A steep rise spends the arm's room

An arm that lifts against the cam can beat a slide found that the direction an arm swings decides how large a cam its flat face needs. On the quick programme, a rise of 20 over 90° and a return over 170°, an arm pivoted 90 from the cam’s centre and lifting against the cam’s turning needs a base of 7.18 with its face stood off at its best. A sliding face needs 32.85. The reason is second order. The cam’s curvature under a flat face depends on the deceleration the cam must turn, divided by the square of the rate at which the face’s normal turns relative to the cam, and an arm that lifts against the cam turns its face faster.

That essay ended by asking whether the advantage survives a steeper rise. A steeper rise raises the deceleration as the inverse square of its span, and it also swings the arm faster, which raises the rate that divides the deceleration. If the two cancel, there is a steepness that only an arm can follow with a flat face on a cam of reasonable size.

They do not cancel, but the arm’s least base does grow far more slowly than the slide’s. From a 90° rise to a 45° rise the slide’s base grows 5.6 times and the arm’s 3.2 times. The surprise is what the arm pays for that. Its least base on a steep rise sits on a knife edge: the return is about to go concave as well as the rise, and the base can barely be raised before it does. Ask the arm’s cam for any working room and its base grows nearly as fast as the slide’s.

The slide’s base grows as the inverse square of the rise #

A flat face is pushed at right angles to itself, so a sliding face never has a pressure angle to limit it. A flat face asks for a convex cam found what limits it instead. The cam’s radius of curvature at the contact is the base plus the displacement plus the displacement’s second derivative, $R_0 + s + s''$, and the cam must stay convex, so the base must exceed the most negative value of $s + s''$. That value comes on the decelerating half of the rise, where $s''$ is large and negative.

A cycloidal rise of lift L over β radians has a peak deceleration of 2πL/β². Halving the span quadruples it. The measurement keeps the quick programme’s return and its dwell, keeps the lift at 20, and shortens the rise from 90° to 75°, 60°, 45° and 35°, adding the time saved to the dwell after the rise.

The slide’s base grows from 32.85 at 90° to 55.23, 96.45, 185.56 and 318.59. The closed form agrees with the measured base to within 0.001 at every span. From 90° to 45° the base grows 5.6 times, more than the factor of four in the deceleration. The base is the deceleration term less the displacement where it peaks, and the displacement does not grow: four times the deceleration less the same displacement is more than four times the difference.

The arm’s least base grows slowly #

The arm is pivoted 90 from the cam’s centre and lifts against the cam’s turning. Its face can stand off its pivot by any distance, and at each rise the search finds the stand-off that gives the least base. That search scans stand-offs from −80 to 20 in steps of two, finds the least convex base at each, and then refines about the best.

The least base is 7.18 at 90°, 10.92 at 75°, 15.91 at 60°, 23.26 at 45° and 29.80 at 35°. From 90° to 45° it grows 3.2 times, against the slide’s 5.6. The arm’s share of the slide’s base falls from 22% to 13%. At 35° the arm needs 29.8, less than the slide needs for a 90° rise. On these numbers, the earlier essay’s question has an answer that favours the arm: the two growths do not cancel, but the arm’s is much the slower.

The mechanism is the one the question proposed. The face’s normal turns relative to the cam at one plus the arm’s angular rate, and a steeper rise swings the arm faster. The cam’s curvature term is the deceleration divided by the square of that rate, so part of the deceleration’s growth is divided away. The best stand-off moves, too, from −51.5 at 90° to −39.2 at 35°: the steeper the rise, the less the face can stand off before the cam fails to clear the pivot or the return fails to stay convex.

Why the return sets a ceiling #

The same mechanism that helps the rise hurts the return, and the earlier essay drew it: lifting against the cam eases the steep rise and taxes the gentle return. On the rise the arm swings against the cam’s turning, so its face turns relative to the cam at one plus the arm’s rate, and the rise’s deceleration is divided by the square of a number greater than one. On the return the arm swings back, with the cam’s turning. The face then turns relative to the cam at one less the arm’s rate, and the return’s deceleration is divided by the square of a number less than one. The return’s deceleration is gentle, because the return takes 170°, but it is magnified.

