The Double Pendulum: Seemingly Simple Chaos th
Tuesday, March 15 , 2011 East Hall Room 30- 10:20 a.m. Objective: The objective of this experiment is to obtain a better understanding of a chaotic system through experimentation. To understand this experiment, one needs to understand the basic principles behind Chaos Theory. Specifically, in this experiment, the sensitivity of initial initial conditions will be examined. Hypothesis: Based on the research which I have have conducted, I have come to an understanding that if the initial conditions vary slightly, a big difference can occur in final results, whether the results are position, velocity, angle, or other variable variable units of measurement. In most most experiments, a hypothesis tries to draw conclusions conclusions based on similar data. dat a. But in this case, I am not looking for a pattern in data. In fact, I am looking for the exact opposite. I was looking looking for there not to be detectable patterns in data; signs of chaos. I believe that if the double pendulum system runs for long enough, starting with the same initial conditions, the patterns in motion will eventually diverge and differ. Software: Interactive Physics Version 3.0.2 Copyright
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Knowledge Revolution, Inc. Portions
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Portions of guide by guideWorks, LLC
Apparatus: There are two main portions to this experiment which much first be explained. The first set of observations will be done in a higher energy state, where masses two and three start at a higher position, having more potential energy. The second part of the experiment will use a lower energy state, where the masses will start at a lower position. In all trials, masses one and three will be equal, as well as masses two and four. The initial conditions in the beginning will be the same, and we will be observing the system to notice differences over time. Procedure: 1) Open interactive physics. 2) Use the menu of shapes on the side and select a circle. 3) Create a fixed point and a line. Attach them to the circle. 4) Use another line to connect another circle to the first one. For the first part of this experiment, set this second circle higher up on the screen. This will ensure that the system is starting in a higher energy state.
5) Double-click on each circle to add mass. 6) Click and drag to highlight the whole system. Copy. 7) Hit “Ctrl V” to paste an exact copy of your system to the screen. 8) Grab the newly copied system with the mouse and drag it away from the initial system. Make sure the two separate systems are far enough apart so that they will not collide at any point in time. Note: Now you have two identical systems! 9) Click the world menu at the top. Set gravity to vertical, as well as turn off electrostatics. 10) Click Run and record the system and observations over time using a stop watch. 11) Click Stop to stop the motion. Record observations. 12) Try to determine when the system first goes into chaos; when both systems appear to have started to differ in motion. Many trials are strongly encouraged. 13) Repeat this process two more times: once with a low energy configuration, and once for an extended period of time. Data: Table 1. Observations of Initial High Energy System Time (s) 5 10 15 20 25 30 35 36 37 38 39 40 41 42 43
Trial One I I I I I I I I I I I I I O O
Trial Two I I I I I I I I I I I I O O O
I= In Sync O= Out of Sync Trial Three I I I I I I I I I I I O O O O
Table 2. Observations of Initial Low Energy System Time (s) 5 10 15 20 25 26 27 28 29 30 31 32 33 34 35 36 37
Trial 1 I I I I I I I I I I I I I I I O O
Trial 2 I I I I I I I I I I I I I O O O O
I= In Sync O= Out of Sync Trial 3 I I I I I I I I I I I I I I O O O
Conclusions: After examining the observations collected, it has definitely been confirmed that a double pendulum is an example of a chaotic system. In the Higher Energy State Experiment, between 40 and 43 seconds, the systems would start to behave differently. At this point in time, the systems looked very different, as can be seen in the sketches provided within our data. Also, in the Lower Energy State Experiment, the systems behaved chaotically between 34 and 36 seconds. In Part Three, the long term experiment, the systems still seemed to behave erratically after three minutes, a slightly longer period in time. In comparison with my hypothesis, this is in a sense what I expected to see. I cannot fit any graphical pattern to this data, because there is no equation that governs exactly how a double pendulum behaves. This demonstrates that even a computer, which may be accurately precise up to eight decimal places, can still show chaos. If there was a difference after many decimal places, the observations collected demonstrate that sensitivity to initial conditions may indeed be a factor which leads to chaos. There may have been a few sources of error while conducting this experiment. For example, time was measured using a stop watch and not directly on the computer. There is most likely a way to do this within the Interactive Physics, but the software was not cooperating. Therefore, pushing stop on a stop watch and on the computer screen may have resulted in a time lag between when the clock was stopped and what screen picture the observation sketch was based upon. This experiment could be improved by more accurate timing, perhaps within the program. This experiment still shows that even with seemingly “simple” motion, chaos can still prevail.
