Dynamics of a hybrid vibro-impact oscillator: canonical formalism

Abstract

Hybrid vibro-impact (HVI) oscillations is a strongly nonlinear dynamical regime that involves both linear oscillations and collisions under periodic, impulsive, or stochastic excitation. This regime arises in various engineering systems, such as mechanical components under tight rigid constraints, seismic-induced sloshing in partially-filled liquid storage tanks, and more. The adaptive nonlinearity of the HVI oscillator is used by the HVI-nonlinear energy sink as an effective vibration mitigation solution for broad energy and frequency range. Due to the extreme nonlinearity of this regime, traditional analytical methods are inapplicable for the description of its transient dynamics. In the current work, we model the HVI oscillator by a forced particle in a truncated quadratic potential well with infinite depth. The slow flow dynamics of the system in the vicinity of primary resonance is described by canonical transformation to action-angle (AA) variables and the corresponding reduced resonance manifold (RM). Two types of bifurcation are examined. The former is associated with transition between linear oscillations and the HVI-regime and vice versa, and the latter with reaching a chosen maximal transient energy level. The transition boundaries on the forcing parameters plane associated with both bifurcation types are obtained analytically. The maximal transient energy level obtained for any given set of forcing parameters is described analytically as well. The energy jumps associated with the bifurcation of type I and crossing the corresponding transition boundary are obtained. Two underlying dynamical mechanisms that govern the occurrence of bifurcations are identified. They correspond to two distinct scenarios: in the first scenario, the energy of the slow flow gradually reaches the threshold energy level and is thus referred to as the ”maximum” mechanism. The second, potentially more dangerous scenario, involves abrupt transitions of the system’s energy response from a relatively small value to the threshold energy level. This pattern is related to the passage of the slow-flow phase trajectory through the saddle point of the RM, and thus is referred to as the “saddle” mechanism. Both mechanisms are universal for systems that undergo escape from a potential well. All theoretical results are in complete agreement with full-scale numerical simulations.