ISEGORIA / MATH ENCYCLOPEDIA
Gears and mechanisms: transmitting motion exactly
How toothed wheels pass on motion without a slip. The involute tooth profile keeps the speed ratio exactly constant while the contact slides along a straight line, even when the centres are pulled apart; a watch’s going train multiplies one turn of the barrel into thousands of turns of the escape wheel so that the fourth wheel turns exactly once a minute; and epicyclic gears, where axes themselves go round, give reductions, differentials and the tourbillon.
Before you begin: Plane geometry, angular velocity and rigid bodies
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. The involute: a constant ratio from a rolling string
Unwind a taut string from a circle, the base circle, and its end traces an involute. Two gears whose teeth are involutes of their base circles touch along a single straight line, the line of action, tangent to both base circles; the push is always along that line and it always crosses the line of centres at the same pitch point, so the speed ratio is exactly the ratio of the base radii at every instant. That is the fundamental law of gearing. Press Play and follow the gold contact points sliding along the line of action on the right. Then pull the centres apart: the teeth get backlash and the pressure angle grows, but the ratio does not change at all, because the base circles did not. The contact ratio counts how many tooth pairs share the load on average; below 1 the mesh would lose contact.
Worked example. Two gears of module \(m=1\) with \(z_1=18\), \(z_2=30\) and \(\alpha_0=20^\circ\) have pitch radii 9 and 15, base radii \(8.457\) and \(14.095\), tip radii 10 and 16, and centre distance 24. The contact ratio is \((\sqrt{100-71.52}+\sqrt{256-198.67}-24\sin20^\circ)/(\pi\cos20^\circ)=(5.337+7.572-8.208)/2.952=1.59\): between one and two pairs of teeth are always in contact. Pull the centres apart by 0.3 and \(\cos\alpha^{\prime}=24\cos20^\circ/24.3\), so \(\alpha^{\prime}=21.9^\circ\), and \(\varepsilon\) drops to 1.31.
Watch out. Below about 17 teeth at \(20^\circ\) a standard involute pinion is undercut by the cutter that makes it; the drawing ignores this and draws the full profile. Watches mostly do not use involutes at all: their wheels drive tiny pinions of 6 to 12 leaves, where involutes would undercut badly, so watch trains use cycloid-derived ogival profiles that tolerate the small pinions and run with almost no friction on approach, at the price of a speed ratio that varies slightly within each tooth. The contact points are computed exactly from the involute geometry; the teeth are drawn without backlash at the standard centre distance.
Why does pulling the centres apart leave the speed ratio unchanged for involute gears?
The common normal at the contact is always the internal tangent to the two base circles. It crosses the line of centres at a point dividing the centre distance in the ratio of the base radii, \(r_{b1}:r_{b2}\), by similar triangles, whatever the centre distance is. The speed ratio equals the inverse ratio of the distances from that point to the centres, so it is \(r_{b2}/r_{b1}\), fixed by the base circles alone. Moving the centres only tilts the tangent line, which is the change in pressure angle, and shifts which part of each tooth touches.
2. The going train of a watch
The mainspring barrel turns about once every seven hours; the escape wheel, at the other end of the train, several times a minute. In between, each wheel drives the pinion of the next, and each mesh multiplies the speed by the number of teeth on the wheel divided by the leaves on the pinion. The centre wheel must turn once an hour to carry the minute hand and the fourth wheel once a minute to carry the seconds hand, so the escape end of the train is fixed by the balance: the escape wheel advances one tooth per oscillation. Change the beat rate or the escapement’s tooth counts and read whether the seconds hand still keeps time. The animation runs at real speed or faster; the escape wheel moves in steps, half a tooth per beat.
Worked example. At 28,800 vph the balance makes \(f=4\) oscillations a second, so a 20-tooth escape wheel turns at \(4/20=0.2\) rev/s. An 8-leaf escape pinion driven by a 96-tooth fourth wheel gives \(T_4=20\times96/(4\times8)=60\ \mathrm s\). The third and centre meshes, \(75/10\) and \(80/10\), multiply by 60, so the centre wheel turns once an hour, and a barrel of 84 teeth on a 12-leaf centre pinion turns once in 7 h: six and a half barrel turns give 45.5 h of power reserve.
Watch out. The tooth counts here are illustrative, chosen to satisfy the constraints, not those of a particular calibre; real trains vary. The seconds hand keeps time only if \(z_{\rm esc}z_4/(f\,p_{\rm esc})\) is exactly 60, and the minute hand only if the whole train multiplies to 3600; an integer mismatch cannot be regulated away, because the regulator changes the balance frequency, which would put the seconds hand right only by making the watch run fast or slow. The barrel’s torque also falls as the spring unwinds, which lowers the amplitude and, through the escapement error of the Airy experiment, the rate.
Why does the escape wheel advance one tooth per oscillation, not per beat?
The pallet fork has two pallet stones, entry and exit. On one beat the entry stone releases a tooth and the wheel moves until the exit stone catches the next tooth; on the following beat, in the other direction of swing, the exit stone releases and the entry stone catches. The two locks are half a tooth pitch apart around the wheel, so each beat is half a tooth and a full oscillation, two beats, is one tooth.
3. Epicyclic gears and the tourbillon
In an epicyclic train the planets’ axles are carried round by an arm, the carrier, while they mesh with a central sun and an outer ring. Relative to the carrier it is an ordinary gear train, which is Willis’s trick: subtract the carrier’s rotation from everything, apply the tooth ratio, add it back. Fix the ring and drive the sun and the carrier creeps round, a large reduction in a small space; fix the carrier and the ring turns backwards; fix the sun and drive the carrier, and each planet spins faster than the carrier turns. The last case is the tourbillon: the whole escapement is mounted in a cage that turns once a minute, its escape pinion rolling round a fixed wheel, so the balance takes every vertical position in turn. On the right, the velocity diagram: across each planet the speed varies linearly, from the sun’s pitch speed to the ring’s.
Worked example. Sun 24, planets 18, ring 60. With the ring fixed, \(\omega_c/\omega_s=z_s/(z_s+z_r)=24/84=2/7\): seven turns of the sun give two of the carrier. With the carrier fixed, \(\omega_r/\omega_s=-24/60=-0.4\). With the sun fixed, \(\omega_c/\omega_r=60/84\). In a tourbillon with an 80-tooth fixed wheel and an 8-leaf escape pinion, a cage turning once a minute makes the pinion turn \(1+80/8=11\) times a minute absolutely, 10 times relative to the cage.
Watch out. Equal spacing of \(n\) identical planets is possible only when \((z_s+z_r)/n\) is an integer; otherwise the teeth of the second planet would not line up with the sun and ring, and the readout flags it. A tourbillon averages out the vertical positional errors of the previous topic only while the watch stays in one vertical position; in the dial-up position, where a wristwatch spends much of the night, it does nothing, and the cage adds mass and inertia the mainspring must drive.
Why is the velocity across a planet a straight line?
A planet is a rigid body moving in the plane, so at each instant it rotates about one point, its instantaneous centre, and the speed of each point is proportional to its distance from that centre. Along the diameter that joins the sun’s pitch point to the ring’s, the speeds therefore lie on a straight line through zero at the instantaneous centre. With the ring fixed the instantaneous centre is the ring’s pitch point and the planet rolls; the carrier’s speed, at the planet’s centre, is the average of the two ends.
In practice
Where this mathematics and physics is at work, in explainers that take the real thing apart.