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Chaotic Motion of a Double Pendulum

Matthew Beckler

Lab Partners:

Kevin Klutzke, Ryne Lehman, Mike Minehart

December 7, 2004

1 Introduction

When studying systems of motion in the real world, a common catchphrase is chaoticmotion. Many people use this term to describe the seeming random behavior of some-

times surprisingly simple objects. An easily created example of this type of system isa double pendulum system, which consists of a pair of rods, the upper rod connectedto a fixed point, the lower rod connected only to the upper, that are free to swing.We investigated the chaotic motion of this system, the effect that changes in the ini-tial conditions had in the end result, and methods of mathematically representing thependulums.

2 Predictions

With a normal pendulum, a small change in the initial angle will only change the final(after a set amount of time) angle by approximately the same result. In other words, alinear change in an initial condition will produce a linear change in the resulting motion.

With the introduction of a second pendulum, all of our intuition must be disregarded.In the chaotic motion exhibited by the double pendulum system, we predict that asmall change will result in two completely different behaviors. We expect this changeto become visible almost immediately after starting the system.

3 Description of Experiments

Since we did not have access to a double pendulum system, we decided to make modi-fications to the basic two pendulum simulation that is provided with the visualpythonmodule. We decided that it would make more sense to have the masses and length ofeach pendulum the same. The two factors that we chose to study were the two angles,1 and 2. See Figure 1.

We conducted essentially two experiments, the first exploring the end result ofthe system when only 1 was changing, and the second when only 2 was changing.

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Figure 1: Double Pendulum System

When we finished with those two experiments, we decided we wanted to investigate theresults over a longer time period when 1 or 2 was changed very slightly. We presentthe results of these longer period experiments after the original experiments

I modified the sample simulation, adding the capability to output the status ofthe pendulum system, after every iteration, to a file located on the hard disk. Thisallowed us to import thousands of data points into a spreadsheet without ever havingto manually key them into the computer. The original simulation had the rate ofsimulation reduced as to allow for easier viewing of the behavior by the operator.Since we only needed the data points, I essentially removed the rate limitation. Thishad no effect on the data collected, because it only increased the speed with which wecould simulate and collect data points.

Experiment 1 1 = {1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3}, 2 = 0, 0 < t < 15

Experiment 2 1 = 2, 2 = {0.2, 0.225, 0.25, 0.275, 0.3}, 0 < t < 10

Experiment 3 1 = {1.7, 1.71}, 2 = 0, 0 < t < 30

Experiment 4 1 = 2, 2 = {0.25, 0.2505}, 0 < t < 30

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We wanted to investigate the long-term behavior with small changes in the initialconditions. The results of these two experiments are below.

Experiment 3:

1 = 2, 2 = {1.7, 1.71}Time 2 = 1.7 2 = 1.71

29.91 0.00 -1.2029.92 0.01 -1.2029.93 0.02 -1.2029.94 0.03 -1.2029.95 0.05 -1.1929.96 0.06 -1.1929.97 0.07 -1.1829.98 0.08 -1.1729.99 0.09 -1.1630.00 0.10 -1.15

Figure 4: 1 Two Lines - Long Duration

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Experiment 4:

1 = {0.25, 0.2505}, 2 = 0Time 1 = 0.25 1 = 0.250529.90 -1.33 1.4429.91 -1.31 1.4629.92 -1.29 1.47

29.93 -1.26 1.4929.94 -1.23 1.5029.95 -1.20 1.5129.96 -1.17 1.5129.97 -1.13 1.5229.98 -1.10 1.5229.99 -1.06 1.5230.00 -1.02 1.52

Figure 5: 2 Two Lines - Long Duration

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5 Results & Analysis

Based on the data that we obtained, and the charts that we created using that data,we decided that the double pendulum system really did exhibit chaotic motion. Sincepart of our assignment was to understand the physics behind our simulation, here isan explanation using Lagrangian mathematics.

