Rock, Rattle and Slide bifurcation theory for piecewise-smooth systems Alan Champneys Department of Engineering Mathematics, University of Bristol Mario di Bernardo, Chris Budd, Piotr Kowalczyk Arne Nordmark Harry Dankowicz, Gabor Licsko, Csaba Bazso ... Milan 4/6/09 – p. 1
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Rock, Rattle and Slidebifurcation theory for piecewise-smooth systems
Alan Champneys
Department of Engineering Mathematics, University of Bristol
Mario di Bernardo, Chris Budd, Piotr Kowalczyk
Arne Nordmark Harry Dankowicz, Gabor Licsko, Csaba Bazso . . .
Milan 4/6/09 – p. 1
Contents
1: Nonsmoothness and discontinuity-inducedbifurcation
Milan 4/6/09 – p. 2
Contents
1: Nonsmoothness and discontinuity-inducedbifurcation
2: grazing bifurcation in impacting systemsEx. i. a rattling heating valve
Milan 4/6/09 – p. 2
Contents
1: Nonsmoothness and discontinuity-inducedbifurcation
2: grazing bifurcation in impacting systemsEx. i. a rattling heating valve
3: grazing & corner bifurcation in PWS systemsEx. ii. extended model for stick-slip
Milan 4/6/09 – p. 2
Contents
1: Nonsmoothness and discontinuity-inducedbifurcation
2: grazing bifurcation in impacting systemsEx. i. a rattling heating valve
3: grazing & corner bifurcation in PWS systemsEx. ii. extended model for stick-slip
4: sliding bifurcation in Filippov systemsEx. iii. relay controller
Milan 4/6/09 – p. 2
Contents
1: Nonsmoothness and discontinuity-inducedbifurcation
2: grazing bifurcation in impacting systemsEx. i. a rattling heating valve
3: grazing & corner bifurcation in PWS systemsEx. ii. extended model for stick-slip
4: sliding bifurcation in Filippov systemsEx. iii. relay controller
5: nonsmooth impact laws with frictionEx. iv. Painlevé paradox of falling rod
Milan 4/6/09 – p. 2
Contents
1: Nonsmoothness and discontinuity-inducedbifurcation
2: grazing bifurcation in impacting systemsEx. i. a rattling heating valve
3: grazing & corner bifurcation in PWS systemsEx. ii. extended model for stick-slip
4: sliding bifurcation in Filippov systemsEx. iii. relay controller
5: nonsmooth impact laws with frictionEx. iv. Painlevé paradox of falling rod
6: Conclusion
Milan 4/6/09 – p. 2
smooth bifurcation theory
x = f(x, µ), x ∈ D ⊂ Rn, µ ∈ R
p, f smooth
Generates semiflow Φµ(x, t) andphase portrait = set of all trajectories {Φ(x, ·), ∀ x ∈ D}.
Milan 4/6/09 – p. 3
smooth bifurcation theory
x = f(x, µ), x ∈ D ⊂ Rn, µ ∈ R
p, f smooth
Generates semiflow Φµ(x, t) andphase portrait = set of all trajectories {Φ(x, ·), ∀ x ∈ D}.
Two notions of bifurcation:
Milan 4/6/09 – p. 3
smooth bifurcation theory
x = f(x, µ), x ∈ D ⊂ Rn, µ ∈ R
p, f smooth
Generates semiflow Φµ(x, t) andphase portrait = set of all trajectories {Φ(x, ·), ∀ x ∈ D}.
Two notions of bifurcation:
Analytic Branch of invariant sets Γ(µ). Bifurcation is aµ-value where Implicit Function Theorem (IFT) fails.⇒ Branching (Lyapunov-Schmidt reduction)
Milan 4/6/09 – p. 3
smooth bifurcation theory
x = f(x, µ), x ∈ D ⊂ Rn, µ ∈ R
p, f smooth
Generates semiflow Φµ(x, t) andphase portrait = set of all trajectories {Φ(x, ·), ∀ x ∈ D}.
Two notions of bifurcation:
Analytic Branch of invariant sets Γ(µ). Bifurcation is aµ-value where Implicit Function Theorem (IFT) fails.⇒ Branching (Lyapunov-Schmidt reduction)
Topological Bifurcation is a µ-value where there isnon-structurally stable phase portrait.⇒ local bifurcations Hopf, fold, flip, torus,. . .⇒ global bifurcations homoclinic, tangency, crisis . . .Classification by co-dimension
Milan 4/6/09 – p. 3
smooth bifurcation theory
x = f(x, µ), x ∈ D ⊂ Rn, µ ∈ R
p, f smooth
Generates semiflow Φµ(x, t) andphase portrait = set of all trajectories {Φ(x, ·), ∀ x ∈ D}.
Two notions of bifurcation:
Analytic Branch of invariant sets Γ(µ). Bifurcation is aµ-value where Implicit Function Theorem (IFT) fails.⇒ Branching (Lyapunov-Schmidt reduction)
Topological Bifurcation is a µ-value where there isnon-structurally stable phase portrait.⇒ local bifurcations Hopf, fold, flip, torus,. . .⇒ global bifurcations homoclinic, tangency, crisis . . .Classification by co-dimension
impulsive positive slip : u > 0. Full friction λT = −µλN .
impulsive negative slip : u < 0. Full friction λT = µλN .
impulsive stick : u = 0, |λT | < µλN . Only possible if |B| < µA.
