Negative effective mass in acoustic metamaterial with nonlinear mass-in-mass subsystems

•Nonlinear mass-in-mass system is basic subsystem of the acoustic metamaterial.•The excitation of the system is in the form of the Jacobi elliptic function.•Nonlinearity is of the Duffing type.•Integrals of Jacobi elliptic function are determined analytically.•Negative effective mass ratio is calculated.•Frequency gap for vibration absorption is significant. In this paper the dynamics of the nonlinear mass-in-mass system as the basic subsystem of the acoustic metamaterial is investigated. The excitation of the system is in the form of the Jacobi elliptic function. The corresponding model to this forcing is the mass-in-mass system with cubic nonlinearity of the Duffing type. Mathematical model of the motion is a system of two coupled strong nonlinear and nonhomogeneous second order differential equations. Particular solution to the system is obtained. The analytical solution of the problem is based on the simple and double integral of the cosine Jacobi function. In the paper the integrals are given in the form of series of trigonometric functions. These results are new one. After some modification the simplified solution in the first approximation is obtained. The result is convenient for discussion. Conditions for elimination of the motion of the mass 1 by connection of the nonlinear dynamic absorber (mass - spring system) are defined. In the consideration the effective mass ratio is introduced in the nonlinear mass-in-mass system. Negative effective mass ratio gives the absorption of vibrations with certain frequencies. The advantage of the nonlinear subunit in comparison to the linear one is that the frequency gap is significantly wider. Nevertheless, it has to be mentioned that the amplitude of vibration differs from zero for a small value. In the paper the analytical results are compared with numerical one and are in agreement.