Determination of drag coefficients in automatic ball balancers at low Reynolds numbers

摘要

The precise calculation of drag forces in the technical application of automatic balancing of rotating machinery provides important information about efficiency and stability. With increasing geometric complexity of the design, this poses a challenge that can be solved with computer-aided fluid dynamic approaches. In rolling element bearings, the influence of drag induced by the lubricant is predominantly considered in the context of efficiency loss estimations, whereas the movement of the rolling elements is mainly constrained by the contact with the bearing rings and, if present, the cage. Automatic ball balancers, which are installed in rotating machinery to reduce unbalance excitation, are in design very similar to fully lubricated ball bearings missing the cage, the inner ring and the majority of the balls. Inherent to the functional principle, the balancing efficiency and stability are significantly influenced by the choice of lubricant and resulting drag forces. Therefore, the estimation of the drag coefficient based on the geometry and lubrication of automatic ball balancers plays an important role in the engineering process. With a focus on the Stokes flow regime, the drag coefficient for a single sphere in an annular flow domain is determined numerically with finite volume discretization and the SIMPLE steady state solution scheme. Based on a parameter study utilizing the presented solution approach, a simple empirical relation between the design of the automatic ball balancer and resulting drag coefficients is derived. As a result, a drag force formulation based on the balancer geometry and the lubrication fluid properties is presented, which helps to supplement a large number of published kinetic models regarding the analysis of automatic ball balancer stability and transient behavior, giving a better understanding of the influences of design decisions regarding geometry and lubricant.