A cam machine is cut at its handovers

Four cuts serve the whole cell asked how many cuts a whole machine needs before its region test stops raising false alarms. A region test sweeps each part over an arc of the drive and flags any ground two parts both cover within that arc. Over the whole drive, two parts that pass through the same place at different times are flagged although they never meet. Three arcs, cut in the right places showed how to cut the drive so that each pair’s visits to shared ground fall into different arcs, and how to prove the cuts are the fewest.

On Peaucellier’s cell the answer was four cuts, the same four that its worst single pair needs. The six pairs of parts that share ground without touching would need sixteen cuts if each were cut alone. They need only four together, because every pair crosses its shared ground on either side of the same moments. The essay put that down to the machine’s symmetry. Every part of the cell is driven by one linkage, and its crossings happen together.

That leaves an obvious test. A machine whose parts are driven by separate cams on one shaft keeps several independent schedules. Each part moves when its own cam says, and nothing makes the moments line up. If the machine’s count approaches the sum of its pairs’, a region test on such a machine needs a partition per pair. If it stays at the worst pair’s, a common drive shaft is enough to make parts take turns.

It does neither. A cam machine’s fewest cuts are its handovers: the number of times in a turn that the shared ground passes from one part to another. For four tools taking turns at one station, that is three, between the worst pair’s one and the sum’s six. Each cut has a window, the time the shared ground stands empty, and that window is the cam diagram’s timing margin, measured to the degree.

A station four tools take in turn #

The machine is a work station of the kind a press or an assembly cell has. Four tools on straight guides point at a common centre, at 0°, 90°, 180° and 270°: a feed, a clamp, a punch and an ejector. Each is a bar 0.3 wide and 1.5 long. Retracted, its front stands 1.2 from the centre. At full stroke its front passes 0.15 beyond the centre, so any two full-stroke tools cover the same square at the middle, 0.3 on a side.

Each tool has its own cam. Its programme is a dwell retracted, a cycloidal rise of 30°, a dwell of 20° at full stroke, and a cycloidal return of 30°. The four programmes start 90° apart, at 5°, 95°, 185° and 275°, which leaves a gap of 10° between one tool’s return and the next one’s rise. The cycloidal law is the one the law that costs least found to have finite jerk everywhere, at a price in peak acceleration, and nothing here depends on it except the closed forms.

No tool ever touches another. That is what the schedule is for: the station is a resource the tools share by taking turns. A region test over the whole turn cannot see that. Each tool’s dashed region contains the shaded square, so all four regions overlap there, and a test over the whole turn flags all six pairs.

The figure shows only the ground; the turns are in the timing diagram, which is how a machine designer draws a cam machine. Each tool’s lift is plotted against the cam angle in its own lane. The bar under each lane marks when the tool’s front is inside the shared square.

The square is never held by two tools at once. Between one tool leaving it and the next arriving, there is a stretch when it is empty. A cut anywhere in that stretch separates the two tools’ visits, and the region test on either side of the cut sees only one of them.

Each pair needs one cut and the machine needs three #

The measurement is the same as on the linkages. Every tool is rasterised at each of the turn’s 360 degrees on one grid of 200 cells a side, so that any two tools’ cells line up. For each pair of tools, each cell both of them ever cover yields an interval: from one tool’s last visit before the other arrives, to the other’s first visit. Any partition that clears the pair must put a cut inside every one of its intervals. The fewest cuts that stab a set of intervals are found greedily, and the intervals the greedy rule cut at are disjoint, which proves no partition does with fewer.

Every pair shares the same hundred cells of the centre square, and every pair needs exactly one cut. The feed and the clamp need a cut between 65° and 115°. The feed and the punch need one anywhere between the feed leaving and the punch arriving, a longer interval that contains the clamp’s whole visit. Cut separately, the six pairs take six cuts.

Together they take three, at 114.5°, 204.5° and 294.5°. The greedy rule places the first at the end of the feed-and-clamp interval. That cut also stabs every interval that starts with the feed’s exit, so it serves the feed’s pairs with the punch and the ejector as well. The second cut serves every pair whose interval starts with the clamp’s exit, and the third every pair whose interval starts with the punch’s. The witnesses are the feed-clamp, clamp-punch and punch-ejector intervals, which do not overlap, so three is the least.

