rock, rattle and slide - politecnico di...

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


2: grazing bifurcation in impacting systemsEx. i. a rattling heating valve

Milan 4/6/09 – p. 2

Contents



3: grazing & corner bifurcation in PWS systemsEx. ii. extended model for stick-slip

Milan 4/6/09 – p. 2

Contents




4: sliding bifurcation in Filippov systemsEx. iii. relay controller

Milan 4/6/09 – p. 2

Contents





5: nonsmooth impact laws with frictionEx. iv. Painlevé paradox of falling rod

Milan 4/6/09 – p. 2

Contents





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


x = f(x, µ), x ∈ D ⊂ Rn, µ ∈ R

p, f smooth


Two notions of bifurcation:

Milan 4/6/09 – p. 3


x = f(x, µ), x ∈ D ⊂ Rn, µ ∈ R

p, f smooth



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


x = f(x, µ), x ∈ D ⊂ Rn, µ ∈ R

p, f smooth




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


x = f(x, µ), x ∈ D ⊂ Rn, µ ∈ R

p, f smooth




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

IFT & struct. stabilty need continuity & smoothness . . .

Milan 4/6/09 – p. 3

three types of nonsmoothness

Impacting systems:

F

x

Milan 4/6/09 – p. 4


Impacting systems:

F

x

Piecewise smooth continuous systems:

Nonlinear Nonsmooth

Milan 4/6/09 – p. 4


Impacting systems:

F

x


Nonlinear Nonsmooth

Y Y YX X X

T

PQ

a

ug

tlz

R

al

ug

b

T

tgz

P

Q

Milan 4/6/09 – p. 4


Impacting systems:

F

x


Nonlinear Nonsmooth

Filippov systems:

F

x

F1

F2

x=0

Milan 4/6/09 – p. 4

a motivating example

Oscillations of a pressure relief valve. Licsko, C. & Hös

noise at ∼ 375Hz at a range of flow speeds

0100

200300

400500

0

10

20

300

1

2

3

4

x 104

n [1/s] f [Hz]

Am

plitu

de

0100

200300

400500

0

5

10

150

0.5

1

1.5

2

x 106

n [1/s] f [Hz]

Am

plitu

de

Milan 4/6/09 – p. 5

a simple (dimensionless) model

y1 = y2

y2 = −κy2 − (y1 + δ) + y3

y3 = β (q −√y3y1)

y1 > 0 valve displacement; y2 valve velocity, y3 pressure

β valve spring stiffness; δ valve pre-stressq, flow rate; κ, fluid damping

at y1 = 0 apply a Newtonian restitution law:

y2(t+∗) = −ry2(t

−

∗)

Low κ ⇒ limit cycles between 2 Hopf bifs q = qmin, qmax.

Milan 4/6/09 – p. 6

brute force numerics

κ = 1.25, β = 20, δ = 10 (representative of experiment)

01

23

45

6 −10

−5

0

5

100

10

20

30

40

50

vel

disp

pres

−20

24

68

10 −10

0

10

20

0

10

20

30

40

50

60

70

80

vel

disp

pres

−20

24

68 −10

0

10

200

10

20

30

40

50

60

vel

disp

pres

−10

12

34

5 −10

−5

0

5

10

0

5

10

15

20

25

30

35

40

vel

disp

pres

−0.5

0

0.5

1 −4

−2

0

2

40

5

10

15

20

veldisp

pres

q

Chaotic rattling due to Grazing events at q ≈ 7.54, 5.95

Milan 4/6/09 – p. 7

more realistic PDE model

Bazso, C. & Hös

1.38 1.4 1.42 1.44 1.46 1.48 1.5

x 106

0

0.5

1

1.5

2

2.5

3

3.5x 10

−3

pv [Pa]

x [m

]

1.38 1.385 1.39 1.395 1.4

x 106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−3

pv [Pa]

x [m

]

Similar results including chattering at low pressureMilan 4/6/09 – p. 8

(in)Formalisms for nonsmooth system

A piecewise smooth (PWS) system is set of ODEs

x = Fi(x, µ), if x ∈ Si,

Milan 4/6/09 – p. 9




discontinuity set Σij := Si ∩ Sj is R(n−1)-dim manifold

⊂ ∂Sj ∪∂Si. Each Fi smooth in Si generates flow Φi(x, t)

x(t)

