## 6 Conclusions  
This work is intended to control the undesired vibrations of a non-linear Jeffcott rotor system utilizing a new control strategy. The introduced control system is a combination of both a radial PD-controller and an eight-poles active magnetic bearings actuator. It is worth mentioning that this is the first time the non-linear interaction between both a Jeffcott rotor model and the eight-pole magnetic actuator that are coupled together via a radial PD-controller has been investigated.. Based on the principle of classical mechanics, the whole mathematical model that governs the rotor-actuator interaction has been derived. Then, the asymptotic analysis has been employed to obtain an analytical solution for the obtained mathematical model. According to the derived analytical solution, the non-linear interaction between the Jeffcott system and the eight-pole actuator has been explored. In addition, the performance of the introduced control technique in mitigating the undesired non-linear oscillation of the Jeffcott system has been explored via plotting the different response curves. Finally, numerical confirmations for all obtained analytical results have been illustrated. Based on the above discussions, we can conclude with the following remarks:  
1. The uncontrolled Jeffcott rotor system responds as a linear dynamical system with a mono-stable periodic solution regardless of the system angular velocity as long as the shaft eccentricity \(E\leq 0.01\) as shown in Figure 4.
2. At the large disc eccentricity (i.e., \(E>0.01\)), the nonlinearities dominate the response curves of the uncontrolled Jeffcott system, where the rotor may oscillate by one of two periodic solutions depending on the initial conditions as shown in Figure 4.  
Figure 23: Time-response and the corresponding frequency-spectrum of the Jeffcott-rotor system before and after control according to Figure 22 at \(\sigma=0.0\), \(E=0.075\), \(\lambda=0.05\), \(\delta_{1}=0.83\), \(\delta_{2}=0.02\) when turning the controller on at time \(t=600\): (**a**,**b**) the normalized temporal oscillations \(u(t)\) and the corresponding frequency spectrum in \(X\) direction, (**c**,**d**) the normalized temporal oscillations \(v(t)\) and the corresponding frequency spectrum in \(Y\) direction. 3. The uncontrolled Jeffcott system has symmetric bifurcation characteristics and symmetric stable periodic motion in both the horizontal and the vertical direction regardless of the shaft eccentricity and the rotor spinning speed. 4. The coupling of the radial PD-control algorithm to the Jeffcott rotor via an eight-pole magnetic actuator results in a new dynamical system with completely different oscillatory behaviors and bifurcation characteristics than the uncontrolled Jeffcott system. 5. The proportional gain \(\delta_{1}\) of the radial-PD-controller can be used as a control key to change the natural frequency of the controlled Jeffcott system \(\omega_{control}^{2}\) (i.e., \(\omega_{control}^{2}=-3-8\cos^{2}(\beta)+8\cos(\beta)\delta_{1}+4\delta_{1}\)) to avoid the high oscillation amplitudes of the rotor system at resonance conditions as illustrated in Figures 13 and 14. 6. The derivative gain \(\delta_{2}\) of the radial-PD-controller can be utilized to increase the damping ratio of the controlled Jeffcott system \(\mu_{control}\) (i.e., \(\mu_{control}=\mu+4\delta_{2}(1+2\cos(\beta))\) in order to mitigate the lateral oscillations of the rotor system at the strong eccentricity magnitudes as shown in Figures 15-17. 7. The optimal selection of the control parameters \(\delta_{1}\) and \(\delta_{2}\) (i.e., \(\delta_{1}=0.83\) and \(\delta_{2}=0.02\)) can eliminate the non-linear bifurcation behaviors and force Jeffcott-system to behave like the linear system regardless of the eccentricity magnitude as depicted in Figures 20-23. 8.