3D Dynamics Deduction and the Spatial Geometric Characteristics of Ball Balance Positioning in Automatic Ball Balancers

In order to explore the geometric characteristics of the dynamic balance positions of balls in any orbit in space, this paper first deduces the dynamic equilibrium equations of an eccentric rotor system with the automatic ball balancer (ABB) skew, mounted with a deflection angle from the perspective of three-dimensional (3D) dynamics. The results obtained are consistent with those derived from the Euler–Lagrange equations. It is exciting that the spatial dynamics method reveals the spatial-geometric characteristics of dynamic balance positioning of the balls when the system is balanced with vibration suppression. Then, through spatial geometric inference, it is proved that the balance positioning of any ball in an arbitrary curved track is characterized. This characteristic indicates that the perpendicular line from any balance position to the rotating spindle of the system must pass through the central axis of curvature of the track at that position. Finally, spatial graphics and numerical analysis are used to verify the theoretical correctness. The characteristics are general rules that can explain the phenomena of the stable equilibrium positions of the balls in previous studies under ideal assumptions.