Varying stiffness and load distributions in defective ball bearings: Analytical formulation and application to defect size estimation

Abstract

This paper presents an analytical formulation of the load distribution and varying effective stiffness of a ball bearing assembly with a raceway defect of varying size, subjected to static loading in the radial, axial and rotational degrees of freedom. The analytical formulation is used to study the effect of the size of the defect on the load distribution and varying stiffness of the bearing assembly. The study considers a square-shaped outer raceway defect centered in the load zone and the bearing is loaded in the radial and axial directions while the moment loads are zero. Analysis of the load distributions shows that as the defect size increases, defect-free raceway sections are subjected to increased static loading when one or more balls completely or partly destress when positioned in the defect zone. The stiffness variations that occur when balls pass through the defect zone are significantly larger and change more rapidly at the defect entrance and exit than the stiffness variations that occur for the defect-free bearing case. These larger, more rapid stiffness variations generate parametric excitations which produce the low frequency defect entrance and exit events typically observed in the vibration response of a bearing with a square-shaped raceway defect. Analysis of the stiffness variations further shows that as the defect size increases, the mean radial stiffness decreases in the loaded radial and axial directions and increases in the unloaded radial direction. The effects of such stiffness changes on the low frequency entrance and exit events in the vibration response are simulated with a multi-body nonlinear dynamic model. Previous work used the time difference between the low frequency entrance event and the high frequency exit event to estimate the size of the defect. However, these previous defect size estimation techniques cannot distinguish between defects that differ in size by an integer number of the ball angular spacing, and a third feature of the vibration response is therefore required to distinguish between such defects. It is hypothesized and validated through simulations that when the static load remains constant, the third distinguishing feature is the characteristic frequencies of the low frequency event.