Dimensioning Wooden Gears

Interestingly the easiest way to construct wooden gears is not cutting notches into discs, as these will be weakened by the grain of the wood and, what's worse, are hard to replace once broken. Our forefathers therefore settled on another construction, namely replaceable pegs seated in holes of a wooden disk.

While building of such gear wheels is easy, the correct dimensioning seems to be some black magic; to lift the veil a bit the following text hopefully will provide enough understanding to get them first time right in your next project.
Let's concentrate on the basic configuration depicted to the right: The turning axles of the two wheels intersect in a right angle. For a working gear train three requirements must be met:

The pegs of one wheel must fit into the gap between the pegs of the other wheel and vice versa with minimum play.
Each peg must not touch any other peg than the two pegs defining its gap on its way into and out of its gap.
The peg carriers must not touch.

Somehow the dimension and pitch of the pegs must be determined for each gear ratio.
Concentrate on the two colored pegs and mentally turn the wheels, keeping the colored pegs in close contact:

Relative to the green peg, the red peg as a whole just moves up and down.

Relative to the red peg, the green peg primarily moves back and forth.

This basic consideration is all we need to define the constraints for two pegs

If the center of the red peg is above the top of the green peg, the pegs will not intersect if in the projection onto the vertical plane (front view) the distance between the center of the red peg and any corner of the green peg is longer than half the diameter of one peg.

The pegs will not intersect as long as the horizontal distance of their centers is longer than their diameter.

If the center of the green peg is in front of the front of the red peg, the pegs will not intersect if in the projection onto the horizontal plane (top view) the distance between the center of the green peg and any corner of the red peg is longer than half the diameter of one peg.

Next step is the description of the motion of the pegs.

The fundamental idea behind all kinds of gearing is the equivalence between two ratios: The ratio of the numbers of teeth is inversely proportional to the number of turns for every pair of meshing gears. Using this rule, the coordinate system shown right and looking at the pitch circles (the circle the centers of the pegs are located on), we can write down the equations for each peg:

(1) m1(x,y,z) = c1 + |r1|*(cos a1, sin a1, 0) (center of base of red peg)

(2) m2(x,y,z) = c2 + |r2|*(cos a2, 0, -sin a2) (center of base of green peg)

For the relevant pegs the angles (a1, a2) must be changed in parallel to stay in mesh, mathematically spoken a parameter (t) is needed:

(3) a1(t) = (0.75 - t / N1) * 2 * PI (N1 is number of pegs in vertical wheel)

(4) a2(t) = (0.25 + (t+0.5) / N2) * 2 * PI (N2 is the number of pegs in the horizontal wheel)

If the centers c1 and c2 are chosen so that the passing pegs (of diameter d) just touch the carrier disc once, the effective length L of each each peg must be at least d/2 to reach the situation in the middle column of the table above.

The pitch circle radius (r) is also determined by the number (n) and diameter of the pegs:

(5) r = (d + e/2) / sin( PI / n )

"Play" e is needed to give a little room in the gap for the entering and leaving peg, will be elaborated further down the page.

The diagram to the left now shows the distance between the red and green pegs during rotation for different values of length L and for a gap size of a little more (e = 7%) than peg diameter d. The longer the peg, the smaller the distance becomes during entry into and exit from the gap.

Some closer study results in an equation for the function plotted above, valid around the lower extremes (e and L given in % of d):

Using standard math (dy/dt) we can derive an equation turning 0 for the parameter t of the local minima:

The next graph shows the dependency of the minimum distance and the peg length for different amounts of play but equal sized wheels. To use it, pick the play you want to use and then see whether the distance is still positive for the intended peg length.

The graph to the left shows the variation of the allowed peg length depending on the ratio of the two gears. It is understandable that the peg must be shorter to pass the other wheel's pegs if that wheel is bigger.

Finally let's look at the dependency between possible peg length and play e.

Unfortunately there is no simple formula linking N1, N2, L and e. But the calculation sheet gives a good approximation, using interpolated values based on accurately calculated tables (requires JavaScript enabled in your browser).