Accurate integration of nonlinear systems using newmark explicit method

Abstract

In the step-by-step solution of a linear elastic system, an appropriate time step can be selected based on analytical evaluation results. However, there is no way to select an appropriate time step for accurate integration of a nonlinear system. In this study, numerical properties of the Newmark explicit method are analytically evaluated after introducing the instantaneous degree of nonlinearity. It is found that the upper stability limit is equal to 2 only for a linear elastic system. In general, it reduces for instantaneous stiffness hardening and it is enlarged for instantaneous stiffness softening. Furthermore, the absolute relative period error increases with the increase of instantaneous degree of nonlinearity for a given product of the natural frequency and the time step. The rough guidelines for accurate integration of a nonlinear system are also proposed in this paper based on the analytical evaluation results. Analytical evaluation results and the feasibility of the rough guidelines proposed for accurate integration of a nonlinear system are confirmed with numerical examples.