## 2 Methodology
### Uncertain Gravity Response Analysis Based on CIM  
Due to the manufacturing error, material properties dispersion, and the wear and tear in the process of operation, uncertainties widely exist in the material properties of the rotor and the supporting bearing properties. These uncertainties can lead to an uncertain gravity response of the generator rotor system. To evaluate the uncertain gravity response, CIM is employed. In interval method, the uncertainties are treated as interval variables, which are described by their upper bounds and lower bounds. The interval uncertainties involved in the uncertain generator rotor system can be expressed as:  
\[\epsilon^{I}=\left\{\epsilon:\underline{\epsilon}<\epsilon<\overline{\epsilon} \right\}=\left[\epsilon^{c}-\epsilon^{d},\epsilon^{c}+\epsilon^{d}\right]= \epsilon^{c}+x^{I}\epsilon^{d} \tag{13}\]  
where \(\epsilon^{I}\) is the interval vector, which is composed of the interval variables of all the uncertain factors; \(\underline{\epsilon}\) and \(\overline{\epsilon}\) are the lower bound and upper bound of the interval vector; \(\epsilon^{c}=\left(\underline{\epsilon}+\overline{\epsilon}\right)/2\) is the nominal values of the interval vector; \(\epsilon^{d}=\left(\overline{\epsilon}-\underline{\epsilon}\right)/2\) is the radius of the interval vector; and \(x^{I}\) is the standard interval vector, of which each term is the interval value bounded by \(-1\) and \(1\). The expression in Equation (13) transforms the general interval vector into a standard interval vector. The aim of CIM is to take the place of FEM analysis by a Chebyshev polynomial expansion. Then, with the help of a Chebyshev polynomial expansion, the bounds of the gravity response can be obtained by evaluating the uncertain gravity response under the interval variables. The equations for obtaining the uncertain gravity response based on FEM can be expressed as:  
\[\textbf{Q}\left(x^{I}\right)=\textbf{L}\left(x^{I}\right)^{-1}\textbf{G}\left( x^{I}\right) \tag{14}\]  
where:  
\[\textbf{L}\left(x^{I}\right)=\left[ \begin{array}{cc}\overline{\textbf{K}} \left(x^{I}\right)-\Omega^{2}\overline{\textbf{M}}\left(x^{I}\right)&\Omega \overline{\textbf{C}}\left(x^{I}\right)\\ -\Omega\overline{\textbf{C}}\left(x^{I}\right)&\overline{\textbf{K}}\left(x^{ I}\right)-\Omega^{2}\overline{\textbf{M}}\left(x^{I}\right)\end{array} \right] \tag{15}\]  
\[\textbf{Q}\left(x^{I}\right)=\left\{ \begin{array}{c}\textbf{Q}_{c}\left(x^ {I}\right)\\ \textbf{Q}_{s}\left(x^{I}\right)\end{array} \right\}\textbf{G}\left(x^{I} \right)=\left\{ \begin{array}{c}\overline{\textbf{G}}_{c}\left(x^{I}\right) \\ \overline{\textbf{G}}_{s}\left(x^{I}\right)\end{array} \right\} \tag{16}\]  
The solution of uncertain gravity response based on a Chebyshev polynomial expansion can be expressed as:  
\[\textbf{Q}\left(x^{I}\right)=f\left(x^{I}\right)=\sum_{i_{1}=0}^{r+1}\cdots \sum_{i_{n}=0}^{r+1}a_{i_{1},\cdots,i_{n}}\left(x^{I}\right)C_{i_{1},\cdots,i_ {n}}\left(x^{I}\right)=\textbf{a}\left(x^{I}\right)^{T}\textbf{C}\left(x^{I}\right) \tag{17}\]  
where **a** is the vector that is composed of coefficients of the Chebyshev polynomial expansion; **C** is the vector that is composed of the Chebyshev polynomial; \(n\) is the dimension of the uncertain problem, which is equal to the number of uncertain factors; and \(r\) is the order of the Chebyshev polynomial expansion. The Chebyshev polynomial is defined as:  
\[C_{i_{1},\cdots,i_{n}}\left(x^{I}\right)=\cos(i_{1}\theta_{1})\cos(i_{2}\theta _{2})\cdots\cos(i_{n}\theta_{n}) \tag{18}\]  
where \(\theta_{i}\)= \(\arccos(x^{I}_{i})\), \(i=1,2,\cdots,n\). The core of CIM is to find the coefficients in Equation (17). First, a standard procedure for finding the coefficients is introduced.