Si

Sj

Σij

Milan 4/6/09 – p. 9




discontinuity set Σij := Si ∩ Sj is R(n−1)-dim manifold

⊂ ∂Sj ∪∂Si. Each Fi smooth in Si generates flow Φi(x, t)

x(t)

Si

Sj

Σij

Degree of smoothness of x ∈ Σij is order of 1stnon-zero term in Taylor expansion of Φi(x, t) − Φj(x, t)

Milan 4/6/09 – p. 9

impacting systems: deg. 0 need reset map

x 7→ Rij(x, µ), if x ∈ Σij

Milan 4/6/09 – p. 10



PWS continuous systems: deg. ≥ 2

i.e. Fi(x) = Fj(x) but ∃k ≥ 1 s.t. dkFi

dxk 6= dkFj

dxk

Milan 4/6/09 – p. 10



PWS continuous systems: deg. ≥ 2

i.e. Fi(x) = Fj(x) but ∃k ≥ 1 s.t. dkFi

dxk 6= dkFj

dxk

Filippov systems deg. 1. Have possibility of slidingmotion. E.g. if Σij := {H(x) = 0},

(HxF1) · (HxF2) < 0.

(a) (b)

F1

F2

Σ

Milan 4/6/09 – p. 10

bifurcation

All smooth bifurcations can occur in PWS systems(because Poincaré map is typically analytic!)

Milan 4/6/09 – p. 11

bifurcation


Also discontinuity induced bifurcations (DIB) whereinvariant sets have non-structurally stable interactionwith a Σij .

Milan 4/6/09 – p. 11

bifurcation



Can lead to classical (topological) bifurcation or not(a) (b)

S1

S2

S3

S4

S1

S2

S3

S4

Milan 4/6/09 – p. 11

bifurcation



Can lead to classical (topological) bifurcation or not(a) (b)

S1

S2

S3

S4

S1

S2

S3

S4

idea topological DIB ⇐ PW structural stabilityMilan 4/6/09 – p. 11

types of DIB

Boundary equilibrium bifurcations

mlz

b

mezmgz

Milan 4/6/09 – p. 12

types of DIB


Grazing bifurcations of limit cyclesc

mezmlz

mgz

Milan 4/6/09 – p. 12

types of DIB


Grazing bifurcations of limit cycles

Sliding and sticking bifurcations

Milan 4/6/09 – p. 12

types of DIB




−0.10

0.1 12

3

0

0.005

0.01

t

dx/dt

x

Milan 4/6/09 – p. 12

types of DIB




Invariant tori bifurcations

Milan 4/6/09 – p. 12

types of DIB





Σ

Milan 4/6/09 – p. 12

types of DIB





Possible global bifurcations

Milan 4/6/09 – p. 12

types of DIB





Possible global bifurcations

(c)(a) (b)

Milan 4/6/09 – p. 12

this talk: periodic orbit DIBs

Goal: Catalogue & unfold codim-1 possibilities.E.g. ‘grazing bifurcation’: [Nordmark]

Milan 4/6/09 – p. 13



Derive map close to DIB as composition of smoothPoincaré map Pπ and discontinuity mapping PDM

Pπ

Πp(t)

Σ : {H(x) = 0}PDM

Milan 4/6/09 – p. 13




Pπ

Πp(t)

Σ : {H(x) = 0}PDM

Use results on border collisions of maps to classifydynamics [Feigin] [Yorke, Banergee et al]

Milan 4/6/09 – p. 13




Pπ

Πp(t)

Σ : {H(x) = 0}PDM

Use results on border collisions of maps to classifydynamics [Feigin] [Yorke, Banergee et al]

Nb. piecewise linear (PWL) flow 6⇒ PWL mapMilan 4/6/09 – p. 13

2. Grazing bifurcation in impact systems

cf. Theory of impact oscillators: [Peterka] 1970s,[Thompson & Ghaffari], [Shaw & Holmes] 1980s,[Budd et al], [Nordmark] 1990s.

Consider single impact surface Σ := {H(x) = 0}with impact law:

x+ = R(x−) = x− + W (x−)HxF (x−)

W is smooth function and HxF (x−) is ‘velocity’. e.g.

