  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 the 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 and frequency response curves
are obtained analytically. Two underlying dynamical mechanisms that govern the
occurrence of bifurcations are identified: the "maximum" and the "saddle"
mechanisms. The latter, a potentially more dangerous scenario, involves abrupt
transitions of the system's energy response from a relatively small value to
the threshold energy level. 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.
