Where on the sine the arm swings

A flat face on an arm is worse put a flat-faced cam follower on a swinging arm and found that the arm charges for it. The follower’s programme is its own rotation ψ, but the cam has to produce $d\sin\psi$, the pivot’s distance resolved onto the face’s normal. The sine distorts the programme, and the distortion costs convexity. A sliding face on the standing cam programme — a rise and return of 20 over 120° each — needs a base circle of 10.664. On an arm pivoted 300 away it needs 14.063, at 90 it needs 28.679, and an arm pivoted 60 away was reported to have no convex cam “whatever else is changed”.

The explanation given for why nothing can be done about it was a sign argument. Differentiating $d\sin\psi$ twice brings down a term $-d\sin\psi\,(\mathrm{d}\psi/\mathrm{d}\alpha)^2$, a square of the follower’s angular speed. A pressure angle can be aimed, because an arm’s ramp has a slope and a sign. A square, the argument went, has only one sign.

That argument quietly assumed something that the face’s geometry forced and nothing about arms requires. The square is multiplied by $\sin\psi$, and $\sin\psi$ has the sign of where the arm is.

A face on a bracket #

The one design variable held fixed so far is where the face sits relative to its own pivot. It was taken to be a line through the pivot, parallel to the arm. Let it instead stand off by a constant c, as though mounted on a bracket, still parallel to the arm and still turning with it.

The change is small in the algebra. The face’s distance from the cam’s centre is now

$$p = d\sin\psi + c,$$

and nothing else moves. Its normal still turns as $\alpha = \psi - \theta - \pi/2$, because the bracket turns with the arm. The radius of curvature of the profile is still $p + \mathrm{d}^2p/\mathrm{d}\alpha^2$, as for any family of lines. A constant adds itself to p and nothing to the second derivative.

What the constant does change is where the arm is. On the low dwell the face must touch the base circle, so $d\sin\psi_0 + c = R_0$ and the swing runs from $\arcsin[(R_0 - c)/d]$ to $\arcsin[(R_0 + L - c)/d]$. With the face through the pivot, $\psi_0 = \arcsin(R_0/d)$, which is positive for every real cam. The face’s distance from the cam is $d\sin\psi$, and that must be positive. So the face through the pivot is confined to the half of the sine where the harmful term is harmful. The bracket removes that confinement. The base circle and the lift fix the length of the swing along the sine, and the stand-off decides where along the sine it sits.

Both panels above deliver the same programme on the same arm. On the left the arm swings from 18.6° to 32.7° and needs a base circle of 28.679. On the right the face stands 33.6 off the pivot, the arm swings from −4.7° to 8.0°, and the smallest convex base is 26.136, about 9% less. The pivot clears the cam in both.

The sign of a square times a sine #

Written out, the second derivative of the support is

$$\frac{\mathrm{d}^2 p}{\mathrm{d}\alpha^2} = d\cos\psi\,\frac{\mathrm{d}^2\psi}{\mathrm{d}\alpha^2} - d\sin\psi\left(\frac{\mathrm{d}\psi}{\mathrm{d}\alpha}\right)^2.$$

The first term is the programme’s acceleration, carried through the cosine. On a slide it would be $s''$ and would decide everything. The second term is the one the arm brings, and it is the only place where the side of the sine matters.

At zero stand-off the heavy stretch sits well up the positive side of the sine. The square is subtracted everywhere, and it grows with $\sin\psi$ as the arm climbs. That is the picture the earlier essay drew, and on its own terms it was right: for a face through its pivot the square has one sign.

Increasing the stand-off moves the stretch down the curve. By a stand-off of about 34 it straddles ψ = 0. Over the first part of the rise, while the arm’s angle is still negative, the term adds to the radius of curvature. Over the rest it takes away, but from a sine that is now small. Along the rise and return of the best-placed arm at 90, the term is positive at 44% of the cam angles, where with the face through the pivot it is positive at none.

Two things stop this from becoming a free lunch. First, the sine’s own curvature is not what matters most. The programme’s deceleration, carried through $d\cos\psi$, is the dominant term, as it is for a sliding face, and the stand-off changes it only through the cosine. Second, pushing the stretch too far down the sine puts the whole swing on the negative side. There the face’s distance from the cam has to come from the bracket rather than the arm, and the sine starts to flatten on that side as well. The slider shows the base falling from 28.68 at no stand-off to its least near 34, then rising again. By a stand-off of 60 it is back above 27.

One base circle, four stand-offs #

The clearest way to see the stand-off is to hold the base circle still and watch the profile’s curvature where it is worst.

