Dynamics of a Two Degree of Freedom Vibro-Impact System With Multiple Motion Limiting Constraints

Abstract

We consider the dynamics of impact oscillators with multiple degrees of freedom subject to more than one motion limiting constraint or stop. A mathematical formulation for modelling such systems is developed using a modal approach including a modal form of the coefficient of restitution rule. The possible impact configurations for an N degree of freedom system are considered, along with definitions of the impact map for multiply constrained systems.We consider sticking motions which occur when a single mass in the system becomes stuck to an impact stop, and discuss the computational issues related to computing such solutions.Then using the example of a two degree of freedom system with two constraints we describe exact modal solutions for the free flight and sticking motions which occur in this system. Numerical examples of sticking orbits for this system are shown and we discuss identifying the region, S in phase space where these orbits exist. We use bifurcation diagrams to indicate differing regimes of vibro-impacting motion for two different cases; firstly when the stops are both equal and on the same side (i.e. the same sign) and secondly when the stops are unequal and of opposing sign. For these two different constraint configurations we observe

Nonlinear dynamics and triboelectric energy harvesting from a three-degree-of-freedom vibro-impact oscillator

“…Chatter is characterized by many consecutive impacts and might therefore be exploited in triboelectric energy harvesting from vibro-impact systems. In the study of a multi-degree-of-freedom vibroimpact system, Wagg [56,61] discussed the chattering and sticking behaviour which appeared at low forcing frequencies, and showed that a variety of non-smooth events could possibly occur. Here, a numerical simulation is conducted to determine whether chatter can appear in the present system at low forcing frequencies.…”

Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator

“…Equation (5) has the well known exact solution for under-damped oscillations 0 < ζ j < 1 [21], so that we can solve for each mode exactly, and hence find the total displacements x 1 , x 2 -see [13,22] for further details. Without loss of generality, this formulation can be considered in either a dimensional or a nondimensional form [22] -in the following work we assume all parameters and variables are nondimensional.…”

“…Without loss of generality, this formulation can be considered in either a dimensional or a nondimensional form [22] -in the following work we assume all parameters and variables are nondimensional.…”

“…There is also a dual sticking regime when both x 1 = s 1 and x 2 = s 2 simultaneously, but this will not be considered here. Each regime has a reduced set of governing equations with explicit solutions [22].…”

“…For sticking to exist the condition F i s i > 0 must apply, which in this case is the region on the positive side of the boundary ∂S [22].…”