Bow Physics

This page talks about the physics behind bow design. You don’t need to know any of this to shoot a bow, but it is interesting.

Longbow

When drawing a longbow, the further back you pull it the harder it gets. If you were to draw a graph with how hard it is to pull going from bottom to top, and how far back you are pulling it going from left to right, then it would look something like this:

Longbow Force-Draw Curve

This is called a Force-Draw Curve (fdc.) You can see that the weight you are holding increases fairly linearly as you draw the bow back. Interestingly the energy stored in the bow, and therefore imparted to the arrow, is precisely the area under this curve.

Recurve

The design of a recurve bow, with cleverly laminated limbs, significantly improves things. For a recurve, the graph will look more like this (the dotted line is the previous longbow graph.)

Recurve Force-Draw Curve

The design of the limbs means that although you still need more force to pull further back, the amount of force you need to add gradually lessens. Also the shape of the line means that the energy stored in the bow is higher, and therefore the maximum draw weight can be lower than for a longbow while still delivering an equal or greater impetus to the arrow (to be honest, a good longbow can approach this type of curve too, but the effect is much more marked for recurves.)

Compound

For a compound bow, the situation is much more complicated:

Compound force-draw curve

This diagram shows two different compound force-draw curves. The lighter blue is for a hard cam with an agressive and small let-off, The second darker blue curve is for a soft cam with a smooth and large let-off. The fdc of a typical recurve is shown for comparison.

Compound bows are designed to give a constant draw weight throughout the draw, ending with a let-off so that you are actually holding less weight at full draw than the maximum draw weight. For this reason the maximum draw weight can be much higher than either a longbow or a recurve, and that fact, combined with the constant draw weight means that the energy stored in the bow is much greater.

The difference between a soft cam and a hard cam is just how quickly the bow reaches its maximum draw weight. A softer cam is much more comfortable to draw, but as you can see from the figure it does sacrifice some energy, and hence arrow speed. You need a very steady draw to shoot a hard cam, otherwise your arrows will be falling off the arrow rest with embarrasing regularity.

The valley is an archers term for the point of minimum weight beyond the peak draw weight. The wall is just the bow up against the stops, you cannot pull past it. The let-off technically, is the difference beween the weight in the valley and the peak weight, divided by the peak weight and expressed as a percentage. Most bows allow the amount of let-off to be modified, as in the figure. The same principle of “soft” vs. “hard” cams applies to the difference between an “agressive” let-off and a “smooth” let-off: a smooth let-off is easier to draw but sacrifices energy.

How Compound Bows Work

Tthe basic idea behind any compound bow is a simple block and tackle. You may remember this from school physics lessons but here’s a picture anyway.

Block and Tackle

When you pull down on the rope attached to the outer wheel, both wheels rotate clockwise. Because the inner wheel has a smaller radius, the rope attached to the inner wheel does not move as far, but does so with equal energy, and so can pull that much harder. The ratio between the outer and the inner radii is a gear ratio. The lower that ratio, the less the inner wheel’s rope moves and the harder it can pull.

If we replace ropes with strings and cables, and put one of these wheels on each end of a bow, we get the following twin cam setup:

Twin cam System

The cable attached to each inner wheel runs to the centre of the opposing wheel, and the single string connects both outer wheels.

This explains how a compound bow is able to pull its sprung steel limbs, but it doesn’t explain that constant draw weight from the graph above. That requires one last refinement: for a compound bow, the wheels aren’t circular.

Here’s a picture of the upper cam of a very simple compound bow at rest:

Compound cam at rest

The outer cam has an oval shape, and is oriented so that at rest the gear ratio is quite low. The actual ratio at any moment is measured as the ratio between the radii at the point each cable makes contact with its wheel.

As the bow is drawn, the cam rotates clockwise so that a wider and wider radius applies to the outer wheel, and therefore a higher and higer gear ratio is in force. This compensates for the increased tension in the limbs, until at full draw it actually over-compensates, which provides the let-off:

Compound cam at full draw

The diagrams above are not intended to be scientifically accurate or measured, they’re just to give the idea. Particularly, the precise shapes of the cams in a compound bow are designed to produce a specific draw weight curve, rather than being simple parabola, and the diameter of the inner cam is often made to vary too.