A base of 27 lies between the two answers in the table below. It is too small for the face through its pivot, which needs 28.68, and larger than the best stand-off’s 26.14. The face through the pivot folds, reaching −1.40 at 80° of cam rotation. At a stand-off of 15 the worst radius is 0.23 and the cam is barely convex. At 33.6 the worst is 0.86, with margin. At 60 it folds again, at −0.14.

The fold sits late in the rise, around 80° of a 120° rise, where the cycloidal law decelerates hardest. That is the same place a sliding face’s $R_0 + s + s''$ is least. The four curves agree almost everywhere else in the turn, because on the dwells the follower does not move and the square has nothing to square. The stand-off acts only where the follower is moving, which is also the only place its effect can reach a fold.

As in the sliding face’s own essay, a negative radius here is not a dent. It is a place where the family of face lines has no boundary: the drawn envelope doubles back into a swallowtail, and a cam cut to it would need the cutter to cross its own path. The threshold is found two ways. One is the curvature formula swept round the turn. The other never uses the formula: it draws the envelope and asks whether the contact ever runs backwards along the face it is touching, with the face’s direction now turning with the arm. At a pivot of 90 with no stand-off the two agree to 4 × 10⁻⁴; stood off 30, to 2 × 10⁻³; and at a pivot of 150 stood off 20, to 1 × 10⁻⁴.

The chart over both numbers #

The unanswered question was a two-parameter chart: over the pivot’s distance and the stand-off together, find the smallest convex base, and ask whether any combination beats the slide.

Each line is a shallow bowl. The bowls are deep on short arms and nearly flat on long ones, and none reaches the dashed line. On an arm pivoted 150 away the best stand-off, 19.3, lowers the base from 18.46 to 18.38. That is less than half a per cent, and the chart shows no reason to go to the trouble. At 300 the bottom of the line is flat: the best stand-off, −6.2, beats the face through the pivot by less than a thousandth, which makes it a rounding choice rather than a design decision.

The reason is the same sine read the other way. On a long arm the swing is a few degrees at most, and over a few degrees the sine is so nearly straight that it hardly matters where the few degrees are. The face’s distortion is set by the sine’s curvature across the swing, and a short stretch of any smooth curve is nearly a line. The stand-off moves a stretch that was already nearly straight.

The table has the numbers.

The swing column shows the mechanism directly. On the short arms the best stand-off puts the swing across zero: −7.6° to 8.8° at 70, −4.7° to 8.0° at 90, −2.3° to 7.3° at 120. The midpoint is within two and a half degrees of the perpendicular. A rule written before any of this was measured would have said to centre the swing, which means a stand-off of $R_0 + L/2$. At 90 that gives 36.1 against the best 33.6, and the bowl is flat enough that the difference costs less than a hundredth of a unit of base circle. On the long arms the best swing wanders off zero, because there the bowl has no real bottom.

The last column answers the chart’s own question. No combination beats the slide. The best excess over a sliding face is 23.06 at a pivot of 70, 15.47 at 90, 10.31 at 120, 7.71 at 150 and 3.40 at 300. The stand-off shortens the gap and never closes it. The reason is visible in the second-derivative formula: a slide has no square term at all. At best the stand-off turns a term that always hurts into one that helps over part of the stroke. It cannot make an arm’s cam better than a mechanism whose cam has no such term.

How short an arm can be #

The one row of the earlier table that the stand-off really changes is the last. There, an arm pivoted 60 away had no convex cam at any base circle, and “whatever else is changed” was part of the claim.

With the face through the pivot the shortest workable arm is 80.0, four times the lift. Below that, making the cam bigger only pushes ψ further up the sine where the square hurts more. Stood off, the shortest workable arm is 60.5, about three times the lift. Between the two, every arm has a cam when the face is on a bracket and none when it is not. At 70, a cam that cannot exist with the face through the pivot needs a base of 33.72 stood off 43.

The limit is not the sine any more but the pivot. A cam cannot contain the point its own follower turns about, and a short arm’s cam, grown large enough to be convex, reaches out to it. Every cam counted here keeps its farthest point inside the pivot’s distance from the cam’s centre. Relax that and the curves would run further left, but they would describe a machine that cannot be assembled.

So the arm pivoted 60 away still has no cam, and the earlier sentence survives — by half a unit of pivot distance, which is closer than it sounded. Its reason was also incomplete. The earlier essay said that making the cam bigger only makes things worse “because making it bigger means a larger $\psi_{0}$”. A stand-off breaks that link between the size of the cam and the angle of the arm, so “whatever else is changed” held at 60 because the pivot would end up inside the cam, not because of the sine.

