Global Analysis of Impacting SystemsPetri T Piiroinen¹, Joanna Mason²,

Neil Humphries¹¹National University of Ireland, Galway - Ireland

²University of Limerick - IrelandPetri.Piiroinen@nuigalway.ie

A model of gear model with backlash

[Mason et al., JSV, 2007]

2''( ) '( ) 4 4 cos 2 2 sin 2t t t t

+ Impact Law

Dynamics

1( )ax ( )IIIR x1( )ax

( )IR x

2 ( )IIx( )IR x

1( )ax1( )ax

2 ( )Ix

( )IIR x

1( )ax 1( )R x1

2 1

2 1

( ( ))

( ) ( ( ( ( )))) (impact map)

( ( ( )))

a

b a a

a

R x

x x R R x

R x

31 2 0 mod 1x x x t t R

Domains of attraction for a gear model with backlash

Incr

easi

ng e

ccen

tric

ity

Increasing eccentricity

Domains of attraction for a gear model with backlash

[Mason et al., 2008]

Grazing

2x

Domains of attraction for a gear model with backlash

[Mason et al., 2008]

2x

A

B

xC

D

2x

Periodic orbits and manifolds

The manifolds are calculated using DSTool.[Back et al. 1992, England et al. 2004]

A

Bx C

D

Periodic orbits and chaos

2x

B

D

C

A

C1 C2

2''( ) '( ) 4 4 cos 2 2 sin 2t t t t

Bifurcations

B

D

C

A

GrazingBoundary Crisis

2''( ) '( ) 4 4 cos 2 2 sin 2t t t t

G1S2

G1G2

S2S1

S1

PD1

PD1

Domains of attraction

A

B

C

D

P2P2

A

B

C

D

P2P2

2x

2x

After boundary crisisBefore boundary crisis

2''( ) '( ) 4 4 cos 2 2 sin 2t t t t

Manifolds

A

B

C

D

P2P2

2x

2x

A

2''( ) '( ) 4 4 cos 2 2 sin 2t t t t

( )uW A

( )sW A

Bifurcations

A

B

x C

C1

C2 B

D

C

A

C1 C2

Boundary crisis

Bifurcations

A

)sin( txxx

)()( txrtx

[Budd & Dux, 1994]

F(t,ω)

xx

An impact oscillator

)sin( txxx

)()( txrtx

An impact oscillator

An impact oscillator

An impact oscillator

Discontinuity surfaces and manifolds

cos( )x x t

)()( txrtx

( , ; ) cos( ) sin( ) cos( ( ))cx v t A t B t t

2

1cos( ), sin( ),

1A c t B v t

x c

x cF(t,ω)

xx

( , ; )| ( , ; )c cV v t x v t c 3

Solution

Impact surface

Impact surface

Impact surface

cV

Impact surface

Impact surface

Impact surface

Grazing

Grazing

Grazing

Grazing

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!