A gentle return cannot widen a steep rise

A steep rise spends the arm’s room compared a flat face on a slide with a flat face on an arm, pivoted 90 from the cam’s centre and lifting against the cam’s turning. The rise was shortened from 90° to 45°. The slide’s base grew 5.6 times and the arm’s least base only 3.2 times, to 23.26. But the arm’s least base sat where the cam was about to go concave in two places at once, on the rise and on the return. A base raised by a hundredth put the return through nought curvature. A designer who wanted a tenth’s room above the base paid 36.85 for it, and the arm’s advantage over the slide stopped growing.

The return in that comparison was the quick programme’s, a cycloidal return over 170°, inherited rather than chosen. The essay ended on the obvious second knob. If the return is what closes the window, a return chosen for the arm should open it, at the cost of time on the return rather than base. If the window stays narrow whatever the return does, the knife edge belongs to the rise.

The measurement says both, in sequence. A return chosen for the arm lowers the least base by a fifth. Then the window closes on the rise, and after that on the arm’s own reach, and it never opens past a twentieth.

Nine returns under one rise #

The rise is kept exactly as before: a lift of 20 over 45° with a cycloidal law, followed by a dwell, the return, and a final dwell of 60°. The return’s span runs over 120°, 170° and 230°, with the dwell after the rise taking up the difference. Its law is cycloidal, simple harmonic or constant acceleration, the three laws the law that costs least compared on a slide.

For each of the nine returns the search is the earlier essay’s. The face’s stand-off from the arm’s pivot is scanned from −80 to 20 in steps of two, the least convex base is found at each, and the room above it is measured, meaning how far the base can rise before the cam goes concave somewhere. Then the least base overall, the least with a tenth’s room and the least with a quarter’s are found. The quick programme’s row, 170° cycloidal, reproduces the earlier ledger’s 45° row exactly, which the model checks.

A slide does not notice any of this. Its least base is set by the rise’s peak deceleration, which none of these returns approaches, and it stays at 185.56 for all nine. The model checks that it does not move. Whatever the return does to the arm, it does through the arm’s swing.

A gentler return lowers the base #

The first result is the one the question hoped for. A slower return, or a gentler law over the same span, lowers the arm’s least base, and so does it lower the base with a tenth’s room.

The cycloidal return gives 53.81 over 120°, 23.26 over 170° and 18.08 over 230°. Simple harmonic motion gives 29.81 over 120° and 18.08 over both 170° and 230°. Constant acceleration gives 63.22, 33.19 and 18.08. The model checks that a slower return of the same law never needs more base, and that a harmonic return never needs more than a cycloidal one of the same span.

Then the improvement stops. Five of the nine returns give exactly 18.08, at a stand-off of −50.8, and the same five give 33.96 with a tenth’s room. Past a certain gentleness the return has stopped mattering, and whatever sets the base now is not the return.

The best return lowers the least base by 22%, from 23.26 to 18.08. With a tenth’s room it lowers the base by 8%, from 36.85 to 33.96. A designer who needs the margin, which is every designer who intends to cut the cam, gets the smaller of the two gains.

What closes the window now #

The useful question is not the base but what limits it. The window above a design’s base closes at the next failure above it, and each window can be probed just above its top to see what failed. There are three possibilities: the return goes concave, the rise goes concave somewhere else on its flank, or the arm can no longer reach. The last means the face’s furthest position from the pivot would exceed the pivot’s own distance from the cam, and no stand-off can place it.

On the quick return, 170° cycloidal, the least base’s window closes on the return, as the earlier essay found: 1.0% of room. On every return too quick or too harsh to reach 18.08, the window also closes on the return, and it is thinner still, from 0.0% to 0.6%. On the five gentle returns it closes on the rise. The base can be raised by 4.9% before a second place on the rise, just past its middle, goes concave. The model checks that the gentlest returns’ windows close on the rise or the reach and never on the return.

So the earlier essay’s knife edge was two edges, and the return was the lower one. Moving it out of the way exposes the rise’s own second edge, 4.9% above the first. The rise’s floor is set by the deceleration at the end of the rise. Its new ceiling is set near the rise’s middle, where the arm swings fastest and a larger base changes the swing most. Neither depends on the return.

