# Steady-State Response Analysis of an Uncertain Rotor Based on Chebyshev Orthogonal Polynomials
## Appendix A
The model of the rigid disc is given by the following formula:  
\[M_{\rm d}\ddot{x}_{\rm d}+\omega\mathbf{G}_{\rm d}\ddot{x}_{\rm d}=\mathbf{q}_{\rm d} \tag{12}\]  
where the vector \(\mathbf{q}_{d}\) is the unbalanced force caused by eccentric mass, \(\mathbf{M}_{d}\) and \(\mathbf{G}_{d}\) are the mass matrix and gyro matrix of the disc, and  
\[M_{\rm d}= \begin{bmatrix}m_{\rm d}&0&0&0\\ 0&m_{\rm d}&0&0\\ 0&0&I_{\rm d}&0\\ 0&0&0&I_{\rm d}\end{bmatrix}  \tag{13}\]  
\[G_{\rm d}= \begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&I_{p}\\ 0&0&I_{p}&0\end{bmatrix}  \tag{14}\]  
where \(m_{d}\) is the mass of the disc, \(I_{d}\) is the diameter moment of inertia, and \(I_{p}\) is the polar moment of inertia. In Equation (12), the unbalanced force \(\mathbf{q}_{d}\) is given by the following equation:  
\[\mathbf{q}_{\rm d}=\left[m_{e}\mathrm{d}_{e}\omega^{2}cos(\omega t+\phi)m_{e} \mathrm{d}_{e}\omega^{2}sin(\omega t+\phi)\;0\;0\right]^{T} \tag{15}\]  
where \(m_{e}\) and \(d_{e}\) are the magnitude of unbalance and eccentricity, and \(\phi\) represents the initial angular positions of unbalance, as shown in Figure 11.  
Based on finite element theory, the general process for constructing the differential equation of the rotor system is as follows: (1) Assemble the element mass, stiffness, damping, and gyroscopic matrices into a global matrix. (2) Incorporate supports and constraints. (3) Apply external excitation. The resulting differential equation of motion for the entire rotor system can then be expressed in the following form:  
\[\mathbf{M}\widetilde{\mathbf{x}}+(\mathbf{C}+\omega\mathbf{G})\dot{\mathbf{x}}+(\mathbf{K}+\mathbf{K}_{b}) \mathbf{x}=\mathbf{q} \tag{10}\]  
where \(x\) is the node displacement, \(\omega\) is the rotor speed, and \(\mathbf{M}\), \(\mathbf{C}\), \(\mathbf{K}\), and \(\mathbf{G}\) are the overall mass matrix, damping matrix, stiffness matrix, and gyro matrix of the rotor system. \(\mathbf{K}_{\text{b}}\) is the supporting stiffness matrix, and \(\mathbf{q}\) is the unbalanced force acting on the whole rotor system. Considering that the unbalanced force can be written as \(\mathbf{q}=\mathbf{Q}e^{i\omega t}\), the response vector can be written as \(\mathbf{x}=\mathbf{X}e^{i\omega t}\), where \(\mathbf{X}\) is the amplitude of the unbalanced response. Transforming the motion equation of the rotor system shown in Equation (10) into frequency domain, we have  
\[\mathbf{Z}(\omega)\mathbf{X}(\omega)=\mathbf{Q}(\omega) \tag{11}\]  
where  
\[\mathbf{Z}(\omega)=-\omega^{2}\mathbf{M}+i\omega(\mathbf{C}+\omega\mathbf{G})+\mathbf{K}+\mathbf{K}_ {b} \tag{12}\]  
Equation (12) can be employed to calculate the unbalanced response at each node of the rotor system at a specified speed. The unbalanced steady-state response of the rotor at each node follows a distinct trajectory. When the supporting parameters are isotropic, the trajectory is circular, with the radius representing the amplitude of the unbalanced response in the X and Y directions at the rotor cross-section. Conversely, if the bearing parameters differ in the two directions, the trajectory takes the form of an ellipse.