A cam is a conjugate pair

The cams field builds a profile in two steps. The pitch curve is where the roller’s centre goes: base radius plus the follower’s displacement, plotted against the cam angle. The cut surface is that curve offset inward by the roller radius, along its own normal.

That construction is correct, it is what a cam grinder does, and it is not the only route. A roller is a shape; the follower carries it along a slide; the cam turns underneath. Ask what shape stays in contact with that circle under that motion and the answer is the cam — computed by the machinery that generates a gear tooth from a straight rack, with a circle instead of a straight edge and a lift law instead of a rolling condition.

The two constructions #

The offset construction says: take the locus of the roller’s centre, and step inward by $r$ along the normal.

The envelope construction says: the follower’s body carries a circle of radius $r$; its centre sits at $R_b + s(\varphi)$ from the cam’s axis, on a fixed line; the cam turns by $-\varphi$; find every point where the meshing equation holds and map it into the cam’s frame.

They have almost nothing in common as arithmetic. The first differentiates a sampled curve to get a normal and steps along it. The second forms two velocities — one of a sliding body, one of a turning body — subtracts them, dots the difference with the circle’s own normal and solves for a root.

Measured against each other on a cycloidal rise of 12 mm over 120°, with a base radius of 30 and a roller of 8: the worst distance between the two surfaces is $1.6\times10^{-5}$ mm over 721 generated points.

Why a second construction was worth having #

The cams field’s offset carried a sign error for six months. The pitch curve is traversed anticlockwise, so its forward tangent turned by −90° is the outward normal; adding the roller radius along it put the cut surface outside the path of the roller’s own centre. Every cam the site had drawn was eight units too large in radius everywhere, with the roller drawn buried in the metal it was supposed to be riding on.

Nothing caught it, and the reason is worth repeating in this field’s terms. Every check the cam library had was about the pitch curve: the undercut margin reads its curvature, the smallest base radius bisects on that margin, the curvature check fits circles to it at two sampling rates. All of them careful, all of them about the curve the roller’s centre travels on — and the bug was in the only curve that gets manufactured.

The envelope route cannot make that mistake, because it does not have a sign to get wrong. There is no normal to step along and no direction to choose: the contact is where the equation has a root, and the equation contains only positions and velocities. So the agreement above is not a check of one implementation against another. It is a check of a construction with a choice in it against one with no choice in it.

The two errors that construction can make show up as distances with meanings:

• against the pitch curve — the roller-centre path — the generated surface is $8.000$ mm away, which is the roller radius. That is the distance the offset is supposed to move.
• against the outward offset, as it was until it was fixed, the generated surface is $16.000$ mm away: two roller radii, which is what a sign error costs.

A wrong answer at exactly twice the right correction is the signature of a flipped sign, in the same way that a velocity check reporting exactly 2 is the signature of a double negation — the size of the discrepancy names the mistake.

The generation, in the cam’s own terms #

It is worth writing out what is handed to the machinery, because the whole claim of this field is that the same six lines take all of these mechanisms and it is easy to assert that without showing it.

The given shape is a circle of radius $r$, expressed in the follower’s own frame with its centre at the frame’s origin.

The follower’s motion is a translation: its frame’s origin sits at $(R_b + s(\varphi),\,0)$, and its velocity is $(\mathrm{d}s/\mathrm{d}\varphi,\,0)$ — the lift per radian, which is the only place the motion law enters.

The cam’s motion is a rotation by $-\varphi$ about the origin, at unit rate.

Everything else is the same routine as for a rack: form the two material velocities at a candidate contact, subtract, dot with the circle’s normal, and bracket the roots. The roots come in pairs — the near side of the roller and the far side — and the cam is the near one, exactly as a cycloidal disc is the near branch of its pins’ envelope.

What is not handed over is any information about offsets, normals, curve directions or which side the metal is on. That is the whole of the difference between the two routes, and it is why one of them can check the other.

What the cam has that a gear does not #

Two differences between this pair and a gear pair are worth setting out, because they explain why the cams field exists separately at all.

