In recent years, there has been much interest in the use of automatic balancing devices (ABDs) in rotating machinery. Autobalancers consist of several freely moving eccentric balancing masses mounted on the rotor, which, at certain operating speeds, act to cancel rotor imbalance. This “automatic balancing” phenomena 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 to cancel the imbalance. However, due to inherent nonlinearity of the autobalancer, the potential for other undesirable non-synchronous limit-cycle behavior exists. In such situations, the balancer masses do not reach their desired synchronous balanced positions resulting in increased rotor vibration. Although several researchers have explored limit-cycle behavior of single-plane ABD-rotor systems, a limit-cycle analysis of a full three dimensional rigid ABD/shaft/rotor considering transverse deflection, out-plane tilting and gyroscopic effects has not been investigated. This paper considers an approximate harmonic analytical solution to describe the limit-cycle behavior in a three dimensional rigid rotor/ABD system. Essentially, the solutions presented here capture both in-plane transverse deflection and out-plane tilting motion of the system under the limit-cycle condition. Here the whirl speed of the ABD balancer masses is determined via the solution of a non-linear characteristic equation. Also, based upon the limit-cycle solutions, the limit-cycle stability is assessed via a perturbation and Floquet analysis exploring three main parameters; ABD balancer mass, ABD damping, and axial location of ABD along the shaft. The coexistence of the stable balanced synchronous condition and undesired non-synchronous limit-cycle is studied. It is found that for certain combinations of ABD parameters and rotor speeds, the non-synchronous limit-cycle can be made unstable thus guaranteeing global asymptotic stability of the synchronous balanced condition. Finally, the analysis is validated through numerical simulation. The findings in this paper yield important insights for researchers wishing to utilize automatic balancing devices in rotor/shaft systems and limit-cycle mitigation.

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