the dependence of the double pendulum’s chaotic nature upon its angle of release

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The Dependence of the Double Pendulum‟s

Chaotic Nature upon Its Angle of Release

Name: Lloyd James

Candidate Number: 000159-018

Category: Physics

Supervisor: Louay El Halabi

Word Count: 3740

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Abstract.

„Dependence upon initial conditions‟ is defined as the degree to which the future state of a

system will be impacted by a change in its initial configuration. A double pendulum is a device

comprised of one pendulum swinging freely on the end of another. The purpose of this

experiment was to investigate the question: “How does the dependence upon initial release

conditions of a double pendulum‟s motion vary with the angle from which the pendulum is

released?” This was to be measured by comparing the resulting motion paths when the pendulum

was released several times from the same angle, with random, minor deviations in the initial

conditions.

Executing the experiment required the release of the pendulum a set number of times from a

variety of angles and taking photographs of the motion every 0.3 seconds for a total of 5.1

seconds. Each of the resulting 595 photographs required measurements of the two pendulum

angles and of the time elapsed. To synchronise the releases from the same angle, a program was

coded in C++ that would use the equations of motion of the double pendulum, along with the

pendulum‟s measurements, to simulate the first few seconds of motion in a discrete fashion. It

would then compute the elapsed time since release for the first photograph. Then, the motion

paths of the pendulum for each of the trial releases from each of the release angles could be

interpolated, and the average standard deviation between the trials over the 5 second time period

could be calculated for each release angle.

It was concluded that the dependence upon initial release conditions of the double pendulum‟s

motion is constant up to a certain critical angle, which can be calculated based upon the

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pendulum‟s measurements. Thereafter, the dependence of its motion upon release angle

increases exponentially.

Words: 300

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Table of Contents. 1.0 Introduction. ............................................................................................................................................................ 5

1.1 Purpose of Research. ........................................................................................................................................... 5

1.2 Background Information Regarding Double Pendulum Motion. ........................................................................ 5

2.0 Experiment. ............................................................................................................................................................. 7

2.1 Preparation/Setup. ............................................................................................................................................... 7

2.2 Photographing Double Pendulum Motion Paths. ................................................................................................ 8

2.3 Measuring Double Pendulum Motion Paths ........................................................................................................ 9

2.4 Measuring Divergence between Double Pendulum Motion Paths. ................................................................... 10

2.5 Comparing Deviations of Double Pendulum Motion Paths between Release Angles. ..................................... 12

3.0 Analysis. ................................................................................................................................................................ 13

3.1 Analysis. ............................................................................................................................................................ 13

3.1.1 Analysis of Observed Motion Paths. .......................................................................................................... 13

3.1.2 Analysis of Measured Motion Paths .......................................................................................................... 14

3.1.3 Analysis of Standard Deviations between Motion Paths ........................................................................... 18

3.1.4 Analysis of Average Standard Deviation over Time between Motion Paths ............................................. 20

3.2 Evaluation. ........................................................................................................................................................ 21

3.2.1 Release and Recording of Pendulum ......................................................................................................... 21

3.2.2 Measurements of Time and Angle ............................................................................................................. 22

3.2.3 Determination of Standard Deviations ....................................................................................................... 22

4.0 Conclusion. ............................................................................................................................................................ 23

4.1 Conclusion......................................................................................................................................................... 23

4.2. Areas for Further Research............................................................................................................................... 23

5.0 Works Cited ........................................................................................................................................................... 24

Works Cited ................................................................................................................................................................. 24

6.0 Appendices ............................................................................................................................................................ 25

6.1 Deriving Equations of Motion Using Lagrangian Mechanics ........................................................................... 25

6.2 Details of Program “Pendulum.exe” ................................................................................................................. 29

6.3 Determining Uncertainty Values ....................................................................................................................... 33

6.4 Deriving the Formula for Critical Point ............................................................................................................ 34

6.5 Sample Raw Motion Path Data for 90° and 120° (2 out of 7) Releases ............................................................ 35

6.6 Motion Path Graphs of Arm 1 ........................................................................................................................... 39

6.7 Ratios of Arm 1 to Arm 2 over 2-3 Second Period ........................................................................................... 46

6.8 Standard Deviation of Motion Paths of Arm 1 .................................................................................................. 53

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1.0 Introduction.

1.1 Purpose of Research.

A system that experiences chaotic motion is one in

which small differences in initial conditions yield greatly

diverging outcomes, making the system difficult to predict

(Kellert, 1993). A double pendulum, which is essentially one

pendulum attached to the end of another pendulum (Figure 1), is

an example of such a system, with its motion path depending

strongly upon the release angle (one of its initial conditions).

Observation indicates that this dependence is more evident

when the angle of release is greater. The pendulum will appear

to accelerate and decelerate randomly when the pendulum is

released from a higher point, in a way that is not obvious when

the pendulum is released from a lower point. The purpose of the research detailed in this essay

was to investigate the question:

“How does the dependence upon initial release conditions of a double pendulum‟s motion

vary with the angle from which the pendulum is released?”

1.2 Background Information Regarding Double Pendulum Motion.

It is important to make the distinction that the chaotic nature of the double pendulum is

not due to random forces acting on the pendulum (as in Brownian Motion) but is rather an

example of deterministic chaos, which occurs in nonlinear systems with only a few degrees of

freedom (Gitterman, 2010).

Figure 1- Double Pendulum

Arm 1 Arm 2

θ1 θ2

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Due to the forces of constraint acting on the arms of the double pendulum (fixing the

arms in circular paths relative to their axes) and the multiple position coordinates necessary to

describe the system, it becomes very difficult to create equations of motion for the double

pendulum using Newtonian Mechanics. Instead, we must turn to Lagrangian Mechanics.

Lagrangian Mechanics are based upon D‟Alembert‟s Principle and apply to systems regardless

of whether or not they conserve energy or momentum. As well, they allow us to deal only with

scalar, rather than vector quantities and to choose any set of generalized coordinates to describe

the system (Calkin, 1996). Using Lagrangian Mechanics, the equations of motion for the system

can be found by firstly choosing the angles between the pendulum arms and the vertical as

generalized coordinates, then by expressing the kinetic and potential energies of the system using

these coordinates, their time derivatives, and time. From there, the Lagrangian can be formed and

substituted into Lagrange‟s equations and the required differentiations can be performed. This

process is detailed in Appendix 1, on page 26. The results of this analysis are the two equations:

𝜃1 =

6𝑚2𝑔

7𝑚1𝑙1sin 𝜃1

−𝜃2 2𝑙2sin(𝜃1 + 𝜃2)

𝑔 sin 𝜃1 −𝑚1

𝑚2− 2

𝜃2 =

−𝜃2 2𝑙1 sin 𝜃1 + 𝜃2 − 𝑔 sin 𝜃2

2𝑙12

𝑙2+ 2𝑙1 cos 𝜃1 + 𝜃2 +

76 𝑙2

Note: The angle between the inner pendulum arm and the vertical is 𝜃1, and the angle between the outer arm and

the vertical is 𝜃2. 𝑙1, 𝑙2, 𝑚1, and 𝑚2 represent their corresponding lengths and masses, respectively. Newton’s

notation for differentiation is used here, so 𝜃2 represents the angular velocity of 𝜃2, and 𝜃2

represents the angular

acceleration.