How much it is magnified depends on how fast the arm swings, and that depends on the base. The face’s distance from the cam’s centre is the pivot distance times the sine of the arm’s angle, plus the stand-off. A larger base puts the face further out, which turns the arm nearer to square with the line from pivot to cam. There the sine flattens, and the arm has to swing further to move the face the same lift. A flat face on an arm is worse found this flattening making arms worse than slides when they lift with the cam. Lifting against it, the flattening helps the rise and harms the return at once, and past some base the return’s magnified deceleration wins. That base is the window’s ceiling.

A stand-off moves both ends. Standing the face further off, in the negative direction, lowers the floor, because it lets a smaller cam put the face where the arm turns it usefully. It also lowers the ceiling, and the scan shows the ceiling falling faster than the floor. On a gentle rise there is room to spare between them. On a steep one the floor rises with the rise’s deceleration, and the search for the least base drives the stand-off until floor and ceiling meet.

The least base sits on a knife edge #

A slide’s least base is a floor: every larger base gives a convex cam, since raising the base adds the same amount to the curvature everywhere. An arm’s is not. Raising the base changes the arm’s swing as well as the cam’s size, because the arm must reach further, and the change helps the rise and harms the return. So at each stand-off the convex bases form a window, with a floor set by the rise and a ceiling set by the return. The room is how far the ceiling is above the floor, as a share of the floor.

At 90° the room is wide open: at nearly every stand-off the base can be doubled and the cam stays convex. At 60° it is 63% with the face through the pivot and falls as the face stands further off. At 45° it is 19% through the pivot and about 1% at the stand-off of least base. At 35° it is 6.6% through the pivot and less everywhere else.

The stand-off that gives the least base is therefore the one with the least room. At the least base itself the room is about one per cent or less from 75° on: 0.8% at 75°, 0.6% at 60°, 1.0% at 45° and 0.7% at 35°. The search for the smallest cam drove the stand-off to the point where the window closes, because closing the window is what lowering the base costs.

The curvature shows what the knife edge is. Every least base touches nought curvature on the rise, by definition: that is what makes it the least. At the 45° rise’s least base the curvature on the return falls to 1.18 as well, on a cam whose base is 23. The cam is nearly flat in two places, one on the rise and one on the return, and a base raised by a hundredth puts the return through nought. The design with a tenth’s room keeps the return 22 clear, at the cost of a base of 36.8.

Ask for room and the advantage stops growing #

A cam cut exactly at a convexity threshold is not a cam a designer would build. It has a point where its surface is flat to second order, and any error in cutting it or in the follower’s motion makes it hollow there. A designer adds base above the threshold to keep the curvature positive with a margin. On a slide that costs nothing but size. On an arm whose window is a hundredth wide, there is no base to add.

So the fair comparison asks each arm design for room. With at least a tenth’s room above its base, the arm needs 7.18 at 90°, where the room is wide open anyway, then 11.08 at 75°, 17.76 at 60° and 36.85 at 45°. From 90° to 45° that is 5.1 times, against the slide’s 5.6. The arm’s share of the slide’s base falls only from 22% to 20%. At 35° no stand-off from −80 to 20 leaves a tenth’s room, so no design at this pivot gives the 35° rise a convex cam with that margin.

The face through the pivot is the simplest arm and makes the trade visible. It needs 16.13 at 90° and 59.03 at 45°, 3.7 times, and keeps 19% room at 45°. At 35° it needs 65.66 with 6.6% room. It is never the smallest arm, but it degrades more gracefully than the stood-off designs, because it was never pressed against the window’s ceiling.

