Global Analysis of Impacting Systems Petri T Piiroinen¹, Joanna Mason², Neil Humphries¹ ¹ National University of Ireland, Galway - Ireland ²University of Limerick - Ireland

A model of gear model with backlash [Mason et al., JSV, 2007] + Impact Law

Dynamics

Domains of attraction for a gear model with backlash Increasing eccentricity

Domains of attraction for a gear model with backlash [Mason et al., 2008] Grazing

Domains of attraction for a gear model with backlash [Mason et al., 2008]

A B x C D Periodic orbits and manifolds The manifolds are calculated using DSTool. [Back et al. 1992, England et al. 2004]

A B x C D Periodic orbits and chaos B D C A C1 C2

Bifurcations B D C A Grazing Boundary Crisis G1 S2 G1 G2 S2 S1 PD1

Domains of attraction A B C D P2 A B C D After boundary crisis Before boundary crisis

Manifolds A B C D P2 A

Bifurcations A B x C C1 C2 B D C A C1 C2

Boundary crisis

Bifurcations A

[Budd & Dux, 1994] F(t,ω) An impact oscillator

Discontinuity surfaces and manifolds

F(t,ω) Solution Impact surface

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Grazing

Summary & Outlook Global versus local analysis : –Tangencies Grazing bifurcations Boundary crisis –Discontinuous geometry Tells us where grazing bifurcations will happen Tells us how loops in the dynamics are formed

Summary & Outlook There are still many mathematical/numerical problems to tackle in this area: –Manifolds –Discontinuous boundaries –How can the curvature of the impact surface be used for the understanding of smooth bifurcations, manifolds, chattering. –Extend this topological viewpoint to a wider class of Nonsmooth systems, –…and many more.

Thank you!