Three is not the worst pair’s count, which is one, as it would be on a symmetric linkage. It is not the sum either. It is the number of times the station changes hands: feed to clamp, clamp to punch, punch to ejector. The fourth handover, ejector back to feed, falls at the turn’s seam, between 355° and the next turn’s 5°, and the seam is already a cut.

A second route checks this. The handovers are counted from the schedule alone, by walking round the turn and noting each time a tool enters shared ground that a different tool held last. The count is three. The model’s check requires the greedy cuts and the handover count to agree on every schedule, and hands the three cuts back to the grid test, which finds no shared cell inside any arc.

The seam earns its keep only when it falls in an empty stretch. Start the turn at 40° instead, in the middle of the feed’s visit, and the feed’s visit is split between the two ends of the drive. The ejector-to-feed handover is now inside the drive and needs its own cut: four instead of three. On a circle the count is four either way. A periodic machine needs one arc per handover, and the seam is free only if it is placed in an empty stretch.

Why a symmetric linkage was different #

The linkage results look like the cam result once they are read the same way. In Peaucellier’s cell, all eight links are driven by one crank through one loop. When the cell is at one end of its working arc, the parts are all at that end. The six pairs’ shared ground is crossed as the cell moves through a few positions, and each crossing is a handover between two parts. The handovers coincide because the parts move together, so four cuts serve sixteen pairs’ needs.

A cam machine gives each part its own clock. The four tools here are driven by one shaft, so their clocks are locked together. Their phases are free, though, and a designer spreads them out so the tools take turns. Spreading them out is exactly what keeps handovers apart in time. The cuts count the turns because each turn ends at a different moment.

That gives the general rule, of which both machines are cases. The fewest cuts a region test needs is the number of distinct moments at which shared ground passes from one part to another. Where the parts move as one, those moments coincide and the count is the worst pair’s. Where the parts keep their own time, the moments are as many as the schedule’s handovers. The sum of the pairs’ cuts is never reached, since a cut that serves one pair of tools at a handover serves every pair whose interval contains that handover.

The count therefore grows with the number of tools, but slowly: n tools taking turns at one station need n − 1 cuts. The pairs number n(n − 1)/2, so the gap between the sum and the machine widens fast. By the same count, a station with ten tools in turn would have forty-five pairs and need nine cuts.

A tool that comes back #

A tool that visits the station twice in a turn makes its pairs more expensive. In the second schedule, a punch strikes at 5° and again at 185°, a clamp works between them at 95°, and an ejector works at 275°. The three tools are on guides at 0°, 90° and 180°.

The punch and the clamp alternate punch, clamp, punch at the shared square, so that pair alone needs two cuts: one between the punch’s first visit and the clamp’s, and one between the clamp’s and the punch’s second. The worst pair’s count rises from one to two. The punch and the ejector need one cut, as do the clamp and the ejector, so the pairs need four between them.

The machine still needs three. The station changes hands punch to clamp, clamp to punch and punch to ejector, three times, at the same moments as the four-tool schedule. The cuts fall at 114.5°, 204.5° and 294.5° again. The count followed the handovers, not the tools and not the pairs: a schedule with fewer tools needs as many cuts as one with more, because its tools hand over as often.

Tools that never enter the shared ground #

The third schedule is busier on the timing diagram and cheaper on the region test. Two jaws on the 0° and 180° guides close at 5° and hold the work until 265°. While they hold, a punch on the 90° guide strikes at 60°. After they release, an ejector on the 270° guide works at 270°.

The jaws have a shorter lift, 1.0, so their fronts stop 0.2 from the centre, just clear of the square the punch and ejector pass through. They share no ground with any tool or with each other. A jaw and the punch are in the station at the same time for 80°, close but never on common ground, and a region test over the whole turn does not flag them.

The only pair that shares ground is the punch and the ejector, and it needs one cut. The turn holds four strokes, the jaws’ two among them, and sixteen moments at which some cam starts or ends a rise or a return. It holds one handover, and the machine’s count is one. A timing diagram’s events are the wrong things to count. Only the handovers on shared ground matter, and a tool that never enters that ground adds none, however busy its cam.

The window is the timing margin #

Each cut can be moved anywhere inside a window without changing anything. On four tools in turn, the windows run from 65° to 115°, 155° to 205° and 245° to 295°, each 50° wide on the grid. Those windows are not a property of the checker. They are the stretches when no tool is in the shared ground.

