Dynamics of a rotor with an eccentric ball auto-balancing device

Abstract

This paper is concerned with a statically unbalanced rotor equipped an automatic ball balancing device, whose axis of symmetry is different from that of the rotor. A simple model of a Jeffcott rotor is used to derive the motion equations of the system in fixed and rotating coordinate systems, as well as the equations that describe the steady-state modes of motion. The impossibility in principle is shown for a rotor with variable imbalance to have a balanced steady-state mode. For an auto-balancing device consisting of two balls, two types of unbalanced steady-states modes are shown to be feasible. It is proposed to call the steady-state mode with a constant residual vibration, whose amplitude is independent of the angular velocity and is equal to eccentricity of the balancer, half-balanced. A solution corresponding to a half-balanced mode is explicitly constructed and conditions for the existence and stability thereof are proposed. Numerical methods are used to construct two-parameter stability diagrams of a half-balanced steady-state mode and to examine unsteady states of rotor motion in the case where it rotates with steady angular velocity.