The relation is stated, not constant. A gear pair’s two bodies turn in a fixed ratio; a cam’s follower moves according to a law somebody wrote — a rise, a dwell, a return — and that law is the specification. In this field’s terms the cam is a conjugate pair whose relation is an arbitrary function of the input rather than a number, which is exactly what a non-circular gear pair is as well. The difference is that the non-circular pair’s second body turns and the cam’s slides.

One body is a circle, on purpose. A roller is the easiest accurate shape there is, and putting it on a bearing converts the sliding at the contact into rotation of the roller — the trick that removes the sliding without changing the ratio. A gear pair cannot do that, because both of its bodies have to carry many teeth.

The consequence is that the cam’s whole difficulty is in one part, which is the point of the design. The follower is a cylinder and a bearing; the cam is a single closed curve, cut once.

Three constructions, one surface #

The comparison in the table is worth reading as a set rather than as one number, because each of the three distances is a statement about a different kind of mistake.

Nought is agreement: two constructions of the same surface, one with a choice in it and one without.

One roller radius is the distance to the roller-centre path. That is not a mistake anybody makes deliberately, and it is exactly what happens if the offset step is omitted — a cam machined to the pitch curve, which is a cam that is one roller radius too large everywhere and whose follower rises by the wrong amount from the first degree of the cycle.

Two roller radii is the sign error, and it is worth noticing that its size does not depend on the lift law, the base radius or the cam angle. It is always exactly $2r$, everywhere, because the two surfaces are reflections of each other in the pitch curve. A defect whose magnitude is a constant of the tooling rather than a function of the design is a defect that would have been obvious in a single measurement of any cam the site had ever drawn — if anybody had measured the cut surface rather than the curve it was offset from.

The pitch point, in a mechanism with no pitch circles #

Everything in this field has run on the pitch point — the instant centre of the relative motion of the two bodies, through which every common normal passes. A cam and a translating follower still have one, and it is worth locating because it explains a quantity the cams field measures for its own reasons.

The follower does not rotate, so the relative motion of the two bodies is the cam’s rotation alone, and the pitch point is at the distance from the cam’s axis where the follower’s velocity matches the cam’s surface velocity. It sits on the line through the axis perpendicular to the follower’s travel, at a distance equal to $\mathrm{d}s/\mathrm{d}\varphi$ — the lift per radian.

Two consequences follow immediately, and both are already in the cams field under other names.

The pressure angle is the angle between the follower’s direction of travel and the common normal, and since the normal passes through the pitch point, it is $\arctan$ of the lift per radian divided by the contact radius. That is precisely the formula the cams field uses, arriving here as a statement about a point rather than as a derivative.

And a dwell is the case where the lift rate is zero: the pitch point collapses onto the cam’s axis, the normal is radial, the pressure angle is zero, and the contact is momentarily a pure rotation with no sliding of the follower at all. A dwell is not a special case of the mechanism; it is a position of the pitch point.

What the agreement is worth, and what it is not #

Sixteen millionths of a millimetre is small and it is not zero, so it is worth saying what the residue is.

Both curves are sampled. The generated surface is a list of contacts, one per shaft angle over 720 angles; the offset surface is a list of points, one per cam angle over 1,440, each stepped along a normal that was estimated from its neighbours by a central difference. Comparing them means measuring each generated point’s distance to the polyline through the offset points, so the number includes the polyline’s own chord error and the finite-difference error in the normals.

Both of those fall as the sampling is refined, and the agreement improves with them — which is the signature of a discretisation residue rather than of a real disagreement. A genuine difference between the two constructions would not care how finely either curve was drawn.

That distinction is worth making every time two routes are compared. The useful question is not how small is the difference but what does it depend on: a difference that shrinks with the sampling is arithmetic, and a difference that stays put is a finding.

The undercut a cam can have #

A cam’s version of undercutting is the case where the roller is too large for the curvature it is asked to follow, and the envelope route says exactly what goes wrong.

