Strain Energy Approach for Nonlinear Stiffness Coeffcients in the Design of Periodic Structures

Abstract

In this work, we explore the dynamic behaviour of a discrete model of a periodic structure under harmonic input with nonlinear stiffness. The periodic structure has a unit cell with three degrees of freedom. We devise an approach that replaces the linear stiffness characteristic of the structure with a nonlinear one in which the nonlinear stiffness coefficients provide the same strain energy. The effect of this approach on the frequency response is analysed using numerical simulation, focusing on band gaps. The approach to determining nonlinear stiffness coefficients is based on the concept of equivalent elastic strain energy. This is different from the common approach found in the literature of adding a cubic term to the linear one, resulting in an increase in the elastic deformation energy of the system. Once the strain energy of the linear system is determined, a family of possible nonlinear stiffness coefficients is found, parameterised by the ratio between the linear and cubic coefficients. This approach can be used with hardening or softening stiffness characteristics. With the nonlinear stiffness coefficients defined, the dynamic response of the metastructure shows the usual shift to high and low frequencies. In addition, some frequency ranges are shown where vibration levels can be greatly reduced when the ratio of nonlinear stiffness coefficients is increased, compared to the case where there are only linear springs. Also, it is shown that the addition of the nonlinear component in the structure can increase or decrease the distance between the resonant frequencies.