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 li...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.展开更多
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 d...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.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.12072266,12172297,11972287,and 12072262)the Open Foundation of the State Key Laboratory of Structural Analysis for Industrial Equipment of China(No.GZ22106)。
文摘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.
基金Project supported by the National Natural Science Foundation of China (Nos. 10972182,10772147,and 10632030)the National Basic Research Program of China (No. 2006CB 601202)+4 种基金the National 111 Project of China (No. B07050)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (No. GZ0802)the Doctoral Foundation of Northwestern Polytechnical University (No. CX200908)the China Postdoctoral Science Foundation (No. 20090450170)the Northwestern Polytechnical University Foundation for Fundamental Research (No. JC200938)
文摘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.