Escape from potential wells in multi-dimensional systems ...sdross/talks/DynamicsDays-2016-post-movie… · Multi-well multi-degree of freedom systems Examples: chemistry, vehicle

Escape from potential wells in multi-dimensionalsystems: experiments and partial control

Shane Ross

Engineering Mechanics Program, Virginia Tech

Dept. of Biomedical Engineering and Mechanics

shaneross.com

Amir BozorgMagham (Maryland), Lawrie Virgin (Duke), Shibabrat Naik (Virginia Tech)

2016 Dynamics Days (Durham, January 8, 2016)

Intermittency and chaotic transitions

e.g., transitioning across “bottlenecks” in phase space; ‘metastability’

Marchal [1990]

ii

Multi-well multi-degree of freedom systems

• Examples: chemistry, vehicle dynamics, structural mechanics

X (mode 1)

Y (mode 2)

−0.03 −0.02 −0.01 0 0.01 0.02 0.03

−0.01

0

0.01

snap-through

load

x

Y

iii

Transitions through bottlenecks via tubes

Topper [1997]

•Wells connected by phase space transition tubes ' S1×R for 2 DOF— Conley, McGehee, 1960s— Llibre, Mart́ınez, Simó, Pollack, Child, 1980s— De Leon, Mehta, Topper, Jaffé, Farrelly, Uzer, MacKay, 1990s— Gómez, Koon, Lo, Marsden, Masdemont, Ross, Yanao, 2000s

iv

Motion near saddles

� Near rank 1 saddles in N DOF, linearized vector fieldeigenvalues are

±λ and ±iωj, j = 2, . . . , N

v

Motion near saddles

� Near rank 1 saddles in N DOF, linearized vector fieldeigenvalues are

±λ and ±iωj, j = 2, . . . , N� Equilibrium point is of type

saddle× center× · · · × center (N − 1 centers).

2q

2p N

p

qN

x xx

1p

q1

the saddle-space projection and N − 1 center projections — the N canonical planes

v

Motion near saddles

� For excess energy ∆E > 0 above the saddle, there’s anormally hyperbolic invariant manifold (NHIM) of boundorbits

M∆E =

N∑i=2

ωi2

(p2i + q

2i

)= ∆E

vi

Motion near saddles


M∆E =

N∑i=2

ωi2

(p2i + q

2i

)= ∆E

� So, M∆E ' S2N−3, topologically, a (2N − 3)-sphere

vi

Motion near saddles


M∆E =

N∑i=2

ωi2

(p2i + q

2i

)= ∆E

� So, M∆E ' S2N−3, topologically, a (2N − 3)-sphere�N = 2, ω = ω2,

M∆E ={ω2

(p22 + q

22

)= ∆E

}M∆E ' S1, a periodic orbit of period Tpo = 2π/ω

vi

Motion near saddles: 2 DOF

� Cylindrical tubes of orbits asymptotic to M∆E: stable andunstable invariant manifolds, W s±(M∆E),W u±(M∆E),' S1×R

� Enclose transitioning trajectories

vii

Motion near saddles: 2 DOF

•B : bounded orbits (periodic): S1•A : asymptotic orbits to 1-sphere: S1 × R (tubes)•T : transitioning and NT : non-transitioning orbits.

p1

q1

NTNT

T

T

A

A

A

A

B

viii

Tube dynamics — global picture

Poincare Section Ui

De Leon [1992]

�Tube dynamics: All transitioning motion between wellsconnected by bottlenecks must occur through tube• Imminent transition regions, transitioning fractions• Consider k Poincaré sections Ui, various excess energies ∆E

ix

Is this geometric theory correct?

x

Is this geometric theory correct?• Good agreement with direct numerical simulation

x


— molecular reactions, ‘reaction island theory’ e.g., De Leon [1992]

Dow

nloaded 16 Sep 2003 to 131.215.42.220. Redistribution subject to A

IP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

x


— molecular reactions, ‘reaction island theory’ e.g., De Leon [1992]

