Dynamic analysis of an automatic ball balancer with double races

Dynamic behaviors are analyzed for an automatic ball balancer with double races, which is a device to reduce unbalanced mass of rotors. The nonlinear equations of motion are derived by using Lagrange's equations with the polar coordinate system. It is shown that the polar coordinate system provides complete stability analysis while the rectangular coordinate system used in other previous studies has limitations on the stability analysis. For the stability analysis, the equilibrium positions and the linearized perturbation equations are obtained by the perturbation method. Based on the linearized equations, the stability of the system is analyzed around the equilibrium positions; furthermore, to confirm the stability, the time responses for the nonlinear equations of motion are computed by using a time integration method.