There are many reasons why students should go to university and do a mathematical (or mathematically related) degree. A good reason is because of interesting and stimulating  investigations. Take for example my dissertations. For my BSc I looked at pendulums and created  working models of inverted pendulums. For my MSc dissertation I looked into splashes, which included looking at the mechanics of stone-skipping.

Pendulums

My BSc dissertation can be downloaded 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 modeling.

The equation of motion for this pendulum is given by,

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,

,

     where the 1 subscript refer to the higher bob and rod.

The double pendulum is very interesting to watch from a mathematicians 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 make a ring-donut in 4-dimensions. Its 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 make 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 manor.

       

Click on the pictures for videos of the inverted single pendulum, and then the inverted double pendulum.

Stone Skipping

My MSc dissertation can be downloaded here.

Splashes are a very interesting are of fluid dynamics. My MSc looked at the some problems relating to the impact of solid bodies on fluids. The most fun application of this is standing at the beach playing skipping-stones (of course there are more serious examples such as maritime structures and bouncing-bombs). Lets look at a stone hitting the water surface,

The equations of motion for a circular stone are,

              where,

    

     and is the density of the water.

It is interesting to see under which conditions, for the parameters, we can achieve the stone skipping effect. An example of this is shown below.

Clearly though, the only question anybody would be interested in is how to get more skips! Fixing the title of the stone to be about 20° (since this gives a good range of throwing angle) I investigated. I created the graph below which seems to show that the number of skips is (approximately) a linear relation ship with the initial speed the stone is thrown at. This means if you want to achieve more skips then throw it with a bit more speed! Common sense really.

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