Parametrically Excited Vibration in Rolling Element Bearings

Abstract

A defect-free rolling element bearing has a varying stiffness. The variation of stiffness depends on number of rolling elements, their configuration and cage frequency. The time-varying characteristics of the stiffness results in a parametric excitation. This may lead to instability which is manifested as high vibration levels. An FEM simulation is performed to evaluate stiffness in each configuration of rolling elements and is used to study the variation of direct stiffness and cross coupled stiffness. The obtained stiffness variation is expanded into a Fourier series to form the equation of motion for the bearing vibration. As the stiffness varies with cage frequency, stiffness term in the equation of motion is periodic with parametric excitation. Hence, the equation of motion is a 2-DOF coupled Mathieu equation. Based on Mathieu parameters and cage frequency there exists unstable rpm ranges for a particular bearing. Floquet theory is employed to find out the stable and unstable regions. This involves finding out maximum Floquet exponent using Monodromy matrix. The results obtained through Floquet theory are in agreement with the numerical solution of the governing equations.