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2.0 Experiment.

2.1 Preparation/Setup.

The goal of the experiment was to conclude with measurements of the dependence upon

initial conditions of the pendulum‟s motion, over different angles. It was decided that if the

pendulum was released a number of times from the same angle, the variation in the range of

pendulum arm positions that occurred at a specific time after release would potentially provide a

measure of how the initial release conditions could impact the subsequent movements of the

pendulum arms. Completing this test for each of several different angles would provide the

necessary data to answer the research question. It was noticed that for the double pendulum used,

the apparent „unpredictability‟ of the motion would greatly increase around the release angle of

105°, so the angles chosen to provide data were 90° to 120° at increments of 5°.

In order to track the motion path of the double pendulum, it was fixed to a wall, and high-

resolution camera was placed adjacent to it, on a tripod at the same level as the center of the

double pendulum. The camera took photographs with a resolution of 5184x3456, with the

pendulum centered in camera‟s field of view. To ensure the pendulum was released from a

specific angle, a large semi-circular paper scale was placed to one side of the pendulum, and

marked with lines radiating out from the center of the pendulum at precise angles to the vertical

(Figure 1). The double pendulum was purchased for this experiment from

chaoticpendulums.com, and is made of aluminium, with low friction bearings connecting the two

arms, and the one arm to the wall. The inner arm (also to be termed Arm 1) is 17.3 cm in length,

and 3.8 cm in width, with a mass of 275g. The outer arm (Arm 2) is 16.4 cm in length, and also

3.8cm in width, with a mass of 110g. To allow measurement of the angle between the pendulum

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arms and the vertical from a photograph, a straight, visible line was made down the center of

each of the pendulums arms (Figure 1). A stopwatch was fixed to the wall next the pendulum, in

the camera‟s view, to provide a time measurement for each photograph of the pendulum.

It was assumed that because the camera used can only take photographs every 0.3

seconds, it would be impossible to know with sufficient precision the time of release, unless the

release occurred at exactly the same time as the photograph was taken. It was critical however, to

know the time of release in order to synchronise the times for each of the trials. As a solution, a

short program was coded in C++, called „Pendulum.exe‟ that, given the position of the two

pendulum arms and the angle of release as input, would use the equations of motion to find the

time taken for the pendulum to go from release to the given position. It would achieve this by

simulating the motion path in a discrete, rather than continuous fashion (seeing as the equations

of motion for the double pendulum cannot be integrated over time). The details of the program

can be found in Appendix 2, on page 30.

2.2 Photographing Double Pendulum Motion Paths.

For a single release angle, the double pendulum was released five times from approximately the

same location. Just before each release, the wall-mounted stopwatch would be started, and the

camera would begin to take photographs continuously every 0.3 seconds. At release, the double

pendulum would be as straight as possible so that the angles between each arm and the vertical

were equal. Photographs would be taken over a 7 second period, so as not to be in danger of

having insufficient photographs for the 5 second period of time with which the motion path

would be measured. This process was repeated for each of the 7 determined release angles,

resulting in 840 photographs being taken.

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2.3 Measuring Double Pendulum Motion Paths

To measure the motion paths, an open-source computer

program called ImageJ was used. This program has the

function to measure an angle that is drawn onto the

photograph. ImageJ was used to measure the angle between

each of the two pendulum arms and the vertical for all the

photographs up to five seconds from release for each of the

trials for each of the release angles (Figure 2). The time

shown on the stopwatch for each photograph would also be

recorded. A total of 1785 data points were generated through this process. The angles measured

for the first photograph taken after release would then be inputted into Pendulum.exe, which

would output the time since release. This measured time deviation would then be used to adjust

all the times collected from the stopwatch to be synchronised relative to the time of release. The

following data table (Figure 3) is an example of the first half of the data that would be collected

using ImageJ for three of the five trials for one arm for a single release angle.

Figure 3 Release Photo.

1

Photo.

2

Photo.

3

Photo.

4

Photo.

5

Photo.

6

Photo.

7

Photo.

8

Time 1 (s) ± 0.01s 0 0.24 0.55 0.87 1.19 1.45 1.76 2.08 2.34

Angle 1(°) ± 0.6° ≈90.0 5.1 -81.1 17.2 61.2 -69.6 -21.8 79.4 2.7

Time 2(s) ± 0.01s 0 0.30 0.62 0.88 1.19 1.56 1.88 2.14 2.46

Angle 2(°) ± 0.6° ≈90.0 -50.4 -32.9 82.2 4.0 -65.9 53.1 33.9 -77.2

Time 3(s) ± 0.01s 0 0.25 0.57 0.88 1.20 1.47 1.78 2.09 2.41

Angle 3(°) ± 0.6° ≈90.0 -11.3 -72.7 50.0 40.6 -74.9 -5.6 85.5 -28.3

Note: Determination of uncertainty values is detailed in Appendix 3.

When the measurements were completed for all seven release angles, graphs were compiled

using the data. Sine fits were generated in the Logger Pro graphing software to estimate the

motion paths of the double pendulum after release. The following (Figure 4) is an example of

Figure 2 – Pendulum Image Processing

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such a graph for the motion paths of arm 1 for the five trials from a 90° release point. Refer to

Appendix 6 on page 40 for a complete set of motion path graphs for Arm 1 from all release

angles.

2.4 Measuring Divergence between Double Pendulum Motion Paths.

Because the first photograph after the release for each trial varies between 0 and 0.3 seconds, the

measured angles and times do not necessarily align from trial to trial, and so cannot be directly

compared. The solution required an Excel function to find 9 points between each of the existing

data points that fall on the Excel-plotted smooth curves. This vastly increased number of points

on the graph allowed all of the points to be shifted as necessary to align with each other, so that

given a particular time for one trial, comparative points could be found at a comparative time

(within approximately 0.03 seconds) for each of the other trials. This meant that at 160-170

points over a five second period, the standard deviation of the pendulum arm angles at a

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particular time could be found and plotted. The following table (Figure 5) and graph (Figure 6)

are an example of these results. Refer to Appendix 8 on page 54 for more complete results.

Figure 5: Standard Deviations of Arm 1 Angles between the Five Trials from 90° Release

Time (s) ± 0.01s Standard Deviation of Arm Angles (°) ± 1°

0.03 4

0.05 9

0.08 13

0.10 18

0.13 22

0.17 23

0.19 25

0.22 22

0.26 18

0.29 15

Note: Determination of uncertainty values is detailed in Appendix 3, on page 34.

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2.5 Comparing Deviations of Double Pendulum Motion Paths between Release Angles.

