There is much current research interest in nonlinear structures, smart materials, and metamaterials, that incorporate bistable, or snap-through, structural elements. Various applications include energy harvesting, energy dissipation, vibration absorption, vibration isolation, targeted energy transfer, bandgap design and metamaterials. In this dissertation, we explore snap-through structures with nonlinearity and negative linear stiffness. We start with a study of a simple Duffing oscillator with snap-through orbits around the separatrix. Multi-degree-of-freedom snap-through structures are known to convert the low-frequency inputs into high-frequency oscillations, and are called twinkling oscillators. A generalized two-degree-of-freedom (2-DOF) snap-through oscillator is shown to have rich bifurcation structure. The steady-state bifurcation analysis uncovered two unique bifurcations "star" and "eclipse" bifurcations, named due to their structures. The 2-DOF twinkler exhibits transient chaos in the snap-through regime. A fractal basin boundary study provides insight into the regions in the parameter space where the total energy level is predictable in an unsymmetric twinkler. Due to its capacity to convert low frequency to high-frequency oscillations, the snap-through oscillators can be used to harvest energy from low-frequency vibration sources. This idea has led us to explore the energy harvesting capacity of twinkling oscillators. Using magnets and linear springs we built (in collaboration with researchers at Duke university) novel experimental twinkling oscillators (SDOF and 2-DOF) for energy harvesting. When the magnets exhibit high-frequency oscillations through the inducting coil, a current is generated in the coil. This experiment shows promising results both for the SDOF and the 2-DOF twinkling energy generators by validating the frequency up-conversion and generating power from the low-frequency input oscillations. The experimental twinkling oscillator converted a 0.1 Hz input oscillation into 2.5 Hz output oscillation, a 25 times frequency up-conversion. The second part of this dissertation focuses on the dispersive nature of the waves in one dimensional nonlinear chains with weak nonlinearity. For metamaterial design, it is important to study the wave dispersion properties in the material for channeling energy in a desired direction or to build frequency-selective materials. In nonlinear structures there are various design parameters that can be tuned to produce desirable properties. The motivation of the wave propagation analysis is to understand the quadratic and cubic nonlinearity effects on the wave propagation behavior in an uniform periodic chain. Here the dispersion properties are studied through a multiple-scales perturbation approach for weakly nonlinear periodic media. Wave speed, cut-off frequencies, and wave-wave interaction characteristics are presented. The results show significant effect of quadratic nonlinearities in the dispersion characteristics of the waves in the chain.
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