## 2 Differential Equations of Motion
### Nonlinear Oil Film Force  
As shown in Figure 2, the structure of a squeeze film damper primarily consists of a journal outer ring (or squirrel cage elastic support outer ring) and a fixed bearing housing. The damping effect is generated by the fluid dynamic pressure during periodic squeezing of the lubricating oil in the annular clearance between the inner and outer rings, thereby reducing vibration in the rotor system.  
The journal is assumed to maintain parallel alignment with the bearing axis during operation. The short bearing approximation [26; 27] and incompressible Reynolds equation are adopted. Then, the transient oil film pressure \(p\) in the SFD can be expressed as follows:  
\[\frac{\partial}{\partial x}(h^{3}\frac{\partial p}{\partial x})=-12\mu\dot{ \psi}_{sf}\frac{\partial h}{\partial\theta}+12\mu\frac{\partial h}{\partial t} \tag{8}\]  
where \(\mu\) represents the lubricating oil viscosity. \(h=c+e\text{cos}\theta\) is the film thickness. \(\theta\) is the circumferential coordinate, measured in the local reference frame, beginning on the  
Figure 2: Diagram of the squeeze film damper.  
line connecting the bearing center \(O_{b}\) and journal center \(O_{j}\). \(C\) is the oil film clearance, \(e=\sqrt{y^{2}+z^{2}}\) represents the journal eccentricity, and \(y\), \(z\) are the displacements of the squeeze film damper node in the OY and OZ directions. The journal precession angular velocity is given by \(\dot{\psi}_{sfd}=\left(y\dot{z}-z\dot{y}\right)/e^{2}\).  
Through integration, the analytical expressions for radial oil film force \(F_{r}\) and tangential oil film force \(F_{t}\) can be obtained:  
\[F_{r}=\frac{\mu RL_{sfd}^{3}}{C^{2}}(\varepsilon\psi_{sfd}I_{1}+\dot{\varepsilon} I_{2}),F_{t}=\frac{\mu RL_{sfd}^{3}}{C^{2}}(\dot{\varepsilon}I_{1}+\varepsilon\psi_{sfd }I_{3}) \tag{9}\]  
where \(\varepsilon=e/c\) is the dimensionless radial displacement of the oil film, \(L_{sfd}\) represents the axial length of the oil film, \(R\) is the oil film radius. \(I_{1}\), \(I_{2}\), \(I_{3}\) are Sommerfeld integrals [28], expressed as follows:  
\[I_{1}=\int_{\theta_{1}}^{\theta_{2}}\frac{\text{sin}\theta\text{cos}\theta}{( 1+\varepsilon\text{cos}\theta)^{2}}d\theta,I_{2}=\int_{\theta_{1}}^{\theta_{2 }}\frac{\text{cos}^{2}\theta}{(1+\varepsilon\text{cos}\theta)^{3}}d\theta,I_{ 3}=\int_{\theta_{1}}^{\theta_{2}}\frac{\text{sin}^{2}\theta}{(1+\varepsilon \text{cos}\theta)^{3}}d\theta \tag{10}\]  
\(\pi\)-film cavitation model is adopted in this paper, where \(\theta_{1}=\pi\) and \(\theta_{2}=2\pi\), and consequently, the expressions for the radial and tangential oil film forces and their components projected onto the base reference frame are respectively given as follows:  
\[\begin{cases}F_{r}=\frac{\mu RL_{sfd}^{3}}{c^{2}}\left[\frac{2\dot{\pi}_{sfd} }{(1-\varepsilon^{2})^{2}}+\frac{\pi}{2}\left(\frac{1+2\varepsilon^{2}}{(1- \varepsilon^{2})^{5/2}}\right)\right]\\ F_{t}=\frac{\mu RL_{sfd}^{3}}{c^{2}}\left[\frac{\pi}{2}\frac{\dot{\pi}_{sfd}}{(1 -\varepsilon^{2})^{3/2}}+\frac{2\dot{\varepsilon}}{(1-\varepsilon^{2})^{2}} \right]\end{cases} \tag{11}\]  
\[\begin{cases}F_{y}=-F_{r}\text{cos}\psi_{sfd}+F_{t}\text{sin}\psi_{sfd}\\ F_{z}=-F_{r}\text{sin}\psi_{sfd}-F_{t}\text{cos}\psi_{sfd}\end{cases} \tag{12}\]