Correlation of Parameters to Instability and Chaos of a Horizontal Flexible Rotor Ball Bearing System

Abstract

The higher order effects from ball bearing nonlinearities cause complex vibration characteristics in rotor ball bearing systems. The sources of nonlinearities are internal radial clearance, Hertzian contact forces between balls and races and varying compliance effect. The same authors in their earlier work have identified the sets of parameters corresponding to instability and chaos for a horizontal flexible rotor supported on deep groove ball bearing. To the best of author’s knowledge, there is not much work reported in the literature on the dynamic analysis for instability and chaos, which is based on energy functions and bearing loads. Extending the preceding research work in the present paper by using a typical set of parameters and specifications of rotor ball bearing system, a correlation of parameters to instability and chaos is attempted using different energy functions associated with the dynamical system. A generalized Timoshenko beam finite element formulation is used to model the flexible rotor shaft. To achieve the convergence of solution with smaller number of elements, shape functions are derived from the exact solutions of governing differential equations of Timoshenko beam element. The sources of excitation are rotating unbalance and parametric excitation due to varying compliance of ball bearing during motion. For the bearing used in the present paper, the ratio of these excitation frequencies comes out to be an irrational number. Therefore, the dynamic response would be quasi-periodic with time period equal to infinity. To extend the use of non-autonomous shooting method to derive quasi-periodic solution, the fixed point algorithm (FPA) proposed in the literature is used to deduce the time period for non-autonomous shooting algorithm. The shooting method otherwise is used only to derive periodic solutions. Thus the non-autonomous shooting method coupled with fixed point algorithm (FPA) is used to compute the quasi-periodic solution, which also gives the monodromy matrix. The eigenvalues of the monodromy matrix, called Floqoet multipliers, give information about instability. The chaotic nature of the dynamic response is established by the maximum value of Lyapunov exponent. Once the instability and chaos is confirmed based on computed values of Floquet multipliers and Lyapunov exponents, the nature of the work done (positive or negative) by different conservative and non-conservative forces and moments during motion are analyzed and the fundamental causes, which make the system response unstable and / or chaotic, are established.