Dow

nloaded 16 Sep 2003 to 131.215.42.220. Redistribution subject to A

IP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

— celestial mechanics, asteroid escape rates e.g., Jaffé, Ross, Lo, Marsden, Farrelly, Uzer [2002]

0 10 20 30 40

Number of Periods

1

0.9

0.8

0.6

Su

rvival

Pro

ba

bility

0.7

0.5

x


Spacecraft trajectories have been designed and flown

xi


xii


• Experimental verification is needed, to enable applications

xii



•Our goal: We seek to perform experimental verification using a tabletop experiment with 2 degrees of freedom (DOF)

xii




• If successful, apply theory to ≥2 DOF systems, combine with control:

xii




• If successful, apply theory to ≥2 DOF systems, combine with control:• Preferentially triggering or avoiding transitions

— ship stability / capsize, etc.

xii




• If successful, apply theory to ≥2 DOF systems, combine with control:• Preferentially triggering or avoiding transitions

— ship stability / capsize, etc.

• Structural mechanics— re-configurable deformation of flexible objects

xii

Verification by experiment

xiii

Verification by experiment• Simple table top experiments; e.g., ball rolling on a 3D-printed surface• motion of ball recorded with digital camera

xiii

Verification by experiment• Simple table top experiments; e.g., ball rolling on a 3D-printed surface• motion of ball recorded with digital camera

Virgin, Lyman, Davis [2010] Am. J. Phys.xiii

Ball rolling on a surface — 2 DOF

• The potential energy is V (x, y) = gH(x, y),where the surface is arbitrary, e.g., we chose

H(x, y) = α(x2 + y2)− β(√x2 + γ +

√y2 + γ)− ξxy + H0.

−15 −10 −5 0 5 10 15−15

−10

−5

0

5

10

15

3 4 5 6 7 8 9 10 11 12

xiv

Ball rolling on a surface — 2 DOF

• The potential energy is V (x, y) = gH(x, y),where the surface is arbitrary, e.g., we chose

H(x, y) = α(x2 + y2)− β(√x2 + γ +

√y2 + γ)− ξxy + H0.

typical experimental trialxv

Transition tubes in the rolling ball system

−10 −5 0 5 10

−10

−5

0

5

10

xvi


−10 −5 0 5 10

−10

−5

0

5

10

12

34

xvii


−10 −5 0 5 10

−10

−5

0

5

10

12

34

xviii


−10 −5 0 5 10

−10

−5

0

5

10

12

34

xix


−10 −5 0 5 10

−10

−5

0

5

10

xx


−10 −5 0 5 10

−10

−5

0

5

10

transion tube from quadrant 1 to 2

xxi


−10 −5 0 5 10

−10

−5

0

5

10


xxii


−10 −5 0 5 10

−10

−5

0

5

10

5 10−50

−40

−30

−20

−10

0

10

20

30

40

50

(cm)

(cm

/s)


xxiii


−10 −5 0 5 10

−10

−5

0

5

10

5 10−50

−40

−30

−20

−10

0

10

20

30

40

50

(cm)

(cm

/s)


xxiv


−10 −5 0 5 10

−10

−5

0

5

10

5 10−50

−40

−30

−20

−10

0

10

20

30

40

50

(cm)

(cm

/s)


xxv


−10 −5 0 5 10

−10

−5

0

5

10

5 10−50

−40

−30

−20

−10

0

10

20

30

40

50

(cm)

(cm

/s)


xxvi


−10 −5 0 5 10

−10

−5

0

5

10

5 10−50

−40

−30

−20

−10

0

10

20

30

40

50

(cm)

(cm

/s)


xxvii


−10 −5 0 5 10

−10

−5

0

5

10

5 10−50

−40

−30

−20

−10

0

10

20

30

40

50

(cm)

(cm

/s)


xxviii


−10 −5 0 5 10

−10

−5

0

5

10

5 10−50

−40

−30

−20

−10

0

10

20

30

40

50

(cm)

