A follower needs a face

The cams field designs a profile from a stated motion and never asks how big the follower is.

That is a reasonable omission for six essays, because the profile is decided by the motion and by the base circle and by nothing about the follower’s size. It is not a reasonable omission for a machine. A flat-faced follower has to be large enough, and how large is decided by geometry rather than by taste.

Where the contact is #

A flat-faced follower rests on the cam along a straight edge. At any cam angle the contact is the point of the profile furthest along the follower’s axis — the highest point, if the follower is above — and there is no reason for that point to be directly above the follower’s centre.

For a translating flat follower there is a classical and very short answer: the contact sits at a distance

$$x = \frac{ds}{d\theta}$$

from the follower’s axis, with θ in radians. Not the lift, not the base radius, not the profile radius: the velocity of the motion the cam was designed to produce.

It follows from the profile being the envelope of the follower’s face. The face at cam angle θ is the line at height R₀ + s(θ) in the fixed frame; the envelope condition is that the profile touches that line where the family’s derivative with respect to θ vanishes, and carrying the differentiation through puts the touching point at s′(θ) along the line.

So the face must reach from the most negative value of s′ to the most positive, and its width is the difference between them.

What that gives #

For a rise of 10 over 120°, a dwell of 60°, a return of 10 over 120° and a dwell of 60°, on a base circle of 30:

Cycloidal: the contact runs from −9.549 to +9.549, so the face must be 19.10 wide.

Harmonic: from −7.500 to +7.500, a face of 15.00.

Parabolic: from −9.545 to +9.545, a face of 19.09.

Same lift, same spans, same base circle. The faces differ by a quarter, and the ordering is not the one a reader of the law that costs least would predict: the harmonic law needs the narrowest face and it is the worst of the three on peak acceleration at the ends of its rise. A follower’s size and a follower’s smoothness are bought from different pockets.

The base circle does not appear at all, which is worth stating plainly because it is the parameter a designer reaches for first. Growing the base circle reduces the pressure angle, reduces the peak curvature, and cures undercutting — and does nothing whatever to the face width. It is a rare thing on this site: a design quantity with exactly one input.

Why it is the velocity and not something else #

The result is short enough to be worth deriving rather than quoting, because the answer is surprising the first time and obvious afterwards.

Put the cam’s centre at the origin and let the follower translate along the y axis, its face horizontal. At cam angle θ the face is the line y = R₀ + s(θ). As θ varies this is a family of lines, and the cam profile is the curve that touches every member of it: the envelope.

A point on the envelope satisfies two conditions at once — it is on the line, and it is on the line’s neighbour. The first is y = R₀ + s(θ). The second comes from differentiating the family with respect to θ while holding the point fixed, and for a family of rotating lines the derivative brings down a term in x: the line’s own rotation moves points at a rate proportional to their distance from the centre of rotation, and the balance between that and the rise s′ puts the touching point at exactly x = s′(θ).

So the offset is a velocity because the envelope condition compares a rise against a rotation, and the rise per radian is what the comparison is between. It is the same reason the pressure angle of a roller follower involves s′ and not s: every question about where on the cam the contact is is a question about rates, and every question about how far the follower has gone is a question about the motion itself.

That also explains why the base circle is absent. Growing R₀ moves the whole profile outwards without changing how fast the follower rises per radian, so the contact’s offset from the axis is untouched.

The second route #

The closed form above is a differentiation of the motion. The check is a measurement off the drawn profile, and it has no step in common with it.

Build the profile the cams field builds — for each cam angle, the point R₀ + s radially out and s′ tangentially, rotated into the cam’s own frame. Then rotate the whole profile under a fixed follower at 720 positions of the cam, and at each position find the point of the profile furthest along the follower’s axis and read off how far it sits from the axis. Take the extremes.

That is a maximisation over a sampled curve, and the closed form is a derivative of an analytic motion. They agree to 1.8 × 10⁻¹⁵ — the floor of double arithmetic — on the harmonic law, and exactly on the other two.

