Dynamic instability characteristics of rolling element bearings

Abstract

The dynamics of bearing is a classical problem in machinery vibration. It is well known that rolling element bearings are susceptible to large vibration response at suitable parameter values arising due to instability. In the present work, we formulate the governing equations of motion of rolling element bearings. Herein, the rolling elements are modeled as lumped spring elements. With odd number of rolling elements, due to asymmetric effects of the bearing crosssection a parametric excitation effect is introduced in the system of governing equations. Further, due to the load zone effect this system represents a non-smooth dynamical system. The parametric stiffness term flips its sign depending on the sign of the displacement response. As such the governing equations of the system resemble the classical asymmetric Mathieu equation. In the literature, the method of Lyapunov-like exponents has been used to determine the stability boundaries of the asymmetric Mathieu equation. Herein, a positive Lyapunov-like exponent indicates instability whereas a stable response manifests as a negative Lyapunov-like exponent. In the present work, we use this method in detecting the stability and instability characteristics over the different bearing parameters. Stability diagrams are presented which can aid the designers and the user of the bearing in confirming the stability and instability zone. The method is validated by numerically integrating the governing equations. It is verified through numerical analysis that parameter combinations associated with an unstable zone manifest an exponential growth in response. Similarly, the parameter combinations associated with stable zone of the stability diagram shows bounded response.