(cm

/s)


xxix

Analysis of experimental data

12-600

-400

-200

0

200

400

600

800

1000

1200

1400

108640 2

time

• 120 experimental trials of about 10 seconds each, recorded at 50 Hz

xxx


12-600

-400

-200

0

200

400

600

800

1000

1200

1400

108640 2

time

• 120 experimental trials of about 10 seconds each, recorded at 50 Hz• 3500 intersections of Poincaré sections, sorted by energy

xxxi


12-600

-400

-200

0

200

400

600

800

1000

1200

1400

108640 2

time

• 120 experimental trials of about 10 seconds each, recorded at 50 Hz• 3500 intersections of Poincaré sections, sorted by energy• 400 transitions detected

xxxii


12-600

-400

-200

0

200

400

600

800

1000

1200

1400

108640 2

time

• 120 experimental trials of about 10 seconds each, recorded at 50 Hz• 3500 intersections of Poincaré sections, sorted by energy• 400 transitions detected

xxxiii

Poincaré sections at various energy ranges

r (cm)0 5 10 15

pr(cm/s)

-80

-60

-40

-20

0

20

40

60

80

∆E=[-500,-400]

xxxiv


r (cm)0 5 10 15

pr(cm/s)

-80

-60

-40

-20

0

20

40

60

80

∆E=[-400,-300]

xxxv


r (cm)0 5 10 15

pr(cm/s)

-80

-60

-40

-20

0

20

40

60

80

∆E=[-300,-200]

xxxvi


r (cm)0 5 10 15

pr(cm/s)

-80

-60

-40

-20

0

20

40

60

80

∆E=[-200,-100]

xxxvii


r (cm)0 5 10 15

pr(cm/s)

-80

-60

-40

-20

0

20

40

60

80

∆E=[-100,0]

xxxviii


r (cm)0 5 10 15

pr(cm/s)

-80

-60

-40

-20

0

20

40

60

80

∆E=[0,100]

xxxix


r (cm)0 5 10 15

pr(cm/s)

-80

-60

-40

-20

0

20

40

60

80

∆E=[100,200]

xl


r (cm)0 5 10 15

pr(cm/s)

-80

-60

-40

-20

0

20

40

60

80

∆E=[200,300]

xli


r (cm)0 5 10 15

pr(cm/s)

-80

-60

-40

-20

0

20

40

60

80

∆E=[300,400]

xlii


r (cm)0 5 10 15

pr(cm/s)

-80

-60

-40

-20

0

20

40

60

80

∆E=[400,500]

xliii


r (cm)0 5 10 15

pr(cm/s)

-80

-60

-40

-20

0

20

40

60

80

∆E=[500,600]

xliv


r (cm)0 5 10 15

pr(cm/s)

-80

-60

-40

-20

0

20

40

60

80

∆E=[600,700]

xlv

Experimental confirmation of transition tubes

• Theory predicts all transitions (p value of correlation < 0.0001)

xlvi


• Theory predicts all transitions (p value of correlation < 0.0001)• Consider overall trend in transition fraction as excess energy grows

xlvi


• Theory predicts all transitions (p value of correlation < 0.0001)• Consider overall trend in transition fraction as excess energy grows

-300 -200 -100 0 100 200 300 400 500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Experiment

Tra

ns

itio

n f

rac

tio

n

xlvi

Theory for small excess energy, ∆E

• Area of the transitioning region, the tube cross-section (MacKay [1990])Atrans = Tpo∆E

where Tpo = 2π/ω period of unstable periodic orbit in bottleneck

xlvii




• Area of energy surfaceA∆E = A0 + τ∆E

xlvii





where

A0 = 2

∫ rmaxrmin

√−145 gH(r)(1 + ∂H∂r

2(r))dr

xlvii





where

A0 = 2

∫ rmaxrmin

√−145 gH(r)(1 + ∂H∂r

2(r))dr

and

τ =

∫ rmaxrmin

√√√√145 (1 + ∂H∂r 2(r))−gH(r) dr

xlvii


• The transitioning fraction, under well-mixed assumption,

ptrans =AtransA∆E

=TpoA0

∆E(

1− τA0∆E +O(∆E2))

xlviii



ptrans =AtransA∆E

=TpoA0

∆E(

1− τA0∆E +O(∆E2))

