Lyapunov Stability Analysis of Explicit Direct Integration Algorithms Considering Strictly Positive Real Lemma

Abstract

In structural dynamics, direct integration algorithms are commonly used to solve the temporally discretized differential equations of motion. Integration algorithms are categorized into either implicit or explicit. Implicit algorithms require iterations and may face numerical convergence problems when applied to nonlinear structural systems. On the other hand, explicit algorithms do not need iterations by making use of certain approximations, making them appealing for nonlinear dynamic problems. In this paper, a generic explicit integration algorithm is formulated for a nonlinear system governed by a nonlinear function of the restoring force. Based on this formulation, an approach is proposed to investigate the Lyapunov stability of explicit algorithms by means of the strictly positive real lemma. In this approach, the stability analysis is equivalent to investigating the strictly positive realness of the transfer function for the formulated system. Subsequently, a Nyquist plot is used to determine the range of the system stiffness where the explicit algorithm is stable in the sense of Lyapunov. This approach is applied to two commonly used types of explicit integration algorithms applied to single-degree-of-freedom systems with softening or stiffening behavior.