Dimensioning Wooden Gears - continued

Another way to mesh wooden gears is shown to the right:

The turning axles of the two wheels are parallel. It should be obvious that this use case is not ideal, as the point of contact jumps back and forth (we will see this further down), but if just an idler is needed it may be acceptable.

For a working gear train two 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.

Somehow the pitch of the pegs must be determined for each gear ratio. The length of the pegs is rather uncritical; just avoid the pegs of one gear rubbing against he disk of the other gear's disk.

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) = c1 + |r1|*(cos a1, sin a1) (center of base of red peg)

(2) m2(x,y) = c2 + |r2|*(cos a2, 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 upper wheel)

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

The pitch circle radius is also determined by the number (n) and diameter (d) 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.
Another variable is the minimum distance Z of the pitch circles. In the drawing above Z is set to 0 for maximum torque transfer. Due to manufacturing or other mechanical constraints Z however may be slightly larger or smaller.

The two pictures below illustrate the effect of the two variables e and z on the minimum distance between the two pegs marked red and green above.

Again some elementary math will provide the dependency of the minimum distance from N1, N2, e and z. The distance is given by

(6)

with

(7) f = d*(1 + e/200) e given in % of d

(8) g = 2*z / (200 + e) e and z given in % of d

To find the absolute minimum this needs to be differentiated and the roots found. The relevant part of the differential dt is
(9)

Again I couldn't find a simple solution. The following graphs shall give a qualitative understanding of the results, while the approximate value for each set of he four parameters can be requested from the Calculation Sheet (requires Javascript enabled in Your browser).