The face the bracket needs #

A flat face has a second size besides its cam: how wide it must be to hold the contact through the turn. A follower needs a face found that a sliding face must span twice the programme’s peak velocity, 38.20 here. On an arm the contact’s offset along the face is $d\cos\psi\,\psi'/(\psi' - 1)$. The stand-off adds nothing to it directly, since it is a derivative of p, but it moves ψ. On the arm pivoted 90 away the face through the pivot must be 40.61 wide, and the stood-off face 40.09. So the bracket buys a slightly smaller cam and a slightly narrower face together. It is not trading one against the other.

The farthest reach of each cam is its base circle plus the lift, 48.68 and 46.14. The rule that a cam must not reach its own pivot is therefore easy to check by hand, even though the chart needed it computed.

Where the bracket already is #

A flat face standing off its pivot is not an invented mechanism. It is close to what an overhead-camshaft engine’s finger follower does. There, a pad on a short lever pivoted at one end is pressed down by the cam and in turn presses on a valve. The pad sits below the line from the pivot to the valve, so in this essay’s language it stands off its own arm. Where the pad touches followed the valve end of a rocker, where its pad slides across the valve tip, and found that how the rocker was squared up decided how far the pad slid. The chart above is the cam end of the same question: how the pad is set against its pivot decides how small a cam can drive it.

What the chart adds is a direction for the setting. A finger follower is a short arm, because compactness is the reason to have one, and short arms are where the stand-off matters. The rule is to set the pad so the lever is square to the pad’s normal halfway up the lift, and every short arm measured here agrees with that to about a hundredth of a unit of base circle. The same geometry that sets the rocker’s ratio at the valve end sets the cam’s size at this end, and neither is fixed by the cam’s programme alone.

As with the cam that cannot be cut, the condition that decides all of this is invisible in the programme. The motion law is the same in every panel of every figure here. Only the drawn envelope says whether a cam exists.

What a designer does with a bracket #

The flat face on a slide is decided by one number, the base circle, and the flat face on an arm through its pivot by two. A bracket adds a third, and the chart shows how little of it is worth spending.

On a long arm the stand-off is free and useless. At fifteen lifts the best stand-off gains less than a thousandth, and anywhere across the middle of the range costs about a hundredth. The bracket can go wherever the rest of the machine wants it, which is useful in itself: a designer who needs the face somewhere other than on the arm’s line can put it there without paying anything at the cam.

On a short arm the stand-off is the difference between a cam and none. Between three and four lifts, only a face on a bracket works. Here the rule is to centre the swing: choose the stand-off so the arm is square to the face’s normal halfway up the stroke. That comes within 0.002 of the best base at 70, 0.009 at 90 and 0.013 at 120.

On no arm does it beat a slide. An arm is chosen for reasons outside the cam, such as where the valve is or what the rocker has to reach. The cam is then made larger to pay for it, and the bracket reduces that cost without removing it. An arm is an offset that grows with the lift found the opposite trade for a roller, where an arm can beat a slide because a pressure angle is first order and can be aimed. Convexity is second order, and the most a bracket can do with a second-order term is choose its sign over part of the stroke.

What this does not settle #

One programme and one law. 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 numbers scale. The best stand-off is a length and should scale with the cam. The best swing is a pair of angles and should not. Neither has been measured on a second programme.

The bracket’s own moment. A face standing off its pivot turns the contact force into a moment about the pivot through a longer lever. Carried round the cam, the bracket also sweeps a larger circle than the face alone would. Neither is computed, and the second in particular may be the real limit on short arms before the pivot’s clearance is.

The pivot on the other side. The arm here swings one way as the follower lifts. On the other side of the cam the swing reverses, and with the face through the pivot the harmful term is unchanged, since it is a square. With a stand-off, reversing the swing moves the stretch the other way along the sine. Whether that makes the other side better, worse or the same has not been checked.

Still open: the face that is not parallel to its arm #

The bracket here keeps the face parallel to the arm, so the face’s normal turns exactly as the arm does. A bracket can also hold the face at a fixed angle β to the arm. Then the face’s normal is the arm’s angle plus β, and the support becomes $d\sin(\psi - \beta) + c$ with the normal at $\psi + \beta - \theta - \pi/2$.

The distinct argument there would be whether that angle is a third independent choice or only the stand-off in another form. The support’s shape along the sine is moved by β just as by c, but the rate at which the normal turns is not, because the normal still turns at $\psi' - 1$. If the angle buys nothing that a stand-off does not, the chart above is the whole design space of a flat face on a rigid arm. If it does — if tilting the face moves where the fold falls, rather than only where the swing sits — then a tilted face could reach bases the parallel face cannot. The test is the smallest convex base at a pivot of 70, where the parallel face needs 33.72.