In the chapter, we study the nonlinear properties of periodic structures with the same mass and different spring stiffness. According to the linear dispersion curve of this periodic structure, we predict the nonlinear characteristics of the structure, that is, the nonlinear spring can make the acoustic branch cut-off frequency shift when the optical branch cut-off frequency is constant, and the prediction is confirmed by the simulation of the transmission coefficient curve. This conclusion shows that our structure can control the band gap more flexibly and achieve better filtering results. Based on this linear and nonlinear periodic structure, we have designed a non-reciprocal structure. Since the structure breaks the reciprocity principle, so the elastic wave can realize asymmetric transmission in it. The simulation results show that if the input signal is the same, the amplitude ratio output from different directions of propagation can be up to an order of magnitude. And we carry out linear and nonlinear simulation of mass-in-mass acoustic metamaterials. The results show that the band gap corresponding to the transmission coefficient curve conforms to the theory.