An arm that lifts against the cam can beat a slide

A flat-faced follower on a sliding stem needs a cam that is convex everywhere, because a flat face can only touch a cam at points where the cam bulges towards it. A flat face asks for a convex cam turned that into a smallest base circle: below it the cam folds. A flat face on an arm is worse put the same face on a swinging arm with the face passing through the arm’s pivot and found the smallest base larger at every pivot distance. Where on the sine the arm swings then stood the face off its pivot on a bracket, parallel to the arm, and found the stand-off a second knob: it moves the arm’s swing along the sine of its angle until the curvature’s harmful term changes sign halfway up the stroke. On an arm pivoted 90 away it lowered the smallest convex base from 28.68 to 26.14, and on no arm did it beat the slide’s 10.66.

That essay ended with two open questions about the design space. A bracket can hold its face at an angle to the arm as well as at a distance from it, and nothing said whether that angle was a third knob or the stand-off in another form. And the arm had always been set to lift in the direction the cam turns; with the pivot on the other side of the cam it would lift the other way, and nobody had measured whether that mattered.

The first question has a one-line answer and the second a large one.

A line on a turning body has one number #

The arm is a rigid body that turns about its pivot, and the face is a line fixed to that body. Turn the body and the line turns with it, keeping its distance from the pivot. Two lines at the same distance from the pivot are the same line turned about it, so a line fixed to the body is described, apart from where the body happens to be turned to, by one number: its distance from the pivot. The direction of the “arm” — the bracket’s metal, the line from the pivot to where the face is fastened — is not part of that description at all.

So a tilted face is a parallel face with a different bracket. Built out explicitly, with the arm at angle ψ, the bracket’s end at the pivot plus $c$ along the arm’s normal, and the face’s normal turned β from the arm’s, the face’s distance from the cam’s centre is

$$p = d\sin(\psi + \beta) + c\cos\beta,$$

and its normal points at $\psi + \beta - \pi/2 - \theta$ in the cam’s frame, θ being the cam’s angle. That is exactly a parallel face on an arm at angle $\psi + \beta$, standing $c\cos\beta$ off its pivot. The distance from the pivot to the tilted face is $c\cos\beta$, because the bracket of length $c$ meets the face at the angle β.

The hero figure draws both. The dark bracket is 43 long with the face tilted 30° on it; the light bracket is 43 cos 30° = 37.2 long with the face square to it. Both end on the same line, the cam is one cam, the contact is one point. Over a full turn of the cam the tilted face’s support, computed from the vectors rather than from either formula, agrees with the parallel face’s to $2 \times 10^{-14}$, and the direction of its normal to $10^{-15}$ radians. Dragging the tilt swings the dark bracket round and shortens the light one, and the face does not move.

Tilting only shortens the bracket #

The consequence for convexity follows without any further argument, but it is worth seeing measured. A search for the smallest convex base run on the tilted geometry itself, at tilts from 0° to 50°, gives 33.725, 33.729, 33.764, 33.895, 34.244 and 35.064. A separate search on a parallel face at the stand-off 43 cos β gives the same six numbers. A stand-off of 43 was already the best at this pivot, and tilting the face can only shorten the effective stand-off, so the base rises with the tilt; no tilt takes it below 33.725.

That settles the first question. The chart over pivot distance and stand-off in the earlier essay is the whole design space of a flat face on a rigid arm, as far as the face’s line is concerned. The tilt may still be worth something to a designer — a bracket that runs at an angle can clear a housing that a straight one would hit — but nothing about the cam, the contact or the force on the pivot depends on it, since all three are properties of the line.

Which way the arm lifts #

The other open question is not about the bracket. An arm pivoted on one side of the cam lifts, as the follower rises, in the same rotational sense as the cam turns. Put the pivot on the other side and it lifts the other way. Every earlier measurement of a flat face on an arm used the first arrangement.

