Reduced-Order Models With Equivalent Dynamics to Enable Programming of Nonlinear Metastructures

Abstract

Abstract. This study presents a reduced-order modeling framework for nonlinear metastructures with the aim of enabling programmable dynamic behavior. For a metastructure composed of a beam and an array of resonators with cubic nonlinearity, the behavior near the second peak is approximated by an oscillator with a cubic spring (a Duffing oscillator) that exhibits a dynamical behavior equivalent to that of the metastructure beam. Then, a parameter identification technique is adapted to determine the parameters, such as linear and cubic stiffnesses, and damping of the equivalent system. The relationships between the local resonator design parameters and the equivalent system parameters are investigated. Comparisons between the frequency responses of the nonlinear metastructure and the equivalent Duffing oscillator confirm that the reduced-order representation reliably captures key nonlinear phenomena such as resonance shifts, jump behavior, and hysteresis near the second resonance peak of the metastructure. The parameter maps reveal combinations of resonator configurations that yield identical equivalent linear and cubic stiffness values, enabling independent programming of linear and nonlinear contributions to the overall response. By leveraging isocontours of equivalent linear or cubic stiffness, the approach demonstrates how linear and cubic stiffness can be adjusted separately, establishing a pathway for programmable nonlinear metastructures. This methodology offers a versatile tool for tailoring resonance characteristics and nonlinear behavior within a certain frequency range, supporting future advances in vibration control and programmable metastructures.