For the designs with a tenth’s room the story is the same with a different limit. On the gentle returns their windows close on the arm’s reach. At a base of 33.96 and a stand-off of −32.6, a tenth more base puts the face’s furthest position 90 from the pivot, which is the pivot’s distance from the cam. At that point the arm cannot reach the cam at all, whatever the rise or return do.

Where a return hurts an arm #

The order of the three laws is not the order a slide would suggest. On a slide the return’s cost is its peak deceleration. Constant acceleration has the lowest, 4 per unit lift and unit span, against 4.93 for simple harmonic motion and 6.28 for cycloidal. Yet constant acceleration is the worst return for the arm at every span, and simple harmonic the best.

A flat face on an arm is worse and the essays after it explained why an arm and a slide disagree here. On the return the arm swings back with the cam’s turning, so its face turns against the cam at one less the arm’s rate. The cam’s curvature loss is divided by the square of that, a number less than one, and so it is magnified, and magnified most where the arm swings fastest. The arm swings fastest where the follower moves fastest, which is the middle of the return. So what hurts an arm is downward acceleration that persists into the fast part of the return. Written out, the cam’s curvature term on the return carries the follower’s acceleration divided by $(1 - \psi')^2$, and $\psi'$, the arm’s rate, is roughly the follower’s speed over the arm’s lever. At the 170° cycloidal return’s peak speed the arm turns at about a third of the cam’s rate, so the divisor is about four-ninths and the acceleration there counts more than twice. At the start of the return, where the follower is still slow, the same acceleration counts nearly at face value.

Constant acceleration holds its full downward acceleration, 4, from the start of the return right up to its middle. There the speed peaks at 2, the same peak as cycloidal motion, and the deceleration then switches sign instantly. Its cam fails exactly there: at 0.50 of the return on the 170° return’s least-base cam. Cycloidal motion’s downward acceleration peaks early, at a quarter of the return, and falls to nought by the middle, so the product peaks between them. Its cam fails at 0.42. Simple harmonic motion has its downward acceleration largest at the start, where the follower is still slow, and the lowest peak speed of the three, $\pi/2$ against 2. Its 120° cam fails at 0.38, and over 170° it no longer fails on the return at all.

A slide’s designer picks a return law by peak deceleration. An arm’s designer lifting against the cam should pick it by peak speed, and by how much deceleration survives into the fast middle. Simple harmonic motion wins on both, and it reaches the best base on a 170° return, where cycloidal motion needed the longest span tried, 230°. That is 60° of the turn handed back to the rest of the programme.

An offset trades the rise for the return found a roller’s pressure angle traded between the two strokes by an offset. Here the trade is in a second-order quantity, and the return’s law is a better lever on it than its span.

How far the face can stand off #

The base at a given stand-off does not depend on the return at all, wherever the return admits a convex cam, because the rise sets the floor at every stand-off. At a stand-off of −44 the least base is 24.1 on every return that has one there. What the return changes is the room above that base, and how far off the face can stand before there is none.

At −44 the quick return leaves 2.1% of room and the 170° harmonic return 8.0%: same cam size, four times the margin. The quick return’s window shuts just past −44, the 120° harmonic return’s just past −36. The 170° harmonic return’s runs on to about −51, still with 7% of room at −50, before the rise closes it.

The room still falls as the face stands further off, on every return. It crosses a tenth at a stand-off of about −28 on the quick return, −22 on the 120° harmonic return and −32 on the 170° harmonic one. That crossing is where the tenth’s-room design sits, and it moves by only a few units of stand-off, so its base moves by only a few units too: 36.85 against 33.96. The return cannot move the crossing far because, near a tenth, the rise’s own ceiling is already close above the floor. Below a tenth the gentle return’s extra stand-off buys a smaller cam, but the room it buys with it never reaches a tenth.

A quarter’s room is not on offer #

A designer with a less forgiving process, a cam that will be ground rather than hobbed or a face whose contact stress is already high, might want a quarter’s room. No stand-off from −80 to 20 gives it on any of the nine returns. The room peaks at the right-hand end of the convex range, a face standing off by +16, where it reaches 24.5%, and the base there is 69.1, larger than every arm design above and still only three-eighths of the slide’s.

