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Symplectic analysis for regulating wave propagation in a one-dimensional nonlinear graded metamaterial

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Abstract

An analytical method, called the symplectic mathematical method, is proposed to study the wave propagation in a spring-mass chain with gradient arranged local resonators and nonlinear ground springs. Combined with the linearized perturbation approach, the symplectic transform matrix for a unit cell of the weakly nonlinear graded metamaterial is derived, which only relies on the state vector. The results of the dispersion relation obtained with the symplectic mathematical method agree well with those achieved by the Bloch theory. It is shown that wider and lower frequency bandgaps are formed when the hardening nonlinearity and incident wave intensity increase. Subsequently, the displacement response and transmission performance of nonlinear graded metamaterials with finite length are studied. The dual tunable effects of nonlinearity and gradation on the wave propagation are explored under different excitation frequencies. For small excitation frequencies, the gradient parameter plays a dominant role compared with the nonlinearity. The reason is that the gradient tuning aims at the gradient arrangement of local resonators, which is limited by the critical value of the local resonator mass. In contrast, for larger excitation frequencies, the hardening nonlinearity is dominant and will contribute to the formation of a new bandgap.

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Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, China

    Yunping Zhao

  2. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, 710072, China

    Yunping Zhao, Xiuhui Hou, Kai Zhang & Zichen Deng

  3. Ministry of Industry and Information Technology, Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an, 710072, China

    Xiuhui Hou, Kai Zhang & Zichen Deng

  4. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Kai Zhang

Authors
  1. Yunping Zhao
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  2. Xiuhui Hou
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  3. Kai Zhang
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  4. Zichen Deng
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Correspondence to Zichen Deng.

Additional information

Citation: ZHAO, Y. P., HOU, X. H., ZHANG, K., and DENG, Z. C. Symplectic analysis for regulating wave propagation in a one-dimensional nonlinear graded metamaterial. Applied Mathematics and Mechanics (English Edition), 44(5), 745–758 (2023) https://doi.org/10.1007/s10483-023-2985-6

Project supported by the National Natural Science Foundation of China (Nos. 12072266, 12172297, 11972287, and 12072262) and the Open Foundation of the State Key Laboratory of Structural Analysis for Industrial Equipment of China (No. GZ22106)

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Zhao, Y., Hou, X., Zhang, K. et al. Symplectic analysis for regulating wave propagation in a one-dimensional nonlinear graded metamaterial. Appl. Math. Mech.-Engl. Ed. 44, 745–758 (2023). https://doi.org/10.1007/s10483-023-2985-6

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  • DOI: https://doi.org/10.1007/s10483-023-2985-6

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