If the pitch curve’s radius of curvature falls below the roller radius anywhere on a concave stretch, the offset curve crosses itself: the locus of contact points doubles back, and the “surface” it describes bounds no solid. Cut it and the tool takes out a bite, leaving a profile that does not follow the intended law. That is the same failure as a rack’s corner cutting into the flank it generated, and it is the same failure as a gear tooth’s fillet crossing its involute.

The cams field measures it as a margin — the smallest curvature radius on the pitch curve against the roller radius — and bisects for the smallest base circle a given roller can be used on. In this field’s terms: the envelope has a cusp when the given profile is more sharply curved than the motion allows, and past that it self-intersects. Same event, three vocabularies, one criterion.

The follower that is not a roller #

The envelope route generalises in a direction the offset construction does not, and it is worth having because two of the three common followers are not rollers.

A flat-faced follower carries a straight line rather than a circle. Hand that line to the same machinery, with the same translating motion and the same turning cam, and the envelope of the line’s positions is the cam surface. The offset construction has nothing to say here — there is no roller radius to step by — and the classical treatment uses a different formula. The envelope treats it as the same problem with a different given shape.

A knife-edge follower carries a point. A point has no envelope but its own path, so the cam surface is the pitch curve, which is why the knife-edge case is the one every textbook starts with and the one nobody builds.

An oscillating follower carries a roller on an arm rather than on a slide, so its motion is a rotation about a second fixed centre instead of a translation. That is two turning bodies — geometrically a gear pair with a peculiar ratio — and the machinery does not notice the difference at all.

Four followers, one routine, and the only thing that changes between them is which curve is handed in and what the second body is doing. That is the same claim this field’s first essay makes with six rows, arriving again inside a single mechanism.

The shortcut is the special case #

The two routes are presented as a construction and a check on it, and the relationship is worth inverting, because it is the other way round: the envelope is the method and the offset is a shortcut that happens to work for one follower.

Look at what the offset construction needs. Step inward by $r$ along the normal requires an $r$ — a single constant distance, the same at every point — and that exists only because the follower carries a circle. A circle is the one shape whose offset is another curve at a constant distance, which is why the shortcut exists and why it is exactly as general as roller followers are.

Hand it a flat-faced follower and there is nothing to do. There is no roller radius, no constant offset, no inward direction, and the construction has no content. The same for a knife edge, for a curved shoe, for a follower whose face is anything but a circle. The offset is not a general method that becomes awkward; it is undefined.

The envelope route needs none of that. It asks what shape stays in contact with a stated curve carried by a body moving in a stated way, and both of those are inputs. Change the curve and it is a different follower; change the motion and it is an oscillating one rather than a translating one. The routine does not change and nothing in it is specialised to circles.

So the honest description of the field’s arrangement is not a construction with a second route to check it. It is a general method and a fast special case, and the special case is the one every cam grinder uses because rollers are what cams run against. The check is worth having for the reason it always is — the special case had a sign in it and the general one does not — and the generality is what makes the check possible rather than a coincidence.

That also says which route a designer reaches for when the follower is unusual. Not adapt the offset, which cannot be adapted, but hand the envelope routine a different curve. A flat face is a line segment, a curved shoe is an arc of the shoe’s own radius, and in each case the profile comes out of the same computation with one input changed — which is the whole argument for having built the general routine on a field where the special case was already working.

What this earns #

The reason to compute a cam twice is not to have a second picture of it.

The offset construction is efficient and it contains a decision — which way the normal points — that nothing downstream can check, because everything downstream is about the curve the decision was made from. The envelope construction is slower, contains no decision, and can only agree with the offset if the offset went the right way.

That is the shape of a genuinely independent second route, and it is the pattern this whole site is built on: mobility counted and mobility measured, a velocity from the closed form and from a difference, a pole from a construction and from an expression. Each pair agrees to the last digit until the day one of them is wrong, and on that day the disagreement is the finding.

Sixteen millionths of a millimetre is what agreement looks like here. Sixteen millimetres was what disagreement looked like, and it was there to be seen for six months in a field whose every check was pointed at the wrong curve.