Wave propagation in mass-in-mass Duffing type non-linear metamaterial implementing Jacobi's elliptic balance method

A comprehensive strategy for computing the dispersion relationship of a nonlinear metamaterial having Duffing-type cubic nonlinearity at the attached oscillator is presented in this paper. Jacobi's elliptic balance method is implemented to obtain the steady-state response of the non-linear metamaterial unit cell. An ingenious approach of defining the force term in equation motion not only reduces the system to a simplified uncoupled form but also facilitates the incorporation of the effect of amplitude dependency. Furthermore, the dispersion relationship can conveniently be obtained by applying Bloch's theorem. Under undamped conditions, the precise response of both the inner and outer mass of the nonlinear metamaterial is determined; whereas, a set of algebraic nonlinear equations is established and numerically solved for the damped system. The attenuation band shifts to the higher frequency range with increasing nonlinearity and excitation amplitude. The effect of excitation amplitude is predominant as compared to the nonlinearity of the spring in altering the dispersion plots. As damping increases, the sharp attenuation drop becomes blunt, and further increment of damping leads to the resonator and outer mass oscillating in nearly perfect harmony and thereby causing the dispersion plots to resemble those of a monoatomic chain.