Post-grazing dynamics of a vibro-impacting energy generator

The motion of a forced vibro-impacting inclined energy harvester is investigated in parameter regimes with asymmetry in the number of impacts on the bottom and top of the device. This motion occurs beyond a grazing bifurcation, at which alternating top and bottom impacts are supplemented by a zero velocity impact with the bottom of the device. For periodic forcing, we obtain semi-analytical expressions for the asymmetric periodic motion with a ratio of 2:1 for the impacts on the device bottom and top, respectively. These expressions are derived via a set of nonlinear maps between different pairs of impacts, combined with impact conditions that provide jump discontinuities in the velocity. Bifurcation diagrams for the analytical solutions are complemented by a linear stability analysis around the 2:1 asymmetric periodic solutions, and are validated numerically. For smaller incline angles, a second grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For larger incline angles, period doubling bifurcations precede this bifurcation. The converted electrical energy per impact is reduced for the asymmetric motions, and therefore less desirable under this metric.Comment: 24 pages, 9 figures, Submitted to Journal of Sound and Vibratio

Fundamental competition of smooth and non-smooth bifurcations and their ghosts in vibro-impact pairs

A combined analysis of smooth and non-smooth bifurcations captures the interplay of different qualitative transitions in a canonical model of an impact pair, a forced capsule in which a ball moves freely between impacts on either end of the capsule. The analysis, generic for the impact pair context, is also relevant for applications. It is applied to a model of an inclined vibro-impact energy harvester device, where the energy is generated via impacts of the ball with a dielectric polymer on the capsule ends. While sequences of bifurcations have been studied extensively in single- degree-of-freedom impacting models, there are limited results for two-degree-of-freedom impacting systems such as the impact pair. Using an analytical characterization of impacting solutions and their stability based on the maps between impacts, we obtain sequences of period doubling and fold bifurcations together with grazing bifurcations, a particular focus here. Grazing occurs when a sequence of impacts on either end of the capsule are augmented by a zero-velocity impact, a transition that is fundamentally different from the smooth bifurcations that are instead characterized by eigenvalues of the local behavior. The combined analyses allow identification of bifurcations also on unstable or unphysical solutions branches, which we term ghost bifurcations. While these ghost bifurcations are not observed experimentally or via simple numerical integration of the model, nevertheless they can influence the birth or death of complex behaviors and additional grazing transitions, as confirmed by comparisons with the numerical results. The competition between the different bifurcations and their ghosts influences the parameter ranges for favorable energy output; thus, the analyses of bifurcation sequences yield important design information.</p

Metastability for discontinuous dynamical systems under Lévy noise: Case study on Amazonian Vegetation

Abstract For the tipping elements in the Earth’s climate system, the most important issue to address is how stable is the desirable state against random perturbations. Extreme biotic and climatic events pose severe hazards to tropical rainforests. Their local effects are extremely stochastic and difficult to measure. Moreover, the direction and intensity of the response of forest trees to such perturbations are unknown, especially given the lack of efficient dynamical vegetation models to evaluate forest tree cover changes over time. In this study, we consider randomness in the mathematical modelling of forest trees by incorporating uncertainty through a stochastic differential equation. According to field-based evidence, the interactions between fires and droughts are a more direct mechanism that may describe sudden forest degradation in the south-eastern Amazon. In modeling the Amazonian vegetation system, we include symmetric α-stable Lévy perturbations. We report results of stability analysis of the metastable fertile forest state. We conclude that even a very slight threat to the forest state stability represents L´evy noise with large jumps of low intensity, that can be interpreted as a fire occurring in a non-drought year. During years of severe drought, high-intensity fires significantly accelerate the transition between a forest and savanna state

Improving the performance of a two-sided vibro-impact energy harvester with asymmetric restitution coefficients

We study the influence of asymmetric restitution coefficients in a model of a two-sided vibro-impact energy harvester (VI-EH), considering the dynamical behavior and the implications for energy output. In the VI-EH, a ball moves freely within a forced cylinder and collides with a compliant dielectric polymer on either end, thus converting the motion into output voltage. We develop (semi-)analytical results for 1:1 periodic solutions, with alternating impacts on either end, focusing on the case of asymmetric restitution coefficients on the top and bottom of the cylinder. New types of 1:1 periodic solutions are found, with energy output clearly different from the symmetric setting. The analysis covers non-intuitive results, including the non-monotonic dependencies of the energy output on the asymmetric restitution coefficients. We find unexpected parameter ranges with improved levels of energy output, as well as stability results indicating that this output is robust to parameter fluctuations or external perturbations. Furthermore, by identifying parameter combinations that limit performance through asymmetries, we show how asymmetric restitution coefficients can counteract these detrimental effects. The analysis is based on maps for the dynamics between impacts, leading to a series of conditions for stable 1:1 periodic solutions in terms of the system parameters. We compare stability and bifurcation structure obtained analytically and numerically. The analysis shows possible regions of bi-stability between different behaviors that may not be captured by numerical approaches.</p