Fields
Loops
A closed chain of links, and the question every one of them starts with: can this move at all, and if so how. Five fields, from counting freedoms to platforms with three legs.
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.
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.
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.
Out of the plane
In space a body has six freedoms and a pin takes five away, so a closed loop needs seven joints before it moves at all. The mechanisms that move with four are not curiosities — one of them is in every car ever built.
Several legs, one platform
Everything else here has one path from the ground to the moving part. Give it three, and the easy problem and the hard one change places — and a new kind of singularity appears in the middle of the workspace, where the machine can move with every motor locked.
Motion asked for, rather than allowed
A linkage gives the motion its lengths permit. Teeth and cams give the motion specified — and the cost is paid in accelerations nobody asked for and in profiles that cannot be cut.
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.
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.
The problem backwards
Given the lengths, find the motion is the reader's problem. Given the motion, find the lengths is the designer's — and how many answers it has is a question about polynomials.
The problem backwards
Given the lengths, find the motion is the reader's problem. Given the motion, find the lengths is the designer's, and it is why coupler-curve atlases were printed and sold. The constructions are exact; whether what they produce can be built is a separate question.
How many answers
Newton finds a configuration. It cannot tell you how many there are, and a search that has stopped finding new ones is not a proof that there are no more. Written as polynomials, a mechanism's closure conditions have a number of solutions that is a property of its shape — and this is where the site stops counting by looking.
As built, and as taught
Every length here has been a number and every pin a point. Give them ranges and holes, put the result in a machine somebody owns, and see which of the published claims survive — then measure the finished machine and ask which of its numbers a measurement could ever have recovered.
As built
Every length here has been a number and every pin a point. Give the lengths ranges and the pins holes and the answers change shape: a curve becomes a band, and some of the mechanisms this site admires most stop working altogether.
Machines you have met
A suspension, a steering linkage, a scissor lift, a cabinet door, a latch, a chain and a rocker arm. Every one of them is sold with a number, and every one of those numbers is a claim this site can check — which is how the field is built: take the mechanism people have actually met, solve it, and see what kind of number the catalogue was quoting.
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.
Numbers that were measured
Every field before this one takes the numbers on the drawing as given: chosen by a designer, cut by a machinist, and thereafter known. They are not known. Run the same kinematics with the parameters as the unknowns and the motion as the data and a new question appears with an exact answer — which of them can be recovered at all. A four-bar read by a protractor is a three-parameter machine however long you measure it, a coupler curve is drawn by three different linkages, a length is not toleranced because it is two holes that are, and the standard description of a robot arm breaks where the arm does not.
Where the premise is tested
The fields whose object breaks one of the assumptions the loop fields share: no loop at all, a state with a discrete part, an input that is not one number, a constraint that does not integrate, no mechanism in the description at all, a shape that is not a choice, a member that is not rigid, an assembly whose size is a parameter, a constraint that only pushes where what a part may do is a cone rather than a subspace — and the connections themselves as the unknown, where the answer is a census rather than a configuration, and the joint itself as the object, where what a pair permits turns out to be a group rather than a number, and the demand itself given as an equation, where the mechanism is compiled from a polynomial instead of searched for, and the links themselves given a width, where the constraint is an inequality between parts rather than an equation in the loop and the plane the mechanism is drawn in turns out to be a stack.
One path to the tool
Every mechanism here so far has been a loop, and a loop is why a configuration on this site is a solve. An arm has no loop: its pose is a product of six transforms with nothing to converge and nothing to refuse. The difficulty does not go away — it moves to the other end, where one tool pose has eight answers and a straight line can cost more than the machine has.
Motion that stops
Every other mechanism here moves whenever its input does. A ratchet, an indexer and an escapement are still for most of a turn and moving for the rest, and what decides which is not an equation running out of answers but a tooth arriving at a face. Whether one holds is the sign of a lever arm; what it wastes is two thirds of its own travel.