The difference is in how fast the face’s normal turns relative to the cam. The cam turns at one radian per radian of its own angle, and the face’s normal turns with the arm, at ψ′ per radian of cam. When the arm lifts with the cam’s turning, the normal turns relative to the cam at $1 - \psi'$ in size during the rise; when it lifts against it, at $1 + \psi'$. The cam’s radius of curvature is the face’s distance plus the second derivative of that distance with respect to the normal’s angle, so it carries the rise’s deceleration divided by the square of the relative rate. An arm lifting against the cam spreads the rise over more turning of the normal and divides the deceleration by more. During the return, with ψ′ negative, the advantage passes to the other sense.

On the symmetric programme those two effects cancel exactly. Its rise and return are mirror images, so running it backwards — which is what reversing the arm’s sense amounts to, as seen from the cam — gives the same programme. The smallest convex base at the best stand-off is 33.725 at a pivot of 70, 26.136 at 90, 20.972 at 120, 18.377 at 150 and 14.062 at 300 whichever way the arm lifts, equal to every digit computed. That is why the question could go unasked: on the programme every earlier essay used, it has no answer to give. A cam that holds its follower both ways found the quick-rise programme below failing a reflection rule on every count — a fast rise, a slow return, unequal dwells — because it is far from its own mirror image. The same distance from symmetry is what the arm’s sense can exploit.

A quick rise splits the two #

The quick-rise programme is the asymmetric one the earlier cam essays already used: a cycloidal rise of 20 over 90°, a dwell of 40°, a return over 170°, a dwell of 60°. Its rise is nearly twice as steep as its return, and a flat face asks for a convex cam found that a sliding face needs a base of 32.85 to keep its cam convex on it, three times what the symmetric programme needs.

An arm that lifts with the cam is worse than the slide, as the earlier essays found arms to be. With its face through the pivot it has no convex cam that clears its own pivot until the pivot is 150 away, where it needs 54.60; at 300 it needs 40.93. Stood off at its best it needs 59.24 at 120, 52.02 at 150 and 40.88 at 300, and at 70 and 90 nothing works.

An arm that lifts against the cam is better than the slide at every pivot, and by a large factor. With its face through the pivot it needs 12.85 at a pivot of 70, 16.13 at 90, 19.43 at 120, 21.64 at 150 and 26.69 at 300: from two fifths of the slide’s base to four fifths. Stood off at its best it needs 7.88 at 70, 7.18 at 90, 6.66 at 120, 6.55 at 150 and 10.65 at 300, a quarter of the slide’s or less. An arm is an offset that grows with the lift found a roller follower’s arm beating a slide because a pressure angle is first order and can be aimed. Here the thing being aimed is second order, and it can still be aimed, because it is divided by the square of a rate the arm’s sense decides.

At long arms the two senses straddle the slide. An arm 300 long with its face through the pivot needs 26.69 one way and 40.93 the other, against the slide’s 32.85: the slide is the limit of an infinitely long arm, which turns its face not at all, and a finite arm turns its face one way or the other. The turning helps the rise in one sense and hurts it in the other, and on a rise this steep the rise is what decides.

How short the arm can be #

The earlier essay asked how short an arm can be and still have a convex cam that clears its own pivot, and found the stand-off shortening it. On the quick programme the sense matters more than the stand-off does. Lifting with the cam, no arm with its face through the pivot works at any length up to 120, the longest the search tried, and with the best stand-off the shortest is 93. Lifting against the cam, the face through the pivot works down to an arm of 48, and with a stand-off every arm down to 40, the shortest tried, has a convex cam. An arm the other sense cannot use at all is, in this one, the better choice of pivot.

The two cams of one arm #

Drawn at their own smallest bases, the two cams for one arm 150 long are 21.64 and 54.60 in base radius, and the second is two and a half times the size of the first. Both are convex, both carry the same programme to the same face, and the only difference in the machine is which side of the cam the pivot sits.

The best stand-offs in the favourable sense are large and negative: −26.4 at a pivot of 70, −51.5 at 90, −85.4 at 120, −117.6 at 150 and −266 at 300. A negative stand-off puts the face on the far side of the pivot from where the earlier essay’s brackets put it, and these ones push the arm’s swing high up the sine, to between 29° and 51° on the 70 arm and between 67° and 82° on the 300 arm. There the sine is flat, so the arm must swing through a larger angle to deliver the same lift — 21.5° on the 70 arm and 14.2° on the 300 arm, where the face through the pivot swings 17.4° and 3.8° — and a larger swing is a faster-turning normal. The stand-off is doing what it did before, choosing where on the sine the arm works, and in this sense the best place is near the top.

