Scaling factors

If you were to double the size of a boat would you get twice as much of everything? The answer is no and this article explains why.

It’s important to use scaling factors when comparing boats of different sizes. This is because we come face to  face with the law of ‘mechanical similitude’ – a conversation stopper of a phrase that describes a scaling law that’s central to our understanding of how boats work and what we can expect from them. The principle is best explained by example.

Let’s take a boat of any length and scale it up to exactly twice its original size. It would be tempting to think that its various dimensional properties would simply double and, indeed, some do. But not all.

  • Linear measurements such as length, beam and draught vary in proportion to the scaling factor. The results of doubling the size being: twice the length, twice the beam and twice the draught
  • Those values that involve areas, such as wetted surface and sail areas, vary as the square.
  • Anything that has a volume, like displacement, varies with the cube. So, incidentally, does the heeling effect of wind velocity on the sails.
  • Stability varies by the power of four.

    So, what does this mean numerically? Well, if say we double the size of a boat we get:

  • Twice the length, beam and draught (x2)
  • Four times wetted surface and sail areas. (2x2 = 4)
  • Eight times the displacement and heeling effect. (2x2x2 = 8)
  • Sixteen times the stability (2x2x2x2 = 16)

To understand the principles better, let’s imagine taking a single matchbox and being asked to build a cube of matchboxes exactly twice the size in every dimension. We would need a stack of eight matchboxes to accomplish this – exactly what the arithmetic tells us.

Of course, it would be absurd to scale up a design in such a simplistic manner. After all, we can expect people who sail small boats to be roughly the same height as those who sail larger ones, so there’s no need to have twice the headroom. But these dimensional relationships give us an important, though inexact, means of comparing the properties of boats of different sizes. If say, comparing a 10m boat with one of 20m we might deduce that the larger boat would have roughly four times the accommodation area and be sixteen times more stable – two very significant advantages that go with size. And, since build costs roughly follow displacement, we could guess that the larger vessel is also likely to be eight times more expensive!

Incidentally, it was the law of mechanical similitude that helped kill off commercial sailing ships. Faced with a growing threat from steam, they struggled to compete by building bigger and bigger ships – only to hit the scaling factors wall head on. With displacement advancing by the cube and sail area only by the square, they soon found they couldn’t set enough sail to propel all that added weight. Bizarre five- and six-masted schooners were their last desperate attempts. Meanwhile, steamers were discovering that they could carry more cargo (cube) for relatively less wetted surface area (square) so, for them, big was definitely beautiful. Game, set and match to mechanisation and a sad end to a glorious maritime era.

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