CONDITIONS FOR BALANCING A ROTATING BODY IN AN ISOLATED SYSTEM WITH AUTOMATIC BALANCERS I. I. Filimonikhina and G. B. Filimonikhin UDC 531.36:534.1:62-755 Conditions are established for automatic balancers with rigid bodies to balance a rotating body that undergoes spatial motion and is a component of an isolated system. It is discovered that a rotating oblate body can be statically balanced when the balance plane is rather close to the center of mass of the system. It is also established that if the rotating body is prolate, the nutation angle can be initially decreased by balancing the body; however, it will then increase because of the dissipation of energy in the system Keywords: rotating body, automatic balancers, nutation angle, rotating oblate body, rotating prolate body, energy dissipation Introduction. Ball-type and pendulum-type automatic balancers (ABs) and ABs with two coupled rigid bodies statically balance a rotating carrying body [1, 4, 6] that undergoes plane-parallel motion and is a part of an isolated system. To identify steady motions of the system—primary motions in which the body is balanced and rotating about the required axis and secondary motions in which this does not occur—and to analyze them for stability, we will use the energy approach [2, 9]. To evaluate how quickly the system attains the primary motion, we will use the first Lyapunov’s method. The energy approach was used in [7] to study the dynamic balancing of a rotating body undergoing spatial motion and being a component of an isolated system with four pendulums. The primary and secondary motions of the system were identified and analyzed for stability. It was established that oblate carrying bodies cannot be dynamically balanced and that the case of a prolate carrying body should be examined additionally. In this paper, we will establish generalized conditions for an AB of any type to balance a spatially rotating body in an isolated system. Two cases will be considered: static balancing by one AB and dynamic balancing by two ABs. To establish these conditions, we will use the energy approach and the empirical method proposed in [3] for rotor systems. Note that the dynamics of an orbital structure with elastic elements, optimization of interorbital transfers, and stability of the principal axis of inertia of inhomogeneous bodies are addressed in [5, 8, 10]. 1. Description of the System. Basic Assumptions. Consider an isolated material system consisting of a rotating carrying body (Fig. 1a) and attached bodies (AB). Ideally, the carrying body would rotate about some axis z rigidly fixed to it. This is not so, however, because of the imbalance, either static or dynamic, of the body about the axis z. The AB bodies can move relative to the carrying body, completely balancing it in a certain position. The type of AB does not matter for further discussion and, thus, is not indicated. Since the system is isolated, its center of mass G moves uniformly and rectilinearly. We assume, without loss of generality, that it is fixed. The momentum of the system is conserved: K G = const. According to the general theory of passive ABs, the system must have at least one primary steady motion in which the carrying body is balanced by the AB bodies and rotates about the axis z [3]. If the system undergoes primary motion, then the axis z is aligned with the vector K G . Let the principal centroidal axes of inertia of the system Oxhz in this motion be rigidly fixed to the carrying body ( O G = in the primary motion). The system has the principal moments of inertia A, B, and C about these axes. International Applied Mechanics, Vol. 43, No. 11, 2007 1276 1063-7095/07/4311-1276 ©2007 Springer Science+Business Media, Inc. Kirovograd National Technical University, Kirovograd, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 43, No. 11, pp. 113–120, November 2007. Original article submitted April 6, 2007.