To compare the deviations among the motion paths between the seven release angles, all the

standard deviation measurements over the five second time periods were averaged together,

resulting in 7 data points for each double pendulum arm. The two graphs of these sets were then

combined together by multiplying all the values for Arm 1 (which had a lower mean averaged

deviation) by a calculated value, so that the mean averaged deviation for Arm 1 was the same as

the mean averaged deviation for Arm 2. This effectively scaled the graph of Arm 1 to be

comparable to the graph of Arm 2. The two graphs were then combined to create an overall

representation of the effect of release angle on the pendulum‟s dependence upon its initial release

conditions. This is shown, along with its component curves, in Figure 7. The best curve to fit the

data is the logistic function, which experiences no change in y-value up to a certain point, then

experiences a rapid increase up to another point, after which the increase tapers off. Then there

is no further increase.

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3.0 Analysis.

3.1 Analysis.

There are four main aspects of the experiment that can be analysed to produce a unified

conclusion: the observed motion paths of the double pendulum, the measured motion paths, the

calculated standard deviations between the motion paths, and the averages of these standard

deviations.

3.1.1 Analysis of Observed Motion Paths.

As previously detailed when discussing the choice of initial release angles, visual observation of

the double pendulum in motion reveals that for all release angles up to 105°, after release the

pendulum will simply swing back and forth in a predictable fashion, and Arm 2 never does a full

rotation around its axis. However, from 110° and higher Arm 2 starts to rotate unpredictably

around its axis and Arm 1 speeds up and slows down in a random manner. 110.5° was also

calculated as the theoretical angle at which a released pendulum would start with enough

gravitational potential energy for Arm 2 to be able to rotate 360°(refer to Appendix 4, on page 35

for this calculation). From this, a possible physical explanation for the rapid increase in

dependence upon initial conditions around 110° emerges. From angles of around 110° and

above, the pendulum would have enough energy for its Arm 2 to flip completely over, and this

rotation would cause a significant deviation from the typical range of ratios between the two

angles. The equations of motion for the double pendulum indicate that the acceleration of the

arms is dependent on both the absolute and relative positions of the two pendulum arms. This

deviation would therefore cause Arm 1 to experience a different acceleration than it would

experience in the same position when released from a different angle, further increasing the

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differences in angle ratio and in angle position. This could result in a feedback loop, rapidly

driving initially close pendulum positions further apart.

3.1.2 Analysis of Measured Motion Paths

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Looking at the measurements made of the double pendulums motion paths, over 5 trials from

each of 7 different angles, there is a clear trend that emerges. From angles 90° to 105° (Figure 8),

all of the motion paths follow a regular (not diverging over time), almost sinusoidal path, and

between trials there is very little divergence between the motion paths. At 110° however (Figure

9), the double pendulum‟s motion follows the same path between trials for one period of

oscillation, but thereafter the motion paths appear to slowly diverge. Some of the trials have

motion paths that are more regular, and some have motion paths that are irregular. At 115°

(Figure 10), this divergence of the motion paths of the trials happens considerably sooner, and is

considerably more noticeable. These observations of the measured motion paths appear to agree

with the analysis of the observed pendulum motion path, where it was found that 110° was the

release angle at which chaotic motion became visually evident, and that 110° is the closest angle

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of release to the point at which the pendulum has enough energy to flip Arm 2, which has been

hypothesised to be the cause of the chaotic motion.

These measurements made of the motion paths can be used to calculate the ratio between the

angle of Arm 2 to the angle of Arm 1. When these ratios are found for all points of a specific

release angle, and all are combined into a graph of arm angle ratios against time, without regard

for which points were done for which trial, the previous conclusions are reinforced. This is

verified because for points from release angles below 110° (Figure 11, below), the ratio of the

angle of Arm 1 to the angle of Arm 2 stays within a small range from 0.5 – 0.8. This pattern

repeats periodically, broken up by segments where the ratio becomes much greater in magnitude,

positive or negative, where the pendulum reaches the bottom of its swing. There, small

differences in angle, as between 0.05° and -0.25° can result in atypically large ratios.

Also, for points from release angles above 110° (Figure 12), there is a significantly larger spread

of angle ratios, at a certain time and over time. This, together with the previous observation,

supports the notion that the degree to which the pendulum‟s motion is chaotic depends on the

ratio between the two arms. This in turn can be thrown off its normal path by a flipping of the

outer arm caused by having a higher release angle. This idea is further supported in Figure 13,

where it is shown that from a release angle of 110°, there is a period of approximately 1.4

seconds where the ratio of the second angle to the first angle stays within a limited range.

However after 1.4 seconds there is a significant increase in the range of these ratios. This

„explosion‟ in the range of ratios corresponds closely with the point in the graph of the motion

paths of the releases from 110° at which the motion paths of the different trials begin to diverge.

This divergence can be seen to occur almost exactly 1.4 seconds into the motion. Refer to

Appendix 7 on page 47 for graphs of this ratio over time for each of the release angles.

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3.1.3 Analysis of Standard Deviations between Motion Paths

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Analysis of the standard deviations between the motion paths over time reveals little more than

has already been determined through analysis of the measured and observed motion paths. What

is does show though is that the difference between the pendulum angles oscillates over time,

being highest when the pendulum approaches 0° to the vertical and lowest when it approaches its

initial release position, or the position opposite to it. This analysis also supports the previous

conclusion in that applying a linear fit to the graph shows that the standard deviation between the

trials changes little up to 110°, and is equally likely to decrease as it is to increase. At and above

110° however, the rate of change of standard deviation increases greatly as the release angle

increases.

3.1.4 Analysis of Average Standard Deviation over Time between Motion Paths

Analysis of the data represented in Figure 7 shows that there is no initial increase of the

dependence upon initial release conditions of the pendulum as the release angle is increased, up

to 105°. This initial phase corresponds with the group of release angles with which there is no

flipping of Arm 2, and hence a small range of ratios between the two angles. Between 105° and

115° there is a rapid increase of the sensitivity as the release angle increases. This period

corresponds with the period in which the pendulum experiences flipping of Arm 2. The rapid

increase in sensitivity is likely due to the increased release angle and consequent increased

kinetic energy causing more flips of Arm 2, and causing the flips to happen sooner. Between

115° and 120° the rate at which the sensitivity increases as the angle increases tapers off. At this

point, the sensitivity will be approaching the maximum measurable using this method, because

from this release angle, after a few seconds the distribution of the arms would be almost random.

It can be seen that for both Arms 1 and 2, there is a dip in the dependence upon initial conditions

at the 100° release. This data point represents a significant anomaly, considering that the points

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on either side of it have almost the same value. However it must be the product of random error

considering that there is no physical explanation for a simple system to become less chaotic

when starting with a greater potential energy.

3.2 Evaluation.

There are three parts to the experiment in which error could have accumulated: the release and

recording of the pendulum, the measurement of the angles and of time, and the determination of

the standard deviation of the angles between trials at one particular time.

