Lyapunov Stability Analysis of Explicit Direct Integration Algorithms Applied to Multi-Degree-of-Freedom Nonlinear Dynamic Problems
Publication: Journal of Engineering Mechanics
Volume 142, Issue 12
Abstract
In 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.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The research was supported by Caltrans (Contract # 65A0454) for the project “Guidelines for nonlinear seismic analysis of ordinary bridges.” The authors thank Caltrans for this support. The authors recognize Ms. Minghui Zheng for her valuable suggestions and comments.
References
Bathe, K. J., and Wilson, E. L. (1972). “Stability and accuracy analysis of direct integration methods.” Earthquake Eng. Struct. Dyn., 1(3), 283–291.
Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory, SIAM, Philadelphia.
Boyd, S., and Vandenberghe, L. (2004). Convex optimization, Cambridge University Press, Cambridge, U.K.
Cains, P. E. (1989). Linear stochastic systems, Wiley, New York.
Chopra, A. K. (2006). Dynamics of structures: Theory and applications to earthquake engineering, 3rd Ed., Pearson Prentice Hall, Upper Saddle River, NJ.
Chung, J., and Hulbert, G. M. (1993). “A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-a method.” J. Appl. Mech., 60(2), 371–375.
CVX Research. (2011). “Matlab software for disciplined convex programming.” 〈http://cvxr.com/cvx〉 (Dec. 23, 2015).
Franklin, G. F., Powell, J. D., and Emami-Naeini, A. (2015). Feedback control of dynamic systems, 7th Ed., Pearson Higher Education, Upper Saddle River, NJ.
Günay, S., and Mosalam, K. M. (2013). “PEER performance-based earthquake engineering methodology, revisited.” J. Earthquake Eng., 17(6), 829–858.
Haddad, W. M., and Chellaboina, V. (2008). Nonlinear dynamical systems and control: A Lyapunov-based approach, Princeton University Press, Princeton, NJ.
Hughes, T. J. R. (1976). “Stability, convergence and growth and decay of energy of the average acceleration method in nonlinear structural dynamics.” Comput. Struct., 6(4–5), 313–324.
Hughes, T. J. R., Pister, K. S., and Taylor, R. L. (1979). “Implicit-explicit finite elements in nonlinear transient analysis.” Comput. Methods Appl. Mech. Eng., 17(18), 159–182.
Hulbert, G. M., and Chung, J. (1996). “Explicit time integration algorithms for structural dynamics with optimal numerical dissipation.” Comput. Methods Appl. Mech. Eng., 137(2), 175–188.
Kapila, V., and Haddad, W. M. (1996). “A multivariable extension of the Tsypkin criterion using a Lyapunov-function approach.” IEEE Trans. Autom. Control, 41(1), 149–152.
Khalil, H. K. (2002). Nonlinear systems, 3rd Ed., Pearson Prentice Hall, Upper Saddle River, NJ.
Kottenstette, N., and Antsaklis, P. J. (2010). “Relationship between positive real, passive dissipative, & positive systems.” Proc., American Control Conf., Baltimore.
Kuhl, D., and Crisfield, M. A. (1999). “Energy conserving and decaying algorithms in non-linear structural dynamics.” Int. J. Numer. Methods Eng., 45(5), 569–599.
Kunimatsu, S., Sang-Hoon, K., Fujii, T., and Ishitobi, M. (2008). “On positive real lemma for non-minimal realization systems.” Proc., Int. Federation of Automatic Control (IFAC), Seoul.
Lee, L., and Chen, J. (2003). “Strictly positive real lemma and absolutely stability for discrete-time descriptor systems.” IEEE Trans. Circuits Syst. I, 50(6), 788–794.
Liang, X., Günay, S., and Mosalam, K. M. (2014). “Integrators for nonlinear response history analysis: revisited.” Proc., Istanbul Bridge Conf., Istanbul, Turkey.
Liang, X., and Mosalam, K. M. (2015). “Lyapunov stability and accuracy of direct integration algorithms in nonlinear dynamic problems and considering the strictly positive real lemma.”, Univ. of California, Berkeley, CA.
Liang, X., and Mosalam, K. M. (2016a). “Lyapunov stability analysis of explicit direct integration algorithms considering strictly positive real lemma.” J. Eng. Mech., 04016079.
Liang, X., and Mosalam, K. M. (2016b). “Lyapunov stability and accuracy of direct integration algorithms applied to nonlinear dynamic problems.” J. Eng. Mech., 04016022.
Liang, X., Mosalam, K. M., and Günay, S. (2016). “Direct integration algorithms for efficient nonlinear seismic response of reinforced concrete highway bridges.” J. Bridge Eng., 04016041.
Lyapunov, A. M. (1892). “The general problem of the stability of motion.” Ph.D. dissertation, Kharkov National Univ., Ukraine (in Russian).
Mosalam, K. M., Liang, X., Günay, S., and Schellenberg, A. (2013). “Alternative integrators and parallel computing for efficient nonlinear response history analyses.” Proc., 4th ECCOMAS Thematic Conf. on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2013), Kos Island, Greece.
Newmark, N. M. (1959). “A method of computation for structural dynamics.” J. Eng. Mech. Div., 85(3), 67–94.
Tamma, K. K., Zhou, X., and Sha, D. (2000). “The time dimension: A theory towards the evolution, classification, characterization and design of computational algorithms for transient/dynamic applications.” Arch. Comput. Methods Eng., 7(2), 67–290.
Xiao, C., and Hill, D. J. (1999). “Generalizations and new proof of the discrete-time positive real lemma and bounded real lemma.” IEEE Trans. Circuits Syst. I, 46(6), 740–743.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 142 • Issue 12 • December 2016
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Oct 5, 2015
Accepted: Jul 7, 2016
Published online: Sep 1, 2016
Published in print: Dec 1, 2016
Discussion open until: Feb 1, 2017
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.