The arm’s advantage over the slide is real at every rise: even with a tenth’s room it needs a fifth of the slide’s base at 45°. What does not hold is the widening. The earlier essay’s suggestion, that arms might be worse than slides only on gentle programmes, is half right. An arm beats a slide on a steep rise by about the same factor as on a gentle one, and the extra the least base seems to promise is bought with room a designer would not give up.

Read across, the table gives a designer three choices for a steep rise on an arm pivoted 90. The least base is the smallest cam and has no room above it. The design with a tenth’s room costs more than half as much base again at 45° and keeps the return clear. The face through the pivot costs more still and needs no stand-off, which makes it the simplest arm to build, and its room is the last to close. An arm is an offset that grows with the lift found a roller’s arm beating a slide by aiming a first-order quantity, the pressure angle, at the stroke that needed it. Here the aimed quantity is second order, and aiming it at the rise spends it on the return.

Checked on the drawn cam #

The thresholds come from the curvature formula. As a second route, each design with a tenth’s room is drawn as the envelope of the face’s positions over a turn, 4,320 of them, and the drawn outline is tested for turning one way only. Inside its window, halfway to the ceiling, every drawn cam turns one way. A tenth below its base, every one doubles back. The slide’s bases are checked against the closed form from the cycloidal law, as above.

The least base at 90° is the earlier essay’s 7.18 at a stand-off of −51.5, found here again by a separate search over the stand-off. The cam that cannot be cut warned that a cam has two reasons to be large and they are different reasons. Here the two reasons are on the same cam: the rise sets the floor and the return sets the ceiling.

What the rise does not settle #

Other pivots. Every arm here is pivoted 90 from the cam’s centre. A longer arm turns its face more slowly and behaves more like a slide, and a shorter one swings further for the same lift. The earlier essay found the best stand-off’s base nearly flat between pivots of 90 and 150 on the 90° rise; whether the window closes at the same rate on longer arms is not measured.

Other laws. Every rise here is cycloidal. The law that costs least found that constant acceleration has a lower peak deceleration than cycloidal motion and an impulsive jerk. A law with a lower peak deceleration raises the slide’s floor less, and it might leave the arm’s window wider. A lift is a size and a law is a shape says the comparison between laws carries over between sizes; the comparison between followers is a different question.

The return. The return is held at 170° throughout. The window’s ceiling is set by the return, so a quicker return would close it sooner and a slower one would open it. An offset trades the rise for the return found the same trade for a roller’s pressure angle, and here it reappears as a trade between the two ends of the window.

How sharply the least base is located. The least base sits where a window closes, and the base against the stand-off has a corner there, not a smooth minimum. A scan of the stand-off in steps of four, refined the same way, finds 16.15 at 60° and 23.64 at 45° in place of the 15.91 and 23.26 that steps of two find. The difference is a few per cent and does not change any conclusion here, but it is the same knife edge seen from the search’s side: a design that sits on a corner is hard to find and harder to hold.

Contact stress. A tenth’s room is a stated margin, not a stress calculation. What a real flat face needs depends on its load and material, and a cam with more room than this may still be too sharply curved somewhere for the face it drives.

Still open: a return chosen for the arm #

The window’s ceiling comes from the return. On every steep rise here the least base’s cam is nearly flat on the return as well as the rise, and the return’s span and law were inherited from the quick programme, not chosen for the arm. A designer who can change the return has a second knob that this measurement held fixed.

The distinct argument would be the return that maximises the arm’s room. Keep the 45° rise and its least-base stand-off, and vary the return’s span from 120° to 230° and its law between cycloidal, harmonic and constant acceleration. The measurement would be the room above the least base and the least base with a tenth’s room at each. If a slower or gentler return opens the window wide, the steep rise is an arm’s after all, and the cost is time on the return, not base. If the window stays narrow whatever the return does, the knife edge belongs to the rise and the arm’s advantage on steep rises is fixed at about a fifth of the slide’s base.