A flat face on an arm is worse

An arm is an offset that grows with the lift carried a cam’s roller on a swinging arm and found the pressure angle obeying the offset follower’s formula exactly, with the offset replaced by the roller’s distance from the cam’s centre measured across its own line of motion — a quantity that changes as the arm swings. The conclusion was favourable: a long arm beats an offset, because an offset is a constant subtracted from a velocity that changes sign between the strokes, and an arm’s is a ramp that can be aimed at one stroke.

A flat face asks for a convex cam then took the roller away. A flat face has no pressure angle at all — it is pushed square to itself whatever the cam is doing — and what replaces it as the thing that limits the design is convexity. The profile is the envelope of the face’s own line, and an envelope of lines is a real curve only where its radius of curvature is positive; for a sliding face that is $R_0 + s + s'' > 0$, and the smallest workable base circle is the largest amount by which $s + s''$ goes negative.

Both essays ended by naming the same question neither answered: a flat face on a swinging arm. The roller’s result suggests it should help. It does not.

Two lines and a support function #

A flat-faced follower is a line, and a cam driving one is a family of lines — one for each cam angle — whose envelope is the profile. So everything about it is decided by two functions of the cam angle: the direction the line’s normal points, and how far the line is from the cam’s centre along that normal.

For a sliding face, the normal never turns in the world, so in the cam’s frame it turns at exactly the cam’s own rate: $\alpha = -\theta - \pi/2$. The distance is the base circle plus the lift: $p = R_0 + s(\theta)$. The envelope’s radius of curvature is $p + \mathrm{d}^2p/\mathrm{d}\alpha^2$, which with those two substitutions is $R_0 + s + s''$, and the contact’s distance along the face from the foot of the perpendicular is $\mathrm{d}p/\mathrm{d}\alpha$, which is $s'$.

For a face on an arm pivoted at $Q$ a distance $d$ from the cam’s centre, with the face parallel to the arm and passing through the pivot, both change. Let $\psi$ be the arm’s angle. The face’s normal is perpendicular to the arm, so it turns with the follower as well as with the cam:

$$\alpha = \psi(\theta) - \theta - \tfrac{\pi}{2},$$

and its distance from the cam’s centre is the pivot’s distance resolved onto that normal:

$$p = d\sin\psi(\theta).$$

That second equation is the whole essay. The follower’s programme is its own rotation $\psi$; what the cam has to produce is $d\sin\psi$. A sliding follower’s programme and the cam’s support function are the same quantity; an arm’s are not, and the difference is a sine.

Both of those statements are about the same object read two ways, which is what a profile is: the cam is not a curve that happens to touch the follower, it is the boundary of everything the follower’s own shape leaves room for, and a support function is the tidiest way to write that boundary down for a family of lines. A roller follower has no such tidy form, because a roller is a circle and its family’s envelope is an offset curve rather than an envelope of lines — which is why the two follower types need different apparatus and why a result about one does not carry to the other.

The contact’s offset, with the arm in it #

The two derivatives follow, and the first is the one the earlier essay named and did not compute.

The contact sits at $\mathrm{d}p/\mathrm{d}\alpha = p'/\alpha'$ along the face from the foot of the perpendicular, and with the two substitutions above that is

$$\frac{d\cos\psi \cdot \psi'}{\psi' - 1},$$

which has the pivot’s distance in it and reduces to $s'$ as the arm goes off to infinity — the sliding formula, recovered as a limit.

The curve drawn for the arm comes from differentiating the envelope itself and never uses that formula; the two agree to $2.1 \times 10^{-9}$ at every cam angle. That is the check that the algebra belongs to the drawn curve rather than to a derivation beside it.

The offset is also what sizes the face: it has to be long enough to hold the contact at every cam angle, and the arm needs 38.36 against the slide’s 38.20 at a pivot three hundred out, rising to 40.61 at a pivot ninety out. That is a small cost and it is the smaller of the two the arm charges.

The face’s width matters more than its cost suggests, because it is where the flat face’s own difficulty lives. A follower needs a face measured the span a sliding flat face must have and found it twice the programme’s peak velocity — a quantity with no base circle in it, so it does not shrink when the cam is made bigger. On an arm the span is that plus a term the pivot controls, and the term is small, so the flat face’s standing complaint about its own size is essentially unchanged by the pivot.

