## 1 Introduction  
Non-linear vibration is a common phenomenon in different types of rotating machinery which arises due to various reasons such as the rotating shafts' imbalance, the misalignment between two coupled shafts, the crack propagation along the rotating shaft, the looseness of the bearings, the asymmetry of the rotating shafts, and the wear between the bearing balls, etc. Accordingly, many scientific research papers have been publishedannually to explore the main causes of these non-linear vibrations as well as to find top-timal ways to suppress or control these undesired oscillations. [PERSON] et al. [1; 2; 3] investigated the influence of the bearing's clearance on the non-linear oscillations of the rotating machinery at both the primary and subharmonic resonance conditions, and the authors reported that the rotor system may exhibit an unstable periodic response at the primary resonance condition. [PERSON] et al. [4] studied theoretically and experimentally the non-linear dynamics of an asymmetric Jeffcott rotor system with radial clearance when subjected to rub-impact force with a snubber ring. They concluded that the occurrence of impact force between the rotor and the snubber ring is one of the undesired phenomena that may encounter industrial applications of the rotating machines. On the other hand, the dynamic characteristics of the Jeffcott system with non-linear stiffness coefficients have been investigated extensively [5; 6; 7; 8; 9; 10; 11; 12; 13]. [PERSON] and [PERSON] [5] investigated the bifurcation characteristics of a Jeffcott rotor system at 1/3-order subharmonic resonance conditions. [PERSON] et al. [6; 7] explored the oscillatory motion of the same model that has been studied in [5], but in the 1/2-order subharmonic resonance case. [PERSON] et al. [8] and [PERSON] and [PERSON] [9] demonstrated numerically and experimentally that the rotor system with non-linear restoring force can exhibit either quasi-periodic or chaotic oscillations besides periodic motion. [PERSON] and [PERSON] [10] explored the bifurcation characteristics of a non-linear Jeffcott system in the vicinity of critical speed, and when the angular speed is twice and three times the critical speed at 1:1 internal resonance. The authors reported that the system may exhibit a more complex response curve than that of the single resonance condition. [PERSON] [11] studied the free vibrations of a Jeffcott system having non-linear spring properties. [PERSON] et al. [12] and [PERSON] et al. [13] investigated the nonlinear oscillations of a horizontally suspended Jeffcott system having non-linear spring characteristics. They demonstrated that the rotor system can perform either forward or backward whirling oscillation depending on the shaft angular speed. In addition, the rotating shafts asymmetry [14], the crack's propagation [15; 16], and the small shaft imbalance [17; 18; 19; 20; 21; 22; 23; 24; 25] may cause large lateral vibration amplitudes, which ultimately results in rub and/or impact forces between rotor and stator. Lateral vibration of the rotating shafts is an unwanted phenomenon that affects the efficiency and performance of the rotating machines. Therefore, many control methodologies have been introduced to eliminate or at least mitigate this undesired phenomenon [26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38]. [PERSON] and [PERSON] [26] introduced a linear dynamic absorber utilizing four electro-magnetic poles to control the lateral vibrations of a non-linear Jeffcott system. [PERSON] et al. [27; 28] applied a proportional-derivative controller via a four-pole electro-magnetic actuator to reduce the lateral oscillation of two different types of rotating shafts. [PERSON] [29] and [PERSON] and [PERSON] [30] used a four-pole electro-magnetic actuator with PD-controller to suppress the non-linear vibrations of the Jeffcott system. They illustrated that the existence of time delays in the control loop may destabilize the stable motions of the rotor system.