Experimental verification of criteria for escape

from a potential well in a multi-degree of

freedom system

Shane Ross

Biomedical Engineering and Mechanics, Virginia Tech

shaneross.com

with Amir BozorgMagham (Univ. of Maryland) and Lawrie Virgin (Duke Univ.)

SIAM Conference on Applications of Dynamical Systems (Snowbird, May 17, 2015)

Intermittency and chaotic transitions

e.g., transitioning across “bottlenecks” in phase space

Marchal [1990]

ii

Multi-well multi-degree of freedom systems

• Examples: chemistry, vehicle dynamics, structural mechanics

x

y

snap

load, P

x ~ sag

y ~ angle

(a) (b)

Sag, X

Angle,Y

X

Yincreasing

load

α β φ

λ

λV(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

Is this geometric theory correct?• Good agreement with direct numerical simulation — molecular re-

actions, ‘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

v

Is this geometric theory correct?

• but experimental verification has been lacking

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

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

— re-configurable deformation of flexible objects— adaptive structures that can bend, fold, and twist to provide advancedengineering opportunities for deployable structures, mechanical sensors

• vehicle stability— capsize problem, etc.

vi

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

vii

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

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

M∆E ={ω2

(p22 + q

22

)= ∆E

}M∆E ' S1, a periodic orbit of period Tpo = 2πω

viii

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

ix

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

x

Tube dynamics

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

xi

Verification by simulation

� Structured transition statistics in chemistry, etc 3+ DOF

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

18

20

Loops Around the Nuclear Core

Ele

ctr

on S

cattering P

robabili

ty (

%)

ε = 0.58

0

1.0

2.0

3.0

4.0

5.0

6.0

0 10 20 30 40 50 60 70 80 90 100

ε = 0.58

Time

selcitraP de ret tac S fo rebmu

N

1 2 3 4 5 6 70

10

20

30

40

50

60

70

80

Loops Around the Nuclear Core

Ele

ctr

on S

cattering P

robabili

ty (

%)

ε = 0.6

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70 80 90 100

ε = 0.6

Time

selcitraP dere ttacS fo rebmu

N

Gabern, Koon, Marsden, Ross [2005]

xii

Verification by experiment

• Simple table top experiments; e.g., ball rolling on a 3D-printed surface

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)− V0,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)− V0,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

Transition tubes in the rolling ball system

−10 −5 0 5 10

−10

−5

0

5

10

12

34

xvii

Transition tubes in the rolling ball system

−10 −5 0 5 10

−10

−5

0

5

10

12

34

xviii

Transition tubes in the rolling ball system

−10 −5 0 5 10

−10

−5

0

5

10

12

34

xix

Transition tubes in the rolling ball system

−10 −5 0 5 10

−10

−5

0

5

10

xx

Transition tubes in the rolling ball system

−10 −5 0 5 10

−10

−5

0

5

10

transion tube from quadrant 1 to 2

xxi

Transition tubes in the rolling ball system

−10 −5 0 5 10

−10

−5

0

5

10

transion tube from quadrant 1 to 2

xxii

Transition tubes in the rolling ball system

−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)

transion tube from quadrant 1 to 2

xxiii

Transition tubes in the rolling ball system

−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)

transion tube from quadrant 1 to 2

xxiv

Transition tubes in the rolling ball system

−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)

transion tube from quadrant 1 to 2

xxv

Transition tubes in the rolling ball system

−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)

transion tube from quadrant 1 to 2

xxvi

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

xxvii

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• 3500 intersections of Poincaré sections, sorted by energy

xxviii

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• 3500 intersections of Poincaré sections, sorted by energy• 400 transitions detected

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• 3500 intersections of Poincaré sections, sorted by energy• 400 transitions detected

xxx

Poincaré sections at various energy ranges

xxxi

Experimental confirmation of transition tubes

• Theory predicts > 95% of transitions• 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

xxxii

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

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

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

xxxiii

Theory for small excess energy, ∆E

• The transitioning fraction, under well-mixed assumption,

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

)

xxxiv

Theory for small excess energy, ∆E

-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

xxxv

Theory for small excess energy, ∆E

-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

xxxvi

Theory for small excess energy, ∆E

-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%

xxxvii

Next steps — structural mechanics

Buckling, bending, twisting, and crumpling of flexible bodies

xxxviii

Next steps — 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

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

X

snap-through

non-snap-through

X

Y

Y.

snap-through

load

x

Y

xxxix

Final words

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

• Unobserved unstable periodic orbits have observable consequences

• Future work: control of transitions in multi-DOF systemse.g., triggering and avoidance of buckling in flexible structures, capsizeavoidance for ships in rough seas and floating structures

• For more, see Lawrie Virgin’s talk tomorrow, 3:45pm, in‘CP25 Topics in Classical and Fluid Dynamical Systems’

• also Isaac Yeaton’s talk tomorrow, 4:45pm (CP25)Snakes on An Invariant Plane: Dynamics of Flying Snakes

Paper in preparation; check status at:shaneross.com

xl

Intermittency and chaotic transitionsMulti-well multi-degree of freedom systemsTransitions through bottlenecks via tubesIs this geometric theory correct?Is this geometric theory correct?Motion near saddlesMotion near saddlesMotion near saddles: 2 DOFMotion near saddles: 2 DOFTube dynamicsVerification by simulationVerification 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 systemAnalysis of experimental dataAnalysis of experimental dataAnalysis of experimental dataAnalysis of experimental dataPoincaré 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, ENext steps — structural mechanicsNext steps — structural mechanicsFinal words