3.2.1 Release and Recording of Pendulum

This stage of the experiment is not likely to be a significant source of error. While there was no

significant attempt to release the pendulum from exactly the same spot, this lack of attempt was a

necessary part of the experiment, at it resulted in slight deviations of the initial release conditions

which enabled the later, larger deviations to be measurable. What may have had an effect on the

experiment, however, is the range in initial angles that resulted from this. A larger range of

angles would have directly affected the result by increasing the standard deviations between the

motion paths of the different trials. Considering that no attempt made to keep the range smaller

than normal or larger than normal, it is unlikely that range of release angles would be especially

large for one particular set of trials. However, such an anomaly could be a possible explanation

of the dip in the graph of average standard deviation of motion paths against release angle

(Figure 7) identified earlier. The camera used had a high definition and was not moved

throughout the experiment, and so was unlikely to be a significant source of error.

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3.2.2 Measurements of Time and Angle

Although the angles were measured using computer software, which has a high precision of

angle measurement, this is not representative of the actual precision achieved, because the angle

must be drawn with a mouse onto the image along a line that is slightly thicker. So, repeated

measurements will likely yield slightly different results. The uncertainty on this process was

calculated by making measurements on one set of data twice, and using the highest difference

between a measurement in one set and a measurement in the other as the uncertainty. (See

Appendix 3 for details) The time measurement, recorded with a stopwatch iPhone application, is

given to two decimal places, and can be assumed to have an uncertainty equal to one in the

lowest decimal place given (0.01 s in this case).

3.2.3 Determination of Standard Deviations

The primary issue here was that, because interpolated data points were required to be able to

compare the motion paths at approximately the same time, and so a function had to be used that

would attempt to draw a smooth curve between all of the data points. While this method

certainly provides a good approximation of the motion path, it will be inexact as it will not be

able to recognise if the outer arm of the pendulum flips, for example, or it may not be continued

to the farthest angle reached by the pendulum during a swing if data points are given just before

and then just after reaching the peak angle, at which its kinetic energy is 0. The interpolated

points do still capture the motion path for the most part however, so the final result should not be

affected significantly.

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4.0 Conclusion.

4.1 Conclusion.

Based on the results achieved by this experiment, it can be concluded that when a double

pendulum is released, the degree to which the resulting motion depends upon its initial release

conditions is determined by whether the release angle is greater or lower than a specific critical

angle. This critical angle is equal to the release angle at which the pendulum has enough starting

potential energy to, later in its motion, have enough kinetic energy to experience a 360° rotation

of its outer arm. If the release angle is less than this, then the motion of the pendulum will be

regular and easily predictable and the ratio of the angle of one arm to the other will be kept

tightly within a small range. If the release angle is greater than this critical angle, the pendulum

will become chaotic, unpredictably changing velocity, and having the ratio of the one arm to the

other vary greatly. Additionally, by equating the kinetic energy required to flip the pendulum‟s

second arm with the potential energy of the pendulum when raised to a certain angle, the

equation relating a pendulum‟s critical angle and the masses and lengths of that pendulum‟s arms

can be found to be (Refer to Appendix 4, on page 35 for detail):

𝜃𝑐 = cos−1 −𝑙2𝑚2

12𝑚1𝑙1 +𝑚2𝑙1 +

12𝑚2𝑙2

4.2. Areas for Further Research.

There are two points of further research that are important to clarify the results of this

experiment. The first course of action should be to ascertain that the dip in the graph of average

standard deviation of motion paths against angle (Figure 7) is in fact the result of random error,

24

and not the result of some underlying physical factor not being considered here. The second

required investigation would be to confirm that the critical angle varies with a change in the

masses and lengths of the pendulum arms as predicted by the established formula. This could be

confirmed by repeating this experiment multiple times with pendula of different arm lengths and

masses, and confirming that each time the critical angle conforms to the formula.

5.0 Works Cited

Works Cited Calkin, M. G. (1996). Lagrangian and Hamiltonian Mechanics. Singapore: World Scientific Publishing

Co. Pte. Ltd.

Gitterman, M. (2010). The Chaotic Pendulum. Singapore: World Scientific Publishing Co. Pte. Ltd.

Kellert, S. H. (1993). In the Wake of Chaos: Unpredictable Order in Dynamical Systems. Chicago:

University of Chicago Press.

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6.0 Appendices

6.1 Deriving Equations of Motion Using Lagrangian Mechanics

Due to the difficulty of using Newtonian Mechanics to find the equations of motion of a double

pendulum, it becomes necessary to used Lagrangian Mechanics to determine them. The

Lagrangian of the system is defined as the difference between the kinetic and potential energy of

the system.

𝐿 = 𝐸𝐾 − 𝐸𝑃

With the double pendulum, the kinetic energy is the sum of the linear kinetic energy of the center

of mass of each of the two pendulum arms

1

2𝑚1𝑣1

2 +1

2𝑚2𝑣2

2

and the rotational kinetic energy around the center of mass of each rod, which is defined as the

product of the moment of inertia and half the rotational velocity.

1

2𝐼𝜔2 =

1

2𝐼𝜃1 2 +

1

2𝐼𝜃2 2 =

1

2𝐼(𝜃1

2 + 𝜃2 2)

The potential energy, relative to a 0 potential energy level at the axis around which Arm 1

rotates, is defined as the sum of the products of the mass of each pendulum arm, the acceleration

due to gravity, and the y axis component of the center of mass of each arm

𝑚1𝑔𝑦1 +𝑚2𝑔𝑦2

Therefore, the Lagrangian can be stated as:

𝐿 =1

2𝑚1𝑣1

2 +1

2𝑚2𝑣2

2 +1

2𝐼 𝜃1

2 + 𝜃2 2 − 𝑚1𝑔𝑦1 −𝑚2𝑔𝑦2

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Knowing that the linear velocity of a center of mass of a pendulum arm is equal to the product of

the distance from that center of mass to the axis of rotation and the rotational velocity of that arm

around the axis,

𝑣1 = 𝑟𝜔 =1

2𝑙1 ∙ 𝜃1

; 𝑣2 = 𝑟𝜔 = 𝑙12 +

1

4𝑙2

2 − 𝑙1𝑙2cos(𝜋 − 𝜃1 − 𝜃2) ∙ 𝜃2

*note: the radical in the above formula is found using the law of cosines on an imagined triangle formed between the axis of

rotation, the connection between the two pendulum arms, and the midpoint of the second arm of the pendulum.

that the y components of the position of each center of mass are

𝑦1 =−𝑙1

2cos 𝜃1 ;𝑦2 = −𝑙1 cos 𝜃1 −

𝑙22

cos 𝜃2

and that each pendulum arm, resembling a slender rod with an axis through its midpoint, has a

moment of inertia of

𝐼1 =1

3𝑚1𝑙1

2; 𝐼2 =1

3𝑚2𝑙2

2,

the formula for the Lagrangian can be rewritten as:

