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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

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

slide2

+ Impact Law

A model of gear model with backlash

[Mason et al., JSV, 2007]

slide4

Domains of attraction for a gear model with backlash

Increasing eccentricity

Increasing eccentricity

slide5

Grazing

Domains of attraction for a gear model with backlash

[Mason et al., 2008]

slide7

A

C

x

B

D

Periodic orbits and manifolds

The manifolds are calculated using DSTool.

[Back et al. 1992, England et al. 2004]

slide8

C2

C1

C

B

A

D

Periodic orbits and chaos

A

C

x

B

D

slide9

Boundary Crisis

Grazing

Bifurcations

S2

C

G1

G1

B

G2

S1

A

D

PD1

S2

PD1

S1

slide10

Domains of attraction

After boundary crisis

Before boundary crisis

P2

P2

P2

P2

A

A

C

C

B

B

D

D

slide11

Manifolds

P2

P2

A

A

C

B

D

slide12

C2

C1

C

B

A

D

Bifurcations

C2

A

C

x

B

C1

slide14

A

Bifurcations

slide15

F(t,ω)

An impact oscillator

[Budd & Dux, 1994]

slide20

F(t,ω)

Impact surface

Solution

Impact surface

slide22

t

Impact surface

slide23

t

Impact surface

slide24

t

Impact surface

summary outlook
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 outlook1
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.