Development of a family of explicit algorithms for structural dynamics with unconditional stability
A new family of explicit integration algorithms is developed based on discrete control theory for solving the dynamic equations of motion. The proposed algorithms are explicit for both displacement and velocity and require no factorisation of the damping matrix and the stiffness matrix. Therefore, for a system with nonlinear damping and stiffness, the proposed algorithms are more efficient than the common explicit algorithms that provide only explicit displacement. Accuracy and stability properties of the proposed algorithms are analysed theoretically and verified numerically. Certain subfamilies are found to be unconditionally stable for any system state (linear elastic, stiffness softening or stiffness hardening) that may occur in earthquake engineering of a practical structure. With dual explicit expression and excellent stability property, the proposed family of algorithms can potentially solve complicated nonlinear dynamic problems.
KeywordsExplicit algorithm Nonlinear structural dynamics Stability Computational efficiency Discrete transfer function
This research is financially supported by the National Natural Science Foundation of China (Nos. 51179093, 91215301 and 41274106) and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130002110032). The authors express their sincerest gratitude for these supports.
- 2.Newmark, N.M.: A method of computation for structural dynamics. J. Eng. Mech. Div. ASCE 85(3), 67–94 (1959)Google Scholar
- 3.Wilson, E.L.: A computer program for the dynamic stress analysis of underground structures. Report UC SESM 68–1. University California, Berkeley (1968)Google Scholar
- 17.Franklin, G.F., Powell, J.D., Emami Naeini, A.: Feedback Control of Dynamic Systems. Prentice Hall, Upper Saddle River (2002)Google Scholar
- 18.Ramirez, M.R.: The numerical transfer function for time integration analysis. Proceedings of New Methods in Transient Analysis, ASME, New York, PVP, vol. 246, pp. 79–85 (1992)Google Scholar
- 21.Chen, C., Ricles, J.M.: Stability analysis of direct integration algorithms applied to nonlinear structural dynamics. J. Eng. Mech. ASCE 134, 703 (2008) Google Scholar
- 23.Ogata, K.: Discrete-Time Control Systems. Prentice Hall, Upper Saddle River (1995)Google Scholar
- 24.Chopra, A.K., Naeim, F.: Dynamics of Structures: Theory and Applications to Earthquake Engineering. Prentice Hall, Upper Saddle River (2007)Google Scholar
- 26.MathWorks, Inc.: Matlab, 2006b. MathWorks, Inc., Natick, MA (2006)Google Scholar