## 1 Introduction
Detailed investigations of the non-linear dynamics of the active bearing system with different configurations can be found in [31; 32; 33; 34; 35]. However, [PERSON] et al. [36; 37; 38] introduced a novel active control strategy to resolve the drawbacks of the available control techniques, either passive methods like balancing or active techniques such as the active magnetic bearing's actuator. They introduced a new adaptive controller that uses the bearing forces as inputs to control the active actuator displacements. Based on the introduced extensive investigations, the authors concluded theoretically and experimentally that the proposed control technique can eliminate the bearing forces and the rotor resonance. In addition, they proved that the controller is always stable. It is worth mentioning that the applied control algorithm in [36; 37; 38] is one of the newest techniques in the field of rotor vibration control, which may get special attention from us in the near future. Within this article, the radial PD-control algorithm along with the eight-pole active magnetic bearings actuator has been introduced as a novel control strategy to suppress the resonant vibrations of the Jeffcott rotor system. The introduced PD-control algorithm is designed such that the instantaneous horizontal and vertical vibrations of the rotor system (i.e., \(x(\tau)\) and \(y(\tau)\)) be measured using appropriate position sensors. The measured signals (i.e., \(x(\tau)\) and \(y(\tau)\)) are fed into a digital controller, which manipulates them to obtain the corresponding cartesian velocities (i.e., calculates \(\dot{x}(\tau)\) and \(\dot{y}(\tau)\)). Then, the radial position and radial velocities of the rotor system in the direction of each pole are calculated according to the geometry of the eight-pole actuator using a predefined mathematical law implemented on the digital controller. Based on the calculated radial positions and radial velocities, the proposed control algorithm generates the control signals in the form of eight control currents that are applied to the eight-poles of the magnetic actuator. Finally, the magnetic actuator applies a controllable attractive force on the rotating shaft to eliminate the rotor's unwanted lateral vibrations \(x(\tau)\) and \(y(\tau)\). According to the proposed control strategy, the nonlinear equations of motion that govern the dynamic interactions between the Jeffcott system and the eight-pole actuator have been derived and then analyzed utilizing asymptotic analysis. The effects of the different control parameters on both the system dynamics and bifurcation behaviors have been explored. Based on the introduced investigations, it is demonstrated that the optimal design of the control gains can eliminate the non-linear bifurcation behaviors and force the Jeffcott-system to behave like linear dynamical systems regardless of the eccentricity magnitude. However, it is found that the controlled Jeffcott system may lose its stability and perform a chaotic or quasi-periodic motion if the control parameters have not been designed properly. Compared with the previously published works concerning rotor vibration control using magnetic bearing actuators, the non-linear vibration control of the Jeffcott rotor system has been tackled extensively before utilizing the four-pole magnetic actuator with different control algorithms [24, 25, 26, 27, 28, 29, 30]. [PERSON] et al. [24, 25] applied the cartesian PD-controller along with the four-pole magnetic actuator to eliminate the non-linear vibrations of the different rotor models. The authors have included the rub-impact force between the rotor and the pole-housing in the studied models. Based on the introduced analysis, they approved the efficiency of the introduced control technique in mitigating the system's resonant vibrations. In addition, they showed that the system may lose its periodic vibrations to respond with quasi-periodic or chaotic motion only when the rub and/or impact occurs between the rotor and the pole housing.