• For small ∆E, growth in ptrans with ∆E is linear, with slope∂ptrans∂∆E

=TpoA0

xlviii



ptrans =AtransA∆E

=TpoA0

∆E(

1− τA0∆E +O(∆E2))

• For small ∆E, growth in ptrans with ∆E is linear, with slope∂ptrans∂∆E

=TpoA0

• For slightly larger values of ∆E, there will be a correction term leadingto a decreasing slope,

∂ptrans∂∆E

=TpoA0

(1− 2 τA0∆E

)

xlviii


-300 -200 -100 0 100 200 300 400 500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Experiment

Tra

ns

itio

n f

rac

tio

n

xlix


-300 -200 -100 0 100 200 300 400 500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Experiment

Tra

ns

itio

n f

rac

tio

n

Theoretical

slope

l


-300 -200 -100 0 100 200 300 400 500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Experiment

Tra

ns

itio

n f

rac

tio

n

Theoretical

slope

Slopes agree

within 5%

li

Combine with control

• Since geometric theory provides the routes of transition / escape, cancombine with control to trigger or avoid transitions

lii



•We’ll consider partial controlSabuco, Sanjuán, Yorke [2012]; Coccolo, Seoane, Zambrano, Sanjuán [2013]

lii



•We’ll consider partial controlSabuco, Sanjuán, Yorke [2012]; Coccolo, Seoane, Zambrano, Sanjuán [2013]

— avoid a transition in the presence of a disturbance which is largerthan the control

lii

Partial control - safe set S

liii


Region to be avoided in white

liv


Region to be avoided in white - could include holes

lv


lvi


lvii


lviii


lix


lx


lxi


lxii


Control smaller than disturbance

lxiii

Ship motion and capsize

lxiv


lxv


•Model built around Hamiltonian,H = p2x/2 + R

2p2y/4 + V (x, y),

where x = roll and y = pitch arecoupled

V (x, y)

lxvi


•Model built around Hamiltonian,H = p2x/2 + R

2p2y/4 + V (x, y),

where x = roll and y = pitch arecoupled

V (x, y)

−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

-1.5 -1 -0.5 0 0.5 1 1.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

E < Ec E > Eclxvi

Tubes leading to capsize(a) (b) (c)

Poincare S-O-S, U1transition tube

periodic orbit

(d)

Figure 10: E↵ective potential energy V (x, y) in configuration space and the tubes leading to es-cape.Fig. 10(b) and Fig. 10(c) show the Hill’s region and energetically forbidden realm (EFR) fore > ecritical and e < ecritical where ecritical denotes the critical energy. The white regions are ener-getically accessible and bounded by the zero velocity curve (the boundary of M(e)). Beyond thezero velocity curve, the shaded region, lies the energetically forbidden region where kinetic energyis negative and motion is impossible.

(Eqn. (11)) and apply tube dynamics (for application in celestial mechanics see Ross [2004], Koonet al. [2000]) to organize trajectories exhibiting bounded and unbounded motion. A related tech-

10

lxvii

Partial control to avoid capsize

(a) (b)

Figure 16: Initial set and its image that defines the set on the Poincaré Surface-Of-Section we wantthe dynamics to stay.

(a) Iteration # 1 (b) Iteration # 2 (c) Iteration # 3

(d) Iteration # 4 (e) Iteration # 5 (f) Iteration # 6

Figure 17: Safe set computed for a significant wave height, Hs = 0.1 m and return time to thePoincaré Surface-Of-Section as the feedback partial control time which gives a discrete disturbance,⇠0 = 0.277915. This system is seemingly hard to partially control as even for a small wave height,there are no safe sets below a safe ratio ⇢0 = 0.825.

