## 1 Introduction  
For the vibration suppression of eccentric rotors, the passive balancing devices that are more widely applied and researched currently are Automatic Ball Balancers (ABBs) at present. The main components are composed of a circular orbit mechanism, mounted perpendicular to the spindle of the rotor and revolving with the rotor, and several balls that can freely move inside the track. In 1975, [PERSON] [1] established an ideally mounted two-ball balancer of a plane rotor system from the 2D viewpoint and derived the equations of motion to analyze stability. [PERSON] [2] in 1996, [PERSON] [3] in 2006, and [PERSON] [4] in 2009, etc., have explored the problems and phenomena of ABBs' suppressing vibration in the model without assembly deviations. However, when assembling the ABBs and eccentric rotor system, assembly deviations may include the deviations of the vertical and horizontal positions of the ABBs' centroid and the centroid of the eccentric rotor as well as the deflections of the mounted angle. Regarding the discussions of vertical position deviation, [PERSON] [5] in 2005 and [PERSON] [6] in 2007 investigated the dynamic behavior of the system when the ABB is mounted at a small distance below the centroid of the plane eccentric rotor theoretically and experimentally, respectively. Numerical analysis and experimental results show that the system can still the approximate ideal-mounted (ABB is coplanar with the planar rotor) balance approach. The radial residual vibration under steady-state operation and the tilting angle during rotor balancing can be effectively reduced by positioning the balls at appropriate angular positions [5], where the radial amplitude can almost be suppressed, yet the tilting angle during balancing exists in a finite amount [6]. Furthermore, regarding the discussions of horizontal position deviation, [PERSON] in 2014 [7] and 2018 [8] proposed that the ABB and the planar rotor are coplanar, but the orbital center of the mounted ABB deviates horizontally from the rotor centroid along the rotor plane. In cases where the working speed is greater than the critical speed of the system, there are two types of incomplete balanced steady-state modes. Take the double ball ABB as an example, one is that two balls are gathered on the opposite side of the mass center of the rotor, and the other is half-balanced mode [7] or called almost balanced mode [8]. The second one is that two balls are spread by an appropriate amplitude spread angle to suppress (but not completely suppress) the eccentric imbalance of the planar rotor. As for the deflection of the mounting angle, in 2022, [PERSON] [9] first established an angular deviation model between the plane of ABBs' track and the plane of the eccentric rotor within the vibration suppression system. The examination principle of the skew-mounted ABB is proposed, comparing the differences in vibration suppression efficiency between skew-mounted and ideal-mounted systems, as well as the conditions of perfect balance. For the research on ABBs' vibration control system, across the past decade, many studies have proposed using springs or partitions along the track circumference to improve the ABBs' performance or reduce the system amplitude at low speeds. Researchers such as [PERSON] [10; 11], [PERSON] [12], and [PERSON] [13], etc., take into account the tangential force along the track, which in turn limits the movement of the balls in the track. However, in most of the literature, it is still assumed that the balls can rotate freely in the track except for viscous damping, and the balls in dynamic balance without any external forces acting along the tangential direction of the track. Additionally, the self-weight of the balls and friction in the track are ignored.