The width has a closed form. A tool’s front enters the shared square when its lift reaches 1.2 − 0.15 = 1.05 of its 1.35, which on the cycloidal rise happens 19.5° into the 30° rise. It leaves at the mirror point of the return, 10.5° into it. So the square stands empty from 10.5° into one tool’s return to 19.5° into the next tool’s rise. That is the gap between them plus the 19.5° left of the return and the 19.5° already spent in the next rise: the gap plus 39.0°. With the 10° gap, the empty time is 49.0°. The grid measures 50°, because it counts between sampled degrees.

The grid’s windows and the closed form agree to within one sampled degree at every schedule tried. Squeeze the schedule by moving each tool’s stroke earlier, so the gap shrinks and goes negative, and the window shrinks degree for degree.

At a gap of −38°, the strokes overlap by 38° and the station still has a degree to spare. At −42° the closed form is negative: the next tool enters the square before the previous one has left. The grid finds 57 cell visits where two tools are in the same cell at the same sampled degree. Those are zero-length intervals, which no cut can stab. They are not false alarms; they are collisions, and the region test’s conservatism has nothing to do with them.

So the region test’s partition and the designer’s timing diagram measure the same margin. A designer laying out a cam machine checks that each tool is clear of the shared ground before the next arrives, and the slack in that check is how many degrees the cams may drift, wear or be mistimed before the tools meet. The partition finds exactly that slack as the room each cut has to move. Its least window is the machine’s tightest handover. A sweep that missed nothing showed a sampled check stepping over a collision it never looked at. This margin is the quantity such a check has to resolve, and it comes in the drive’s own units.

A plane the whole turn would add #

The partition matters for building, too, as it did on the linkages. A plane is a colour assigns the parts of a machine to parallel planes so that no two parts that conflict share one. Read the conflicts from a region test over the whole turn and every pair of tools at the four-tool station conflicts, since their regions all contain the centre square. Four mutually conflicting parts need four planes.

Four planes would be a station whose tools cannot reach a common workpiece, since each would work at a different height. The tools are built in one plane because they take turns, and the partition is what tells a checker so. Read from the machine’s three cuts, no pair of tools conflicts and the station needs one plane. The other schedules gain planes the same way: three for the tool that strikes twice and two for the jaws’ station, each cut back to one.

On a linkage, the regions overlap and the parts never meet found the false alarm, and the essay on Peaucellier’s cell priced it: a plane each, which is a thicker machine. On a cam station the same false alarm costs the machine its purpose, because the shared ground is where the work is done. A whole-turn region test is not merely conservative for a machine of this kind. It cannot describe it.

What this station leaves aside #

Tools on guides that are not radial. Every tool here points at the centre, so the shared ground of every pair is the same square. Tools approaching at other angles, or reaching different points of the work, would share different ground pair by pair. The count would still be the handovers of each piece of shared ground, but the pieces would have different owners and the cuts might not coincide. That is not measured here.

Swinging followers. The tools slide. A follower on a pivoted arm sweeps an arc, and its region over its stroke is a fan, not a strip. The room a machine sweeps measured such regions. The handover rule does not care about the shape of the region, only about the times shared cells are visited, so it should carry over. The windows’ closed form would not.

Speed. The window is measured in degrees of cam rotation, which is the right unit for a timing diagram and the wrong one for a collision at speed. At higher speed a follower held on by a spring can leave its cam, and a cam that holds its follower both ways is the design that stops it. A follower that leaves its cam is no longer where the programme puts it, and the margin here assumes the follower follows.

A tool’s own body outside the square. Every tool’s retracted body is far from the others, so the only shared ground is the centre. A station built more tightly, with the tools’ bodies passing near one another on the way in, would add shared ground away from the centre, and with it handovers whose windows are set by bodies rather than fronts.

Still open: handovers that are not the same for every pair #

The station here is the simplest case: one piece of shared ground, owned in turn. A real assembly cell has several, each shared by a different subset of tools, and the handovers on one need not line up with those on another. The four-tool count was three because the only shared ground was the centre and the station handed it round once.

The distinct argument would be a cell with two work points, a press at one and a transfer arm visiting both, each tool’s ground overlapping a different set of others. The measurement would be the machine’s fewest cuts against the handovers counted separately on each piece of shared ground, and against their total. If the machine’s count is the total, the rule is per piece of ground, and a region test on a cell is a sum over its stations. If it is less, handovers at different stations can share a cut when they happen together, and laying out the cams to make them do so would make a cell easier to check without changing what it does.