20

y-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

v y

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

initial set Q safe set S

• Safe set shown when disturbance (red) is random ocean waves andsmaller control (green) is via steering or control moment gyroscope

• Could inform control schemes to avoid capsize in rough seas

lxviii

Partial control to avoid capsize

(a) (b)





20

(a) (b)





20

initial set Q safe set S

• Safe set shown when disturbance (red) is random ocean waves andsmaller control (green) is via steering or control moment gyroscope

• Could inform control schemes to avoid capsize in rough seas

lxix

Next steps — structural mechanics

Buckling, bending, twisting, and crumpling of flexible bodies

• adaptive structures that can bend, fold, and twist to provide advancedengineering opportunities for deployable structures, mechanical sensors

lxx


X (mode 1)

Y (mode 2)

−0.03 −0.02 −0.01 0 0.01 0.02 0.03

−0.01

0

0.01

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

X

X

Y

Y.

snap-through

load

x

Y

lxxi


X (mode 1)

Y (mode 2)

−0.03 −0.02 −0.01 0 0.01 0.02 0.03

−0.01

0

0.01

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

X

X

Y

Y.

snap-through

load

x

Ynon-snap-through

lxxii


X (mode 1)

Y (mode 2)

−0.03 −0.02 −0.01 0 0.01 0.02 0.03

−0.01

0

0.01

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

X

snap-through

X

Y

Y.

snap-through

load

x

Ynon-snap-through

lxxiii


X (mode 1)

Y (mode 2)

−0.03 −0.02 −0.01 0 0.01 0.02 0.03

−0.01

0

0.01

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

X

snap-through

X

Y

Y.

snap-through

load

x

Y

lxxiv

Final words

• 2 DOF experiment for understanding geometry of transitions

lxxv

Final words

• 2 DOF experiment for understanding geometry of transitions— verified geometric theory of tube dynamics

lxxv

Final words


• Unobserved unstable periodic orbits have observable consequences

lxxv

Final words


• Unobserved unstable periodic orbits have observable consequences• Future work:

– analysis and control of transitions in other multi-DOF systemse.g., triggering and avoidance of buckling in flexible structures, capsizeavoidance for ships in rough seas and floating structures

lxxv

Final words




Thanks to: Lawrie Virgin, Amir BozorgMagham, Shibabrat Naik

lxxv

Final words




Thanks to: Lawrie Virgin, Amir BozorgMagham, Shibabrat Naik

Papers in preparation; check status at:shaneross.com

lxxv

Intermittency and chaotic transitionsMulti-well multi-degree of freedom systemsTransitions through bottlenecks via tubesMotion near saddlesMotion near saddlesMotion near saddles: 2 DOFMotion near saddles: 2 DOFTube dynamics — global pictureIs this geometric theory correct?Is this geometric theory correct?Is this geometric theory correct?Verification by experimentBall rolling on a surface — 2 DOFBall rolling on a surface — 2 DOFTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemTransition tubes in the rolling ball systemAnalysis of experimental dataAnalysis of experimental dataAnalysis of experimental dataAnalysis of experimental dataPoincaré sections at various energy rangesPoincaré sections at various energy rangesPoincaré sections at various energy rangesPoincaré sections at various energy rangesPoincaré sections at various energy rangesPoincaré sections at various energy rangesPoincaré sections at various energy rangesPoincaré sections at various energy rangesPoincaré sections at various energy rangesPoincaré sections at various energy rangesPoincaré sections at various energy rangesPoincaré sections at various energy rangesExperimental confirmation of transition tubesTheory for small excess energy, ETheory for small excess energy, ETheory for small excess energy, ETheory for small excess energy, ETheory for small excess energy, ECombine with controlPartial control - safe set SPartial control - safe set SPartial control - safe set SPartial control - safe set SPartial control - safe set SPartial control - safe set SPartial control - safe set SPartial control - safe set SPartial control - safe set SPartial control - safe set SPartial control - safe set SShip motion and capsizeShip motion and capsizeShip motion and capsizeTubes leading to capsizePartial control to avoid capsizePartial control to avoid capsizeNext steps — structural mechanicsNext steps — structural mechanicsNext steps — structural mechanicsNext steps — structural mechanicsNext steps — structural mechanicsFinal words

Documents

Escape from potential wells in multi-dimensional systems ...sdross/talks/DynamicsDays-2016-post-movie… · Multi-well multi-degree of freedom systems Examples: chemistry, vehicle