The pair a catalogue sells

Every mechanism this site draws declares its joints, and the practice field’s job is to ask what happens when the declaration meets a part somebody bought. A clearance is a link; which pin to buy is a tolerance question with a stack-up behind it; taking up the play is what a designer does about it.

This rung asks a question one step earlier: is the part on the shelf the joint on the drawing?

The answer is often no, the difference is usually one freedom, and the freedom is usually the axial one — which in a planar mechanism points out of the page, where no analysis on this site has been looking.

The joint is decided by the surfaces #

A lower pair’s permitted set is the symmetry group of its own surface, and there are six such groups. Nothing about a part’s name, its price or its catalogue section enters.

So the question about a bought part is always the same one: which surfaces are in contact, and what symmetry do they have together? Not which surfaces exist — a bearing has dozens — but which pair of them is doing the constraining.

That question has a clean answer surprisingly often, and where it does not, the answer is usually that the part is doing something else as well.

Where a bearing and a joint agree #

A plain journal in a bore, with no shoulders. A circular cylinder against a circular cylinder. The symmetry group of a circular cylinder is a rotation and a translation on the same axis: a cylindrical pair, two freedoms.

A journal with two thrust faces. The same cylinder with two annuli, which is a compound surface — and the annuli’s normals forbid the slide. One freedom, a revolute pair. That is the shaft with collars in the census, and the computation sees one surface and reports one freedom without being told it is in two pieces.

A taper bearing. A cone against a cone. The symmetry group of a cone is a rotation about its axis, so a taper is a revolute pair — the same joint as the shaft with collars, from a surface that shares nothing with it. The location is done by the taper rather than by a shoulder, which is a bearing decision and not a kinematic one.

A spherical plain bearing. A sphere against a sphere: a spherical pair, three freedoms.

A linear guide with a non-circular section. A prism against a prism: a prismatic pair, one freedom. The moment the section becomes round the rotation appears, which is why a round shaft in a round bushing is a cylindrical pair however it is drawn.

A thread and nut. A helicoid against a helicoid: a helical pair, one freedom, and the pitch is the thread’s own lead. That the pitch is part of the joint rather than a setting of it is worth remembering when a lead screw is swapped for one of a different pitch — every displacement the joint permits has changed, not merely a ratio in a controller.

The one that resists a clean answer is a rolling-element bearing, and it resists for an instructive reason. Its two rings do not touch each other at all; they touch a set of balls, and each contact is a point. So a ball bearing is not a lower pair in any sense — it is an assembly of higher pairs — and the reason it behaves as a revolute is that the assembly, taken as a whole, permits one rotation and nothing else. What the group view says about it is that the assembly has a group even though none of its contacts does, which is the same statement it makes about a chain and is why a bearing can be treated as a joint at all.

Where the count goes wrong on a bill of materials #

The failure this rung exists to name: counting mobility from a parts list.

A mechanism drawn with four revolutes and built with four plain journals has four cylindrical pairs, not four revolutes. In space that is four freedoms rather than four constraints removed at five each — the count changes by one per bearing, and a mechanism that was mobility one becomes mobility five.

In a planar mechanism it does not show, because the axial freedom is out of plane and the planar count never sees it. That is exactly why it is easy to miss: the drawing is planar, the count is planar, and the extra freedom is real and lives in the third dimension where nobody is looking. It shows up as a shaft walking, as a coupler drifting sideways, or as a load path nobody designed — and it is a kinematic fact rather than a manufacturing defect.

The lattice says exactly what the substitution costs: $R \subset C$, so a cylindrical pair permits everything a revolute does and one thing more. The mechanism does not lose anything; it gains a freedom, and the gain is what the count missed.

The same mistake in the other direction is rarer and worth naming. A mechanism drawn with a cylindrical pair and built with a located bearing has lost a freedom, and a loop that was mobility one becomes a structure. That one announces itself: the assembly binds, and somebody files something. The extra-freedom version does not announce itself at all, which is why it is the one worth checking for.