W = −(1 + r)Hx ⇒ Newton’s ‘restitution law’

More complex impact laws are possible, e.g. impactwith friction (see later)

Milan 4/6/09 – p. 14

discontinuity mapping (PDM)

−0.6 0 0.4−0.1

0

0.1

HxF

H

x2 x3

x5

x1

x4

x0

x6

Σ

Π

PDM: x1 7→ x5 maps Poincaré sectionΠ = {HxF (x) = 0} to itself

Computes correction to trajectory as if Σ were absent

Milan 4/6/09 – p. 15

explicit form of PDM

cf. [Fredrickson & Nordmark]

x 7→{

x if H(x) ≥ 0

x + β(x, y)y if H(x) < 0

}

where β = −√

2a

(

W − (HxF )xW

aF

)

+ O(y2),

where y =√−H and

a(x) = d2H/dt2 = (HxF )xF = HxxFF + HxFxF

⇒ square root map

Milan 4/6/09 – p. 16

Proof is by Taylor expansion of flow in (x, y) and IFT

Milan 4/6/09 – p. 17


Use PDM to correct non-grazing Poincaré map Pπ :PN = Pπ ◦ PPDM

PN (x, µ) = M1x + Nµ + O(x2, µ2) if H(x) > 0

= M2x + Nµ + B√

|H(x)| + O(x2, µ2) H(x) < 0

Milan 4/6/09 – p. 17




= M2x + Nµ + B√

|H(x)| + O(x2, µ2) H(x) < 0

Conditions on M1,2, B, C for given periodic orbit to exist

Milan 4/6/09 – p. 17




= M2x + Nµ + B√

|H(x)| + O(x2, µ2) H(x) < 0


Attractor for µ > 0 depends on linearisation of orbit forµ < 0.

Milan 4/6/09 – p. 17




= M2x + Nµ + B√

|H(x)| + O(x2, µ2) H(x) < 0


Attractor for µ > 0 depends on linearisation of orbit forµ < 0.

Simplest case:λ1 real leading eigenvalue of M1 . . ., then dynamics isdetermined by 1D map:

Milan 4/6/09 – p. 17

dynamics of 1D map

f(x) =√

µ − x + λ1µ x < µ, f(x) = λ1x x > µ,

1. 2/3 < |λ1| < 1: robust chaotic attractor size ∼ √µ.

2. If 1/4 < |λ1| < 2/3 alternating series of chaos andperiod-n orbits, n → ∞ as µ → 0.

3. 0 < |λ1| < 1/4: just period-adding cascade

Milan 4/6/09 – p. 18

return to Ex.i: valve rattle

Two grazing bifurcation events q = 7.54, q = 5.95

q = 5.95: λ1 < 0 ⇒ discontinuous jump in attractorq = 7.54: λ1 = 0.8537 ⇒ jump to chaos. Iterate map

7.4 7.45 7.5 7.55 7.6 7.65−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

y[−

]

Milan 4/6/09 – p. 19

3. DIBs in PWS continuous systems

Simplest case: grazing bifurcationAnalyse using discontinuity mapping: PDM

��

��

��

��

��

��

��

��

��

��

��

δ∆

Σx x

fx

0xε

01tst 2t ft

S

S

ε

DiscontinuityMap

0

-

+

ΠΠ 1

2

Π

-

Milan 4/6/09 – p. 20

general results for PDM

Map discontinuity

degree F jump Uniform Case Non-uniform0 δ-function x square-root1 bounded F - O(1/2)

2 C0 Fx O(3/2) O(3/2)-type3 C1 Fxx O(5/2) O(3/2)-type

F (x) continuous at Σ ⇒ no immediate jump in attractor

Milan 4/6/09 – p. 21

1D map with O(3/2) singularity

Consider [Halse, di Bernardo et al]

x 7→{

νx − µ x ≤ 0

νx + ηx3/2 − µ x > 0

0 < ν < 1 ⇒ simple fixed point. No bifurcation at µ = 0.

but with η < 0 get nearby fold at µ = −4(1−ν)3

3η2 (muchcloser than smooth fold if ν ≈ 1)

Milan 4/6/09 – p. 22

Also, get period-adding cascades. E.g. for γ = 3/2, η = −1.Then stable Lk−1R orbits exist for

−8(νk + 1)3 − 12(1 − νk)(1 + νk)2

27ν2(k−1)(1 + ν + ν2 + . . . + νk−1)< µ < −

(

νk − 1

νk−1(ν − 1)

)2

.

case γ = 2:

