Abstract
From viewpoint of nonlinear dynamics, the model reduction and its influence on the long-term behaviours of a class of nonlinear dissipative autonomous dynamical system with higher dimension are investigated theoretically under some assumptions. The system is analyzed in the state space with an introduction of a distance definition which can be used to describe the distance between the full system and the reduced system, and the solution of the full system is then projected onto the complete space spanned by the eigenvectors of the linear operator of the governing equations. As a result, the influence of mode series truncation on the long-term behaviours and the error estimate are derived, showing that the error is dependent on the first products of frequencies and damping ratios in the subspace spanned by the eigenvectors with higher modal damping. Furthermore, the fundamental understanding for the topological change of the solution due to the application of different model reduction is interpreted in a mathematically precise way, using the qualitative theory of nonlinear dynamics.
Similar content being viewed by others
References
Seydel, R.Practical Bifurcation and Stability analysis: From Equilibrium to Chaos[M]. Springer-Verlag, New York, 1994, 322–326.
Zhang Jiazhong, Xu Qingyu, Zheng Tiesheng. A method for determing the periodic solution and its stability of a dynamic system with local nonlinearities[J].Acta Mechanica Sinica, 1998,30(5): 572–579 (in Chinese).
Friswell M I, Penny J E T, Garvey S D. The application of the IRS and balanced realization methods to obtain reduced models of structures with local nonlinearities[J].Journal of Sound and Vibration, 1996,196(4): 453–468.
Slaats P M A, de Jongh J, Sauren A A H J. Model reduction tools for nonlinear structural dynamics[J].Computers & Structures, 1995,54(6): 1155–1171.
Fey R H B, van Campen D H, de Kraker A. Long term structural dynamics of mechanical system with local nonlinearities[J].ASME Journal of Vibration and Acoustics, 1996,118(2): 147–163.
Temam R.Infinite-Dimensional Dynamical System in Mechanics and Physics[M]. Springer-Verlag, New York, 1997, 20–23.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by LU Chuan-jing
Project supported by the National Natural Science Foundation of China (No. 10272089)
Rights and permissions
About this article
Cite this article
Jia-zhong, Z., Yan, L. & Dang-min, C. Error estimate for influence of model reduction of nonlinear dissipative autonomous dynamical system on long-term behaviours. Appl Math Mech 26, 938–943 (2005). https://doi.org/10.1007/BF02464244
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02464244