What a catalogue’s word carries #

A catalogue’s name for a part is a statement about how it is made and how it is loaded, and it is only accidentally a statement about which pair it gives.

“Bearing” says a rotation is intended and says nothing about whether the axial direction is located. A deep-groove ball bearing locates axially and gives a revolute; a needle roller without collars does not and gives a cylindrical pair; a cylindrical roller bearing of the type with one loose ring is deliberately cylindrical, because the machine wants the shaft to grow with temperature.

“Rod end” and “spherical bearing” say a spherical pair, three freedoms.

“Linear bearing” says a prismatic pair if the section is keyed and a cylindrical one if it is a ball bushing on a round shaft — and the second is a genuinely common source of an unintended freedom.

“Bushing” says nothing at all about the pair and has to be looked at.

None of that is a criticism of catalogues. They are organised by what the part has to survive, and the kinematic identity is a different axis of classification that happens to be the one a mobility count needs.

There is a second reason the two axes do not line up, and it is worth stating because it is not a matter of terminology. A catalogue’s classification is about load — radial, axial, moment — and load is exactly what this site does not have. A bearing that carries an axial load is one whose surfaces forbid axial motion, so the two classifications agree on that one point and nowhere else: a taper and a deep-groove ball bearing carry axial load very differently and give the same pair, and two needle rollers with and without collars carry radial load identically and give different ones.

Two bearings on one shaft #

The case worth working through, because it is where the group view earns its place in the practice field.

A shaft carried in two bearings is not one joint. It is two, in parallel — the shaft is connected to the housing by two routes at once — so what the shaft may do is the intersection of what the two permit.

Two cylindrical pairs on the same axis intersect in a cylindrical pair: still two freedoms, and the shaft can still walk. Add a shoulder at one end and one of them becomes a revolute; the intersection is a revolute, one freedom, and the shaft is located.

Two cylindrical pairs whose axes are not quite the same — which is every real machine — intersect in nothing at all: two non-coaxial cylindrical groups meet in the identity. That is the overconstraint that makes an assembly bind, and it is the reason a hinge works only because its clearances let it: the ideal mechanism is a structure and the built one moves on its play.

The freedom nobody ordered #

It is worth putting a number on the thing this rung is warning about, because “one freedom per bearing” sounds abstract until it is a distance.

A shaft in two plain journals, with no thrust face anywhere, can move axially by as much as the housing allows — millimetres, not microns. Nothing in the kinematics stops it. In a planar mechanism that motion is out of plane and every planar analysis on this site is silent about it: the mobility count is planar, the constraint rank is computed from planar constraints, and the coupler curve is a planar curve.

The consequences are real and they are the sort that get diagnosed as something else. A coupler that drifts sideways puts its tracing point off the curve the site would draw. A link that walks along its pin loads a face nobody designed to be loaded. And a mechanism assembled with the shaft at one end of its travel behaves differently from one assembled at the other, which reads as unit-to-unit variation.

Every one of those is the difference between a cylindrical pair and a revolute pair, and the two are indistinguishable in every count this site ran before the essays on joints — because both have the same planar freedom count, and the difference lives in a dimension the planar analysis does not have.

What clearance does to a group #

The group view says a joint’s permitted set is exact. A real joint has clearance. The two have to be reconciled and the reconciliation is the practice field’s standing one.

A pin in a hole with clearance $c$ does not permit a rotation and nothing else; it permits a rotation plus a bounded excursion in every other direction, of size $c$. The permitted set is a neighbourhood of the group rather than the group, and it is not closed under composition either — composing two displacements each within $c$ of the group can land $2c$ away.

So the honest statement is that a real revolute pair’s permitted set is within $c$ of a one-dimensional group, and every kinematic consequence of the group holds to within $c$. A clearance is a link is the same statement made operational: model the play as a short link and solve, and the result is the group’s answer plus a band.

That is also the right reading of Sarrus’s exact straight line as built. The group argument makes it exactly straight; six joints with clearance make it straight to the accumulated play; and the difference between $9.8 \times 10^{-16}$ and a few hundredths of a millimetre is the whole content of this field.

