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Journal of Vibration Testing and System Dynamics 1(1) (2017) 53--63 | DOI:10.5890/JVTSD.2017.03.004

Yan Liu$^{1}$, Jiazhong Zhang$^{2}$, Jiahui Chen$^{2}$, Yamiao Zhang$^{2}$

$^{1}$ School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, P. R. China

$^{2}$ School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, P. R. China

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Abstract

In light of Approximate Inertial Manifolds stemmed from nonlinear dynamics and Mode Analysis Technique in linear structural dynam- ics, a combined method is presented to reduce the second order in time nonlinear dissipative autonomous dynamic systems with higher dimension or many degrees-of-freedom to lower dimensional systems, and the ensuing error estimate is investigated theoretically. By this method, the system is studied in the phase space with a distance definition which can describe the distance between the original and reduced systems, and the solution of the original system is then pro- jected onto the complete space spanned by the eigenvectors of the linear operator of the governing equations. With the introduction of an Approximate Inertial Manifolds, the interaction between lower and higher modes or the influence of higher modes on the long-term behaviors, which will be ignored if the traditional Galerkin proce- dure is used to approach the governing equation, is considered to improve the distance between the original and reduced systems. Ad- ditionally, the error estimate for the approximation to the attractor is presented, and an explicit iterative scheme is then proposed to ap- proach the Approximate Inertial Manifolds. Finally, a comparison between the traditional Galerkin method and the method presented has been given and discussed. The results show that the method presented can provide a better and acceptable approximation to the long-term behaviors of the second order in time nonlinear dissipative autonomous dynamic systems with many degrees-of-freedom, espe- cially for the numerical analysis of complex bifurcation and chaos in complicated dynamical systems.

Acknowledgments

The research is supported by the National Natural Science Foundation of China (Grant No. 51305355), the National Fundamental Research Program of China (973 Program, Grant No. 2012CB026002) and the National Key Technology R&D Program of China (Grant No. 2013BAF01B02).

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