What is still outside

Every field on this site has a boundary and most of them state it in passing. This one states it on purpose.

The site computes kinematics: where a mechanism can be, and how those positions change. It does not compute dynamics: what forces are needed, what they do to the parts, or what the parts do back.

That sentence has appeared, in one form or another, at the end of eight essays. It was honest each time and it accumulated into something that needed a page of its own — partly because a limit repeated eight times starts to look like an evasion, and partly because this field moved the line, and a line that moves should be redrawn rather than left to be inferred.

What moved inside #

Two of the four things the site kept disclaiming turned out to be geometry.

A tolerance is a set of geometries rather than one. Every technique the site already had applies to each member of the set, and the answers become intervals instead of numbers. Nothing new was needed except the willingness to solve sixteen mechanisms where one had been solved before.

A clearance is a short link with a free direction. It is a mechanism, made of links and revolutes, and the same solver positions it, the same formula counts it and the same Jacobian measures it. Backlash in a gear mesh is the same quantity computed from tooth thicknesses rather than from pin diameters.

Neither of those required a force, and that is why they belong here. The test for whether something is inside the site’s boundary is not whether it is a “practical” consideration; it is whether it is a question about where the mechanism can be.

What is still outside, and what each would cost #

Four things, in increasing order of how much they would change what a figure is.

Friction #

The site computes the transmission angle, which says how much of an applied force becomes useful output torque and how much goes into the bearings. It does not say whether the mechanism moves.

With friction, a sufficiently poor transmission angle produces genuine locking: the mechanism will not move however hard it is pushed. That is the situation the 40° design rule exists to keep clear of, and this site can compute the 40° and not the rule behind it.

Bringing friction in would need a coefficient — a material property, measured rather than derived, varying with lubricant, surface finish, speed and temperature. It would turn every figure from a computation into a computation-with-a-parameter, and the parameter would be the least reliable number on the page.

Elasticity #

Every link here is rigid. Real links bend, and a mechanism’s actual position is its kinematic position plus a deflection under load.

For most linkages the deflection is small compared with the clearances, which is why this field’s numbers are useful without it. For some — a long slender coupler, a high-speed mechanism, anything where the loads are large — it is not, and the deflection can exceed the whole tolerance band computed in the first essay of this field.

Bringing elasticity in would change what a link is: not a distance constraint but a spring, whose length depends on the force through it, which depends on the configuration, which depends on the length. The solve stops being a geometric one and becomes a coupled one, and the whole site’s premise — that a configuration is the output of a constraint solve — would need restating.

Inertia #

The site computes accelerations. The cam field differentiates a follower’s displacement three times and reports jerk, and every number in it is right.

What it does not do is multiply by a mass. So it can say that constant-acceleration motion has an impulsive jerk and cycloidal does not; it cannot say what that impulse does to the follower, whether the spring holds the follower on the cam at speed, or how much torque the shaft needs.

This is the largest of the four holes and the one with the clearest boundary. Bringing inertia in changes what a figure is: a position is a solve, and a motion under load is an integration. A figure would stop being a function of a configuration and become a function of a history — of the initial conditions, the elapsed time and the driving torque. That is a different site.

Wear #

A clearance is a number on a drawing when the machine is new. It grows.

Everything in the clearance ladder applies unchanged to a worn joint — the kinematics of 0.03 of play is the kinematics of 0.03 of play, whether it was made that way or arrived there. What the site cannot say is how long it takes, and that depends on contact pressure, sliding distance, hardness, surface finish and lubrication, of which only the sliding distance is kinematic.

What can be said kinematically is which joints matter, and that is worth having: the sensitivity ranking that allocates tolerances also says where a clearance costs the most accuracy. Combining it with a wear rate would give a mechanism’s accuracy over its life. This site supplies exactly one half of that product.

Two things that sound outside and are not #

The boundary is easier to hold if the near misses are named, because both of these get disclaimed by reflex and both are geometry.

Assembly. Whether a mechanism can be put together at all is a purely geometric question — it is whether the constraint equations have a real solution — and this site answers it constantly. A synthesised linkage that reaches three poses but cannot be driven through them is refused by a sweep, not by a stress calculation; a corner linkage in a tolerance box that fails to solve is a part that will not go together. Neither involves a force.

Interference. Whether two links collide is also geometry, and it is the one obvious geometric question this site does not answer, because a link here is a distance constraint and has no width. That is a real gap rather than a boundary: it needs no new physics, only a body attached to each link and a collision test, and it would catch a class of design error the current figures cannot see. Naming it as a gap rather than as an exclusion is the honest filing.

What the eight disclaimers actually said #

Since this page exists to replace them, it is worth recording what they were, because they were not all the same claim.

Three said the site does not model friction — in the transmission angle, in jamming and in the pressure-angle essays. Those were about whether a mechanism moves under load.

Two said it does not model clearance and backlash, in Peaucellier and in the mesh essays. Those are now wrong, and this field is why: they are computed here, and the essays that said otherwise were describing a limit of the site rather than a limit of kinematics.

Two said it does not model inertia, in the cam field. Those remain exactly true.

One said it does not model wear. That one remains true and is the least likely to change, because wear is the only one of the four with no geometric formulation at all.

Getting from eight scattered sentences to that four-way split is most of what this page is for. A limit that is stated the same way in eight places looks like one limit; it was four, one of them has moved, and knowing which is which is the difference between a reader who trusts the site’s numbers and one who discounts all of them equally.

Why the line is where it is #

It would be possible to add a friction coefficient and a mass and call the site more complete. The reason not to is a matter of what a figure is for.

