An assessment of the newmark method for solving chaotic vibrations of impacting oscillators

Abstract

The fourth-order Runge-Kutta method has been the preferred numerical integration scheme for solving chaotic problems in non-linear systems. This method is very accurate, but requires very small time-steps and four equation solutions per time-step. These drawbacks hinder the solution of chaotic problems in multi-degree-of-freedom (MDOF) systems. This paper presents the solution of the chaotic problem of impacting single-degree-of-freedom (SDOF) oscillators, using the Newmark method which is computationally efficient and unconditionally stable. The scheme incorporates an equilibrium iteration and variable time-stepping algorithm based on a convergence criteria which ensures that solution errors are minimized at each step. The results are compared with those obtained from the fourth-order Runge-Kutta method. It is concluded that the Newmark method with an adequate check on the solution accuracy could give qualitatively the same results as the Runge-Kutta method. The method has the advantage of an extension to MDOF real-life problems of chaos which could be solved using numerical techniques like FEM with limited computing effort. Such an extension is being pursued separately.