𝐿 =1

2𝑚1

1

2𝑙1 ∙ 𝜃1

2

+1

2𝑚2 𝑙1

2 +1

4𝑙2

2 − 𝑙1𝑙2cos(𝜋 − 𝜃1 − 𝜃2) ∙ 𝜃2

2

+1

2

1

3𝑚1𝑙1

2 𝜃1 2

+1

2

1

3𝑚2𝑙2

2 𝜃2 2 − 𝑚1𝑔

−𝑙12

cos 𝜃1 − 𝑚2𝑔 −𝑙1 cos 𝜃1 − 𝑙22

cos 𝜃2

=1

8𝑚1𝑙1

2𝜃1 2 +

1

2𝑚2 𝑙1

2 +1

4𝑙2

2 − 𝑙1𝑙2cos(𝜋 − 𝜃1 − 𝜃2) 𝜃2 2 +

1

6𝑚1𝑙1

2𝜃1 2

+1

6𝑚2𝑙2

2𝜃2 2 +

1

2𝑚1𝑔𝑙1 cos 𝜃1 +𝑚2𝑔𝑙1 cos 𝜃1 +

1

2𝑚2𝑔𝑙2 cos 𝜃2

27

Knowing that the difference between the time derivative of the partial derivative of the

Lagrangian with respect to the time derivative of a generalized coordinate and the partial

derivative of the Lagrangian with respect to a generalized coordinate is 0,

𝑑

𝑑𝑡 𝜕𝐿

𝜕𝜃1 −

𝜕𝐿

𝜕𝜃1= 0;

𝑑

𝑑𝑡 𝜕𝐿

𝜕𝜃2 −

𝜕𝐿

𝜕𝜃2= 0

to find the equations of motion of the double pendulum one can find the required derivatives.

𝜕𝐿

𝜕𝜃1

=1

4𝑚1𝑙1

2𝜃1 +

1

3𝑚1𝑙1

2𝜃1 =

7

12𝑚1𝑙1

2𝜃1

𝑑

𝑑𝑡 𝜕𝐿

𝜕𝜃1 =

𝑑

𝑑𝑡

7

12𝑚1𝑙1

2𝜃1 =

7

12𝑚1𝑙1

2𝜃1

𝜕𝐿

𝜕𝜃1=

𝜕

𝜕𝜃1

1

2𝑚2𝜃2

2𝑙12 +

1

8𝑚2𝜃2

2𝑙22−

1

2𝑚2𝜃2

2𝑙1𝑙2cos(𝜋 − 𝜃1 − 𝜃2) +1


+𝑚2𝑔𝑙1 cos 𝜃1

= −1

2𝑚2𝜃2

2𝑙1𝑙2sin(𝜋 − 𝜃1 − 𝜃2) −1

2𝑚1𝑔𝑙1 sin 𝜃1

− 𝑚2𝑔𝑙1 sin 𝜃1 =1

2𝑙1𝑚2𝑔 sin 𝜃1

−𝜃2 2𝑙2sin(𝜃1 + 𝜃2)


𝑚2− 2

𝑑

𝑑𝑡 𝜕𝐿

𝜕𝜃1 −

𝜕𝐿

𝜕𝜃1= 0 →

7

12𝑚1𝑙1

2𝜃1

=1

2𝑙1𝑚2𝑔 sin 𝜃1

−𝜃2 2𝑙2sin 𝜃1 + 𝜃2


𝑚2− 2 → 𝜃1

=6𝑚2𝑔

7𝑚1𝑙1sin 𝜃1

−𝜃2 2𝑙2sin(𝜃1 + 𝜃2)


𝑚2− 2

28

𝜕𝐿

𝜕𝜃2

= 𝑚2 𝑙12 +

1

4𝑙2

2 − 𝑙1𝑙2cos(𝜋 − 𝜃1 − 𝜃2) 𝜃2 +

1

3𝑚2𝑙2

2𝜃2

= 𝑚2𝜃2 𝑙1

2 +1

4𝑙2

2 + 𝑙1𝑙2 cos 𝜃1 + 𝜃2 +1

3𝑙2

2

𝑑

𝑑𝑡 𝜕𝐿

𝜕𝜃2 =

𝑑

𝑑𝑡 𝑚2𝜃2

𝑙12 +

1

4𝑙2

2 + 𝑙1𝑙2 cos 𝜃1 + 𝜃2 +1

3𝑙2

2

=1

2𝑚2𝜃2

𝑙2 2𝑙1

2

𝑙2+ 2𝑙1 cos 𝜃1 + 𝜃2 +

7

6𝑙2

𝜕𝐿

𝜕𝜃2=

𝜕

𝜕𝜃2

1

2𝑚2𝜃2

2𝑙12 +

1

8𝑚2𝜃2

2𝑙22

+1

2𝑚2𝜃2

2𝑙1𝑙2 cos 𝜃1 + 𝜃2 +1


= −1

2𝑚2𝜃2

2𝑙1𝑙2 sin 𝜃1 + 𝜃2 −1

2𝑚2𝑔𝑙2 sin 𝜃2

= −1

2𝑚2𝑙2 𝜃2

2𝑙1 sin 𝜃1 + 𝜃2 + 𝑔 sin 𝜃2

𝑑

𝑑𝑡 𝜕𝐿

𝜕𝜃2 −

𝜕𝐿

𝜕𝜃2= 0 →

1

2𝑚2𝜃2

𝑙2 2𝑙1

2

𝑙2+ 2𝑙1 cos 𝜃1 + 𝜃2 +

7

6𝑙2

= −1

2𝑚2𝑙2 𝜃2

2𝑙1 sin 𝜃1 + 𝜃2 + 𝑔 sin 𝜃2

→ 𝜃2 =

−𝜃2 2𝑙1 sin 𝜃1 + 𝜃2 − 𝑔 sin 𝜃2

2𝑙12

𝑙2+ 2𝑙1 cos 𝜃1 + 𝜃2 +

76 𝑙2

29

6.2 Details of Program “Pendulum.exe”

Note: Technical variables and functions are hidden to conserve space.

void Accelerations();

//This function will take the length and mass variables, along with the arm 2 angular velocity, as input. The output will be the angular

accelerations of arms 1 and 2.

void Calculate(float timestep);

//This function will discretely calculate the velocities and positions of the two arms over given timesteps, until the position of the first arm

reaches the target value.

float angacc1; //Angular acceleration of arm 1

float angacc2; //Angular acceleration of arm 2

float angvel1; //Angular velocity of arm 1

float angvel2; //Angular velocity of arm 2

float ang1; //Angle of arm 1

float ang2; //Angle of arm 2

float mass1; //Mass of arm 1

float mass2; //Mass of arm 2

float length1; //Length of arm 1

float length2; //Length of arm 2

float g = 9.81; //Acceleration due to gravity

float targetang1; //The arm 1 angle up to which the simulation and time counting is performed.

float targetang2; //The arm 2 angle up to which the simulation and time counting is performed.

int main()

{

system("cls");

mass1 = 0.275;

mass2 = 0.110;

length1 = 0.173;

length2 = 0.164;

//Measurements of the pendulum

time1reached = false;


finished = false;

cout << "Angle 1A: " << endl;

cin >> input1;

ang1 = atof(input1.c_str());

//Input of the starting angle of arm 1

cout << "Angle 2A: " << endl;

cin >> input2;

ang2 = atof(input2.c_str());

//Input of the starting angle of arm 2

cout << "Angular Velocity 1A: " << endl;

cin >> input3;

angvel1 = atof(input3.c_str());

//Input of the starting velocity of arm 1

cout << "Angular Velocity 2A: " << endl;

cin >> input4;

angvel2 = atof(input4.c_str());

//Input of the starting velocity of arm 2

cout << "Angle 1B: " << endl;

30

cin >> input5;

targetang1 = atof(input5.c_str());

//Input of the target angle of arm 1

cout << "Angle 2B: " << endl;

cin >> input6;

targetang2 = atof(input6.c_str());

//Input of the target angle of arm 2

//The below process will simulate the motion path, and measure the time, over three different time steps, which grradually get smaller.

