Low Order Modeling of Load Distribution and Friction in Ball Bearings and Ball Screws - with Application to Electric Power Assisted Steering

Abstract

Rolling ball machine components like ball bearings and ball screws are used in a wide range of machineries for load bearing and motion transmission. Ball bearings and ball screws are also the two key mechanical components in rack-type electric power assisted steering (rack EPAS) gears in motor vehicles. They are subjected to large multi-directional loads and manufacturing errors in rack EPAS, making it hard to accurately calculate load distribution (i.e., the load borne by individual balls) for the purpose of sizing. Rack EPAS gear also suffers from “stick-slip” (i.e., sticky feel sensed by the driver) mainly due to the friction variation of ball bearings and ball screws, which adversely affects driving experience. Motivated by the industrial application, the objective of this doctoral thesis is to develop low order load distribution and friction models for ball bearings and ball screws to aid analysis, optimal design and manufacturing tolerance specification of rolling ball machine components (used in EPAS). A low order static load distribution model for ball screw is first proposed incorporating geometric errors and elastic deformation effects. A new way of describing the ball screw groove surfaces with geometric errors using multivariate functions is proposed. A ball-to-groove contact model based on Hertzian Contact Theory including geometric error effects is developed. The proposed load distribution model is validated against high order Finite Element Analysis (FEA) models created in ANSYS and is shown to be accurate but computationally much faster. It is thus applicable to ball screws with multi-directional loading and geometric errors, like those in EPAS. Two sources of contact-induced friction variation in ball bearings and ball screws are investigated and modeled. Based on a sensitivity analysis of ball-to-groove contact friction due to rolling, sliding and spin motions, the transition between four-point contact operation and two-point contact operation is shown to give rise to significant friction variation in a four-point contact ball bearing. Another source of friction variation is ball-to-ball contact. In this work, low order numerical models for ball-to-ball contact friction in linear ball bearings and ball screws are proposed, both of which are validated favorably against ANSYS FEA models while being computationally much faster. Based on friction analysis and simplifying approximations, an analytical model for ball-to-ball contact friction in four-point contact linear ball bearings is derived. The insights gained from the analytical model are leveraged to mitigate ball-to-ball contact and thus significantly reduce friction variation. The developed load distribution and friction models are applied to rack EPAS gear in a few realistic scenarios and proven to be useful for industrial applications. Important insights are derived for ball bearing and ball screw design, inspection and manufacturing tolerance specification based on the developed models.