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Abstract In this work we propose the Step Matrix Multiplication based Path Integration method (SMM-PI) for nonlinear vibro-impact oscillator systems. This method allows the efficient and accurate deterministic computation of the time-dependent response probability density function by transforming the corresponding Chapman–Kolmogorov equation to a matrix–vector multiplication using high-order numerical time-stepping and interpolation methods. Additionally, the SMM-PI approach yields the computation of the joint probability distribution for response and impact velocity, as well as the time between impacts and other important characteristics. The method is applied to a nonlinear oscillator with a pair of impact barriers, and to a linear oscillator with a single barrier, providing relevant densities and analysing energy accumulation and absorption properties. We validate the results with the help of stochastic Monte-Carlo simulations and show the superior ability of the introduced formulation to compute accurate response statistics. 

Abstract 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. 

Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic “Bristol book” [di Bernardo et al., Piecewise-smooth Dynamical Systems. Theory and Applications (Springer-Verlag, 2008)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied. 

Vibro-impact phenomena in engineering systems, considered an adverse effect in some settings, are an intrinsic part of the mechanism in others. In energy harvesting, a vibro-impact component is often intentionally introduced to increase the power output or the system’s bandwidth. The impacts can be treated as “hard” for instantaneous impacts or “soft” for compliant materials. Since both types of models exhibit complex dynamics, a comparison is non-trivial. We develop a soft impact model for a vibro-impact energy harvester, calibrating it with the relevant hard impact model for large stiffness, and systematically compare the different phenomena and dynamics in various compliant regimes. Numerical results are used in two different parametric analyses, considering the bifurcation diagrams in terms of device size and external forcing parameters. Varying the natural frequency of the membranes that form the impact boundaries, we observe shifts in the bifurcation structure that promote period-1 orbits for increased softness parameters, often generating higher power output, but also introducing parameter sensitivities for increased softness. Complementary analytical results reveal unstable periodic orbits and co-existing behaviors, potentially missed by computational methods, that can influence the bifurcation structure and in turn the energy output. A non-dimensional formulation highlights the significance of ratios of external and natural frequencies in delineating soft and hard impact scenarios parametrically. The soft impact model exhibits new symmetry breaking bifurcations related to key quantities that characterize the soft impact dynamics, such as the effective restitution coefficients, the impact phase, and the contact time interval, not captured by hard impact models.