The Onset of Chaos in a Two-Degree-of-Freedom Impacting System

2006

Nonlinear Dyn

Abstract.The behaviour of a multi-degree-of-freedom vibro-impact system is studied using a two degree-of-freedom impact oscillator as a motivating example. A multi-modal model is used to simulate the behaviour of the system, and examine the complex dynamics which occurs when both degrees of freedom are subjected to a motion limiting constraint. In particular the chattering and sticking behaviour which occurs for low forcing frequencies is discussed. In this region a variety of nonsmooth events can occur, including newly studied phenomena such as sliding bifurcations. In this paper the multiple nonsmooth events which can occur in the two degree-of-freedom system are categorised, and demonstrated using numerical simulations.

“…Several authors have considered two constraints placed an equal distance either side of an oscillating mass, e.g. [2][3][4].…”

Section: Introductionmentioning

confidence: 99%

Multiple Non-Smooth Events in Multi-Degree-of-Freedom Vibro-Impact Systems

2006

Nonlinear Dyn

“…This said, some authors have considered two constraints placed an equal distance either side of an oscillating mass, e.g. Shaw & Shaw [1989], Hogan & Homer [1999]. In this study we will consider the more general case of variable constraints applied to a number of the masses in the system.…”

Section: Introductionmentioning

confidence: 99%

“…For example, such systems have been studied in relation to impact damper systems Masri [1972], Chatterjee et al [1995], with respect to bifurcations and the onset of chaotic motion [Shaw & Shaw 1989], and the dynamics of rotor bearings [Neilson & Gonsalves 1993].…”

Section: Introductionmentioning

confidence: 99%

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

Int. J. Bifurcation Chaos

Bishop

2004

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

“…Two-degree of freedom impact oscillators have been studied in relation to impact damper systems (Masri 1972;Chatterjee et al 1995), which are essentially a two degree of freedom system with a primary mass and an additional impacting mass. The two-degree of freedom impact oscillator has also been considered by Shaw & Shaw (1989), who studied bifurcations, and the onset of chaotic motion in such systems, while Neilson & Gonsalves (1993) considered the dynamics of rotor bearings using a two degree of freedom model.…”

Section: Introductionmentioning

confidence: 99%

Chatter, Sticking and Chaotic Impacting Motion in a Two-Degree of Freedom Impact Oscillator

Int. J. Bifurcation Chaos

Bishop

2001

We consider the dynamics of a two degree of freedom impact oscillator subject to a motion limiting constraint. These systems exhibit a range of periodic and non-periodic impact motions. For a particular set of parameters, we consider the bifurcations which occur between differing regimes of impacting motion and in particular those which occur due to a grazing bifurcation. Unexpected resonant behaviour is also observed, due to the complexity of the dynamics. We consider both periodic and chaotic chatter motions and the regions of sticking which exist. Finally we consider the types of chaotic motion that occur within the parameter range. We discuss the possibility in relating successive low velocity impacts, especially in respect to possible low dimensional mappings for such a system.