Stability Analysis of Direct Integration Algorithms Applied to Nonlinear Structural Dynamics

Direct integration algorithms are often used to solve the temporally discretized equations of motion for structural dynamic problems. Numerous studies have been conducted to investigate the stability of integration algorithms for linear elastic structures. Studies involving the stability analysis of integration algorithms for nonlinear structures are limited. This paper utilizes discrete control theory to investigate the stability of direct integration algorithms for nonlinear structural dynamics. The direct integration algorithms are represented by a closed-loop block diagram, where the nonlinear restoring force of the structure is related to a varying feedback gain. The root locus method is used to analyze the stability of the closed-loop system for various degrees of nonlinear structural behavior. The well-known methods of the Newmark family of integration algorithms and the Hilber–Hughes–Taylor $\alpha$ method, as well as a newly developed integration algorithm, referred to as the CR integration algorithm, are analyzed using the proposed method. It is shown that the stability of an integration algorithm under nonlinear structural behavior is dependent on the poles and zeros of its open-loop discrete transfer function. An unconditionally stable integration algorithm for linear elastic structures is shown not to always remain stable under nonlinear structural behavior.