0 0.2 0.40

50

100

150

200

250

300

350

400

450

500

α

βµ

k=2

k=3

k=4

k=5

k=6 k=7

Milan 4/6/09 – p. 23

Ex.ii: A realistic stick-slip oscillator

Dankowicz 1999

U = 1

y1

y3

y5

y1 - horizontal displacement; y2 = y1

y3 - vertical displacement; y4 = y3

y5 - shear deformation of asperitiesbelt velocity U = 1

Milan 4/6/09 – p. 24

equations of motion

y1 = y2,

y2 = −1 +[

1 − γU |1 − y4|y2 + βU2(1 − y4)2√

K(y1)]

ey1−d,

y3 = y4,

y4 = −sy3 +

√gσ

Ue−d

[

µ(y5e−y1 − 1) + αU2S(y1, y4)

]

,

y5 =1

τ[(1 − y4) − |1 − y4|y5],

where K(y1) = 1 − y1−d∆ ,

S(y1, y4) = (1 − y4)|1 − y4|K(y1)e−y1 − 1 + d

∆ .

⇒ PWS continuous across discontinuity boundary y4 = 1.

Milan 4/6/09 – p. 25

grazing bifurcation analysis

Dankowicz & Nordmark 20003 successive zooms of bifurcation diagram:

Milan 4/6/09 – p. 26

grazing bifurcation analysis

Dankowicz & Nordmark 2000Simulation (left) and iteration of DM (right)

in local map co-ordinates ∼ y4 × 10−4

Milan 4/6/09 – p. 26

4. Sliding DIBs in Filippov systems

Kowalczyk, Nordmark, diBernardoFour possible DIB involving collision of limit cycle withsliding boundary ∂Σ−; see Mike Jeffrey’s talk

(a)

S

S

+

−

B AC

crossing sliding(b)

S+

C

B

A

grazing sliding

(c) S−

S+

A

C

B

switching sliding(d) C

S+

B

A

adding slidingMilan 4/6/09 – p. 27

Unfold with discontinuity mapping Di Bernardo, Kowalczyk,Nordmark

Bifurcation type DM leading-order term Map singularity

crossing sliding ε2 + O(ε3) 2grazing sliding ε + O(ε3/2) 1

switching sliding ε3 + O(ε4) 3adding sliding ε2 + O(ε5/2) 2

Maps are non-invertible on one side

Only grazing sliding ⇒ jump in attractor

Milan 4/6/09 – p. 28

Ex.iv: a relay control system

x = Ax − Bsgn(y), y = CT x,

A =

0

B

B

@

−a1 1 0

−a2 0 1

−a3 0 0

1

C

C

A

, B =

0

B

B

@

b1

b2

b3

1

C

C

A

, CT =

0

B

B

@

1

0

0

1

C

C

A

T

.

Complex dynamics:

−2

0

2

−4

0

4−0.1

0

0.1

x1

x2x3 z

z

z

0p1

m4

p2

m2

p4m0p1

b = (1,−2, 1)T , a31 = −5 a21 = −99.3, and(a) a11 = 1.206, 1.35, periodic; (b) nearby, chaotic

Milan 4/6/09 – p. 29

bifurcation diagram

A grazing-sliding cascade

0.02 0.025 0.03 0.035 0.04 0.045 0.05

2.4

2.5

2.6

SN

GS5GS4

GS3

GS2

GS1

AS

−2−1

01

2

−5

0

5−0.1

0

0.1

x2

x3

x1

x1

x2x3 −2

0

2

−4

0

4−0.1

0

0.1

x2

x3

x1

x1

x2x3

near grazing−sliding behaviour sliding segment born

in grazing−sliding

Milan 4/6/09 – p. 30

5. Impact with friction

Dankowicz Nordmark & C.

q ∈ Rn, with rigid contact in 2D + Coulomb friction

M (q, t) q = f(q, q, t) + λT cTu (q, t) + λNcT

v (q, t),

Scalar constraint y ≥ 0, y ∈ R normal distance;λN ≥ 0, λT ∈ R normal and tangential forces;

Coulomb friction, |λT | ≤ µλN , λT = −sign(u)µλN if u 6= 0

e.g. rod & table Painlevé 1905, Brogliato et al.

