## Pendulums

My BSc dissertation can be viewed here.

Many people have discovered it is possible to 'balance' a pendulum in an upright position by moving its bases right and left (think of balancing a broom on your fingers). What is not commonly know is that it is also possible to make a pendulum stand upright by moving the base-pivot up and down. If the pendulum is made to stand up right in this manor then it is also very stable and can be pushed so that it will effectively swing upside-down! The picture below shows what we are modelling.

 The equation of motion for this pendulum is given by, $${ d^2 \alpha \over dt^2 } + \Bigl( - { g \over l } + { \alpha \omega^2 \over l } \cos ( \omega t ) \Bigr) \sin \alpha = 0$$ which can be manipulated into a form similar to Mathieu’s equation. After a detailed Fourier analysis included in the dissertation I am able to find a region of stability for the inverted pendulum. This is shown on the graph opposite. Choosing parameters within the shaded region will mean it is possible to invert the pendulum.

With some help from a friend in the Engineering department we then set about making a model, the video of which is available below. However, its a little more exciting than that.

 Let us introduce the double pendulum. The equations of motion (which need to be solved simultaneously) are, $$l_2 { d^2 \beta \over d t^2 } + l_1 { d^2 \alpha \over d t^2 } \cos ( \beta - \alpha ) + l_1 \Bigl( { d \alpha \over d t } \Bigr)^2 \sin( \beta - \alpha ) + g \sin ( \beta ) = 0$$ $$l_1 { d^2 \alpha \over d t^2 } + m l_2 { d^2 \beta \over d t^2 } \cos ( \beta - \alpha ) - m l_2 \Bigl( { d \beta \over d t } \Bigr)^2 \sin( \beta - \alpha ) + g \sin ( \alpha ) = 0$$ $$m = { m_2 \over m_1 + m_2 }$$ where the 1 subscript refer to the higher bob and rod, and 2 for the lower.
 The double pendulum is very interesting to watch from a mathematician’s point of view because of the type of motion it shows. Many people are quick to shout at chaotic motion, but what I find interesting is the quasi-periodic motion it can show. The best description of quasi-periodic motion I have heard is 'periodic but not quite'. With this motion the phase path (see the dissertation) wraps itself around all the dimensions in a torus shape. In normal speak for a double pendulum this means it makes a ring-donut in 4-dimensions. It’s quite hard to draw in 4D so the picture opposite is a projection of this in 2D but makes the point well. Importantly, in the full amount of dimensions none of the lines ever cross each other. As well as making good pictures, the double pendulum is also interesting because it to can be made to have a stable upright position in the same way as the single pendulum does. We also tried this in our experiment, and the video is shown below. In the videos the pendulum collapses because the rig broke. This happened several times and we did not have time to make another and so we could not expand the experiment. We were going to attach a third because in theory you could make any amount of pendulums become inverted. If you imagine an infinite amount of short pendulums attached to each other they you effectively have a flexible wire, this too can be made to stand on end in the same manner.