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Lyapunov Stability Analysis of Explicit Direct Integration Algorithms Applied to Multi-Degree-of-Freedom Nonlinear Dynamic Problems
AbstractIn nonlinear structural dynamics, direct integration algorithms are used to solve the differential equations of motion after they are temporally discretized. Explicit algorithms do not require iterations and thus avoid numerical problems of convergence by making use of certain approximations. Thus, they are appealing for multi-degree-of-freedom (MDOF) nonlinear dynamic problems. In this paper, the study previously conducted by the authors for nonlinear single-degree-of-freedom systems is extended to MDOF ones to investigate the Lyapunov stability of explicit algorithms. For this purpose, a generic-explicit integration algorithm is formulated for generic MDOF nonlinear systems with softening or stiffening behavior governed by nonlinear functions of the restoring forces. This approach transforms the stability analysis of the formulated nonlinear system to investigating the strictly positive realness of its corresponding transfer function matrix. Furthermore, this is equivalent to a problem of convex optimization that can be solved numerically. Using this approach, a sufficient condition that the bounds where the explicit algorithm is stable in the sense of Lyapunov for the MDOF nonlinear system can be obtained. This approach is applied to two commonly used explicit integration algorithms for a bridge structure and also demonstrated by a generic nonlinear multistory shear building.
Lyapunov Stability Analysis of Explicit Direct Integration Algorithms Applied to Multi-Degree-of-Freedom Nonlinear Dynamic Problems
AbstractIn nonlinear structural dynamics, direct integration algorithms are used to solve the differential equations of motion after they are temporally discretized. Explicit algorithms do not require iterations and thus avoid numerical problems of convergence by making use of certain approximations. Thus, they are appealing for multi-degree-of-freedom (MDOF) nonlinear dynamic problems. In this paper, the study previously conducted by the authors for nonlinear single-degree-of-freedom systems is extended to MDOF ones to investigate the Lyapunov stability of explicit algorithms. For this purpose, a generic-explicit integration algorithm is formulated for generic MDOF nonlinear systems with softening or stiffening behavior governed by nonlinear functions of the restoring forces. This approach transforms the stability analysis of the formulated nonlinear system to investigating the strictly positive realness of its corresponding transfer function matrix. Furthermore, this is equivalent to a problem of convex optimization that can be solved numerically. Using this approach, a sufficient condition that the bounds where the explicit algorithm is stable in the sense of Lyapunov for the MDOF nonlinear system can be obtained. This approach is applied to two commonly used explicit integration algorithms for a bridge structure and also demonstrated by a generic nonlinear multistory shear building.
Lyapunov Stability Analysis of Explicit Direct Integration Algorithms Applied to Multi-Degree-of-Freedom Nonlinear Dynamic Problems
Liang, Xiao (author) / Mosalam, Khalid M
2016
Article (Journal)
English