P1

P2

¸N

¸T

X

Y

Sx

Sy

Rµ

(x; y)

¹

u

¡¹

¡¸T =¸N

Milan 4/6/09 – p. 31

contact dynamics

Project Lagrangian onto u and v directions:

u = a (q, q, t) + λT A (q, t) + λNB (q, t) ,

v = b (q, q, t) + λT B (q, t) + λNC (q, t) ,

A = cu · M−1· cT

u , B = cu · M−1· cT

v , C = cv · M−1· cT

v ,

positive definite M ⇒ A > 0 C > 0, AC − B2 > 0

special case B = 0 ⇒ “independent” normal andtangential motion ⇒ can use Newtonian restitutionv → −rv at impact (well posed)

what if B 6= 0?, e.g. for rod example (l = 2, m = 2):A = 1 + 3 sin2 θ,B = 3 sin 2θ, C = 1 + 3 cos2 θ

Milan 4/6/09 – p. 32

modes of sustained motion

free flight : y > 0. No contact forces:

(λT , λN ) = (0, 0).

positive/negative slip : y = 0, v = 0, λN > 0, u 6= 0. Full frictionλT = −sign(u)µλN .

(λT , λN ) =b

C − sign(u)µB(sign(u)µ,−1).

stick : y = 0, v = 0, λN > 0, u = 0, |λT | < µλN .

(λT , λN ) =1

AC − B2(bB − aC, aB − bA)

Milan 4/6/09 – p. 33

impacts

Def: impact phase infinitesimal time intervals in whichλN and λT are impulses (distributions)

key idea: re-scale τ = t/ε, ΛN,T = ελN,T = O(1) andlet ε → 0.

impact-phase dynamics: q′ = 0 and

u′ = AΛT + BΛN , v′ = BΛT + CΛN

(A,B,C are constant during impact since q′ = 0.

integrating IN,T =∫

impact ΛN,T dτ gives:

(IT , IN ) = 1AC−B2 (C∆u − B∆v,A∆v − B∆u).

Change in q is then: ∆q = M−1(cTu IT + cT

v IN )

Milan 4/6/09 – p. 34

but how to compute∆u, ∆v?

u′ = AΛT + BΛN , v′ = BΛT + CΛN ,

⇒ 3 modes of impulsive motion:

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

λN (y, v) =(1 + r2) − (1 − r2) tanh

(v

δ

)

2

(

−y

ε

)

for small δ, ε

−0.1 0 0.1 0.2 0.3−0.4

−0.2

0

0.2

0.4

0.6

C−µ B

b

resolvable (ongoing work) ⇒(?) impact always occurs

Milan 4/6/09 – p. 44

b. transition into stick or chatter

e.g. nearby initial conditions with b < 0

0 0.5 1 1.5 2−0.04

−0.02

0

0.02

0.04

t

v

0 0.5 1 1.5−1

−0.5

0

0.5

1x 10−6

t

v

. . . Define multiplier e: v → ev after impact + lift off.

u1 u0u2

v2v0

v1

Milan 4/6/09 – p. 45

analysis of chatter

Find parameter regions in which e > 1 (reverse chatter)despite r < 1 - even in the “non-Painlevé” case

Milan 4/6/09 – p. 46

analysis of chatter


then we would have “infinite” non-uniqueness inforwards time -:(

Milan 4/6/09 – p. 46

analysis of chatter



but can such transitions occur?

Milan 4/6/09 – p. 46

analysis of chatter




analysis of smoothed “stiff” systems suggest yes . . .

Milan 4/6/09 – p. 46

analysis of chatter





it depends how you take the smoothing -:(

Milan 4/6/09 – p. 46

analysis of chatter





it depends how you take the smoothing -:(

ongoing work . . .

Milan 4/6/09 – p. 46

6. Conclusion

used piecewise-smooth as formalism.

Milan 4/6/09 – p. 47

6. Conclusion


⇒ degree of smoothness case by case

Milan 4/6/09 – p. 47

6. Conclusion



⇒ discontinuity induced bifurcationclassification with DMs ⇒ sudden jumps to chaos, etc.

Milan 4/6/09 – p. 47

6. Conclusion




provides natural explanation of observed behaviour

Milan 4/6/09 – p. 47

6. Conclusion





much on-going work, e.g.impact + friction Dankowitz, Nordmark & C.catastrophic sliding bifurcations ∼ canards Jeffrey,C. di Bernardo, Shaw, Moehlis

Milan 4/6/09 – p. 47

6. Conclusion





much on-going work, e.g.impact + friction Dankowitz, Nordmark & C.catastrophic sliding bifurcations ∼ canards Jeffrey,C. di Bernardo, Shaw, Moehlis

Piecewise-Smooth Dynamical Systems: Theory &Applications diBernardo, Budd, C. & KowalczykSpringer Jan 08. . . + SIAM review Dec 08

Milan 4/6/09 – p. 47

rock, rattle and slide - politecnico di...

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