## 2 Control Equations for Euler-Bernoulli Rotor
### Derivation of the Squeeze Film Damper (SFD) Oil Film Force
The radial oil film reaction force for a squeeze film damper with a cavity is given by  
\[F_{r}=\int_{-\dfrac{L}{2}}^{\dfrac{L}{2}}\int_{\pi}^{2\pi}p(\theta,Z)T\cos \theta d\theta dZ\]  
Substituting Equation (42) and the formula \(h=C(1-\varepsilon\cos\theta)\) into the aforementioned formula, and integrating with respect to \(Z\), yields  
\[F_{r}=-\dfrac{\mu\Omega RL^{3}}{C^{2}}\int_{\pi}^{2\pi}p(\theta,Z)T\cos\theta d \theta dZ \tag{43}\]  
The integral in the equation can be computed to obtain  
\[\int_{\pi}^{2\pi}\dfrac{\varepsilon\sin\theta\cos\theta}{(1+ \varepsilon\cos\theta)}d\theta=\int_{\pi}^{2\pi}\dfrac{\cos\theta}{(1+ \varepsilon\cos\theta)}\dfrac{\varepsilon\sin\theta}{(1+\varepsilon\cos \theta)^{2}}d\theta\] \[=\int_{\pi}^{2\pi}\bigg{[}\dfrac{1}{\varepsilon}-\dfrac{1}{ \varepsilon(1+\varepsilon\cos\theta)}\bigg{]}\dfrac{\varepsilon\sin\theta}{(1 +\varepsilon\cos\theta)^{3}}d\theta\] \[=-\dfrac{1}{\varepsilon}\int_{\pi}^{2\pi}\dfrac{-\sin\theta d \theta}{(1+\varepsilon\cos\theta)}+\dfrac{1}{\varepsilon}\int_{\pi}^{2\pi} \dfrac{-\varepsilon\sin\theta d\theta}{(1+\varepsilon\cos\theta)^{2}}=\dfrac {2\varepsilon^{2}}{\left(1-\varepsilon^{2}\right)^{2}}\]  
Thus, the radial oil film reaction force for a short bearing squeeze film damper with a cavity is derived as  
\[F_{r}=-\dfrac{\mu\Omega RL^{3}}{C^{2}}\Bigg{[}\dfrac{2\varepsilon^{2}}{\left( 1-\varepsilon^{2}\right)^{2}}\Bigg{]} \tag{44}\]  
Following the analogy of Equation (43), the circumferential oil film reaction force for a squeeze film damper with a cavity can be similarly obtained as  
\[\frac{\partial h}{\partial\theta}=-C\epsilon\sin\theta\]0  
Applying the method of integration by parts, one can deduce that  
\[\frac{\partial h}{\partial\theta}=-C\epsilon\sin\theta\]1  
The Sommerfeld boundary condition is employed, which is given by  
\[\frac{\partial h}{\partial\theta}=-C\epsilon\sin\theta\]2The characteristic of this substitution is that \(\alpha\) and \(\theta\) share the same value at 0, \(\pi\), and \(2\pi\) on both sides of the equation. From Equation (47), it can also be inferred that  
\[ \begin{array}{l}1+\varepsilon\cos\theta=\frac{1-\varepsilon^{2}}{1-\varepsilon \cos\alpha}\\ d\theta=\frac{\left(1-\varepsilon^{2}\right)^{1/2}}{\left(1-\varepsilon\cos \alpha\right)}d\alpha\end{array} \]  
Substituting this into Equation (46) yields  
\[\int_{\pi}^{2\pi}\frac{\varepsilon\sin^{2}\theta d\theta}{\left(1+\varepsilon \cos\theta\right)^{3}}=-\frac{1}{2\varepsilon\left(1-\varepsilon^{2}\right)^ {1/2}}\int_{\pi}^{2\pi}d\alpha+\frac{1}{2\varepsilon\left(1-\varepsilon^{2} \right)^{3/2}}+\int_{\pi}^{2\pi}\left(1-\varepsilon\cos\alpha\right)d\alpha= \frac{\pi\varepsilon}{2\left(1-\varepsilon^{2}\right)^{3/2}}\]  
Further substituting into Equation (45) results in the circumferential oil film reaction force for a squeeze film damper with a cavity under short-bearing conditions:  
\[F_{t}=-\frac{\mu\Omega RL^{3}}{C^{2}}\Bigg{[}\frac{\pi\varepsilon}{2\left(1- \varepsilon^{2}\right)^{3/2}}\Bigg{]} \tag{48}\]  
For an oil film without a cavity, integration from 0 to \(2\pi\) is necessary. Utilizing the same approach, the radial and circumferential oil film reaction forces for a squeeze film damper without a cavity can be derived as  
\[ \begin{array}{l}F_{r}=0\\ F_{t}=-\frac{\mu\Omega RL^{3}}{C^{2}}\Bigg{[}\frac{\pi\varepsilon}{\left(1- \varepsilon^{2}\right)^{3/2}}\Bigg{]}\end{array}  \tag{49}\]  
Proceeding with a brief overview of the squeeze film damper (SFD) element, a schematic of the SFD oil film force is depicted as follows, with the inner circle representing the shaft journal and the outer circle representing the SFD outer race, and the space between them filled with the oil film. The direction of the oil film force acting on the journal is as illustrated in the expression. The restoring force exerted on the rotating shaft due to the oil film force is  
\[\left\{