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Applied Mathematics and Mechanics (English Edition) ›› 2010, Vol. 31 ›› Issue (11): 1371-1382.doi: https://doi.org/10.1007/s10483-010-1369-7

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Symplectic analysis for wave propagation in one-dimensional nonlinear periodic structures

侯秀慧1 邓子辰1,2 周加喜3   

  1. 1. Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, P. R. China;
    2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, Liaoning Province, P. R. China;
    3. College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, P. R. China
  • 收稿日期:2010-04-20 修回日期:2010-07-19 出版日期:2010-11-01 发布日期:2010-11-01

Symplectic analysis for wave propagation in one-dimensional nonlinear periodic structures

HOU Xiu-Hui1, DENG Zi-Chen1,2, ZHOU Jia-Xi3   

  1. 1. Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, P. R. China;
    2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, Liaoning Province, P. R. China;
    3. College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, P. R. China
  • Received:2010-04-20 Revised:2010-07-19 Online:2010-11-01 Published:2010-11-01

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摘要： The wave propagation problem in the nonlinear periodic mass-spring structure chain is analyzed using the symplectic mathematical method. The energy method is used to construct the dynamic equation, and the nonlinear dynamic equation is linearized using the small parameter perturbation method. Eigen-solutions of the symplectic matrix are used to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass-spring chain, arising from the nonlinear spring stiffness effect, has profound effects on the overall transmission of the chain. The wave propagation characteristics are altered due to nonlinearity, and related to the incident wave intensity, which is a genuine nonlinear effect not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to closing of transmitting gaps. Comparison with the normal recursive approach shows effectiveness and superiority of the symplectic method for the wave propagation problem in nonlinear periodic structures.

Abstract: The wave propagation problem in the nonlinear periodic mass-spring structure chain is analyzed using the symplectic mathematical method. The energy method is used to construct the dynamic equation, and the nonlinear dynamic equation is linearized using the small parameter perturbation method. Eigen-solutions of the symplectic matrix are used to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass-spring chain, arising from the nonlinear spring stiffness effect, has profound effects on the overall transmission of the chain. The wave propagation characteristics are altered due to nonlinearity, and related to the incident wave intensity, which is a genuine nonlinear effect not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to closing of transmitting gaps. Comparison with the normal recursive approach shows effectiveness and superiority of the symplectic method for the wave propagation problem in nonlinear periodic structures.

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