The mathematics of a boat.

Did you leave school and think you’d never need mathematics again? I didn’t. I’ve been doing various engineering projects over the years including 5 self designed, self built sportscars, one of which was battery powered. I built a suspension fork for my Proflex mountain bike around 18 months ago. I built a timber framed extension on our house a few years ago and around a year ago I built a timber framed building to house our forge. Now I’m building a boat. All these projects required the use of mathematics. Boat building has surprised me with the depth of maths and science I’ve needed to explore. I’ve discovered a lot and read some scientific papers in detail to find some of the answers I needed. This need for information has amazed me but admittedly I haven’t made things easy for myself.

As an enthusiastic cyclist and more specifically a mountain biker I wanted my boat to be pedal powered. The obvious way to accomplish this would be to spin a screw propeller behind the boat but this isn’t without it’s problems. You need to turn your pedalling energy through 90 degrees and pierce the back of the boat for the propeller shaft to run through a seal. Other alternatives are to use a paddle wheel like an old steamer. This can be quite efficient but requires a large paddle wheel to achieve this. After looking at the options for hull design I have decided to build a mini boat, which are usually less than 8 feet in length. This 8 foot length is determined by the 8′ X 4′ sizes of plywood, metal and other sheet materials, suited to build small boats. It will also be small and light enough for me to get it down to the water. I’m choosing an unusual form of propulsion in the form of a flapping foil. This is how fish, dolphins and whales have moved around for millions of years so it must work. It’s rather an experimental method which adds extra challenge to the project but I’d convinced myself that I could make it work. The mathematics of this type of propulsion is interesting.

I’ve watched some Youtube videos of experimental boats using flapping foil propulsion but they seem to me to often have a fatal flaw. They move the foil side to side like the tail of a tuna fish, which has the unfortunate effect of waggling the back of the boat around. Why not align the foil horizontally and move it up and down? The boat may still bob a little but will surely be far more stable than by arranging the movement the other way. I’ve also looked at how fish and marine mammals tails move and it isn’t just a matter of rotating it around a pivot at the body end. The tails move whilst maintaining a fairly constant angle of attack to the water, flipping at the ends of the stroke to move the opposite way. I’ve therefore designed a mechanism to achieve this. The angle of attack will be adjustable to find the best figure by trial and error once I’m on the water. One question was how big should my fish tail be and how far should it move? Mathematics has come to my rescue.

Flapping foils, like the wings of a bird or tail of a fish, were first described mathematically in 1878 by Czech physicist Vincenc Strouhal. He discovered a dimensionless quantity known now as the Strouhal Number (St) which is given by a simple equation…..

St=fA/U

Where f is the frequency of flapping, A is the amplitude of the movement and U is the speed of travel. Experiments have shown that efficiency is at it’s highest when St lies in the range 0.2 to 0.4. When fish like tuna or dolphins swim the figures have been measured and they also maintain their own St within the same range of efficiency. Who’d have imagined!

I’ve discovered that boats have an effective speed limit which is dependent on the length of the hull at the waterline. The speed can only be overcome if the hull rises up and planes, that is if it skims along the top of the water. This requires a big step up in energy so is not likely to be happening with a pedal powered boat so I’m afraid I’ll be limited to little over 4mph or 1.9 meters per second. My frequency will be the same as the pedal rotations on a bicycle which usually falls within the range of 1 to 2 hertz, or rotations per second. So how will things look at a reasonable cruising speed of 1.5 meters per second with a relaxed frequency of 1 hertz? I initially decided, so that I didn’t need deep water to actuate the fish tail to make my sweep (amplitude) 25 cm or 0.25 meters.

St=1 X 0.25 / 1.5 = 0.16666666

Well 0.1666666666 doesn’t look bad but is outside the range for best efficiency so how can I improve matters? I need increase the figures on top of the equation so I could pedal faster, or I could increase the length of each flap. Pedalling at a faster 1.5 hertz gives me….

St= 1.5 X 0.25 / 1.5 = 0.25

That’s right on the money but I can do better with a longer stroke of 30 cm, at a more relaxed 1 hertz…

St= 1 X 0.3 / 1.5 =0.2

In the zone again and by increasing the pedalling rate I’ll always stay in the efficient band. This is exciting to me. I may be alone in this excitement, I fully accept, but as I sail down the Leeds/Liverpool canal I can comfort myself with the thought that I’m not wasting my energy.

I’m starting with a big fish tail and if I can’t pedal quickly enough to get in the efficient area I’ll make it smaller, whilst still keeping the shape of a tuna fish tail. Launch is still some time away but the tension is building.

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