## 2 Differential Equations of Motion  
### Systematic Equations Under Base Motion Excitation  
As shown in Figure 1, to describe the dynamic behavior of a rotor system on a moving base, three Cartesian reference frames are introduced: the inertial reference frame \(O_{g}x_{g}y_{g}z_{g}\) fixed to the ground, a non-inertial reference frame \(Oxyz\) fixed to the rigid base, and a rotor reference frame \(O_{x}x_{y}y_{z}x_{\text{}}\) fixed to the center of mass of the rigid disk or shaft section. \(x_{0}\), \(y_{0}\), \(z_{0}\) denote the translational displacement components of the base projected onto the reference frame \(Oxyz\). The angular velocities \(\omega_{bx}\), \(\omega_{by}\), and \(\omega_{bz}\) represent the base rolling, pitching and yawing motions about the \(O\mathbf{x}\), \(O\mathbf{y}\), and \(O\mathbf{z}\) axis, respectively. The axial motion of the system is neglected in this study. Through the derivation of the motion relationships between different coordinate systems, expressions of kinetic and potential energy can be obtained for both the rigid disk and the shaft element based on Timoshenko beam theory. Utilizing [PERSON]'s equations, the equations of motion for the flexible rotor system under base motion can be derived [20]:  
\[\mathbf{M}_{s}\bar{\mathbf{q}}+(\mathbf{C}_{s}-\Omega\mathbf{G}+\mathbf{C}_{base})\dot{\mathbf{q}}+(\mathbf{ K}_{s}+\mathbf{K}_{base})\mathbf{q}=\mathbf{F}_{mu}+\mathbf{F}_{base}+\mathbf{F}_{sfd} \tag{1}\]  
where \(\mathbf{q}\), \(\mathbf{q}\), and \(\bar{\mathbf{q}}\) represent the system's generalized displacement, velocity, and acceleration vectors, respectively; \(\Omega\) denotes the rotational speed of the rotor system. \(\mathbf{M}_{s}\), \(\mathbf{C}_{s}\), \(\mathbf{K}_{s}\) are the system mass, damping, and stiffness matrices, respectively; \(\mathbf{G}_{s}\) is the gyroscopic matrix; \(\mathbf{C}_{base}\) and \(\mathbf{K}_{base}\) are the additional damping and stiffness matrices induced by base motion; \(\mathbf{F}_{mu}\) represents the unbalanced force; \(\mathbf{F}_{sfd}\) denotes the nonlinear oil film force; and \(\mathbf{F}_{base}\) represents the inertial force terms introduced by base motion. All the matrices and vectors are listed in reference [20]. Taking the disk element as an example, the matrices and external force related to base motion are expressed as follows:  
\[\mathbf{C}_{base}= \begin{bmatrix}0&-2m_{i}\omega_{bx}&0&0\\ 2m_{i}\omega_{bx}&0&0&0\\ 0&0&0&J_{i}^{p}-2J_{i}^{d}\\ 0&0&2J_{i}^{d}-J_{i}^{p}&0\end{bmatrix}  \tag{2}\]  
\[\mathbf{K}_{base}= \begin{bmatrix}-m_{i}\big{(}\omega_{bx}^{2}+\omega_{bz}^{2} \big{)}&-m_{i}\Big{(}\dot{\omega}_{bx}+\omega_{by}\omega_{bz}\Big{)}&0&0\\ m_{i}\Big{(}\dot{\omega}_{bx}+\omega_{by}\omega_{bz}\Big{)}&-m_{i}\Big{(} \omega_{bx}^{2}+\omega_{by}^{2}\Big{)}&0&0\\ 0&0&J_{i}^{p}\Omega\omega_{bx}+J_{i}^{pd}\big{(}\omega_{bx}^{2}-\omega_{bz}^ {2}\big{)}&-J_{d}\dot{\omega}_{bx}+J_{i}^{pd}\omega_{by}\omega_{bz}\\ 0&0&-J_{i}^{pd}\Big{(}\dot{\omega}_{bx}-\omega_{by}\omega_{bz}\Big{)}&J_{i}^{ p}\Omega\omega_{bx}+J_{i}^{pd}\Big{(}\omega_{bx}^{2}-\omega_{by}^{2}\Big{)} \end{bmatrix}  \tag{3}\]  
\[\mathbf{F}_{base}= \begin{bmatrix}-m_{i}\big{(}\ddot{y}_{0}+\dot{x}_{0}\omega_{bz} -\dot{z}_{0}\omega_{bx}\big{)}-m_{i}x_{d}\Big{(}\dot{\omega}_{bz}+\omega_{bx} \omega_{by}\Big{)}\\ -m_{i}\Big{(}\ddot{z}_{0}+\dot{y}_{\phi}\omega_{bx}-\dot{x}_{0}\omega_{by} \Big{)}+m_{i}x_{d}\Big{(}\dot{\omega}_{by}-\omega_{bx}\omega_{bz}\Big{)}\\ -J_{i}^{p}(\Omega+\omega_{bx})\omega_{bz}-J_{i}^{d}\Big{(}\dot{\omega}_{by}- \omega_{bx}\omega_{bz}\Big{)}\\ J_{i}^{p}(\Omega+\omega_{bx})\omega_{by}-J_{i}^{d}\Big{(}\dot{\omega}_{bz}+ \omega_{bx}\omega_{by}\Big{)}\end{bmatrix}  \tag{4}\]  
where \(m_{i}\), \(J_{i}^{p}\), and \(J_{i}^{d}\) represent the mass, polar moment of inertia, and diametral moment of inertia of the i-th disk element, respectively. \(J_{i}^{pd}\) is defined as the difference between polar and diametral moments of inertia (\(J_{i}^{pd}=J_{i}^{p}-J_{i}^{d}\)), and \(x_{d}\) denotes the axial position of the disk.