The Lagrangian is an equation that relates the kinetic and potential energy of asystem. It is described as:

L(1, 2, 1, 2) = T1 + T2 V1 V2 (1)

Where:

T1 = Kinetic energy of the first rod

T2 = Kinetic energy of the second rod

V1 = Potential energy of the first rod

V2 = Potential energy of the second rod

For the purposes of this explanation:

1 =1

t, 2 =

2

t, 1 =

21

t, 2 =

22

t

If we decide that the potential energy is equal to zero at the top, then:

V1 = m1gy1, V2 = m2gy2

Using the diagram, a bit of trigonometry, we can write the cartesian coordinates ofthe centers of mass of the two rods in terms of a,b,1, 2. We can then combine theseexpressions into our potential energy formulas to produce:

V1 = m1ga

2cos 1, V2 = m2g(a cos 1 +

b

2cos 2)

Now we can move on to find the kinetic energy. If we start with the standard forof the kinetic energy of a rigid body:

KE =1

2mv2

cm+

1

2Icm

2

Using a large amount of messy algebra, many time derivatives, and a few trigidentities, we can find the kinetic energy of each rod:

T1 =1

2m1(

a2

41

2

) +1

2(

1

12m2a

2)12

T2 = 12

m2(a21

2+ b

2

42

2+ ab12 cos(1 2)) + 1

2( 1

12m2b

2)22

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If we take the kinetic energies and potential energies from both rods, and plug themback into the lagrangian (Equation 1), we get:

(1

6m1a

2 +1

2m2a

2)12

+1

6m2b

222

+1

2m2ab12 cos(1 2)+

+(

1

2 m1ga + m2ga)cos 1 +

1

2 m2gb cos 2

Using a bit more algebraic substitution and differientation with respect to time, weget the following relationships:

(1

3m1a

2 + m2a2)1 +

1

2m2ab2 cos(1 2)

1

2m2ab2

2

sin(1 2) + (1

2m1ga + m2ga)sin = 0 (2)

1

3m2b

22 +1

2m2ab1 cos(1 2)

1

2m2ab1

2

sin(1 2) +1

2m2gb sin 2 = 0 (3)

Since Equations 2 and 3 are nowhere near linear, it is very difficult to solve theequations symbolically. Using the visualpython modules and the number crunchingability of the computer, it is possible to numerically iterate the equations over a set timeinterval. This is what we used to obtain all of our data, as well as the demonstrationat the end of our presentation.

6 Error Attribution

As mentioned in the previous section, the lagrangian equation for the two pendulum

system is nearly impossible to solve algebraically except for a select few special cases.When using the computer to numerically approximate the angles and energies involved,error must be introduced. Also, we set the simulation to iterate by thousandths of asecond, and we recorded the data every hundredth of a second. Ideally, we wouldbe able to handle and work with data with an even higher resolution, however thelimitations of our minds to handle hundereds of thousands of pieces of data was animportant factor, as well as the computers ability to update the chart of points inrealtime, when dealing with that many points.

7 Estimating Uncertainties

Estimating the uncertainties and error is different for this lab report, because we didnot actually use a real-world experiment. As mentioned in the previous section, thelagrangian must be numberically approximated by the computer, and this adds a very

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small amount of error, but since the computer is able to use approximately 15 decimaldigits through all calculations, the error is negligible. If we were going to try andfit a function to our data, we would perhaps want to use more points for a higherresolution. However, on the graphs contained in this document, the data points areso closely spaced that they look more like a smooth curve than a collection of uniquepoints. For the purposes of this lab report, the accuracy that we obtained is more than

enough.

8 Conclusion

In the course of our numeric simulation, we affirmed many aspects of our prediction.The system does indeed exhibit chaotic motion, and small changed in the initial con-ditions actually do produce very large and varied end results. The only area in whichwe were slight mistaken was in the time duration that was required to produce a noti-cible discrepancy between the two closely started pendulum systems. We had initiallythought that the difference would become evident quite quickly, but as our data andgraphs showed, the pendulums were relatively close for quite a while. This is evidentin the demonstration at the end of the presentation, where we simulate two double-

pendulums swinging at the same time, with the same upper pivot position.

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