The agreement is the point rather than the number. A face width computed only from s′ would be an assertion about a formula; a face width read only off a profile would be an assertion about a sampling. Together they say that the profile the site draws really is the envelope the site claims it is, measured at the one place the two constructions can disagree.

What the sampling decides here, and what it does not #

The measured route is a maximisation over a sampled curve at a sampled set of cam angles, so it inherits the field’s standing question about sampling — and this is one of the few places where the answer is comfortable.

The contact offset s′ is a smooth function of the cam angle on every one of these programmes, with its extremes at interior points of the rise and the return rather than at the dwell seams. A smooth function’s maximum is found by sampling to second order: halving the spacing quarters the error. At 720 samples per turn the error is far below the fifteenth decimal place the two routes agree to.

That is a different situation from the swept gap, whose minimum can sit at a corner and can hide in a narrow interval, and the difference is worth naming. A smooth extremum is cheap to find and a piecewise-smooth one is not, and which of the two a quantity is depends on whether it is a min over features or a value of one function. The face width is the second kind.

The one place it stops being smooth is a programme with a discontinuous velocity — a rise that starts abruptly, which is a motion law nobody should use and which this site’s programme refuses on other grounds. There the face width is set by a jump rather than by a maximum, and the sampled route would find the jump’s larger side while the closed form finds the same thing, so the two still agree.

The version that was wrong, and what it looked like #

The measured route came out at 25.98 against a closed form of 19.10 on its first run — 36% too large — and the cause is worth recording because the picture looked perfectly sensible.

In the fixed frame the follower’s face is the line y = R₀ + s and the contact is at (s′, R₀ + s). The cam-frame point is that, rotated back by the cam angle. The first version rotated the cam-frame point forward instead — the same two numbers, composed the wrong way round — which produces a closed curve of about the right size, in about the right place, with a lobe where the lobe should be.

Nothing about the drawn profile said it was wrong. What said it was wrong was a second number disagreeing with it, which is the entire argument for having one.

What the face is for, and what it is not #

It is not a strength calculation. How thick the follower has to be, what the contact stress is and how the face wears are force questions and are outside this site.

It is not the follower’s whole size. A face 19.1 wide is the minimum span of contact surface; a real follower has material round it, a stem, a guide and a retainer, and all of those are larger.

And it is not symmetric in general. For these programmes it is, because the rise and the return are the same law over the same span, so s′ is odd about the dwells and its extremes are equal and opposite. A rise-and-quick-return programme has a face that overhangs one side by more than the other, and cutting it symmetrically wastes as much material as the asymmetry.

A follower is a body, which is the general point #

This essay belongs to two fields at once and that is deliberate.

The cams field supplies the profile, the motion law, the envelope and the velocity. The bodies field supplies the question — is this part big enough for what it has to touch — which is exactly the class of question that opens when a part stops being a point or a line.

The pattern is the same one that runs through the whole of that field. A pin is not a point, so two of them on one link have a minimum spacing. A link is not a distance, so two of them cannot be in the same place. A slider block is not a point on a line, so a guide has to be longer than the stroke. And a follower is not an edge, so its face has a width — each of them a quantity that has been sitting inside a construction this site has drawn for phases, waiting for somebody to give the part a size.

The width against the lobe #

There is a comparison worth drawing because it is the one that makes the number feel wrong.

The cam here has a base circle of 30 and a lift of 10, so its lobe rises 10 above a circle of radius 30 — a bump a third the size of the base. And the face it needs is 19.1 wide, nearly two thirds of the base circle’s radius and nearly twice the lift.

That is not an error and it is not an unusual proportion. A follower on a cam of this kind is a broad flat plate, much wider than the lobe is tall, and the reason is that the contact traverses it: over one turn the touching point sweeps across the face from one end to the other and back, twice.

