# Steady-State Response Analysis of an Uncertain Rotor Based on Chebyshev Orthogonal Polynomials
## Appendix A  
A schematic representation of the finite element model established for the rotor system is shown in Figure 11.  
In Figure 11, the shaft of the rotor system is modeled using one-dimensional Timoshenko beam elements, divided into 25 segments. The discs are represented by concentrated mass elements, while the red triangles indicate the bearing elements. \"Brg Type 3\" specifies that the bearing has constant stiffness and damping, and no rotational stiffness. The load-deformation relationship of the bearing is typically nonlinear and closely related to the rotational speed. To simplify the rotor dynamic analysis, the load-deformation relationship of the supports is often approximated as linear. Considering only the lateral bending vibration, the shaft beam element can be described by 8 degrees of freedom. The analysis of beam bending can be approached similarly to that of bar extension. Figure 14 shows a typical element, along with its local coordinates. To guarantee the continuity of both deflection and slope at the element boundaries, the local coordinates incorporate the translations and rotations at the ends of the element. In Figure 14, \(l_{e}\) is the length of the axis unit, and each node has four degrees of freedom, which, respectively, represent the node displacement along the x and y directions (represented by \(u\) and \(v\) respectively) and the cross-sectional rotation around the x and y axes (represented by \(\alpha=v/z\) and \(\beta=u/z\) respectively). Therefore, eight coordinate components are needed to describe the displacement \(x_{e}\) of the shaft unit, namely  
\[\mathbf{x}_{e}=[u_{1}v_{1}\alpha_{1}\beta_{1}u_{2}v_{2}\alpha_{2}\beta_{2}]^{T} \tag{1}\]  
The shaft element model is given by the following formula:  
\[\mathbf{M}_{e}\ddot{\mathbf{x}}_{e}+(\mathbf{C}_{e}+\omega\mathbf{G}_{e})\big{\{}\dot{\mathbf{x}}_{ e}\big{\}}+\mathbf{K}_{e}\mathbf{x}_{e}=\mathbf{q}_{e} \tag{2}\]  
where the vector \(\mathbf{q}_{e}\) is the force applied to each node on the shaft unit, and \(\omega\) is the rotational speed of the rotor. \(\mathbf{M}e\), \(\mathbf{G}e\), \(\mathbf{K}e\), and \(\mathbf{C}e\) are the mass matrix, gyro matrix, stiffness matrix, and damping matrix of the shaft element, respectively. The density, cross-sectional area, elastic modulus, and moment of inertia of the shaft element are, respectively, denoted as \(\rho_{e}\), \(A_{e}\), \(E_{e}\), and \(l_{e}\), and the specific forms of the element matrices \(\mathbf{M}e\), \(\mathbf{G}e\), and \(\mathbf{K}e\) are as follows:  
\[\mathbf{M}_{e}=\frac{\rho_{e}A_{e}l_{e}}{420}\left[ \begin{array}{cccccccc}156&0& 0&22l_{e}&54&0&0&-13l_{e}\\ 0&156&-22l_{e}&0&0&54&13l_{e}&0\\ 0&-22l_{e}&4l_{e}&0&0&-13l_{e}&-3l_{e}^{2}&0\\ 22l_{e}&0&0&4l_{e}^{2}&13l_{e}&0&0&-3l_{e}^{2}\\ 54&0&0&13l_{e}&156&0&0&-22l_{e}\\ 0&54&-13l_{e}&0&0&156&22l_{e}&0\\ 0&13l_{e}&-3l_{e}^{2}&0&0&22l_{e}&4l_{e}^{2}&0\\ -13l_{e}&0&0&-3l_{e}^{2}&-22l_{e}&0&0&4l_{e}^{2}\\ \end{array} \right] \tag{3}\]\[K_{e}=\frac{E_{e}l_{e}}{l_{e}^{3}}\left[ \begin{array}{cccccccc}12&0&0&6l_{e}&-12&0& 0&6l_{e}\\ 0&12&-6l_{e}&0&0&-12&-6l_{e}&0\\ 0&-6l_{e}&4l_{e}^{2}&0&0&6l_{e}&2l_{e}^{2}&0\\ 6l_{e}&0&0&4l_{e}^{2}&-6l_{e}&0&0&2l_{e}^{2}\\ -12&0&0&-6l_{e}&12&0&0&-6l_{e}\\ 0&-12&6l_{e}&0&0&12&6l_{e}&0\\ 0&-6l_{e}&2l_{e}^{2}&0&0&6l_{e}&4l_{e}^{2}&0\\ 6l_{e}&0&0&2l_{e}^{2}&-6l_{e}&0&0&4l_{e}^{2}\end{array} \right] \tag{10}\]  
\[\mathbf{G}_{e}=\frac{\rho_{e}l_{e}}{15l_{e}} \begin{bmatrix}0&36&-3l_{e}&0&0&-36&-3l _{e}&0\\ -36&0&0&-3l_{e}&36&0&0&-3l_{e}\\ 3l_{e}&0&0&4l_{e}^{2}&-3l_{e}&0&0&-l_{e}^{2}\\ 0&3l_{e}&-4l_{e}^{2}&0&0&-3l_{e}&l_{e}^{2}&0\\ 0&-36&3l_{e}&0&0&36&3l_{e}&0\\ 36&0&0&3l_{e}&-36&0&0&3l_{e}\\ 3l_{e}&0&0&-l_{e}^{2}&-3l_{e}&0&0&4l_{e}^{2}\\ 0&3l_{e}&l_{e}^{2}&0&0&-3l_{e}&-4l_{e}^{2}&0\end{bmatrix}  \tag{11}\]  
Considering the rigid disc as a concentrated mass on the node of the shaft unit can effectively reduce the degree of freedom of the model and improve the calculation efficiency of the finite element model.