Simulation of the Ball Kinetic in Ball-Type Automatic Balancing Devices by Solving the Axisymmetric Navier-Stokes Equations in Annular Cavities

Automatic balancing devices are useful in order to balance rotating systems, which show variable unbalance configurations during operation, e.g. centrifuges, without the need for active components like controllers or actuators. The design of a ball-balancer consists of an annulus symmetric to the axis of rotation. It is filled with a viscous fluid and counterbalancing balls, which can move freely and oppose the rotor unbalance in the plane of the annulus when operated at super-critical speed. In order to determine the time for the balancing effect to be reached once the critical speed of the rotor is surpassed, sufficient modelling depth is needed to describe the movement of the balls during rotor run-up. Derived from transient simulations the influence of the balancer design parameters on the balancing process can be evaluated. A non-linear model of a ball-type automatic balancer is presented with which frictional forces based on Hertzian contact pressure and drag forces induced by the surrounding fluid are considered. Latter are obtained by solving the axisymmetric Navier-Stokes equations in the annular cavity by the method of finite differences. As a consequence, only one friction coefficient has to be quantified empirically. The model is included in multi-body simulations of tabletop centrifuges and the resulting angular movement of the balls is held against experimental data gained from video material of a balancer specimen with a transparent lid. Furthermore, the rotor deflection is compared with the simulation results.