Finite element simulation of chaotic vibrations of a beam with non-linear boundary conditions

Abstract

The solution of the chaotic vibration of non-linear mechanical systems involves the numerical integration of the governing equations of motion over a large time duration. This time duration should be adequately large to ensure that the transients die down and the solution captures the steady-state chaos. This demands that the integration scheme be stable, and accurate. The scheme which has found wide acceptance for chaotic problems is the fourth-order Runge-Kutta method. However, the Runge-Kutta method is not a preferred integration scheme for engineering solutions because it calls for four equation solutions per time-step and a small time-step to get accurate results from 'stiff equations' produced by the engineering structures. These drawbacks have restricted the study of chaos to single-or-limited number of degrees of freedom. This paper is an attempt to solve the chaotic vibration problem of structures with non-linear boundary conditions by the finite element method. The solution is attempted for the cantilever beam with one side-stop for which experimental results are available in the literature. This particular class of non-linearity has been chosen because of its abundance in and significance to the real-life structures. The authors' study shows that the temporally discrete solution of the spatially discrete model could capture the phenomenon of chaos. The authors expect this study to be useful for identifying the chaotic regimes for these structures in the physical coordinates of forcing amplitude and frequency. This, in turn, could be used for a more accurate prediction of fretting wear-limited life of these components.