the swinging spring: regular and chaotic...

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The Swinging Spring: Regular and ChaoticMotion

Leah Ganis

May 30th, 2013

Leah Ganis The Swinging Spring: Regular and Chaotic Motion

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Outline of Talk

I Introduction to Problem

I The Basics: Hamiltonian, Equations of Motion, Fixed Points,Stability

I Linear Modes

I The Progressing Ellipse and Other Regular Motions

I Chaotic Motion

I References


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Introduction

The swinging spring, or elastic pendulum, is a simple mechanicalsystem in which many different types of motion can occur. Thesystem is comprised of a heavy mass, attached to an essentiallymassless spring which does not deform. The system moves underthe force of gravity and in accordance with Hooke’s Law.

x

y

z

rφ

mk


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The Basics

We can write down the equations of motion by finding theLagrangian of the system and using the Euler-Lagrange equations.The Lagrangian, L is given by

L = T − V

where T is the kinetic energy of the system and V is the potentialenergy.


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The Basics

In Cartesian coordinates, the kinetic energy is given by thefollowing:

T =1

2m(x2 + y2 + z2)

and the potential is given by the sum of gravitational potential andthe spring potential:

V = mgz +1

2k(r − l0)2

where m is the mass, g is the gravitational constant, k the springconstant, r the stretched length of the spring (

√x2 + y2 + z2),

and l0 the unstretched length of the spring.


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The Basics

The equations of motion are then given by:

x = − k

m

(r − l0r

)x

y = − k

m

(r − l0r

)y

z = − k

m

(r − l0r

)z − g


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The Basics

This system has two fixed points: one linear center where the bobis hanging straight down (x , y , z) = (0, 0,−l) and one saddle-typefixed point where the bob is poised just above where it is attached.

SaddlePoint

Center-likeFixed Point

This talk will not consider the saddle-type fixed point.


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The Basics

There are two constants of motion, the total energy E of thesystem, and the total angular momentum, h:

E = T + V

h = xy − y x

With only two first integrals and three spacial coordinates, thesystem is not integrable.


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Linear Modes

I Consider very small amplitude motion about the fixed point.Linearizing about the fixed point (0, 0,−l) (i.e. r ≈ l), weobtain the equations for small oscillations:

x = −g

lx , y = −g

ly z = − k

mz

I All three equations are easily solved and are simply sums ofsines and cosines. The x and y components trace out an

ellipse and have frequency ωR =√

gl which is the same as a

rigid pendulum.

I The vertical height varies sinusoidally with frequency

ωZ =√

km which is that of a spring oscillating in one

dimension.


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Linear Modes

I If the ratio ε = ωRωZ

is an integer or rational number, themotion will be periodic.

I Example: ε = 2, the bob will rotate around twice in the x − yprojection before returning to the initial position. For εirrational, the motion will be quasi-periodic.

I Since the equations are completely decoupled in thisapproximation, we can expect no exchange of energy betweenswinging and springing modes. In other words, swinging doesnot induce springing and vice versa.


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Elliptical Motion

Assuming small amplitude motion and dropping all terms of thirdorder order or higher, the equations of motion become thefollowing:

x + ω2Rx = λxz

y + ω2Ry = λyz

z + 4ω2Rz =

1

2λ(x2 + y2)

where λ = l0ω2Z/l and it is assumed ωZ = 2ωR .


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Elliptical Motion

Seek solutions which at lowest order are periodic and elliptical inthe x − y projection:

x = ε(A cosωt) + ε2x2 + · · ·y = ε(B cosωt) + ε2y2 + · · ·z = ε(C cos 2ωt) + ε2z2 + · · ·

where ε is a small parameter, A, B, C constants, andω = ω0 + εω1 + · · ·


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Elliptical Motion

I For A = 0 or B = 0, the solutions will be approximately cupshaped or cap shaped.

I For C = 0, we will have at lowest order that the elasticpendulum sweeps out a cone shape with the height of the bobapproximately constant.

I There are no solutions to the case where A,B,C 6= 0. Insteadchange to a rotating coordinate frame to analyze.


