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Global Analysis of Impacting Systems

Global Analysis of Impacting Systems. Petri T Piiroinen ¹, Joanna Mason ², Neil Humphries¹ ¹ National University of Ireland, Galway - Ireland ² University of Limerick - Ireland Petri.Piiroinen@nuigalway.ie. + Impact Law. A model of gear model with backlash. [Mason et al. , JSV, 2007].

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Global Analysis of Impacting Systems

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

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

  3. Dynamics

  4. Domains of attraction for a gear model with backlash Increasing eccentricity Increasing eccentricity

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

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

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

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

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

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

  11. Manifolds P2 P2 A A C B D

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

  13. Boundary crisis

  14. A Bifurcations

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

  16. An impact oscillator

  17. An impact oscillator

  18. An impact oscillator

  19. Discontinuity surfaces and manifolds

  20. F(t,ω) Impact surface Solution Impact surface

  21. Impact surface t

  22. t Impact surface

  23. t Impact surface

  24. t Impact surface

  25. Grazing

  26. Grazing

  27. Grazing

  28. Grazing

  29. Grazing

  30. 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

  31. 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.

  32. Thank you!

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