⇒ For all modes: u′ = kuλN , v′ = kvλN where
(ku, kv) = (k+u , k+
v ) = (B − µA, C − µB) for pos. slip
(ku, kv) = (k−
u , k−
v ) = (B + µA, C + µB) for neg. slip
(ku, kv) = (k0u, k0
v) = (0,AC − B2
A) for stick
Milan 4/6/09 – p. 35
when is the impact finished?
3 possibilities :
1. Newtonian coefficient of restitution Relate post-impactvelocities to pre-impact: v1 = −rv0
2. Poisson coefficient of restitution (Glocker) Relate normalimpulses during compression and restitution:Ir = −rIc
3. Energetic coefficient of restitution (Stronge) Relatenormal-force work during compression and restitution:Wr = −r2Wc
If impact phase has a single mode ⇒ all 3 agree.But (Stewart) 1 & 2 may increase kinetic energy for r < 1.Hence we use 3 & derive explicit formulae (cf. Stronge)
Milan 4/6/09 – p. 36
impulsive motion follows straight lines
u
v
12
810
k+u > 0
u
v
2
4
5
6
k+v < 0
1
2
79
k¡
u < 0
u
v
1
35
6
k¡
v < 0
u
v
12
3456
<
>
>
k+u
k+v
k¡
u
k¡
v >
0
0
0
0
(a)
(b) (c)
(d) (e)v
Milan 4/6/09 – p. 37
discontinuity-induced bifurcation
dynamics cross region boundary as parameters vary
⇒ hybrid flow map can be C1 (no bifurcation) or C0
(jump in multipliers)
e.g. loss of period-one impacting periodic orbit
0:01
0:005
0
!
0:85 0:87 0:89¹ 0:85 0:87 0:89¹0
1
2
j¸ij
0:17
0:16
0:150:52 0:54 0:56 0:58 ¹
!
0:52 0:54 0:56 0:58 ¹
0
0:2
0:4
0:6
j¸ij
(a) (b)
(c) (d)
rod example with Van-der-pol type forcing: Sx = −k1(x − udrt) − c1(u − udr)
Sy = −k2(y − y0) − c2(y − y0)2 − y21)v R = −k3(θ − θ0) − c3θ
Milan 4/6/09 – p. 38
ambiguities during sustained motion
To to simulate as a hybrid system, need to resolve:
A. Painlev e paradox for slip If y = 0, v = 0, b > 0 andC − µB < 0, u > 0 (or C + µB < 0, u < 0), then motioncould continue with
Sustained free flightSustained positive (negative) slipAn impact with zero initial normal velocity
B. Painlev e paradox for stick If y = 0, v = 0, u = 0, b > 0,|bB − aC| < µ(aB − bA) and C − µB < 0, (orC + µB < 0), then motion could continue with
Sustained free flightSustained stick
Milan 4/6/09 – p. 39
show consistency via smoothing
Introduce constitutive relation λN (y, v) that is “stiff”,“restoring”, and “dissipative”.
Case A slip (WLOG positive slip),
y = v, v = b + (C − µB)λN (y, v).
b > 0, C − µB < 0 ⇒ large negative stiffness,⇒ slipping will never occur, must immediately lift off(y > 0) or take impact (y < 0)
Case B stick v = (bA−aB)+(AC−B2)λN (y,v)A
⇒ always large positive “stiffness” hence verticalmotion is asymptotically stable (evenifb > 0)
Milan 4/6/09 – p. 40
ambiguities at mode transitions
Sustained motion is consistent BUT what about transitions
Case a. approach to the Painlevé boundary(C − µB = 0) during (positive) slip.
previous analysis shows: can’t actually reachC − µB = 0, so what happens instead?
Case b. transitions into stick or chatter
Def: chattering (also known as zeno-ness) isaccumulation of impacts. No contradiction if accumulatein forwards time. But can get reverse chatter.
Milan 4/6/09 – p. 41
a. unfolding C − µB → 0 while slipping
cf. Genôt & Brogliato
Re-scale time t = (C − µB)s ⇒
d
ds
(
C − µB
b
)
=
(
α1 0
α2 α3
)(
C − µB
b
)
Eigenvector (0, 1)T ⇒ trajectory tend to C − µB = 0,only if b = 0
Milan 4/6/09 – p. 42
approaching the singular point
C ¡ ¹B
b
C ¡ ¹B
b
C ¡ ¹B
b
C ¡ ¹B
b
C ¡ ¹B
b
C ¡ ¹B
b
C ¡ ¹B
b
C ¡ ¹B
b
C ¡ ¹B
b
C ¡ ¹B
b
C ¡ ¹B
b
C ¡ ¹B
b
®2 < 0
®3 > ®1 > 0
®1 > ®3 > 0
®2 > 0 ®2 > 0
®1 > 0 > ®3
®3 > 0 > ®1
®2 < 0
0 > ®1 > ®3
®2 > 0
0 > ®3 > ®1
®2 < 0
®2 < 0
®1 > 0 > ®3
®2 < 0
0 > ®1 > ®3
®2 < 0
®1 > ®3 > 0
®2 > 0
®3 > ®1 > 0
®2 > 0
®3 > 0 > ®1
®2 > 0
0 > ®3 > ®1
Milan 4/6/09 – p. 43
what happens after singular point?
could lift off, or take a (zero-velocity) impact.
e.g. simulate example for stiff, compliant contact force