A nonlinear model for structural vibrations in rolling element bearings .1. Derivation of governing equations

This paper, the first of two companion papers, presents a model for investigating structural vibrations in rolling element bearings. The analytical formulation accounts for tangential and radial motions of the rolling elements, as well as the cage, the inner and the outer races. The contacts between the rolling elements and races are treated as nonlinear springs whose stiffnesses are obtained by application of the equation for Hertzian elastic contact deformation. The derivation of the equations of motion is facilitated by assuming that only rolling contact exists between the races and rolling elements. Application of Lagrange's equations leads to a system of nonlinear ordinary differential equations governing the motion of the bearing system. These equations are then solved using the Runge-Kutta integration technique. Using the formulation in the second part- ''A Nonlinear Model for Structural Vibrations in Rolling Element Bearings: Part II--Simulation and Results,'' a number of effects on bearing structural vibrations are studied. This work is unique from previous studies in that the model simulates vibration from intrinsic properties and constituent elements of the bearing, and rakes into account every contact region, within the bearing, representing it by a nonlinear spring.