//This results in a high precision of the final result.

//The program will simulate the motion until at the next time step, the pendulum passes the target angle.

//The program will then shrink the time step, and continue this process.

prevang1 = ang1;

prevang2 = ang2;

prevangvel1 = angvel1;


prevangacc1 = 0;

prevangacc2 = 0;

counter = 0;

Calculate(0.05);

prevang1 = finalang1 [counter];

prevang2 = finalang2;

prevangvel1 = finalangvel1;


prevangacc1 = finalangacc1;


ang1 = finalang1 [counter];

ang2 = finalang2;

angvel1 = finalangvel1;


angacc1 = finalangacc1;


finished = false;



counter = 1;

Calculate(0.01);

prevang1 = finalang1 [counter];

prevang2 = finalang2;





ang1 = finalang1 [counter];

ang2 = finalang2;





finished = false;



counter = 2;

Calculate(0.002);

31

//The result of this process is a measurement of the final velocities of the two arms, and more importantly:

// a measurement of the time taken by the pendulum to reach the target angle.

system("cls");

cout << "Timestep - 0.05: " << endl << "Final Angle: " << finalang1 [0] << endl << "Final Time: " << angle1time [0] << endl << endl;



cout << "Final Angular Velocity of Angle 1: " << finalangvel1 << endl;

cout << "Final Angular Velocity of Angle 2: " << finalangvel2 << endl;

system("pause");

return 0;

}

void Calculate(float timestep)

{

//This part checks to see if the current angle is larger than the target.

if ((targetang1 > ang1))

{

larger1 = true;

}

else

{

larger1 = false;

}

if ((targetang2 > ang2))

{

larger2 = true;

}

else

{

larger2 = false;

}

for (int step = 0; finished != true; step++)

{

//This is the calculation of what the positions and velocities of the arms will be in the next time step.

ang1 = ang1 + angvel1 * timestep + 0.5 * angacc1 * timestep * timestep;

ang2 = ang2 + angvel2 * timestep + 0.5 * angacc2 * timestep * timestep;

angvel1 = angvel1 + angacc1 * timestep;

angvel2 = angvel2 + angacc2 * timestep;

//This is where the function to calculate the accelerations is called.

Accelerations();

//Technical functions:

if ((larger1 == true))

{

if ((time1reached == false))

{

if((targetang1 <= ang1))

{

angle1time [counter] = step * timestep;

finalang1 [counter] = prevang1;

finalangvel1 = prevangvel1;

finalangacc1 = prevangacc1;


time1reached = true;

}

}

}

32

else

{


{

if((targetang1 >= ang1))

{

angle1time [counter] = step * timestep;

finalang1 [counter] = prevang1;





}

}

}

if ((larger2 == true))

{


{

if((targetang2 <= ang2))

{

angle2time = step * timestep;

finalang2 = prevang2;




}

}

}

else

{


{

if((targetang2 >= ang2))

{

angle2time = step * timestep;

finalang2 = prevang2;




}

}

}

if ((time1reached == true))

{

finished = true;

}

prevang1 = ang1;

prevang2 = ang2;



prevangacc1 = angacc1;

prevangacc2 = angacc2;

}

}

void Accelerations()

{

//Derived equation of motion for the acceleration of arm 2

33

float dividend = angvel2 * angvel2 * -1 * length1 * sin(ang1 + ang2) - g * sin(ang2);

float divisor = 2 * length1 * length1 / length2 + 2 * length1 * cos(ang1 + ang2) + 1.16667 * length2;

angacc2 = dividend / divisor;

//Derived equation of motion for the acceleration of arm 1

dividend = 6 * mass2 * g * sin(ang1);

divisor = 7 * mass1 * length1;

float factor1 = dividend / divisor;

dividend = -1 * angvel2 * angvel2 * length2 * sin(ang1 + ang2);

divisor = g * sin(ang1);

float factor2 = dividend / divisor;

float factor3 = mass1 / mass2;

float factor4 = 2;

angacc1 = factor1 * factor2 - factor1 * factor3 - factor1 * factor4;

}

6.3 Determining Uncertainty Values

The uncertainty on all time values throughout the experiment is 0.01s. This value corresponds

with one of the smallest digit given on the electronic stopwatch. This uncertainty does no change

through calculation as no calculations are done on the time itself.

The initial uncertainty on the angles measured from the photographs was found to be 0.6°. This

value was found by doing a set of 34 data points twice, and then looking at the deviations

between the first set of measurements and the second. While the average deviation was only 0.2°,

the largest deviation to occur was 0.6°, and consequently it was thought that this would be the

best value to represent the uncertainty.

When these angle data points are made into the standard deviation between angles, the

uncertainty was elevated up to 1°. This is because during the calculations for the standard

deviation, the only calculation that has an effect on the uncertainty is the addition of two values,

both of which have an uncertainty of 0.6°

34

6.4 Deriving the Formula for Critical Point

It was determined that the „critical point‟ of the double pendulum, the point after which the

dependence upon initial release conditions increases greatly, is equal to the point at which the

double pendulum starts with enough potential energy to be able to have the kinetic energy

required to flip arm 2 360°.

To determine the kinetic energy required to flip arm 2 360°, one simply needs to calculate how

much potential energy the arm has at a 180° angle to the vertical (at which point arm 2 has

enough potential energy to perform a full rotation). This can be measured using the formula:

𝐸𝑝 = −𝑚𝑔ℎ, where m is the mass of arm 2, and h is equal to the length of arm 2 (seen as the

distance between the position of the center of mass of arm 2 at 180° and at 0° is equal to the

length of arm 2). In this specific case, the equation is therefore: 𝐸𝑝 = 𝑚2𝑔𝑙2

To determine the initial potential energy of the pendulum when released from a certain angle, the

formula 𝐸𝑝 = −1

2𝑚1𝑔𝑙1 cos 𝜃1 − 𝑚2𝑔𝑙1 cos 𝜃1 −

1

2𝑚2𝑔𝑙2 cos 𝜃2 (which was found in

appendix 1) can be used. Because at release, the angles of the first arm to the vertical and of the

second arm to the vertical are equal, this equation reduces to: 𝐸𝑝 = −cos 𝜃 𝑔 1

2𝑚1𝑙1 +𝑚2𝑙1 +

1

2𝑚2𝑙2 .