Where the cam stops being a curve #

The other derivative is the profile’s radius of curvature, and it decides whether there is a cam at all.

At a base circle of 14 and a lift of 20, a pivot 5000 away reaches a worst radius of 3.16 and is fine; a pivot 300 away reaches −0.06 and has just failed; a pivot 150 away reaches −4.43 and a pivot 90 away −12.91. All four are being asked for the same programme at the same base circle, and they differ only in where the follower’s pivot is.

A negative radius of curvature is not a cam with a dent in it. It is a place where the envelope has folded into a swallowtail — the family of lines has no boundary there — and a cam cut to that profile would have the tool cross its own path. The sliding flat face’s version of this is measured two ways, by the formula and by watching the drawn envelope run backwards along the face it is touching, and the two agree; the same fold is what is happening here.

The slide is the limit, and it is the best case #

Bisecting on the base circle at each pivot distance gives the design chart.

The sliding follower needs 10.6640. A pivot 20000 away needs 10.7096; 5000 needs 10.8483; 1000 needs 11.6079; 300 needs 14.0644; 150 needs 18.4640; 90 needs 28.6789. The curve is monotonic and it approaches the slide’s value from above.

So the answer to the question the earlier essay left is no, and there is nothing to trade. A pivot cannot make a flat-faced cam convex that a sliding flat face could not, at any arm length, because the sliding follower is the arm’s limit and its best case. The roller’s result does not transfer, and the reason it does not is the sine: a roller follower’s difficulty is a pressure angle, which the arm’s changing offset can be aimed at; a flat face’s difficulty is a support function’s second derivative, and the arm distorts the support function in a way that has no aim to it.

The convergence is itself a check on the whole apparatus, and it is worth saying why it is not automatic. The arm’s formulae were derived from a support function with a sine in it and a normal that turns with the follower; the slide’s were derived long before, from a support function with no sine and a normal that does not. The two derivations share only the envelope’s own curvature rule. That the first reduces to the second as the pivot goes to infinity — 10.7096 at 20000 against 10.6640 — is the arithmetic saying the substitutions were right, and it would have failed loudly had the normal’s rate or the sign of $\psi$ been got wrong, because either mistake changes $\alpha'$ and therefore the whole second derivative.

The excess falls as the reciprocal of the pivot’s distance: the products are 944, 921 and 912 at the three furthest pivots, converging on a constant that is a property of the programme. That gives the design rule its shape. An arm pivoted $n$ base circles away pays about one part in $n$ for it, which is negligible at ten and serious at three.

And the last row is the part a designer must know before drawing anything. An arm pivoted 60 away — three lifts from the cam’s centre — has no convex cam at any base circle. Making the cam bigger does not help, because making it bigger means a larger $\psi_0$, and the sine’s distortion gets worse rather than better as the arm swings further from the perpendicular. The programme simply cannot be delivered by a flat face on an arm that short.

It is worth setting that against the roller’s answer at the same lengths. An arm ninety units out is a good arrangement for a roller: the arm’s ramp is steep there, and a steep ramp is what buys a pressure angle on the rise at the return’s expense. It is the worst arrangement measured here. So the same design decision — bring the pivot in — improves a roller follower and ruins a flat-faced one, and a designer who has learnt the first rule and applies it to the second gets a cam that cannot be cut. That is the same class of trap as the cam that cannot be cut for a roller: a condition that is invisible in the programme and appears only when the profile is drawn.

One more reading of the chart, because it is the shape a designer meets rather than the number. The excess is not a fixed penalty to be budgeted; it is a ratio to the base circle, and the base circle is itself a size the designer is choosing. So the decision is scale-free above the failure point: a pivot at ten base circles costs a tenth of a base circle whatever the cam’s absolute size, and doubling every length in the machine changes nothing. That is the flat face behaving as a shape rather than a size — until the short-arm failure, which is not a ratio at all but a boundary in the same scale-free space, at a pivot of about three lifts.

What the sine is doing #

The mechanism of the failure is worth naming, because it is a general fact about supports and not about cams.

