## 3 Analytical Investigations  
To investigate the control performance of the applied radial PD-controller in suppressing the non-linear vibrations of the considered Jeffcott system, an analytical approximate solution for the system equations of motion (20) and (21) is suggested as follows [41; 42]:  
\[u(t,\varepsilon) =u_{0}(T_{0},T_{1})+\varepsilon u_{1}(T_{0},T_{1})+O(\varepsilon ^{2}) \tag{22}\] \[v(t,\varepsilon) =v_{0}(T_{0},T_{1})+\varepsilon v_{1}(T_{0},T_{1})+O(\varepsilon ^{2}) \tag{23}\]  
Figure 3: Schematic diagram to show the interconnection between the Jeffcott-rotor system, magnetic bearings actuator, and control algorithm. where \(cc\) denotes the complex conjugate terms. To derive a bounded solution of Equations (31) and (32) at the primary resonance condition (i.e., when \(\Omega=1\)), the small divisors and the secular terms must vanish. Accordingly, to find the solvability condition of Equations (31) and (32), let the parameter \(\sigma\) represents the closeness of Jeffcott-rotor angular speed \(\Omega\) to its normalized natural frequency \(\omega=1\) such that:  
\[\Omega=1+\sigma=1+\widetilde{\epsilon}\widetilde{\sigma} \tag{34}\]  
Inserting Equation (34) into Equations (32) and (33), we can obtain the following solvability conditions of Equations (32) and (33):  
\[\  
where \(\varphi_{1}=\sigma t-\theta_{1},\ \varphi_{2}=\sigma t-\theta_{2}\). Substituting Equations (30), (31), (34), and (37) into Equations (22) and (23), we have:  
\[u(t)=a_{1}(t)\cos(\Omega t-\varphi_{1}(t)) \tag{42}\]  
\[v(t)=a_{2}(t)cos(\Omega t-\varphi_{2}(t)) \tag{43}\]  
Equations (42) and (43) represent the periodic solution of Equations (20) and (21), where \(a_{1}(t)\) and \(a_{2}(t)\) are the instantaneous oscillation amplitudes of the controlled rotor in \(X\) and \(Y\) directions, respectively, while \(\varphi_{1}(t)\) and \(\varphi_{2}(t)\) denote the phase angles. In addition, \(a_{1}(t),\ a_{2}(t),\ \varphi_{1}(t)\), and \(\varphi_{2}(t)\) are governed by the autonomous dynamical system given by Equations (38) to (41). So, setting \(\dot{a}_{1}(t)=\dot{a}_{2}(t)=\dot{\varphi_{1}}(t)=\dot{\varphi_{2}}(t)=0\) into Equations (38) to (41), one can obtain a system of non-linear algebraic equations that governs the steady-state vibration amplitudes and phase angles as follows:  
\[ \begin{array}{ll}f_{1}(a_{1},a_{2},\varphi_{1},\varphi_{2})=&-\frac{1}{2} \mu a_{1}+\frac{1}{2}\beta_{2}a_{1}+\frac{1}{8}\beta_{5}a_{1}^{3}+\frac{1}{2} \beta_{10}a_{1}a_{2}^{2}+\frac{1}{8}(\beta_{7}-\beta_{10})a_{1}a_{2}^{2}\cos( 2\varphi_{1}-2\varphi_{2})\\ &+\frac{1}{8}(-\lambda+\beta_{4}-\beta_{8}+\beta_{9})a_{1}a_{2}^{2}\sin(2 \varphi_{1}-2\varphi_{2})+\frac{1}{2}E(1+\sigma)^{2}\sin(\varphi_{1})=0\end{array}  \tag{44}\]  
\[ \begin{array}{ll}f_{2}(a_{1},a_{2}\varphi_{1},\varphi_{2})=&-\frac{1}{2}\mu a _{2}+\frac{1}{2}\beta_{2}a_{2}+\frac{1}{8}\beta_{5}a_{2}^{3}+\frac{1}{4}\beta_ {10}a_{2}a_{1}^{2}+\frac{1}{8}(\beta_{7}-\beta_{10})a_{2}a_{1}^{2}\cos(2 \varphi_{2}-2\varphi_{1})\\ &+\frac{1}{8}(-\lambda+\beta_{4}-\beta_{8}+\beta_{9})a_{2}a_{1}^{2}\sin(2 \varphi_{2}-2\varphi_{1})-\frac{1}{2}E(1+\sigma)^{2}\cos(\varphi_{2})=0\end{array}  \tag{45}\]  
\[ \begin{array}{ll}f_{3}(a_{1},a_{2},\varphi_{1},\varphi_{2})=&\sigma-\lambda a _{1}^{2}-\frac{1}{4}\lambda a_{2}^{2}+\frac{1}{2}\beta_{1}+\frac{3}{8}\beta_ {5}a_{1}^{2}+\frac{1}{4}\beta_{4}a_{2}^{2}+\frac{3}{8}\beta_{6}a_{1}^{2}+\frac{ 1}{4}\beta_{8}a_{2}^{2}\\ &+\frac{1}{8}(-\lambda+\beta_{4}-\beta_{8}+\beta_{9})a_{2}^{2}\cos(2\varphi_{ 1}-2\varphi_{2})+\frac{1}{8}(-\beta_{7}+\beta_{10})a_{2}^{2}\sin(2\varphi_{1} -2\varphi_{2})\\ &+\frac{1}{2a_{1}}E(1+\sigma)^{2}\cos(\varphi_{1})=0\end{array}  \tag{46}\]  
\[