Illustrated essays on linkages, gears and cams with the motion computed rather than sketched: every mechanism is solved from its loop-closure equations before it is drawn, and every claim about how it moves is checked against the solution. https://www.machinekinematics.xyz/ 2026-08-05T18:09:53.677Z https://www.machinekinematics.xyz/essays/the-four-bar/ 2026-08-05T18:09:53.677Z

The smallest interesting machine there is. Four lengths decide everything about it — which link can turn all the way round, how hard it pushes, where it stops and whether it can be assembled at all — and every one of those is a number that falls out of a solve rather than a judgement about a drawing.

https://www.machinekinematics.xyz/essays/what-decides-whether-it-moves/ 2026-08-05T18:09:53.677Z

Before a mechanism does anything it has to be able to. Two bars pinned to a point cannot move; three can. The count that separates them is one subtraction, it is the first thing anybody computes about a machine, and it can be wrong in a way that no amount of care with the arithmetic will catch.

https://www.machinekinematics.xyz/essays/why-a-tooth-is-an-involute/ 2026-08-05T18:09:53.677Z

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.

https://www.machinekinematics.xyz/essays/prescribing-motion/ 2026-08-05T18:09:53.677Z

A linkage gives the motion its geometry allows. A cam gives the motion it was asked for, which sounds like an improvement and is a trade — the displacement becomes free and the derivatives stop being.

https://www.machinekinematics.xyz/essays/the-ratio-that-is-not-a-number/ 2026-08-05T18:09:53.677Z

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.

https://www.machinekinematics.xyz/essays/what-a-coupler-point-draws/ 2026-08-05T18:09:53.677Z

A point rigidly attached to the coupler of a four-bar traces a curve of degree six. Move the attachment a little and the curve changes a great deal. For most of the twentieth century the practical way to find the linkage that draws a wanted curve was to look it up in a book of printed atlases.

https://www.machinekinematics.xyz/essays/counting-and-measuring-mobility/ 2026-08-05T18:09:53.677Z

Grübler's criterion counts links and joints and never asks how long anything is. The rank of the constraint Jacobian measures the lengths and never asks what a joint is. Two calculations with no inputs in common, producing one number — which is the only arrangement under which agreement is evidence.

https://www.machinekinematics.xyz/essays/grashof-predicted-and-swept/ 2026-08-05T18:09:53.677Z

Add the shortest link to the longest. If the total does not exceed the other two, some link can turn a full revolution. It is a sentence about four numbers, it was published in 1883, and it is the kind of claim this site refuses to print without measuring — so every linkage here is also asked for all 360 positions and required to agree.

https://www.machinekinematics.xyz/essays/the-straight-line-problem/ 2026-08-05T18:09:53.677Z

Before 1800 a long true flat surface was harder to make than almost anything else, so guiding a piston straight without a slide was worth solving. Watt's answer was an approximation. Measuring how good an approximation, over how much of the stroke, turns out to be a more interesting question than whether it is exact.

https://www.machinekinematics.xyz/essays/what-happens-in-a-mesh/ 2026-08-05T18:09:53.677Z

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.

https://www.machinekinematics.xyz/essays/two-things-called-jamming/ 2026-08-05T18:09:53.677Z

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.

https://www.machinekinematics.xyz/essays/the-law-that-costs-least/ 2026-08-05T18:09:53.677Z

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.

https://www.machinekinematics.xyz/essays/the-mechanism-grubler-says-cannot-move/ 2026-08-05T18:09:53.677Z

Three parallel bars between two frames. Five links, six pins, and the criterion every engineering course teaches gives zero degrees of freedom — a structure. It is a mechanism, it is in drafting machines and locomotive coupling rods, and the formula cannot see why.

https://www.machinekinematics.xyz/essays/the-transmission-angle/ 2026-08-05T18:09:53.677Z

The angle at which the coupler meets the rocker decides how much of an applied force becomes useful output torque and how much goes into the bearings. It is pure geometry, it is computed here from every solved position rather than from a formula, and it is the number a linkage is judged by after Grashof has said it turns.

https://www.machinekinematics.xyz/essays/undercutting-and-the-seventeen-tooth-rule/ 2026-08-05T18:09:53.677Z

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.

https://www.machinekinematics.xyz/essays/stopping-thirty-times-a-second/ 2026-08-05T18:09:53.677Z

A Geneva wheel turns continuous rotation into steps, and its one design requirement is that the pin enters the slot along the slot so the driven wheel starts and stops from rest. That fixes every dimension from the slot count. What it does not fix is the acceleration, which is why film sprocket holes tear.

https://www.machinekinematics.xyz/essays/peaucellier-and-the-exact-answer/ 2026-08-05T18:09:53.677Z

Eighty years after Watt settled for an approximation, a French army officer found a linkage that draws an exactly straight line from pin joints alone. It works by inversion in a circle, the product it holds constant is measurable, and on this site it comes out straight to 10⁻¹⁶ of its span.

https://www.machinekinematics.xyz/essays/the-slider-crank/ 2026-08-05T18:09:53.677Z

Replace one pin of a four-bar with a slide and you get the mechanism in every reciprocating engine ever built. Its stroke is exactly twice the crank throw and does not depend on the connecting rod at all. Everything else about the motion depends on the rod, including the part that is always described as a sine wave and is not.

https://www.machinekinematics.xyz/essays/epicyclic-ratios-two-ways/ 2026-08-05T18:09:53.677Z

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.