Which gives a picture of what a flat follower is doing that a static drawing hides. It is not a plate resting on a bump. It is a plate being swept across by a contact that travels nearly 20 units while the follower itself moves 10 — so a great deal more of the follower is doing work than the part directly above the cam’s centre, and the whole of it has to be finished, hardened and supported.

What a designer does with it #

Three readings, all of them immediate.

Choose the law with the face in mind. If space beside the cam is the binding constraint, the harmonic law’s 15.0 against the cycloidal’s 19.1 is a real saving, paid for in acceleration at the ends of the rise. That is a trade the cams field can price on both sides and could not previously state on this one.

Do not scale the face with the lift. It is the commonest rule of thumb and it is not even dimensionally the right quantity: the lift is a displacement and the face width is a displacement per radian. Two programmes with the same lift over different spans have quite different faces — halving every span doubles s′ and doubles the face.

And check the asymmetry. The face width is a range, not a radius, and the two ends are only equal when the programme is symmetric. Quoting one number for a quick-return cam and centring the face on the axis gives a follower that runs off on one side and has material to spare on the other.

Three programmes, and where the face comes from in each #

The table’s three rows are all symmetric rise-and-return programmes with the same lift and spans, so their differences are entirely in the shape of s′ — and reading them side by side says what the face width is actually measuring.

Cycloidal motion has s following a sine-modulated ramp, so s′ is a raised cosine: it starts at zero, rises to a single peak in the middle of the rise, and returns to zero. The peak is 2L/Δ, twice the average velocity, and that factor of two is where 19.10 comes from on a lift of 10 over 120°.

Harmonic motion has s′ a half sine, peaking at πL/2Δ — a factor of π/2 rather than 2 — which is why its face is 15.00 and a quarter narrower.

Parabolic motion accelerates uniformly for half the rise and decelerates for the other half, so s′ is a triangle peaking at 2L/Δ, exactly the same peak as the cycloidal. Its face is 19.09 against 19.10, and the hundredth is the numerical differentiation rather than a real difference.

So the face width is a peak velocity in disguise, and the ranking of the three laws by face width is their ranking by peak velocity. That is a quantity the cams field already tabulates for a different reason, and this is a second use for it — which is the tidiest possible outcome for a new measurement: it turns out to be an old one, read in different units.

The measurement that was not made #

One thing this essay does not do, and it is worth naming rather than leaving.

A roller follower has an analogous question — how large the roller may be before the profile undercuts — and that one the cams field already answers, because the roller’s radius enters the profile rather than sitting outside it. The flat follower’s face is the opposite case: it does not enter the profile at all, which is why it could be omitted for six essays without any figure being wrong.

The base-circle interaction is also left alone, and it is worth being precise about the claim. The face width does not depend on the base circle. Whether the cam is valid does — too small a base circle and the profile undercuts, at which point there is no cam and the face width is a number about a shape that cannot be cut. So the independence is conditional on the profile existing, which is a condition the cams field already checks by a different measurement, and the two are combined here only by both being true.

The oscillating flat follower, whose face swings as well as translating, has a contact offset that is not simply s′ and needs the same derivation done again with the pivot in it. It is a short calculation and it is not done here.

The face’s width being set by the maximum of $ds/d\theta$ is a design number and it is worth reading as a specification rather than as an observation. The contact wanders by exactly the lift rate, so the face must extend at least that far either side of the follower’s axis — and the lift rate’s extremes are computable from the motion programme before any profile is cut. So the face width is decided at the same moment the programme is chosen, by the same numbers, and it is one of the very few cam dimensions available that early. It also says which programmes are expensive in face width. A fast rise has a large peak lift rate and needs a wide face; a gentle one needs less. So the choice of motion law prices two things at once — the peak acceleration and the follower’s own size — and they are not independent, since both are read off the same derivative. A programme is not only a motion; it is a bill of dimensions, and the face is the item most often left off it.