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Rotating unbalance is a major cause of vibration in rotating machinery. One of the new methods used to control and reduce the unbalance is use of automatic dynamic ball balancer. In previous researches comprehensive studies have been done on the dynamic behavior and stability of automatic ball balancer, however there is no research has been done on determining the optimum values of the system parameters. Harmful forces caused by the unbalance in the system causes unwanted vibration and malfunctions of the system, therefore reducing the time of balancing is necessary. In this study, the effect of damping ratio and the mass of balls of the automatic dynamic ball balancer on the stability and the balance of system have been investigated. Moreover, for the first time the optimum values of these parameters to minimize the balance time are obtained. The results show that the optimal choice of the system parameters reduces the balance time considerably.

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The unbalancing is a destructive phenomenon and is a major cause of undesired vibrations in rotating machinery. One of the new methods used to reduce the imbalance is the implementing of automatic dynamic ball balancer. In previous studies the dynamic behavior of automatic ball balancer has been investigated. These studies indicate numerous advantages of automatic ball balancer. However, the traditional automatic ball balancer has two major deficiencies: First, the rotor vibration amplitude is larger than that of a rotor without an automatic ball balancer in speeds below the first critical speed and, the second deficiency is that it has a limited stable region of the perfect balancing configuration. In this paper, a new design of a three-ball automatic balancer is introduced. The governing equations of motion are derived using the Lagrange's equations, and the balanced stable region is obtained. It is shown that this type of automatic ball balancer can prevent from increasing the vibrations of the rotor at the speed range below the first critical speed. Moreover, the new type of balancer increases the balance stable region of the system. Reducing the vibration amplitude in the mentioned range causes the life time of the system to be increased. Moreover, increasing the balanced stable range makes the new design of balancer can balance the systems with a wider range of parameters.

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Unbalance in rotating machines causes malfunction of the system operation and it may leads to its failure. Therefore, the sources for imbalance should be investigated, identified, and measured to solve the mentioned challenges. Rotating unbalance appears when the geometric and the inertia axes of the rotor do not coincide, and as a result this causes self- excited vibrations. One of the methods to control and reduce the unbalances is utilizing automatic ball balancer (ABB). In previous studies, the stability and the dynamic behavior of ABB have been mostly investigated by using numerical methods, and the perturbation methods are applied only for stability analysis. Because of the advantages of the analytical methods in studying the dynamics of the systems, in the present study, for the first time the dynamic behavior as well as the stability of a rotor equipped with an ABB is analyzed by the multiple scales method. To this end, nonlinear equations of the systems are derived using the Lagrange’s equations and firstly, the multiple scales method is applied to investigate the stability of system and then the response of the system is achieved considering one and two terms of approximation. The results demonstrate that the stability analysis using the multiple scales method and the first method of Lyapanov lead to the same results. Moreover, the responses obtained by the multiple scales method and the mostly used numerical method, Rung-Kotta technique, are in a good agreement.