Where the steep rise folds #

The curvature over a turn shows where the senses part. Through the dwells and the slow return the two curves are close, the return a little worse for the arm lifting against the cam and a little better for the one lifting with it, as the sign of ψ′ there says it should be. On the steep rise they separate widely. At the base of 21.64 that makes the favourable arm’s cam just convex, its radius of curvature touches nought at 70° of cam rotation, three quarters of the way up the rise, where the cycloidal law’s deceleration is greatest. The arm lifting with the cam, at the same base, falls to −30.8 at 63°: its cam folds back on itself across a large part of the rise.

So the reversal is not a small effect amplified by a threshold. It moves the worst point of the cam’s curvature by thirty units of radius on a base of twenty, and it does it by dividing the rise’s deceleration by a larger square.

Checked on the drawn cam #

A result this large deserves the second route the earlier essay built for the threshold. Instead of the radius-of-curvature formula, draw the envelope of the face’s line over a turn and find the base below which the contact point ever runs backwards along the face — the visible form of a cam that folds. For the arm lifting against the cam on the quick programme, the two routes give 7.8928 and 7.8942 at a pivot of 70 and stand-off −26.4, 7.1912 and 7.1927 at 90 and −51.5, and 16.1285 and 16.1284 at 90 with the face through the pivot. The differences are a thousandth of a unit or less, the resolution of the drawn envelope, and every cam the formula calls convex draws convex.

What a designer gains #

The tilt is free, and buys nothing for the cam. A bracket can be bent to any angle to clear the machine around it, and the cam is designed for the one number that matters, the face’s distance from the pivot.

On a symmetric programme, the side the pivot sits on does not matter to the cam. The two senses are mirror images to every digit, and a lift is a size and a law is a shape suggests the statement survives any rescaling of the programme, since mirror symmetry is a statement about shape.

On a programme with a steep rise, put the pivot where the arm lifts against the cam’s turning. On the quick-rise programme that turns an arm from a liability into the smallest cam a flat face can have: 6.55 at a pivot of 150 with a stand-off of −117.6, against 32.85 for a slide and 52.02 for the same arm on the wrong side. A designer whose programme returns steeply and rises slowly should do the reverse.

What the arm’s sense does not settle #

One law and one quick programme. The rise is cycloidal and 90° long throughout. A steeper rise should favour the reversed sense more and a gentler one less, but only one programme was measured.

The bracket’s own size. The best stand-offs in the favourable sense are long: −117.6 on a pivot 150 away puts the face nearly as far from the pivot as the pivot is from the cam. The bracket that holds it, and the moment the contact force exerts on the pivot through it, are not computed.

The force. A flat face’s contact force acts along the face’s normal through the contact, so its moment about the pivot is set by how far along the face the contact sits from the foot of the pivot’s perpendicular. The sense changes the cam, and with it where the contact sits; neither that moment nor how far the contact travels along the face, which sets how long the face must be, was measured for the reversed sense.

Still open: the rise an arm can take #

The slide’s smallest convex base grows as its rise gets steeper, because the deceleration the cam must turn grows. The arm lifting against the cam divides that deceleration by the square of a rate that also grows with the rise’s steepness, since a steeper rise swings the arm faster.

The distinct argument would be whether those two growths cancel. The law that costs least ranked motion laws by what a steeper rise costs a follower; a reversed arm may change that ranking for flat faces, since it charges the deceleration at a rate a law’s own shape partly sets. The measurement would shorten the quick programme’s rise from 90° to 60° and 45°, keep the lift, and find the smallest convex base for the slide and for the arm lifting against the cam at each pivot. If the arm’s base grows more slowly than the slide’s, there is a steepness of rise that only an arm can follow with a flat face on a cam of practical size, and the earlier finding that arms are worse than slides would turn out to be a statement about gentle programmes.