Every essay
What can move
Before a mechanism does anything it must be able to. Mobility is countable, and the count can be wrong — which is how some of the most useful mechanisms ever built were nearly ruled out.
What decides whether it moves
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.
Counting and measuring mobility
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.
The mechanism Grübler says cannot move
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.
Linkages
Four bars and four pins is the smallest interesting machine there is. Everything about which link turns, how hard it pushes and where it jams follows from the four lengths.
Four bars and four pins
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.
Grashof, predicted and then swept
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.
The transmission angle
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.
The slider-crank
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.
The paths points trace
A point on a coupler draws a curve of degree six. Choosing the linkage that draws the curve you want is the oldest hard problem in the subject, and straight lines were the hardest of all.
What a coupler point draws
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.
The straight-line problem
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.
Peaucellier and the exact answer
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.
Teeth
A gear tooth is not a shape somebody liked. It is the curve that keeps the velocity ratio constant while the contact point slides, and there is essentially only one answer.
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.
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.
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.
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.
Prescribed motion
A linkage gives the motion its geometry allows. A cam gives the motion you asked for — and the cost is paid in accelerations you did not ask for.
Prescribing motion
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.
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.
Stopping thirty times a second
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.
Drawn wrongly
The mechanisms that are illustrated confidently and incorrectly, the ratios quoted from the wrong formula, and the pictures that would not move if they were built.
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.
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.