At the critical point, the initial potential energy is equal to the required kinetic energy, resulting

in:

35

−cos 𝜃𝑐 𝑔 1

2𝑚1𝑙1 +𝑚2𝑙1 +

1

2𝑚2𝑙2 = 𝑚2𝑔𝑙2 → cos 𝜃𝑐

=−𝑚2𝑔𝑙2

𝑔 12𝑚1𝑙1 +𝑚2𝑙1 +

12𝑚2𝑙2

→ 𝜃𝑐

= cos−1 −𝑚2𝑙2

12𝑚1𝑙1 +𝑚2𝑙1 +

12𝑚2𝑙2

Using this formula, the critical point of a double pendulum with the dimensions of the one used

in the experiment will be:

𝜃𝑐 = cos−1 −0.110𝑘𝑔 ∙ 0.164𝑚

12 ∙ 0.275𝑘𝑔 ∙ 0.173𝑚 + 0.110𝑘𝑔 ∙ 0.173𝑚 +

12 ∙ 0.110𝑘𝑔 ∙ 0.164𝑚

= cos−1 −0.01804𝑘𝑔 ∙ 𝑚

0.0518375𝑘𝑔 ∙ 𝑚 = cos−1 −0.34801 = 110.4°

This calculated result corresponds almost exactly with the release angle after which the motion

of the double pendulum appears chaotic.

6.5 Sample Raw Motion Path Data for 90° and 120° (2 out of 7) Releases

Motion Path Data for Pendulum When Released from 90° - Arm 1 - Photographs 1 - 9 Photograph 1 2 3 4 5 6 7 8 9 Time 1 (s) ± 0.01s 0.00 0.24 0.55 0.87 1.19 1.45 1.76 2.08 2.34 Angle 1(°) ± 0.6° 90.0 5.1 -81.1 17.2 61.2 -69.6 -21.8 79.4 2.7 Time 2(s) ± 0.01s 0.00 0.30 0.62 0.88 1.19 1.56 1.88 2.14 2.46 Angle 2(°) ± 0.6° 90.0 -50.4 -32.9 82.2 4.0 -65.9 53.1 33.9 -77.2 Time 3(s) ± 0.01s 0.00 0.25 0.57 0.88 1.20 1.47 1.78 2.09 2.41 Angle 3(°) ± 0.6° 90.0 -11.3 -72.7 50.0 40.6 -74.9 -5.6 85.5 -28.3 Time 4(s) ± 0.01s 0.00 0.19 0.45 0.82 1.09 1.41 1.72 1.98 2.29 Angle 4(°) ± 0.6° 90.0 39.4 -78.5 -0.7 85.2 -31.8 -49.2 68.8 16.8 Time 5(s) ± 0.01s 0.00 0.14 0.40 0.72 1.03 1.35 1.61 1.93 2.19 Angle 5(°) ± 0.6° 90.0 70.3 -52.8 -35.9 83.9 1.50 -79.7 24.3 66.9

36

Motion Path Data for Pendulum When Released from 90° - Arm 1 -Photographs 10 - 18 Photograph 10 11 12 13 14 15 16 17 18 Time 1 (s) ± 0.01s 2.71 2.97 3.29 3.56 3.87 4.19 4.51 4.82 5.08 Angle 1(°) ± 0.6° -79.3 42.5 49.7 -69.1 -12.9 80.5 -14.3 -59.9 65.1 Time 2(s) ± 0.01s 2.80 3.09 3.41 3.67 3.99 4.30 4.62 4.88 5.20 Angle 2(°) ± 0.6° 2.9 79.6 37.8 -45.5 70.6 9.9 -78.0 21.4 61.4 Time 3(s) ± 0.01s 2.67 2.99 3.30 3.57 3.89 4.20 4.57 4.88 5.15 Angle 3(°) ± 0.6° -58.3 58.1 34.0 -78.2 5.0 73.7 -66.1 -16.7 77.1 Time 4(s) ± 0.01s 2.67 2.98 3.24 3.56 3.88 4.19 4.51 4.82 5.14 Angle 4(°) ± 0.6° -80.3 31.7 56.0 -71.8 -4.1 79.9 -27.4 -49.5 68.5 Time 5(s) ± 0.01s 2.51 2.82 3.09 3.40 3.72 3.98 4.30 4.61 4.93 Angle 5(°) ± 0.6° -60.2 -24.8 79.8 -0.9 -75.9 29.7 55.2 -71.4 -3.7

Motion Path Data for Pendulum When Released from 90° - Arm 2 - Photographs 1 - 9 Photograph 1 2 3 4 5 6 7 8 9 Time 1 (s) ± 0.01s 0.00 0.24 0.55 0.87 1.19 1.45 1.76 2.08 2.34 Angle 1(°) ± 0.6° 90.0 25.8 -123.4 82.5 102.9 -81.0 -84.1 127.3 -16.8 Time 2(s) ± 0.01s 0.00 0.30 0.62 0.88 1.19 1.56 1.88 2.14 2.46 Angle 2(°) ± 0.6° 90.0 -104.9 -84.0 93.9 25.2 -113.2 100.6 74.2 -106.8 Time 3(s) ± 0.01s 0.00 0.25 0.57 0.88 1.20 1.47 1.78 2.09 2.41 Angle 3(°) ± 0.6° 90.0 -42.4 -115.3 90.7 85.1 -114.4 -18.1 89.3 -65.4 Time 4(s) ± 0.01s 0.00 0.19 0.45 0.82 1.09 1.41 1.72 1.98 2.29 Angle 4(°) ± 0.6° 90.0 84.1 -111.4 6.9 92.8 -70.7 -112.5 112.0 45.2 Time 5(s) ± 0.01s 0.00 0.14 0.40 0.72 1.03 1.35 1.61 1.93 2.19 Angle 5(°) ± 0.6° 90.0 102.6 -111.4 -87.5 90.3 22.2 -116.4 78.8 87.2

Motion Path Data for Pendulum When Released from 90° - Arm 2 - Photographs 10 - 18 Photograph 10 11 12 13 14 15 16 17 18 Time 1 (s) ± 0.01s 2.71 2.97 3.29 3.56 3.87 4.19 4.51 4.82 5.08 Angle 1(°) ± 0.6° -85.2 97.3 113.4 -99.2 -45.9 121.6 -72.6 -86.5 91.8 Time 2(s) ± 0.01s 2.80 3.09 3.41 3.67 3.99 4.30 4.62 4.88 5.20 Angle 2(°) ± 0.6° 21.9 94.3 75.8 -106.5 111.5 26.2 -102.5 74.8 97.1 Time 3(s) ± 0.01s 2.67 2.99 3.30 3.57 3.89 4.20 4.57 4.88 5.15 Angle 3(°) ± 0.6° -119.7 113.7 67.5 -101.6 28.3 103.9 -86.1 -71.8 117.0 Time 4(s) ± 0.01s 2.67 2.98 3.24 3.56 3.88 4.19 4.51 4.82 5.14 Angle 4(°) ± 0.6° -108.0 95.8 92.1 -91.3 -31.4 110.1 -65.1 -98.5 112.7 Time 5(s) ± 0.01s 2.51 2.82 3.09 3.40 3.72 3.98 4.30 4.61 4.93 Angle 5(°) ± 0.6° -100.6 -86.6 106.3 0.8 -112.2 94.1 91.7 -88.6 -33.2