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The Progressing Ellipse

Using rotating coordinates α, β, and γ, with Θ varying with timeand assuming the α− β axis rotating with constant angularvelocity Θ = Ω we have:αβ

γ

=

cos Θ sin Θ 0− sin Θ cos Θ 0

0 0 1

xyz

.Denoting the rotation matrix as R, ~α = (α, β, γ)T and~x = (x , y , z)T , we have that ~α = R~x + 2R~x + R~x .


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Differentiating as required above, we get the following set ofequations.

α + (ω2R − Ω2)α− 2Ωβ = λαγ

β + (ω2R − Ω2)β + 2Ωα = λβγ

γ + 4ω2Rγ =

1

2λ(α2 + β2)

Again, seek solutions of the form:

α = ε(A cosωt) + ε2α2 + · · ·β = ε(B cosωt) + ε2β2 + · · ·γ = ε(C cos 2ωt) + ε2γ2 + · · ·


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I Assuming small rotation, i.e. Ω = εΩ1 and plugging the formof the solution into the differential equations, and requiringgrowing terms in the second order equations of ε to go to zero(and after a lot of algebra), a set of algebraic equations forΩ1, ω1,A,B, and C is obtained.

I For fixed values of A and B, the equations can be solvedexplicitly for Ω1, ω1 and C , with two possible solutions.


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Simulating the Progressing Ellipse

Parameter values used are the same as used by Lynch (2002) andare as follows:

I mass: m = 1kg

I equilibrium stretched length: l = 1m

I gravitational constant: g = π2 m s−2

I spring constant: k = 4π2 kg s−1

All simulations were done in MATLAB using ode45 with anabsolute error tolerance of 10−6.


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Initial values used are as follows:

I A = 0.01

I B = 0.005

I C ≈ 0.0237 is given by the following formula derived byLynch(2002):

C = ± A2 − B2

2√

2(A2 + B2)

I ω = ωR + ω1 where ω1 is given by:

ω1 = ∓3√

2(A2 + B2)

16lωR


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The initial position was set to be ~x = (A, 0,C − l)T and the initialvelocity was set to be ~x = (0, ωB, 0)T . The 3-D image of themovement through 211 seconds (corresponding to 90 ofprecession) is shown:


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Simulating the Retrogressing Ellipse

−0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01−0.01

−0.005

0

0.005

0.01−1.003

−1.002

−1.001

−1

−0.999

−0.998

−0.997The Retrogressing Ellipse

X position

Y position

Z po

sitio

n

Student Version of MATLAB


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−0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

X position

Y po

sitio

nX−Y Projection of Progressing Ellipse



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4 5 6 7 8 9 10x 10−3

−1.003

−1.002

−1.001

−1

−0.999

−0.998

−0.997

Radial Distance

Hei

ght



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Another Type of Motion?

I Question: Can we have any other types of motion?

I Answer: Why, yes, yes we can!


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Precession of the Swing Plane

I If the bob is started with almost entirely vertical oscillations,gradually the vertical oscillations subside and a swingingmotion occurs.

I Swinging motion subsides and is replaced by a springingmotion as before, and the process repeats.

I The motion appears planar (but is really elliptical).

I The “swing plane” rotates each time we return to swingingmotion.


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Simulating Precession of the Swing Plane

Initial conditions used leading to a precessing swing plane are asfollows:

I x0 = 0.04m, y0 = 0

I z0 = −l + 0.08m, note this corresponds to 8cm ofcompression.

I x0 = 0, y0 = 0.03427m/s, and z0 = 0.

For the complete derivation, see Lynch (2002).


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Precession of Swing Plane

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

X Position

Y Po

sitio

nX−Y Projection of Precessing Swing Plane



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Precession of Swing Plane

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18−1.08

−1.06

−1.04

−1.02

−1

−0.98

−0.96

−0.94

−0.92

−0.9

Radial Position

Hei

ghtRadial Position versus Height for Precessing Swing Plane



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Chaotic Motion

Can we find chaotic motion in this system? (Yes of course, else this would not be a

very interesting system.)

Main characteristics of chaotic motion:

I Sensitive dependence on initial conditions - nearby trajectoriesdiverge exponentially fast - i.e. positive Lyapunov exponent.

