Gear-Tooth Contact, Leverage and Friction
Ideal tooth shape, contact on the line of centres, and the angles that make a drive easy or laborious
This densely written page, framed by two pairs of large pitch circles and a small toothed pair below, works out the mechanics of gear-tooth contact. Leonardo specifies the ideal tooth (thick through two-thirds to three-quarters of its span, as long as the gap between teeth) and warns that when a tooth tip meets the root of the opposing tooth the drive becomes laborious, because the force acts farther from the wheel's centre. He argues that the true contact should fall on the line joining the two centres, shown on line a n by teeth r t, and analyses the angle a 8 n: an obtuse angle eases motion, an acute one hinders it, tooth friction costing about a quarter of the load handled. A separate proportional note observes that removing equal parts from two quantities in a ratio does not preserve the ratio: from 2 and 4 (a double), remove one from each to get 1 and 3, now a triple.
On this page
Ideal tooth: thickness two-thirds to three-quarters, length equal to the gap
The tooth should be thick through the middle two-thirds of its span, up to three-quarters, and as long as the interval between teeth. The thicker the wheel that carries them, the more durable the teeth, and the interval between them somewhat greater.
Tip-on-root contact makes the drive more laborious
When the tips of the driving wheel's teeth touch the root of the driven wheel's teeth, the motion in the mover becomes more laborious, because the force then acts farther from the centre of the moving wheel.
Contact should fall on the line joining the two centres (a n, teeth r t)
The contact made by the teeth of toothed wheels should lie on the same line that runs from one centre to the other, as shown on line a n by teeth r t, which marks the centre of that contact. When the pinion's teeth touch the wheel's teeth at their root, the motion is harder than when they touch toward the tips.
The angle a 8 n, tooth friction, and where force is born and destroyed
When the angle a 8 n is more obtuse the motion is easier, and more acute, harder. The friction of teeth 8 and 6 equals in resistance a quarter of the load. With wheel n driving wheel a, force is born in angle a 8 n, reaches its peak at r t, and is destroyed in angle a b n; greater disparity between the wheels makes that angle more acute and the motion more laborious.
Removing equal parts from a ratio does not preserve it (2:4 becomes 1:3)
If you take equal parts from two quantities standing in some ratio, the remainder does not keep that ratio. From 2 and 4 (a double), remove one from each: 2 leaves 1, 4 leaves 3. So taking equal parts from a double, from double it becomes triple.
Durable teeth, and wider spacing on wheels far from the first mover
These teeth are noted as more durable than the others. The farther the multiplied wheels lie from the wheel of first motion, the wider the spaces of the teeth that drive the pinion must be made, since any small impediment would otherwise stop them; in the first wheels of the force, however, they are made almost equal.