37

Motion Path Data for Pendulum When Released from 120° - Arm 1 - Photographs 1 - 9 Photograph 1 2 3 4 5 6 7 8 9 Time 1 (s) ± 0.01s 0.00 0.13 0.42 0.74 1.06 1.37 1.63 1.89 2.21 Angle 1(°) ± 0.6° 120.0 104.9 -48.7 -128.6 102.1 57.0 -81.2 44.6 5.1 Time 2(s) ± 0.01s 0.00 0.29 0.62 0.92 1.29 1.71 2.02 2.34 2.62 Angle 2(°) ± 0.6° 120.0 -12.0 -138.9 59.8 77.2 -132.9 -6.9 126.5 -2.0 Time 3(s) ± 0.01s 0.00 0.26 0.58 0.84 1.16 1.42 1.74 2.05 2.37 Angle 3(°) ± 0.6° 120.0 22.4 -137.8 -96.6 75.4 116.1 -27.5 -55.9 101.3 Time 4(s) ± 0.01s 0.00 0.29 0.60 0.92 1.18 1.50 1.76 2.08 2.39 Angle 4(°) ± 0.6° 120.0 -14.5 -139.8 33.9 130.7 -35.9 -107.9 34.3 114.1 Time 5(s) ± 0.01s 0.00 0.29 0.61 0.83 1.14 1.50 1.82 2.13 2.45 Angle 5(°) ± 0.6° 120.0 -8.9 -138.0 46.7 128.9 -68.8 -128.2 77.8 40.2

Motion Path Data for Pendulum When Released from 120° - Arm 1 - Photographs 10 - 18 Photograph 10 11 12 13 14 15 16 17 18 Time 1 (s) ± 0.01s 2.53 2.79 3.14 3.42 3.68 4.00 4.32 4.58 4.89 Angle 1(°) ± 0.6° -76.2 53.3 -16.9 -36.4 62.6 -18.3 -26.9 76.9 -41.1 Time 2(s) ± 0.01s 2.92 3.23 3.55 3.87 4.24 4.55 4.86 5.18 5.44 Angle 2(°) ± 0.6° -126.3 -8.3 112.9 -85.0 -10.6 103.0 -40.3 -96.0 68.3 Time 3(s) ± 0.01s 2.63 2.95 3.21 3.53 3.79 4.10 4.42 4.74 5.00 Angle 3(°) ± 0.6° -6.3 -51.7 93.4 -28.8 -105.1 99.2 64.5 -91.5 -29.5 Time 4(s) ± 0.01s 2.66 2.97 3.29 3.55 3.87 4.18 4.45 4.76 5.06 Angle 4(°) ± 0.6° -20.5 -107.7 21.8 107.5 -76.1 -55.3 122.5 64.0 -77.9 Time 5(s) ± 0.01s 2.87 3.19 3.50 3.76 4.13 4.45 4.71 5.03 5.35 Angle 5(°) ± 0.6° -88.1 85.8 4.6 -111.5 64.8 55.0 -70.2 59.4 -0.3

Motion Path Data for Pendulum When Released from 120° - Arm 2 - Photographs 1 - 9 Photograph 1 2 3 4 5 6 7 8 9 Time 1 (s) ± 0.01s 0.00 0.13 0.42 0.74 1.06 1.37 1.63 1.89 2.21 Angle 1(°) ± 0.6° 120.0 147.2 -125.7 -22.5 -18.2 177.1 54.3 -100.8 102.6 Time 2(s) ± 0.01s 0.00 0.29 0.62 0.92 1.29 1.71 2.02 2.34 2.62 Angle 2(°) ± 0.6° 120.0 -74.0 -34.3 -17.3 178.5 -41.0 -109.4 -10.7 86.4 Time 3(s) ± 0.01s 0.00 0.26 0.58 0.84 1.16 1.42 1.74 2.05 2.37 Angle 3(°) ± 0.6° 120.0 76.6 -68.0 148.2 31.4 159.7 -120.1 139.5 78.6 Time 4(s) ± 0.01s 0.00 0.29 0.60 0.92 1.18 1.50 1.76 2.08 2.39 Angle 4(°) ± 0.6° 120.0 -77.0 -42.3 -1.3 -93.5 13.4 -178.6 62.5 129.1 Time 5(s) ± 0.01s 0.00 0.29 0.61 0.83 1.14 1.50 1.82 2.13 2.45 Angle 5(°) ± 0.6° 120.0 -69.5 -40.7 -20.5 -81.8 76.1 61.4 -55.7 -141.3

38

Motion Path Data for Pendulum When Released from 120° - Arm 2 - Photographs 10 - 18 Photograph 10 11 12 13 14 15 16 17 18 Time 1 (s) ± 0.01s 2.53 2.79 3.14 3.42 3.68 4.00 4.32 4.58 4.89 Angle 1(°) ± 0.6° -84.8 -174.4 80.6 -150.1 44.4 -171.4 85.6 -26.6 -137.1 Time 2(s) ± 0.01s 2.92 3.23 3.55 3.87 4.24 4.55 4.86 5.18 5.44 Angle 2(°) ± 0.6° 54.7 -95.4 -3.8 68.1 -175.4 -162.3 -138.8 -81.2 110.7 Time 3(s) ± 0.01s 2.63 2.95 3.21 3.53 3.79 4.10 4.42 4.74 5.00 Angle 3(°) ± 0.6° -88.9 118.6 -0.6 -152.1 36.6 -39.3 41.1 -159.4 -89.6 Time 4(s) ± 0.01s 2.66 2.97 3.29 3.55 3.87 4.18 4.45 4.76 5.06 Angle 4(°) ± 0.6° -101.9 -174.1 118.7 -140.9 -90.0 -49.0 24.1 -13.6 -153.4 Time 5(s) ± 0.01s 2.87 3.19 3.50 3.76 4.13 4.45 4.71 5.03 5.35 Angle 5(°) ± 0.6° -34.8 107.9 139.2 -71.6 65.5 101.1 -138.2 140.9 -13.3

39

6.6 Motion Path Graphs of Arm 1

Note: Only the

motion paths of

data collected

for Arm 1 of

the pendulum

are shown here,

for brevity and

because those

motion paths

reflect the same

pattern as

these, the only

difference

being that they

are typically

more chaotic

than the motion

paths for the

Arm 1, as Arm

2 experiences a

great deal more

motion.

46

6.7 Ratios of Arm 1 to Arm 2 over 2-3 Second Period

Note: A graph is

shown of these

ratios for a 2

second period for

release angles of

90 – 105 degrees,

as they show the

pattern that exists

and continues for

the remaining time

period. A graph is

shown for a 3

second period for

the remaining

graphs as they

require this time to

demonstrate the

more chaotic

nature of their

motion

53

6.8 Standard Deviation of Motion Paths of Arm 1

Note: Graphs of

standard deviation

are only shown for

Arm 1, for brevity

and because the

Arm 2 graphs

show no additional

patterns that are

not demonstrated

in the Arm 1

graphs. The only

difference is that

the standard

deviation is

generally higher

for Arm 2.

the dependence of the double pendulum’s chaotic nature upon its angle of release

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