## 2 Equation of Motion for the Investigated Rotor  
The studied rotor system consists of a massive shaft that is mounted at both ends within two SFDs with centring springs. The configuration of the rotor system is illustrated in Figure 1. The origin of the stationary coordinate system xyz lies in the centre of the rotor. The rotor is considered as absolutely rigid and rotates with a constant angular speed. SFDs with squirrel cage springs are arranged symmetrically with respect to the central plane. The rotor is excited by an unbalance force. The effect of gravity is mitigated by prestressing the centring springs. In addition to SFD damping, the rotor is also affected by external damping from the environment. It is assumed that the damping is linear and proportional to the velocity. The damping force always acts in the opposite direction to the oscillations. By employing the Lagrange equations of the second kind and under the defined assumptions, the lateral vibrations are governed by a set of two nonlinear equations of motion  
\[\mathbf{M}\ddot{\mathbf{q}}+\mathbf{C}\dot{\mathbf{q}}+\mathbf{K}\mathbf{q}= \mathbf{f}_{\mathrm{D}}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}+\mathbf{f}_{ \mathrm{A}} \tag{1}\]  
\[\mathbf{M}= \begin{bmatrix}m&0\\ 0&m\end{bmatrix} ,\ \ \mathbf{C}= \begin{bmatrix}b&0\\ 0&b\end{bmatrix} ,\ \ \mathbf{K}=2 \begin{bmatrix}k&0\\ 0&k\end{bmatrix} , \tag{2}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the mass, external damping, and stiffness matrix of the rotor system. The elements of the matrices are specified by the parameters of the rotor mass \(m\), damping coefficient \(b\), and stiffness \(k\) of the single squirrel cage spring. The nonlinear force generated by SFD is represented by the horizontal \(F_{\mathrm{Dy}}\) and vertical \(F_{\mathrm{Dz}}\) force components, which are arranged into the vector \(\mathbf{f}_{\mathrm{D}}\) of the hydraulic damping forces  
\[\mathbf{f}_{\mathrm{D}}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}=2 \begin{bmatrix} F_{Dy}\big{(}y,z,\dot{y},\dot{z}\big{)}\\ F_{Dz}\big{(}y,z,\dot{y},\dot{z}\big{)}\end{bmatrix} . \tag{3}\]  
The vector \(\mathbf{q}\) is composed of the horizontal and vertical displacements, \(y\) and \(z\)  
\[\mathbf{q}= \begin{bmatrix}y\\ z\end{bmatrix} ,\ \ \dot{\mathbf{q}}= \begin{bmatrix}\dot{y}\\ \dot{z}\end{bmatrix} ,\ \ \ddot{\mathbf{q}}= \begin{bmatrix}\ddot{y}\\ \dot{z}\end{bmatrix} , \tag{4}\]  
where \((\cdot)\) and \((\cdot)\) denote the first and second derivatives with respect to time, respectively. Figure 1: Scheme of the rigid rotor mounted in SFDs. The unbalance force, the gravity force, and the constant preload force \(F_{\text{PS}}\) of the centring springs are included in the vector \(\mathbf{f}_{\text{A}}\) of the applied forces:  
\[\mathbf{f}_{\text{A}}= \begin{bmatrix}me_{\text{G}}\omega^{2}\cos(\gamma+\beta) \\ me_{\text{G}}\omega^{2}\sin(\gamma+\beta)-mg+2F_{\text{PS}}\end{bmatrix} , \tag{5}\]  
where \(g\) is the gravity acceleration, \(\omega\) is the angular speed of the rotor rotation, \(e_{\text{G}}\) and \(\beta\) is the eccentricity and phase shift of the rotor unbalance, respectively, and \(\gamma\) represents the angular position of the line connecting the rotor and the journal centres. The kinematic parameters of the rotor system under load are illustrated in a simplified cross-section view in Figure 2. Point G denotes the centre of gravity of the rotor. Point O denotes the centre of the damper sleeve and lies in the centre of the yz plane. The point O is the centre of the journal with a radius of \(R\), and it is located on the r-axis of the rotating coordinate system tr, which have the origin located in point O. Additionally, a coordinate system with XYZ axes is introduced for computation of the hydraulic forces in the oil film. The circular centred orbit of the synchronous precession motion is depicted in Figure 2 by the blue dashed line.