More than one input
Every mechanism here so far has had one input, and its output has been a function of it. A differential does not: its cage turns at the mean of two wheels, and neither of them decides anything alone. What a gear train has is not a ratio but a relation — a plane of permitted motions — and every number a gearbox is sold with is that plane cut by a brake, a clutch or a choice of which shaft is driven.
Wheels, and where they may not go
A rolling wheel forbids a velocity, not a position. It may not go sideways and it can still be parked anywhere, at any angle — and the gap between those two sentences is worth an exponent, a shortest path that is a whole circle, and a ball that comes back turned by the area it went round.
The motion, not the mechanism
Everything else here is about a machine. This is about the motion a machine makes, near one instant — where the moving plane is turning, which of its points are going straight, which of them could be replaced by a single pivot and for how long. Two linkages with nothing in common that agree on those numbers are interchangeable, and the mechanism turns out to be one of infinitely many ways of producing a motion rather than the thing the subject is about.
The shape is the unknown
Everywhere else here a body has a shape and the question is where it goes. Put two bodies on fixed centres, require them to stay in contact, and the second shape stops being a choice: it is the envelope of the first one's positions, and there is exactly one of it. One routine cuts a gear tooth out of a straight edge, a cam out of a roller and a rotary engine's rotor out of its housing.
Members that pull
Every link on this site so far has been rigid: it holds two points at a distance, in both directions, and a configuration is the root of an equation. A belt, a chain, a rope and a tendon are none of that. A strand has no shape of its own — where it runs is decided by the bodies it touches — and it constrains one way only, so it does nothing at all until it is taut, and a mechanism made with one has a different mobility in different places.
Many of one thing
Every mechanism here so far is a chain: a handful of links, one or two loops, and a mobility somebody can check on the back of an envelope. A scissor lift, a folded sheet and a deployable ring are one small unit repeated, and three things change at once. The count of bodies is a parameter; mobility becomes the rank of a matrix rather than an arithmetic; and a unit that moves can be rigid the moment it is joined to another of itself.
Contacts that only push
Every constraint here so far has been an equation. A part resting against another part is not one: the contact says do not come closer and nothing at all about going away, so what the part may do is a cone rather than a subspace and a freedom stops being two-sided. Six constraints fix a body in space and six contacts fix nothing — the number is seven — and whether a part is held stops being a rank and becomes a question about where the origin sits inside a hull.
The chain before the lengths
Every field before this one is handed a mechanism and asked what it does. Here the connections are the unknown: which graphs of links and pins are mechanisms at all, how many there are, and which of them are the same mechanism drawn twice. The search space is finite and every quantity is a count — one four-link chain, two six-link, sixteen eight-link, two hundred and thirty at ten — and the counting rule everything else on the site rests on turns out to admit eight graphs for every one that deserves it.
What a joint is
Every field before this one declares its joints and then counts what they take away. A count cannot tell a pin from a slide: both are one, and the two mechanisms you get by swapping them are not related at all. What a joint permits is a set of displacements closed under composition — a group — and the six lower pairs turn out to be the six groups a surface can have as its own symmetry. The same instrument, run on a whole mechanism, separates the two kinds of overconstraint by an integer.
The curve as an equation
Every other field here is handed its demand geometrically: three positions, a sampled path, a ratio. This one is handed a polynomial, and the mechanism is compiled from it rather than searched for — every monomial becomes a cosine, every cosine becomes a link, and the linkage closing is the equation being satisfied. The answer is exact everywhere the machine moves, and what it costs is not accuracy but size.
Links with a width
Every link in every other field is a distance between two points, and a distance cannot collide with anything because it is not anywhere. Give each one a body and the constraint stops being an equation in the loop and becomes an inequality between pairs of parts — not local, not smooth, and not in the plane the mechanism is drawn in. Not one machine here fits in a single plane, a clearance turns out to be a function with corners, a sampled sweep needs a bound to prove it missed nothing, and a four-bar turns all the way round exactly when it cannot be built with a bearing at each pivot.
The other ways in
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