Double Pendulum
This post is one of a series of articles adapted from my university dissertation on data visualisation — see the rest of the series.
The single pendulum is a simple system to model, as it is simply the force of the weight of the bob acting with respect to gravity:
The formula for the angular acceleration is simply given by:
d²θ/dt² = −g·sin(θ) / R
Where g is the gravitational constant, θ is the angle (0 = vertical), R is the length of the rod and d²θ/dt² (the second derivative of θ with respect to time) is the angular acceleration. [Neumann, 2004]
However, the double pendulum (a second rod attached to the bob, with another bob on the end of that) is a more complex system and is much more difficult to model:
This system requires a set of two second order ODEs (Ordinary Differential Equations):
It is possible to convert this system of two second order ODEs into four first order ODEs, so that MATLAB can compute a numerical solution to model the motion of the double pendulum given a set of initial conditions (rod lengths, bob weights, initial angles and angular velocities), and then plot the path the pendulum takes in its motion under gravity (angle 1 vs. angle 2):
This behaviour of the two paths of motion can be described as quasiperiodic. Quasiperiodicity describes a pattern which is almost periodic. Periodic would mean it would take the same path every time, like the single pendulum:
I feel that it's difficult to see how the quasiperiodicity occurs in the above double pendulum plots, because all you see is a twisted continuous loop. In my experience mathematicians and physicists studying systems such as these find it difficult to grasp the concept from these drawn-out plots. Therefore I made a plan to help teach this concept.
Using the MATLAB program I wrote to model the double pendulum, which already plotted the motion of the angles (seen above), I made a simple adjustment: rather than plotting all the data in one go, I ran a for loop which plotted 1 point at a time, pausing for a tiny fraction of a second, so that the observer would be able to see the motion being drawn out in real time. This way they get to see the path being drawn, so they know where it went, the path that it took and exactly what happened.
In examples of quasiperiodicity this helps show how the path taken will differ ever so slightly from the previous run, and then again, just cutting past the previous two paths, always slightly out of its previous path. It's difficult for me to prove the usefulness of this approach on paper, but here are a few plots with times given to give an impression of what the observer would see:
