## 4 Methods Used to Solve the Equation of Motion
### Computation of the Synchronous Response with a Circular Centred Orbit  
In the case of a rotor system, which is not subjected to a constant force and performs a circular motion around the centre position at the same frequency as the rotor rotational \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Symbol** & **Value** & **Dimension** \\ \hline Rotor mass & \(m\) & 285 & kg \\ Stiffness of the single squirrel cage spring & \(k\) & 25 & MN \\ Balance quality grade according to ISO 1940-1 & Q & 6.3 & - \\ Gravity acceleration & \(g\) & 9.81 & m s\({}^{-2}\) \\ Dynamic viscosity of the oil ISO VG 46 & \(\eta\) & 0.03 & Pa s \\ Damper length & \(L\) & 20 & mm \\ Damper radius & \(R\) & 25 & mm \\ Width of the damper gap & \(c\) & 100 & \(\mu\)m \\ Dimensionless unbalance & \(\delta\) & 0.3 & - \\ Dimensionless SFD constant & \(\lambda\) & 0.0503 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the rotor and SFD. speed, this method can be used. It is a powerful technique when applied to the rigid and Jeffcott rotor models, as it requires minimal computational effort. To determine the vibration response, the equations of motion (1) in the fixed coordinate system are transformed into the dimensionless form [8] in the rotating coordinate system  
\[\varepsilon^{\prime\prime}+2\zeta\varepsilon^{\prime}-\varepsilon\gamma^{\prime 2 }+\varepsilon=\delta\Omega^{2}\cos(\beta)+\overline{F}_{\rm Dr}+\overline{F}_{ \rm E}\sin(\gamma), \tag{9}\]  
\[\varepsilon\gamma^{\prime\prime}+2\zeta\varepsilon\gamma^{\prime}+2\varepsilon ^{\prime}\gamma^{\prime}=\delta\Omega^{2}\sin(\beta)+\overline{F}_{\rm Dr}+ \overline{F}_{\rm E}\cos(\gamma). \tag{10}\]  
The relations for the damping ratio \(\zeta\), the dimensionless mass unbalance \(\delta\), the dimensionless expressions for the external force \(\overline{F}_{\rm E}\), radial \(\overline{F}_{\rm Dr}\), and tangential \(\overline{F}_{\rm Dt}\) component of the damper force are introduced:  
\[\zeta=\frac{b}{2\sqrt{2 km}},\delta=\frac{\varepsilon_{\rm G}}{c},\overline{F} _{\rm E}=\frac{2F_{\rm FS}-mg}{mc\omega_{\rm n}^{2}},\overline{F}_{\rm Dr}= \frac{2F_{\rm Dr}}{mc\omega_{\rm n}^{2}},\overline{F}_{\rm Dt}=\frac{2F_{\rm Dt }}{mc\omega_{\rm n}^{2}}. \tag{11}\]  
The dimensionless speed \(\Omega\) and time \(\tau\) are defined with the natural frequency \(\omega_{\rm n}\)  
\[\Omega=\frac{\omega}{\omega_{\rm n}},\tau=\omega_{\rm n}t,\omega_{\rm n}= \sqrt{\frac{2k}{m}}, \tag{12}\]  
and the first (\(\prime\)) and second (\(\prime\prime\)) derivatives with respect to the nondimensional time have the following form:  
\[\varepsilon^{\prime}=\frac{\mathrm{d}\varepsilon}{\mathrm{d}\tau}=\frac{ \dot{e}}{c\omega_{\rm n}},\varepsilon^{\prime\prime}=\frac{\ddot{e}}{c\omega _{\rm n}^{2}},\gamma^{\prime}=\frac{\mathrm{d}\gamma}{\mathrm{d}\tau}=\frac{ \dot{\gamma}}{\omega_{\rm n}},\gamma^{\prime\prime}=\frac{\ddot{\gamma}}{ \omega_{\rm n}^{2}}. \tag{13}\]  
By introducing the nondimensional SFD constant [8]  
\[\lambda=\frac{\eta R}{\sqrt{2 km}}\bigg{(}\frac{L}{c}\bigg{)}^{3}, \tag{14}\]  
four parameters (\(\eta\), \(L\), \(R\), \(c\)) describing the hydraulic force can be reduced into one. This further simplifies the equations of motion in Equations (9) and (10) and reduces the number of physical parameters. The value of the dimensionless SFD constant and unbalance of the investigated rotor is listed in Table 1.  
The following conditions for the steady-state motion are assumed: (i) the circular orbit has a constant relative eccentricity \(\varepsilon=\varepsilon_{\rm C}\), hence \(\varepsilon_{\rm C}^{\prime}=\varepsilon_{\rm C}^{\prime\prime}=0\), (ii) the angular position \(\gamma=\gamma_{\rm C}=\Omega\tau\), therefore \(\gamma_{\rm C}^{\prime}=\Omega\) and \(\gamma_{\rm C}^{\prime\prime}=0\), and (iii) the damper is perfectly centred \(\overline{F}_{\rm E}=0\).