The nonlinear dynamic analysis of the ball-spring automatic balancer by the multiple scales method

Utilizing ball-type auto-balancer in rotating systems is a popular method in industry in order to eliminate the rotational vibration. Recently, by using numerical methods, it is shown that a ball-type auto-balancer equipped with both the radial and peripheral springs, ball-spring automatic balancer (AB), is an improved model. In this paper, due to the advantages of analytical methods over the numerical ones, the complete system dynamics, i.e., the stability analysis and the time response, of the rotor equipped with ball-spring AB is analyzed by the multiple scales method as a unified technique. Therefore, with no need for implementing any other stability theories, the stability analysis is a definite advantage of this method due to the less computation cost. Moreover, the time responses show a good agreement with those obtained through the numerical method. Finally, for the first time, the influence of the peripheral springs on the time responses and the stable equilibrium points of the rotor with ball-spring AB is studied in details. The results show that the peripheral springs decrease the vibration amplitude at the transient and leave some residual imbalance at the steady state and accordingly, there is a compromise between the system working performance at the transient and the steady state.