## 2 Equations of Motion  
The non-linear differential equations that govern the lateral vibrations of the considered Jeffcott system shown in Figure 1, can be written as follows [26; 39]:  
\[m\ddot{x}(\tau)+c\dot{x}(\tau)+F_{RX}=me\omega^{2}\cos(\omega\tau) \tag{1}\]  
\[m\ddot{y}(\tau)+c\dot{y}(\tau)+F_{RY}=me\omega^{2}\sin(\omega\tau) \tag{2}\]  
where \(m\) represents the rotor mass in \(kg\), \(c\) denotes the linear damping in \(N.s/m\), \(F_{RX}\) and \(F_{RY}\) are the shaft non-linear restoring forces in \(N\), \(e\) is the eccentricity of the rotor in \(m\), \(\omega\) is the shaft angular speed in \(s^{-1}\), and \(\tau\) denotes the time in \(s\) (see Figure 1). It is considered that the shaft restoring force \(F_{R}\) is a cubic non-linear function of the shaft radial displacement \(R=\overline{OG}\) away from the geometric center \(O\) as shown in Figure 1. Accordingly, the restoring force \(F_{R}\) can be expressed as follows [12; 13; 26]:  
\[F_{R}=k_{l}R+k_{n}R^{3},\qquad R=\sqrt{x(\tau)+y(\tau)} \tag{3}\]  
where \(k_{l}\) is the linear stiffness coefficient in \(N/m\), and \(k_{n}\) denotes the non-linear stiffness coefficient in \(N/m^{3}\). According to Equation (3), the components of \(F_{R}\) in both \(X\) and \(Y\) directions (i.e., \(F_{RX}\) and \(F_{RY}\)) can be expressed as follows:  
\[F_{RX}=\Big{[}k_{l}R+k_{n}R^{3}\Big{]}\cos(\theta)=\Big{[}k_{l}+k_{n}R^{2} \Big{]}R\cos(\theta)=k_{l}x(\tau)+k_{n}\Big{[}x^{3}(\tau)+x(\tau)y^{2}(\tau) \Big{]} \tag{4}\]  
\[F_{RY}=\Big{[}k_{l}R+k_{n}R^{3}\Big{]}\sin(\theta)=\Big{[}k_{l}+k_{n}R^{2} \Big{]}R\sin(\theta)=k_{l}y(\tau)+k_{n}\Big{[}y^{3}(\tau)+x^{2}(\tau)y(\tau) \Big{]} \tag{5}\]  
Substituting Equations (4) and (5) into Equations (1) and (2), we have  
\[m\ddot{x}(\tau)+c\dot{x}(\tau)+k_{l}x(\tau)+k_{n}\Big{(}x^{3}(\tau)+x(\tau)y^ {2}(\tau)\Big{)}=me\omega^{2}\cos(\omega\tau) \tag{6}\]  
\[m\ddot{y}(\tau)+c\dot{y}(\tau)+k_{l}y(\tau)+k_{n}\Big{(}y^{3}(\tau)+x^{2}(\tau )y(\tau)\Big{)}=me\omega^{2}\sin(\omega\tau) \tag{7}\]  
Figure 1: Uncontrolled Jeffcott rotor system. To mitigate the instantaneous displacements \(x(\tau)\) and \(y(\tau)\) of the considered rotor system, it is suggested to apply the control forces \(F_{CX}\) and \(F_{CY}\) on the rotor system in \(X\) and \(Y\) directions, respectively, via an eight-pole magnetic bearings actuator as shown in Figure 2. Accordingly, Equations (6) and (7) should be modified to:  
\[m\ddot{y}(\tau)+c\dot{y}(\tau)+F_{RY}=me\omega^{2}\sin(\omega\tau) \tag{2}\]0 \[m\ddot{y}(\tau)+c\dot{y}(\tau)+F_{RY}=me\omega^{2}\sin(\omega\tau) \tag{2}\]1  
Based on the geometry of the 8-pole system shown in Figure 2b, one can express the net control forces \(F_{CX}\) and \(F_{Cy}\) such that:  
\[m\ddot{y}(\tau)+c\dot{y}(\tau)+F_{RY}=me\omega^{2}\sin(\omega\tau) \tag{2}\]2  
\[m\ddot{y}(\tau)+c\dot{y}(\tau)+F_{RY}=me\omega^{2}\sin(\omega\tau) \tag{2}\]3  
According to the electro-magnetic theory, the attractive force \(F_{j}\) (\(j=1,2,\ldots,8\)) can be expressed as follows [40]:  
\[m\ddot{y}(\tau)+c\dot{y}(\tau)+F_{RY}=me\omega^{2}\sin(\omega\tau) \tag{2}\]4  
where \(\mu_{0}\) is the air magnetic permeability, \(N\) is the winding number of the \(j\)th electrical coil, \(A\cos(\delta)\) is the actual cross-sectional area of each pole, \(I_{j}\) is the \(j\)th pole electrical current, and \(h_{j}\) is the air-gap size between the rotor and the \(j\)th pole as shown in \[m\ddot{y}(\tau)+c\dot{y}(\tau)+F_{RY}=me\omega^{2}\sin(\omega\tau) \tag{2}\]5 Accordingly, for the small displacements \(x(\tau)\) and \(y(\tau)\) of the considered Jeffcott rotor  
\[m\ddot{y}(\tau)+c\dot{y}(\tau)+F_{RY}=me\omega^{2}\sin(\omega\tau) \tag{2}\]6 (a) Jeffcott-rotor system and magnetic bearings actuator at the nominal position. (b) Jeffcott-rotor system and magnetic bearings actuator at small displacements \(x(\tau)\) and \(y(\tau)\) away from the nominal position. \[