Periodic, aperiodic and chaotic motions of harmonically excited SDOF and MDOF nonlinear dynamical systems

Abstract

Responses of nonlinear dynamical systems with single degree of freedom (SDOF) or multiple degrees of freedom (MDOF) to periodic excitations are investigated in this paper using a numerical scheme. The scheme is developed on the basis of the effective mass, damping and stiffness matrices, the incremental generalized coordinates and the Newmark integration method to solve efficiently and accurately second-order nonlinear ordinary differential equations in the hundreds or more. Using the proposed numerical method, long-term behavior of SDOF and MDOF systems of any type of nonlinearities including the well-known van der Pol nonlinear damping forces, the Duffing type nonlinear spring forces and the time-delayed spring force at any strength level (weak, moderate and strong) can be accurately determined. With the help of an adequate mapping frequency, orders of periodicity of periodic responses can be easily and reliably identified. Numerical results, obtained for three oscillators – an SDOF van der Pol oscillator, a time-delayed SDOF Duffing oscillator, and a five-DOF Duffing oscillator, demonstrate that the proposed scheme is ideally suited for solving large scale nonlinear dynamical problems.