Automatic two-plane balancing for rigid rotors

Abstract

We present an analysis of a two-plane automatic balancing device for rigid rotors. Ball bearings, which are free to travel around a race, are used to eliminate imbalance due to shaft eccentricity or misalignment. The rotating frame is used to derive autonomous equations of motion and the symmetry breaking bifurcations of this system are investigated. Stability diagrams in various parameter planes show the coexistence of a stable balanced state with other less desirable dynamics.

Two-plane automatic balancing: A symmetry breaking analysis

“…In this section we extend a previously considered ABB model [6] so that it includes effects such as support anisotropy and rotor acceleration. These features can be incorporated by appending the appropriate linear rotor equations with the forcing terms that arise from the motion of the balls.…”

“…The mechanical device that we wish to model is illustrated in Figure 1, and is based on a rigid rotor that has been fitted with a two-plane automatic balancer [6,13,14]. The rotor system in the absence of the balancing balls has mass M , principal moments of inertia [J t , J t , J p ], and is mounted on two compliant linear bearings that are located at S 1 and S 2 .…”

“…The equations of motion for a two-plane ABB have been derived in [6,14] using Lagrange's method. We shall write them here with respect to the complex vibrational coordinate vector q, which yields…”

“…Therefore, whilst an ABB can compensate for the imbalance for some highly supercritical speeds, it can also make the vibration levels significantly worse during the rotor run-up. Nevertheless, recent advances in the analysis of simple isotropic ABB systems have led to improved predictions of their regions of stability [5,6]. However, the set-up on a real machine is usually asymmetric.…”

“…However, due to the inherent non-linearity of the mechanism, those techniques are not able to accurately predict the stability boundaries of balanced operation. More recently, Rodrigues et al [6] presented the first non-linear bifurcation analysis of the two-plane ABB. Lagrange's method and rotating coordinates were used to derive an autonomous set of governing equations.…”