Recently, researchers have incorporated topological phases into mechanical metamaterials to facilitate defect-immune elastic wave and vibration manipulation. The topological mechanical metamaterials developed thus far have achieved extraordinary wave control capabilities through the construction of robust elastic waveguides at the boundaries and interfaces of 1D, 2D, and 3D periodic mechanical lattices. Given the overwhelming focus of previous research on traditional integer-dimensional mechanical architectures, an unexplored opportunity exists to investigate the emergence of topological phases in fractal mechanical metamaterials, which have a non-integer dimension and exhibit self-similarity across multiple scales. This research addresses the unexplored opportunity and advances the state of the art through the synthesis of a 1.89D fractal mechanical metamaterial that harnesses higher-order topological phases to enable multifaceted elastic wave and vibration control. The proposed fractal topological mechanical metamaterial is a thin plate with embedded torsional spring-mass resonators that are arranged into the pattern of a 1.89D Sierpiński carpet. A numerical eigenfrequency analysis uncovers coexisting topological corner and edge states that trap wave energy at the myriad corner and edge interfaces available in the 1.89D fractal. The outcomes from this study provide insight into the attainment of higher-order topological states in fractal metamaterials that localize elastic waves and vibrations across various locations and frequencies, opening the door for future research of topological phases in mechanical metamaterials with fractal architectures.
Unlike conventional elastic waveguides, topologically protected wave transmission in topological metamaterials is immune to backscattering and localization from lattice imperfections and sharp corners. Topologically protected waveguides can be formed by breaking space inversion symmetry within the unit cell of a hexagonal lattice, creating an elastic realization of the quantum valley Hall effect. Recent studies have demonstrated the achievement of tunable topological edge states through the application of an external bias, such as a mechanical, thermal, or magnetic load. These initial studies demonstrate the capability to modify topological edge states through oftentimes complex realizations of truss-like lattice structures or external stimuli. However, a comprehensive reconfigurable topological metamaterial that enables real-time adaptation of both frequency and spatial characteristics of topological properties in an easily integrable manner has yet to be developed. Thus, to advance the state of the art, this research introduces an electromechanical metamaterial with the capability to adjust the frequency range for topological edge states and instantaneously create or eliminate topological interfaces through the integration of piezoelectric circuitry with a continuous mechanical substrate. The metamaterial is comprised of inductor circuitry connected to a thin piezoelectric plate in a periodic manner which produces a hexagonal lattice pattern of electromechanical resonators. The plane wave expansion method is used to reveal a tunable Dirac cone in the band structure of the lattice unit cell and indicate how perturbations to the circuit inductance can open topologically distinct bandgaps. Numerical simulations identify edge modes located at frequencies within the topological bandgap and demonstrate adaptive topologically protected elastic wave transmission.
Bistable vibration energy harvesters have been used to achieve strong energy harvesting performance over a wide frequency bandwidth. Performance of bistable energy harvesters is dependent on whether the external excitation is large enough to surpass the minimum threshold to high energy, or ‘snap through’ oscillations. Studies have indicated that lowering the potential energy barrier via an auxiliary unit is an effective way to ensure that high energy orbits are achieved. Recent advancements have shown that directly extracting energy from an auxiliary unit used to dynamically lower the potential barrier of a bistable energy harvester can enhance performance. However, there remains an unexplored opportunity for further improvement by incorporating nonlinearity into the auxiliary harvesting element. Thus, to advance the state of the art, this research introduces an energy harvesting system composed of a bistable cantilever harvester magnetically coupled to an auxiliary nonlinear harvesting element. An analysis of the system potential energy indicates that the additional nonlinear characteristics of the coupled harvesting element can enable tailoring of the potential energy profile such that quad-stability, or multi-directional bistability, can be achieved. Investigation of the quasi-static potential energy trajectory of the proposed device indicates that the number of stable states, height of the potential energy barrier, and snap through amplitude may all be tailored through consideration of the effective linear stiffness of the nonlinear harvesting unit. Numerical simulations of the system dynamics indicate that the additional nonlinearity incorporated into the coupled system improves broadband harvesting performance.
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