Machine Kinematics
solved, then drawn
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The thread: The constant that is not constant

A velocity ratio quoted as a number is a claim that it does not vary through the cycle. For gears that claim is true and is the whole reason the involute exists; for most other mechanisms it is false, and the variation is measurable.
the traced pointtangencybase circle r = 9.4020 teeth, module 1, 20° pressure anglenormal meets the base circle to 2.3e-5 Teeth

Why a tooth is an involute

A gear tooth is not a shape anybody chose for its looks. It is what one requirement forces — that the ratio of the two shafts' speeds stays exactly constant while the contact point slides along the flank. Impose that and the curve is essentially determined.

 -0.25000.2500.5000100200300crank angle (degrees)output ÷ input angular velocitya 20:40 gear pair, 0.500mean 0.000four-bar 4/1/3.5/3-0.40 to 0.33 through one turn Drawn wrongly

The ratio that is not a number

A four-bar's output-to-input speed ratio runs from −0.29 to 0.51 through one turn and changes sign on the way. Quoting a single figure for it quotes the average of that curve, which the mechanism never exhibits. A gear pair is the case where the same phrase is honest, and it is honest by construction.

 pitch pointline of actionmodule 1, 20° pressure angle, centre distance 26contact ratio 1.612 Teeth

What happens in a mesh

Contact between two gear teeth happens only along one straight line, and only over part of it. How much of that line lies between the two tip circles, divided by the base pitch, is the contact ratio — and if it drops below one the drive periodically stops being driven.

 0153045600100200300crank angle (degrees)mechanical advantage (clipped at 60)toggle: advantage 1673worst μ = 46°four-bar 3.4/1.2/3/2.4, 720 solved positions222° apart Drawn wrongly

Two things called jamming

A four-bar's mechanical advantage peaks at 1,673 in one configuration and its transmission angle collapses in another, 222° away. Both get described as the mechanism jamming. One is enormous force output and the other is force disappearing into the bearings, and they are opposite situations.

 05101520050100150lift-0.010-0.00500.0050.010050100150cam angle (degrees)acceleration (per degree²)constant accelerationsimple harmoniccycloidalacceleration differentiated from the displacement above itsmoothest is not gentlest Prescribed motion

The law that costs least is not the smoothest

Constant acceleration gives the lowest peak acceleration of any motion law and an impulsive jerk. Cycloidal motion has finite jerk everywhere and a peak acceleration 57% higher. The trade is real, it is measurable, and the displacement curves that everyone plots give no hint of it.

 10 teethundercut14 teethundercut17 teethundercut18 teethclean24 teethcleanred: the root circle has risen above the base circlethreshold N = 2/sin²α = 17.097 Teeth

Undercutting, and the seventeen-tooth rule

Below a certain tooth count a standard cutter eats into the flank it is supposed to be forming. The count is quoted as seventeen. The formula gives 17.097, which means seventeen undercuts slightly and eighteen is the smallest that does not — the rounding went the convenient way.

 ringsunhold the ringsun in, carrier out4.000 : 1same directionhold the carriersun in, ring out−3.000 : 1output reverseshold the sunring in, carrier out1.333 : 1same directionWillis: (ω_s − ω_c)/(ω_r − ω_c) = −72/24both derivations agree, and the build requires it Teeth

Epicyclic ratios, two ways

An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. The sign errors are notorious, so this site computes every ratio by Willis's equation and by the tabular method and requires them to agree.

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