Understanding the instability phenomena of rotor-shaft and driveline systems incorporating universal joints is becoming increasingly important because of the trend towards light-weight, high-speed supercritical designs. In this paper, a nondimensional, periodic, linear time-varying model with torsional and lateral degrees-of-freedom is developed for a rotor shaft-disk assembly supported on a flexible bearing and driven through a U-joint. The stability of this system is investigated utilizing Floquet theory. It is shown that the interaction between torsional and lateral dynamics results in new regions of parametric instability that have not been addressed in previous investigations. The presence of load inertia and misalignment causes dynamic coupling of the torsion and lateral modes, which can result in torsion-lateral instability for shaft speeds near the sum-type combinations of the torsion and lateral natural frequencies. The effect of angular misalignment, static load-torque, load-inertia, lateral frequency split, and auxiliary damping on the stability of the system is studied over a range of shaft operating speeds. Other than avoiding the unstable operating frequencies, the effectiveness of using auxiliary lateral viscous damping as a means of stabilizing the system is investigated. Finally, a closed-form technique based on perturbation expansions is derived to determine the auxiliary damping necessary to stabilize the system for the least stable case (worst case).

Due to inherent nonlinearity of the autobalancer, the potential for other, undesirable, nonsynchronous limit-cycle vibration exists. In such undesirable situations, the balancer masses do not reach their desired synchronous balanced steady-state positions resulting in increased rotor vibration. Such behavior has been widely studied and is well understood for rotor systems on idealized bearings with symmetric supports. However, a comprehensive study into this nonlinear behavior of an imbalanced planar-rigid rotor/autobalancing device (ABD) system mounted on a general bearing holding asymmetric damping and stiffness forces including nonconservative effects cross-coupling ones has not been fully conducted. Therefore, this research primarily focuses on the unstable nonsynchronous limit-cycle behavior and the synchronous balancing condition of system under the influence of the general bearing support. Here, solutions for rotor limit-cycle amplitudes and the corresponding whirl speeds are obtained via a harmonic balance approach. Furthermore, the limit-cycle stability is assessed via perturbation and Floquet analysis, and all the possible responses including undesirable coexistence for the bearing parameters and operating speeds have been thoroughly studied. It is found that, due to asymmetric behavior of bearing support, the multiple limit cycles are encountered in the range of supercritical speeds and more complicate coexistences are invited into the ABD–rotor system compared to the case with idealized symmetric bearing supports. The findings in this paper yield important insights for researchers wishing to utilize automatic balancing devices in more practical rotor systems mounted on a asymmetric general bearing support.

In recent years, there has been much interest in the use of so-called automatic balancing devices (ABDs) in rotating machinery. Essentially, ABDs or “autobalancers” consist of several freely moving eccentric balancing masses mounted on the rotor, which, at certain operating speeds, act to cancel rotor imbalance at steady-state. This “automatic balancing” 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 limit-cycle behavior exists. In such situations, the balancer masses do not reach their desired synchronous balanced steady-state 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 so-called Alford's force cross-coupling term. The limit-cycle 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 limit-cycle 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 time- and frequency-domain simulation. The findings in this paper yield important insights for researchers wishing to utilize ABDs on rotors having journal bearing support.

This research explores the use of automatic balancing (AB) devices or “autobalancers” for imbalance vibration suppression of flexible shafts operating at supercritical speeds. Essentially, an autobalancer is a passive device consisting of several freely moving eccentric masses or balancer balls free to roll within a circular track mounted on a rotor that is to be balanced. At certain speeds, the stable equilibrium positions of the balls are such that they reduce or cancel the rotor imbalance. This “automatic balancing” phenomenon occurs as a result of the nonlinear dynamic interactions between the balancer balls and the rotor transverse vibration. Thus, autobalancer devices can passively compensate for unknown imbalance without the need for a control system and are able to naturally adjust for changing imbalance conditions. Autobalancers are currently utilized for imbalance correction in some single plane rotor applications such as computer hard-disk drives, CD-ROM drives, machine tools and energy storage flywheels. While autobalancers can effectively compensate for imbalance of planar, disk-type, rigid rotors, the use of autobalancing devices for nonplanar and flexible shafts with multiple modes of vibration has not been fully considered. This study explores the dynamics and stability of an imbalanced flexible shaft-disk system equipped with a dual-ball automatic balancing device. The system is analyzed by solving a coupled set of nonlinear equations to determine the fixed-point equilibrium conditions in rotating coordinates, and stability is assessed via eigenvalue analysis of the perturbed system about each equilibrium configuration. It is determined that regions of stable automatic balancing occur at supercritical shaft speeds between each flexible mode. Additionally, the effects of bearing support stiffness, axial mounting offset between the imbalance and autobalancer planes, and ball/track viscous damping are explored. This investigation develops a new, efficient, analysis method for calculating the fixed-point equilibrium configurations of the flexible shaft-AB system. Finally, a new effective force ratio parameter is identified, which governs the equilibrium behavior of flexible shaft/AB systems with noncollocated autobalancer and imbalance planes. This analysis yields valuable insights for balancing of flexible rotor systems operating at supercritical speeds.

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