To ordinary people, the innocent pendulum is a simple device without too many uses. For physicists and physics students, they are an excellent tool for measuring and predicting periodic motion. They are more complex than most basic devices because modeling their motion requires solving a differential equation. Some variations have no analytic solution, so numerical approximation techniques are required.
Chaos in Double Pendulums
For a final project demonstrating skills I learned in computational physics, I decided to use RK4 to estimate the second order differential equation of a double pendulum. After accurately predicting the motion of a single double pendulum, I modified the code to allow
for multiple in the same window. The motion of double pendulums is considered chaotic meaning the smallest changes in initial conditions result in drastically different outcomes.
At the start, each pendulum has both its angles a thousandth of a radian larger than the previous one.
At first they all appear in sync, but after a couple swings they explode in different directions. This is a simple but obvious demonstration of why small errors are often exaggerated over time and need to be noted when modelling chaotic situations.
Making a Pendulum Wave
In addition to the double pendulums, I had seen real life models of a "pendulum wave."
These demonstrate how the length of a pendulum is the primary predictor for the amount of time it takes to swing in a single period.
I modified the double pendulum code to represent single pendulums and calculated the required length for each pendulum to oscillate one more time than its predecessor during the duration of the simulation.
Code can be downloaded from - https://github.com/Cody-Ratterman/octave-pendulums
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