx\approx \frac{2}{m}\sum\limits_{j=1}^{m}f(x_{j})C_{i}\big{(}x_{j}\big{)}=\frac{2}{m} \sum\limits_{j=1}^{m}f\big{(}\cos\theta_{j}\big{)}\cos i\theta_{j} \tag{18}\]  
Coefficient \(f_{k_{1},\ldots,k_{n}}\) is expressed as  
\[ \begin{array}{l}f_{k_{1},\ldots,k_{n}}=\left(\frac{2}{\pi}\right)^{n}\int_{ -1}^{1}\cdots\int_{-1}^{1}\frac{f(\mathbf{x})C_{k_{1},\ldots,k_{n}}(\mathbf{x })}{\sqrt{1-x_{1}^{2}}\cdots\sqrt{1-x_{1}^{2}}}dx_{1}\ldots dx_{n}\\ \approx\left(\frac{2}{m}\right)^{n}\sum\limits_{j_{1}=1}^{m}\cdots\sum\limits _{j_{n}=1}^{m}f\big{(}\cos\theta_{j_{1}},\ldots,\cos\theta_{j_{n}}\big{)} \cos\big{(}k_{1}\theta_{j_{1}}\big{)}\ldots\cos\big{(}k_{n}\theta_{j_{n}} \big{)}\end{array}  \tag{19}\]  
\[ \begin{array}{l}\theta_{j_{i}}=\frac{2j_{i}-1}{\frac{m}{n}}\cdot\frac{\pi}{ 2},\ j_{i}=1,2,\ldots,m,\ i=1,2,\ldots,n\\ x_{i,j_{i}}=\frac{x_{i}^{L}+x_{i}^{L}}{2}+\frac{x_{i}^{R}-x_{i}^{L}}{2}\cos \theta_{j_{i}},\ j_{i}=1,2,\ldots,m,\ i=1,2,\ldots,n\end{array}  \tag{20}\]  
where, \(m\) is the number of sampling points that need interpolation in each dimension, \(l\) denotes the total number of zeros in subscript \(k_{1},\ldots,k_{n}\). Chebyshev polynomial surrogate model of \(n\)-dimensional function \(f(x)\) are defined as  
\[ \begin{split} f(x)=f(x_{1},x_{2},\cdots,x_{n})\approx p_{p}(x)=\sum \limits_{k_{1}=0}^{p}\cdots\sum\limits_{k_{n}=0}^{p}\Big{(}\frac{1}{2}\Big{)} ^{l}f_{k_{1},k_{2},\cdots,k_{n}}C_{k_{1},k_{2},\cdots,k_{n}}(\mathbf{x}),\\ \mathbf{x}\in E_{\mathbf{x}^{l},\mathbf{W}_{e}}=\Big{\{}\mathbf{x }|\left(\mathbf{x}-\mathbf{x}^{m}\right)^{T}\mathbf{W}_{e}(\mathbf{x}-\mathbf{ x}^{m})\Big{\}}\end{split}  \tag{21}\]  
The differential Equation (10) of the convex uncertain rotor system can be solved based on Equation (21).