Every figure here is generated from a stated rule and every claim is given a test it could fail. That works because the inputs are geometry: four lengths and a crank angle, which are exact, checkable, and the same for anybody who builds the mechanism. A figure computed from a coefficient of friction is a figure whose most important input was guessed, and the assertion attached to it would be an assertion about the guess.

The site would look the same. Every gate would stay green. And the numbers in the captions would have quietly changed from measurements into estimates, with nothing on the page to say which was which — which is the exact failure mode the whole collection exists to make visible.

So the boundary is not a limit of ambition. It is the line at which the site’s own standard of evidence stops being available, and the honest thing is to say where that is rather than to cross it quietly.

What a reader should take from a figure here #

Concretely, so it is usable rather than a disclaimer.

A figure on this site says where the mechanism can be, exactly, and to fourteen decimal places on request. It says how the positions change with the input, how they change with the dimensions, and how much room the joints add.

It does not say whether the mechanism will survive, whether it will move when pushed, how much it will droop, or what it will be like in five years.

Those are four different engineering questions with four different bodies of technique behind them, and every one of them takes the geometry as its input. A stress calculation needs to know where the load is applied, which is a position. A dynamic simulation needs to know the accelerations, which are here. A wear estimate needs the sliding distances, which follow from the motion.

That is the useful way to see the boundary: not as the site being incomplete, but as it being the first of several calculations, and the one the others rest on. Getting it wrong makes the rest wrong invisibly — which is what a sketched mechanism does, and why every position here is solved before it is drawn.

The reason it is not here is worth stating rather than glossing: it needs a datum scheme, and a datum scheme is a decision about how the part will be made rather than a property of the mechanism. This site can compute the sensitivity of the output to any set of parameters whatever, including feature positions; what it cannot do is know which parameters a given factory will hold. So the extension is one function call away and one design decision away, and only the first of those belongs to this site.

The disclaimers were doing real work #

It would be easy to read this page as a tidying exercise. It is worth defending the eight sentences it replaces, because a site with fewer of them would be worse rather than better.

A limit stated at the point where a reader might otherwise over-read a number is doing something no index page can do. Somebody reading about the transmission angle at that moment is thinking about force, and that is exactly the moment to say that the number in front of them is geometry. Moving all of it to a page called “what is outside” would be moving it to the one place a reader who needs it will not be.

So the eight stay, and this page exists alongside them for a different purpose: to say that they are four claims rather than one, that two of them have moved, and that the boundary is drawn where the site’s standard of evidence stops rather than where its ambition does.

The pattern generalises to anything that computes. A limit belongs both at the point of use and in one place where its shape can be seen, and the two say different things. The first stops a misreading. The second is what lets somebody decide whether the whole collection is worth trusting.

What would have to be true to move the line again #

Since the line moved once, it is fair to ask what would move it again, and the answer is a useful test to have.

The tolerance and clearance work qualified because it needed no measured parameter. A tolerance is a number a designer chooses; a clearance is a dimension. Both are exact inputs to an exact computation, and every figure that came out of them can be checked by anybody who builds the mechanism.

Friction, elasticity, inertia and wear all fail that test in the same way: each needs a material property that is measured, varies with conditions, and is quoted with an uncertainty larger than most of the effects on this site. A figure resting on one would be a figure whose most important input could not be verified from the drawing.

So the test is not “is it practical” and not “is it hard”. It is: can every input be read off a drawing? If it can, the site can compute it and assert it and be checked. If it cannot, the honest thing is a sentence saying so — which is what the eight were, and what this page is the map of.

An inventory, so the boundary is checkable #

The clearest form of a boundary is a list, and this is the one this site would want a sceptical reader to hold it to.

Computed and asserted here. Positions, from a constraint solve. Velocities and accelerations, by differentiation. Mobility, by formula and by rank. Assembly and reachability, from whether the solve converges. Screw systems and the constraints reciprocal to them. Tolerance bands and sensitivities. Clearance, lost motion and coupler bands. Backlash in a mesh, from tooth thickness. Solution counts, by homotopy.

Named and not computed. Interference between links, which is geometry and is a gap rather than a boundary. Feature-based tolerancing, described below. Wear rate, friction, deflection, inertia.

Computed elsewhere and taken as input. Nothing. Every number on this site comes out of code in this repository, which is why the boundary can be drawn at all — a site that imported a figure would have to import its uncertainty too.

That inventory is the thing to check a claim against. If a caption here asserts something in the second list, it is wrong and should be reported as such.

The one that got away #

For completeness, one hole this field opened rather than closed.

The tolerance analysis here treats the four lengths as the independent variables. Real parts are not toleranced that way; they are toleranced on features — holes, faces, datums — and the lengths are derived from those. Two pivot holes bored in one setup share an error, so the distance between them is better than either position, and no analysis on the lengths can see it.

Doing it properly means taking the feature positions as the variables and deriving the lengths, which is a straightforward extension of exactly the machinery in this field — the same implicit differentiation, one layer further out. It is not here, it is not hard, and it is the first thing the work after this should do.

The inventory has a property worth pointing at, because it is what makes the boundary defensible rather than merely stated. Every item on it has a price attached — what it would take to bring the thing inside, in machinery rather than in effort. Friction needs a coefficient and a normal force; elasticity needs a stiffness; inertia needs masses and a time; wear needs a rate and a history. Each of those is a number the geometry does not contain, and that is the test the boundary is drawn by: a quantity belongs inside if it can be computed from shapes and positions, and outside if computing it requires a property of a material. That is a sharper criterion than kinematics not dynamics, it decides the awkward cases without appeal to convention — a clearance is inside because it is a length, a preload is outside because it is a force — and it explains why two of the four disclaimers could be brought in and two could not. The line is drawn where the inputs change kind, and an inventory that says what each crossing would cost is a boundary a reader can check rather than one they have to accept.