Stability analysis of a parametrically excited ball bearing system

Abstract

In this paper, the stability analysis of a ball bearing system experiencing varying stiffness coefficients is taken into account. The presence of variable stiffness may cause the system to experience instabilities at given combinations of rotational speed, number, and the dimension of balls, thus complicating the design process. The objective is to obtain the stability boundary curves (SBCs) which separate the stable and unstable regions. The well-known Mathieu equation is adopted as the governing equations of the system in horizontal and vertical directions. In order to calculate the SBCs the equations of motion are solved applying approximate methods such as the harmonic balance method (HBM) and the multiple scales method (MSM). This procedure is straightforward if Uncoupled Mathieu equations, either Damped or Undamped, are considered, however, a realistic bearing system can be effectively described only using two coupled Mathieu equations, thus introducing two dominant frequencies which are not an integer coefficient of each other. This last Damped and Coupled set of equations applied to a bearing system is, for the first time, solved using HBM in place of resorting to cost intensive iterative methods. The accuracy of all investigated cases, Uncoupled–Undamped, Uncoupled–Damped and Coupled–Damped, is ensured by Floquet Theory.