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Abstract Recent work in nonlinear topological metamaterials has revealed many useful properties such as amplitude dependent localized vibration modes and nonreciprocal wave propagation. However, thus far, there have not been any studies to include the use of local resonators in these systems. This work seeks to fill that gap through investigating a nonlinear quasi-periodic metamaterial with periodic local resonator attachments. We model a one-dimensional metamaterial lattice as a spring-mass chain with coupled local resonators. Quasi-periodic modulation in the nonlinear connecting springs is utilized to achieve topological features. For comparison, a similar system without local resonators is also modeled. Both analytical and numerical methods are used to study this system. The dispersion relation of the infinite chain of the proposed system is determined analytically through the perturbation method of multiple scales. This analytical solution is compared to the finite chain response, estimated using the method of harmonic balance and solved numerically. The resulting band structures and mode shapes are used to study the effects of quasi-periodic parameters and excitation amplitude on the system behavior both with and without the presence of local resonators. Specifically, the impact of local resonators on topological features such as edge modes is established, demonstrating the appearance of a trivial bandgap and multiple localized edge states for both main cells and local resonators. These results highlight the interplay between local resonance and nonlinearity in a topological metamaterial demonstrating for the first time the presence of an amplitude invariant bandgap alongside amplitude dependent topological bandgaps. 

Abstract Numerous recent works have established the potential of various types of metamaterials for simultaneous vibration control and energy harvesting. In this paper, we investigate a weakly nonlinear metamaterial with electromechanical (EM) local resonators coupled to a resistance-inductance shunt circuit, a system with no previous examination in the literature. An analytical solution is developed for the system, using the perturbation method of multiple scales, and validated through direct numerical integration. The resulting linear and nonlinear band structures are used for parametric analysis of the system, focusing on the effect of resonator and shunt circuit parameters on band gap formation and vibration attenuation. This band structure analysis informs further study of the system through wavepacket excitation as well as spectro-spatial analysis. The voltage response of the system is studied through spatial profiles and spectrograms to observe the effects of shunt inductance, nonlinearity, and their interactions. Results describe the impact of adding a shunted inductor, including significant changes to the band structure; multiple methods of tuning band gaps and pass bands of the system; and changes to wave propagation and voltage response. The results demonstrate the flexibility of the proposed metamaterial and its potential for both vibration control and energy harvesting, specifically compared to a previously studied system with resistance-only shunt. 

Within the field of elastic metamaterials, topological metamaterials have recently received much attention due to their ability to host topologically robust edge states. Introducing local resonators to these metamaterials also opens the door for many applications such as energy harvesting and reconfigurable metamaterials. However, the interactions between phenomena from local resonance and modulation patterning are currently unknown. This work fills that gap by studying multiple cases of spatially modulated metamaterials with local resonators to reveal the mechanisms behind bandgap formation. Their dispersion relations are determined analytically for infinite chains and validated numerically using eigenvalue analysis. The inverse method is used to determine the imaginary wavenumber components from which each bandgap is characterized by its formation mechanism. The topological nature of the bandgaps is also explored through calculating the Chern number and integrated density of states. The band structures are obtained for various sources of modulation as well as multiple resonator parameters to illustrate how both local resonance and modulation patterning interact together to influence the band structure. Other unique features of these metamaterials are further demonstrated through the mode shapes obtained using the eigenvectors. The results reveal a complex band structure that is highly tunable, and the observations given here can be used to guide designers in choosing resonator parameters and patterning to fit a variety of applications.