Choosing the pair on purpose #

The same reading run forwards is a design method rather than a warning, and it is how bearing arrangements are actually decided in practice — usually without the vocabulary.

A shaft needs to be located axially once. Locating it twice is an intersection of two revolute groups on the same axis, which is a revolute group again if the axes agree exactly and the identity if they do not — so a shaft in two located bearings is a structure held together by its clearances and by whatever the housing does when it warms up. The standard arrangement is therefore one located bearing and one free: a revolute at one end and a cylindrical pair at the other, intersecting in a revolute, with the axial growth taken up by the free end.

That is a group argument, it is the reason every textbook on shaft design gives for the arrangement, and it is normally phrased in terms of thermal expansion. The kinematic version is more general: two joints in parallel give the intersection, and choosing the second one to permit more than the first is how a designer keeps the intersection equal to the first.

The same reasoning names the other standard arrangements. A pair of angular-contact bearings mounted back to back is one revolute; a rod end at each end of a link is two spherical pairs whose intersection is the identity plus the link’s own spin, which is why a two-ball link is a strut with a free rotation about its own axis. Holding an axle still is the applied field’s version of the same question asked about a whole assembly.

The error is always in the same direction #

There is a structural asymmetry running through every case in this rung, and naming it turns a checklist into something that can be reasoned from.

A bought part can be looser than the drawing and never tighter. A plain journal permits a cylindrical pair where the drawing wanted a revolute; a rod end permits a spherical pair where the drawing wanted a revolute; a bushing permits whatever its bore permits. In every case the hardware’s permitted set contains the drawing’s, and the difference is a surplus freedom rather than a missing one.

That is not a coincidence and it is not about catalogues being careless. A component is sold to permit a motion, and permitting is what a surface does; constraining is what the mounting does. A bearing supplies the freedom and the shoulder, the circlip, the locknut, the second bearing or the housing face supplies the constraint — and none of those is the bearing.

So the difference between a drawing and a bill of materials is always a set of freedoms the mounting was supposed to remove, which gives the checklist its underlying question: what removes this freedom, and is it on the drawing? An answer of “the bearing” is always wrong, because the bearing is what permitted it.

The rule also says where to look when a mechanism has a freedom nobody ordered. Not at the parts — they are doing exactly what they were sold to do — but at the interfaces: which face bears against which, which fastener carries which direction, and whether any of them was drawn as a constraint or merely as a part in the right place. A shaft that walks axially has a missing thrust face somewhere, and the thrust face is a feature rather than a component.

And it explains why the one-directional error is the harmless-looking kind. A missing constraint does not stop a mechanism working; it makes it work with an extra freedom that something incidental takes up — a wire, a seal, a slight interference, gravity. Those are not constraints and they do not hold, so the mechanism works until whichever of them was doing it stops. That is a failure with a long delay in it, which is the worst kind to trace and the easiest kind to design out at the drawing stage, from a list of pairs rather than a list of parts.

The checklist #

What this rung leaves a reader with is short.

Ask which surfaces touch. A bearing has many and two of them decide the pair.

Ask whether the axial direction is located, and by what. That is the difference between a cylindrical pair and a revolute, and it is one freedom per bearing.

Count from the pairs, not from the parts list. A mechanism’s mobility is arithmetic on the dimensions of its joints’ groups, and a parts list is arithmetic on how many things were ordered.

And remember that two bearings on a shaft are an intersection. If their axes agree the intersection is the pair intended; if they do not, it is the identity, and the machine works on its clearances or not at all.

The reason this belongs in the practice field rather than in the pairs field is the last of those. Every statement in the pairs field is about exact geometry, and every real bearing arrangement violates exact geometry — two bores are never coaxial, three axes are never parallel, and a clearance is always there. What the group view gives the practice field is not another exact statement but a way of saying which exact statement the built machine is a neighbourhood of, and how wide the neighbourhood is.