Limit Cycle Analysis of Planar Rotor/Autobalancer System Influenced by Alford's Force

Abstract

In recent years, there has been much interest in the use of socalled automatic balancing devices (ABDs) in rotating machinery. Essentially, ABDs or “autobalancersâ€‌ consist of several freely moving eccentric balancing masses mounted on the rotor, which, at certain operating speeds, act to cancel rotor imbalance at steadystate. This “automatic balancingâ€‌ phenomenon occurs as a result of nonlinear dynamic interactions between the balancer and rotor, wherein the balancer masses naturally synchronize with the rotor with appropriate phase and cancel the imbalance. However, due to inherent nonlinearity of the autobalancer, the potential for other, undesirable, nonsynchronous limitcycle behavior exists. In such situations, the balancer masses do not reach their desired synchronous balanced steadystate positions resulting in increased rotor vibration. In this paper, an approximate analytical harmonic solution for the limit cycles is obtained for the special case of symmetric support stiffness together with the socalled Alford's force crosscoupling term. The limitcycle stability is assessed via Floquet analysis with a perturbation. It is found that the stable balanced synchronous conditions coexist with undesirable nonsynchronous limit cycles. For certain combinations of bearing parameters and operating speeds, the nonsynchronous limitcycle can be made unstable guaranteeing global asymptotic stability of the synchronous balanced condition. Additionally, the analytical bifurcation of the coexistence zone and the pure balanced synchronous condition is derived. Finally, the analysis is validated through numerical timeand frequencydomain simulation. The findings in this paper yield important insights for researchers wishing to utilize ABDs on rotors having journal bearing support.