A sliding face’s support is $R_0 + s$, so $\mathrm{d}^2p/\mathrm{d}\alpha^2 = s''$ and the convexity condition asks only about the programme’s acceleration. An arm’s support is $d\sin\psi$, so differentiating twice brings down a $-d\sin\psi\,(\psi')^2$ term alongside the $d\cos\psi\,\psi''$ one. The first is always negative, it is proportional to the square of the follower’s velocity, and it is present even during a stretch of the programme with no acceleration in it at all.

That is why nothing can be aimed. A pressure angle is a first-order quantity and an arm gives a designer a first-order handle — a ramp with a slope and a sign — to set against it. Convexity is second order, the arm’s contribution to it is a square, and a square has one sign.

It also says which way the cost scales with the design. The harmful term is $d\sin\psi \cdot (\psi')^2$; the useful support is $d\sin\psi$; and $\psi' \approx s'/(d\cos\psi)$, so the term goes as $s'^2/(d\cos^2\psi)$ and falls as the pivot goes off. The reciprocal law in the table is that ratio, and the constant it converges to is $\max(s'^2)$ scaled by the programme’s own shape.

What a designer actually decides #

Put the three numbers together and the flat-faced follower’s design space is small enough to state.

A flat face on a slide is decided by one number: the base circle, which must exceed the largest negative value of $s + s''$. Everything else — the face’s width, the contact’s excursion, the profile itself — follows from the programme with nothing left to choose. That is the flat face’s virtue, and it is why it survives in valve gear where a roller would be a bearing to look after.

A flat face on an arm adds one number, the pivot’s distance, and the chart says what it buys: nothing, and it costs about one part in $n$ of base circle when the pivot is $n$ base circles out. So the pivot is chosen for reasons outside the cam — where the valve is, what the rocker has to reach, whether there is room for a slide at all — and the cam is then made a little larger to pay for it.

The one decision the chart forbids is the short arm. Below about four lifts from the cam’s centre there is no base circle that works, and the failure is not gradual: the required base rises steeply and then there is nothing. A designer who wants a compact rocker and a flat face is choosing between them.

What this does not settle #

The face passes through the pivot. A flat face offset from its own pivot, so that the face is parallel to the arm but standing off it, is a two-parameter family this does not explore. It changes $p$ by a constant in the follower’s frame and therefore by a turning quantity in the cam’s, and whether that constant can be spent usefully is an open and answerable question.

One programme. Everything is measured on the symmetric cycloidal programme with a lift of 20. A lift is a size and a law is a shape says which of a cam’s numbers scale and which do not, and the excess-times-pivot constant is a quantity of the first kind squared over one of the second, so it should scale as the lift squared. That is stated and not measured.

The pivot’s side is not swept. The arm here swings one way as the follower lifts. A pivot on the other side of the cam reverses which way $\psi$ moves, and for a roller that choice matters because it flips the ramp’s sign. Here the harmful term is a square and cannot be flipped, so the side ought not to matter; that expectation is not measured.

Convexity only. A cam can be convex and still be a bad cam: the contact stress, the sliding velocity along the face and the moment the offset contact puts on the follower’s own bearing are all quantities a designer weighs, and none is computed. The moment in particular is the flat face’s counterpart of the pressure angle and it is larger here than on a slide, because the offsets are larger.

No yoke. A cam that holds its follower both ways closes two faces round a cam and finds the programme has to be its own reflection. Two faces on one swinging arm measure across the cam along a line that turns, and whether the constant-breadth condition survives that is a separate question this does not touch.

Still open: the arm the face is allowed to be offset from #

The one design variable held fixed throughout is the face’s position relative to its own pivot: the face is taken to pass through $Q$. Letting it stand off by a constant $c$ in the follower’s frame gives $p = d\sin\psi + c$, which changes the support by a constant and therefore changes nothing about $\mathrm{d}^2p/\mathrm{d}\alpha^2$ — but it changes $\psi_0$ for a given base circle, and $\psi_0$ is what the sine’s distortion depends on.

Its distinct argument would be the two-parameter chart: over the pivot’s distance and the face’s stand-off together, the smallest convex base circle, and whether any combination of the two beats the sliding follower. The arithmetic above suggests not, since a stand-off buys base circle at the price of arm swing and the harmful term rises with the swing, but the trade has a sign that has not been computed and the answer could be that a short arm with a large stand-off is workable where a short arm alone is not — which would turn the last row of the table from “no cam” into “no cam of this shape”.