I Aperiodic long-term behavior - not all trajectories settle downto fixed points, or periodic/quasi periodic orbits.

I Trajectories densely fill the space.


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Chaos in 2 Space Dimensions

Let’s consider a simpler version of our original equations - restrictthe motion to be planar.

Hamiltonian in dimensionless coordinates after rescaling length,time, and energy:

H =1

2(p2

1 + p22) + fq2 +

1

2

(1− f −

√q2

1 + (1− q22)

)2

where f =(ωRωZ

)2.


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Energy

Max

imal

Lya

puno

v Ex

pone

nt

0.1 0.3 0.5 0.7

0.015

0.045

0.075

E0

Figure : Graph of maximal Lyapunov exponent, adapted fromNunez-Yepez et al (1989).


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The positive Lyapunov exponent is a good indicator for sensitivedependence on initial conditions, but what about the otherindicators of chaos?

Instead, let’s look at some Poincare sections.


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Low energy:

−0.1 −0.05 0 0.05 0.1

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

q1

p 1



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Slightly higher energy:

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

q1

p 1



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A little higher:

−1.5 −1 −0.5 0 0.5 1 1.5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

q1

p 1



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Cranked up all the way to 11:

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−1.5

−1

−0.5

0

0.5

1

1.5

q1

p 1



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I For certain energy values, periodic or quasi periodic regularmotion vanishes and trajectories tend to fill a 2-D spaceinstead of a smooth curve.

I For very low energy, virtually all trajectories demonstrateregular motion.

I At high energies, there also appears to be regular motion.

What do these chaotic trajectories really look like back in 3-D?


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Chaotic Motion in 3-D

−1−0.5

00.5

1

−1−0.5

00.5

1−1.5

−1.4

−1.3

−1.2

−1.1

−1

−0.9

−0.8

−0.7

−0.6

−0.5

X position

Two trajectories starting 1mm apart

Y Position

Z Po

sitio

n

Trajectory 1Trajectory 2



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−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X Position

Y Po

sitio

nX−Y Projection two trajectories, starting 1mm apart




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0 20 40 60 80 100 120 140 160 180 200−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

X Po

sitio

nX Position of two trajectories, starting 1mm apart




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0 5 10 15−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

Y Po

sitio

nY Position of two trajectories, starting 1mm apart




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Conclusions

I Highly complex dynamics can occur from what seems like avery simple physical system.

I There are many qualitatively different types of regular motion.

I For certain energy values, the system demonstrates all thehallmarks of chaos.


References

References

R Carretero-Gonzalez, H. N. Nunez Yepez, and A. L. Salas-Brito. Regular and chaotic behaviour in an extensiblependulum. Universidad Autonoma Metropolitana, (326), 1994.

R Cuerno, AF Ranada, and JJ Ruiz-Lorenzo. Deterministic chaos in the elastic pendulum: a simple laboratory fornonlinear dynamics. American Journal of Physics, 60(1):73–79, 1992.

Peter Lynch. The Swinging Spring, 2002a. URLhttp://mathsci.ucd.ie/~plynch/SwingingSpring/SS_Home_Page.html.

Peter Lynch. Resonant motions of the three-dimensional elastic pendulum. International Journal of Non-LinearMechanics, 37(2):345–367, March 2002b. ISSN 00207462. doi: 10.1016/S0020-7462(00)00121-9.

HN Nunez Yepez, AL Salas-Brito, and L. Vicente. Onset of chaos in an extensible pendulum. Physics Letters A,145(2), 1990.

Steven H. Strogatz. Nonlinear Dynamics and Chaos. Perseus Books Publishing, Cambridge, 1 edition, 1994. ISBN978-0-7382-0453-6.

A. Vitt and G. Gorelik. Oscillations of an elastic pendulum as an example of the oscillations of two parametricallycoupled linear systems. Journal of Technical Physics, 3(2-3):294–307, 1933.


http://mathsci.ucd.ie/~plynch/SwingingSpring/SS_Home_Page.html

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Special Thanks To...

Special thanks to my lovely fiance Michael for debugging my code.

Figure : Polly the cat, who also helped with the making of thispresentation.


the swinging spring: regular and chaotic...

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