The model checks both negatives: no return gives the least base a tenth’s room, and no return gives any design a quarter’s. On this rise and this arm, generous room is not a matter of choosing the return. It needs a longer arm, a gentler rise or a different follower.

What the arm buys and what it cannot #

The arm still beats the slide by a wide margin. At its best, with a tenth’s room, it needs 33.96 against the slide’s 185.56, under a fifth. The earlier essay’s comparison stands, and a return chosen for the arm improves it by a further 8%.

What the measurement settles is where the remaining limit lives. A flat face asks for a convex cam set the rule that every flat-faced follower must satisfy. An arm that lifts against the cam can beat a slide found the arm’s advantage. The steep-rise essay found the knife edge. This one finds that the knife edge has a second blade on the rise, 4.9% above the first, and that a margined design then meets the arm’s reach. The cam that cannot be cut warned that a cam has more than one reason to be large. On a steep rise driving an arm, the reasons are the rise’s floor, the rise’s ceiling and the arm’s length, and the return is only the first to give way.

A lift is a size and a law is a shape said a law’s comparison carries between sizes. These results scale the same way: double the lift and the pivot, and every base doubles, so the shares and the rankings stand.

What a designer takes from nine returns #

Three practical rules come out of the table, for a steep rise on an arm lifting against the cam.

Choose the return’s law before its span. A simple harmonic return of 170° does everything a cycloidal one of 230° does for the arm, and a constant-acceleration return is worse than either at every span. The law that is kindest to a slide is not the kindest to an arm, and an arm’s designer who borrows a slide’s preference pays for it in base.

Expect a fifth off the least base and a twelfth off the margined one. The gentlest returns take the least base from 23.26 to 18.08 and the base with a tenth’s room from 36.85 to 33.96. The first is worth having only if the cam will be cut with less than 5% of curvature margin, which is not a margin anyone would plan for.

Do not look to the return for room. The window above any of these designs is set by the rise once the return is gentle, and a design that wants more room has to change the rise, the arm’s length or the follower. The return has already given what it can.

That is the answer to the earlier essay’s question. The knife edge belonged to the return first, and to the rise as soon as the return was moved, and the arm’s advantage on a steep rise is fixed at about a fifth of the slide’s base whatever the return does.

An arm is an offset that grows with the lift showed a roller’s arm aiming a first-order quantity at the stroke that needed it. A flat face on an arm aims a second-order quantity, and the aim taken at the rise is spent wherever else the arm swings fast. On a steep rise driven by an arm, the fast places are the middle of the rise and the middle of the return. A gentle return removes the second, and the first is the rise’s own.

What the returns do not settle #

Other rise laws. The rise is cycloidal throughout, and the rise now sets both edges of the window. A rise law with a lower peak speed, simple harmonic for instance, would swing the arm more slowly through the middle of the rise, which is where the second edge sits. The window might widen there for the same reason the harmonic return widened its side.

Longer arms. At a tenth’s room the gentle returns run into the arm’s reach. A pivot further than 90 from the cam would push that limit out, and whether the rise’s own room then allows a quarter is not measured.

Asymmetric and polynomial laws. Only three classical laws were tried. A return law designed for the arm, slow in the middle and quick at the ends, could do better than simple harmonic motion, though the gentlest returns here already take the return out of the problem.

Still open: a rise chosen for the arm #

The return is no longer the limit once it is gentle. The rise is, at both ends of the window: its end sets the floor and its middle the new ceiling. Both are about how fast the arm swings while the rise decelerates.

The distinct argument would change the rise’s law with its 45° span and lift fixed: cycloidal, simple harmonic, and a modified sine, which keeps cycloidal motion’s smooth ends with a lower peak speed. It would measure the least base and its room, with a gentle return held fixed so that the rise alone decides. If a law with a lower peak speed opens the rise’s ceiling, the arm’s window on a steep rise is a choice of law after all, made on the rise rather than the return. If every law’s ceiling sits near 5%, the narrow window is the price of driving an arm against a steep rise at all.