Topological Protection in a Strongly Nonlinear Interface Lattice

Abstract

The application of topological phases in classical mechanical systems is unlocking previously unattainable topologically insulated waves in acoustic, phonic, and elastic media. Mechanical topological insulators are well understood for linear and weakly nonlinear systems, however traditional analysis methods break down for strongly nonlinear systems since linear methods can not be applied in that case. We study one such system in the form of a one-dimensional mechanical analog of the Shu-Schrieffer-Heeger interface model with strong nonlinearity of the cubic form. Two nonlinear half-lattices make a topological interface system with a nonlinear coupling added to the stiff spring, while linear grounding springs are added on all oscillators. The frequency-energy dependence of the nonlinear bulk modes and topologically insulated mode is explored using Numerical continuation of the system's nonlinear normal modes (NNMs), i.e., of standing waves. Moreover, the linear stability of the NNMs are investigated using Floquet Multipliers (FMs) and Krein signature analysis. We find that the nonlinear topological lattice supports a family of topologically insulated NNMs that are parameterized by the total energy of the system and are stable within a range of frequencies. Using numerical simulations, we empirically recover the geometric Zak phase to determine at which energies the nontrivial phase conditions cease to exists. We compare the predictions from FM analysis and the numerical phase analysis by numerically investigating the interface system for harmonic excitation applied to the interface site. It is shown that empirical calculations of the geometric Zak Phase lead to reliable measures for predicting the existence of the topological mode in the nonlinear system, and that while the FMs of the NNMs are dependable predictors of the NNMs stability, they are inferior to the empirical phase calculations for predicting at which energies the lattice supports waves localized at the interface. Furthermore, the results are shown to be robust to parametric perturbations of both chiral and achiral forms. Thus, we provide a new and improved method for analyzing and predicting the existence of topologically insulated modes in a strongly nonlinear lattice.