Non-periodic motion caused by grazing incidence in an impact oscillator

Abstract

The motion of a single-degree-of-freedom, periodically forced oscillator subjected to a rigid amplitude constraint is considered. Using analytical methods, the singularities caused by grazing impact are studied. It is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter, a special type of bifurcation occurs. The motion after the bifurcation may be non-periodic, and a criterion for this based on orientation and eigenvalues is given.

Corner bifurcations in non-smoothly forced impact oscillators

“…Such complex dynamics can arise either through smooth bifurcations, in particular period-doubling bifurcations, or through grazing bifurcations which arise when u has a grazing impact with z with du/dt = dv/dt. Grazing leads to large local stretching of phase space and is described in detail in [9,26,27]. Fig.…”

“…In contrast, the transition as we cross Σ 2 is a grazing event [12,28,29,17], involving a grazing impact between z and u close to t 0 . A more refined analysis (see for example [26]) of the grazing event shows that this has a square-root form. The discontinuous map is an outer approximation to this.…”

“…The obstacle itself may be considered to be moving smoothly, and an important example of an impact oscillator, described in [19], is that of a particle moving under gravity and impacting with a sinusoidally moving table. There is now a comprehensive theory describing the behaviour of these systems, in which we see both periodic and chaotic motions arising, and changing at both smooth bifurcations (saddle-node, period-doubling etc) and non-smooth events such as grazing bifurcations [26,27] and chattering sequences [8].…”

Multi-modal analysis on the intermittent contact dynamics of atomic force microscope

“…X max is the tip amplitude at the steady-state. For the grazing impact, the Poincaré mapping can be analytically derived [3,23]. Therefore, the bifurcation and the transition to chaos can be clearly shown [3].…”

“…van de Water and Molenaar [3] demonstrated that for the grazing impact, the tapping-mode AFM displays the characteristic features of an impact oscillator. However, the grazing impact is a particular impact case with zero impact velocity [22,23]. The AFM studied by van de Water and Molenaar interacts with the rigid surface through a liquid bridge, which offers a mechanism to realize the grazing impact [3].…”