# Dynamic Response of Dual-Disk Rotor System with Uncertainties Based on Chebyshev Convex Method
## 3 The Chebyshev Convex Method for Dynamic Analysis
### The Ellipsoid Convex Method for Uncertain Parameters of Rotor Systems  
The ellipsoid model is the most extensive among the convex models with the advantages of rounded boundaries and compact uncertainty regions. The multi-ellipsoid model is applied to describe the finite-dimensional distribution of the convex model process.  
Figure 2: Schematic representation of a typical dual-disk rotor system.  
The \(n\)-dimensional interval variable is usually denoted as \(\textbf{X}^{\text{I}}=\left[X^{\text{I}}_{1},X^{\text{I}}_{2},\cdots,X^{\text{I}}_ {n}\right]\). \(X\) represents an uncertain parameter. In FEM model, it is the value in the \([M]\), \([C]\), \([K]\) and \([J]\) of Equation (7). The uncertain region of \(X^{\text{I}}\) is expressed as a region contained by an \(n\)-dimensional hyper-ellipsoid as follows:  
\[E_{\textbf{X}^{\text{I}},W_{e}}=\left\{\textbf{X}|\left(\textbf{X}-\textbf{X}^ {m}\right)^{\text{T}}\textbf{W}_{e}(\textbf{X}-\textbf{X}^{m})\leq 1\right\} \tag{8}\]  
The region contained by \(E_{\textbf{X}^{\text{I}},W_{e}}\) is an \(n\)-dimensional ellipsoid model, assuming that all possible values of uncertain parameters are contained in ellipsoid. **X** is the ellipsoid convex variable, \(\textbf{X}^{m}\) is the median value of a convex variable, and \(\textbf{W}_{e}\) is the characteristic matrix of ellipsoid model. It determines the size and shape of the ellipsoid model and can represent the correlation between variables.  
\[\textbf{W}_{e}=\left[\begin{array}{cccc}W_{11}&W_{12}&\cdots&W_{1n}\\ W_{21}&W_{22}&\cdots&W_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ W_{n1}&W_{n2}&\cdots&W_{nn}\end{array}\right] \tag{9}\]  
When the axis of the ellipsoid is along the direction of the coordinate axis, \(\textbf{W}_{e}\) becomes a diagonal matrix.  
As an example, the geometric characteristics of the ellipsoid model are studied, taking the two-dimensional ellipse in Figure 3. The geometric quantity is required to determine an ellipse: elliptic central coordinates \(\left[X^{m}_{1},X^{m}_{2}\right]\); length of semi-major axis \(r_{1}\); length of short half axis \(r_{2}\); and the angle between the major axis of the ellipse and the \(X_{1}\) axis is donated as \(\theta(-45^{\circ}<\theta<45^{\circ})\). To facilitate the solution, the mean value, variance, and covariance of the ellipse are usually used to describe the geometric characteristics of the ellipse [31; 32].  
The ellipsoid convex model describes the uncertain parameters in the rotor system. The differential equation of the convex uncertain rotor system is  
\[M\ddot{U}(\textbf{x},\textbf{t})+(C+G)\dot{U}(\textbf{x},\textbf{t})+KU( \textbf{x},\textbf{t})=p(t),\ \ \textbf{x}\in E_{\textbf{X}^{\text{I}},\textbf{W}_{e}} \tag{10}\]  
From the convex variable **X**, the response interval form of the uncertain rotor system is  
\[\textbf{U}^{I}=\left[\begin{matrix}\textbf{U}&-\\ \textbf{U}&\textbf{U}\end{matrix}\right] \tag{11}\]  
Figure 3: The uncertainty domain of two-dimensional ellipse in the convex model process.  
where  
\[\begin{array}{l}\mathbf{U}=\min\Big{\{}\mathbf{U}\Big{|}\mathbf{U}=\mathbf{U}( \mathbf{x}),\mathbf{x}\in E_{\mathbf{x}^{l},W_{e}}\Big{\}}\\ \mathbf{U}=\max\Big{\{}\mathbf{U}\Big{|}\mathbf{U}=\mathbf{U}(\mathbf{x}), \mathbf{x}\in E_{\mathbf{x}^{l},W_{e}}\Big{\}}\end{array} \tag{12}\]