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

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.