Machine Kinematics
solved, then drawn
Concept

Swept volume — where it appears

The volume a chamber gains and loses over a cycle, computed here as an area times a width. It is a geometric quantity that needs no fluid, and it is what a pump or an engine is sold by.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

The room crank rocker needs. The shaded region is every point any part of this machine occupies at some position of its drive, computed as an occupancy grid over 150 solved configurations and outlined by marching squares — so the outline drawn is the boundary of the set the area was counted from rather than a second object that agrees with it. Its area is 12.78 square units, filling 54% of the box that contains it. The dashed rectangle is the box the joints need, which is what every figure that treats links as lines could have told you; the material needs a box 20% larger in area.

The room a machine sweeps

Every point any part of a machine occupies at some position of its drive. It is a region rather than a curve, its area is an integral computed two ways, and the one shape in the field with a closed form is what the grid is calibrated against.

bodies · Body Every pin position, and what it costs — 3 mm between the doors. One cell per candidate pin position: the door is swung through 95° from each and the worst clearance recorded. Light cells clear; dark cells mean the door would pass through the partition or through the door beside it, by up to 18 mm. The horizontal line is the door's own front face. With the doors touching, the boundary sits exactly on it — because the front corner's sideways rate is the pin's distance behind it, and that is zero only there. Each millimetre of gap buys a few millimetres of depth, and at 3 mm the deepest pin that still clears is 10 mm behind the face.

Where a hinge pin can go

A cabinet door's front corner moves sideways as it opens at a rate equal to how far the pin sits behind it, so with the doors touching, no pin behind the door's face can open one without going through the next. Three millimetres of gap buys nine and a half millimetres of depth, and that is the whole reason a concealed hinge has four bars instead of a pin.

applied · Coupler Three chambers, 15,257 mm² between them. Each chamber is bounded by one flank of the rotor and the arc of housing between the two apexes that seal it, so its area is a polygon integral over two curves that are both known exactly. At this shaft angle they are 6309, 1147, 7801 mm². They add to 15257, and the housing's area less the rotor's is 15258 — a number that cannot depend on where the rotor is, and does not, to 9.2e-6 of itself. The engine's displacement is that constant shared out differently, and the three chambers are one chamber counted at three phases of the same cycle. positioned by solving, not by drawing.

Three chambers and a constant

A rotor with three corners divides its housing into three chambers, and as the shaft turns they trade area: one grows exactly as fast as the other two shrink. The total does not move — and the fact that it does not is a measurement of whether the corners are actually touching the wall.

meshing · Meshing The drive in 2 arcs. Chebyshev's two arms, whose regions over the whole drive overlap though the parts never come within 0.380 of each other. The outlines are each part's region over the whole drive. The filled ground is what a test that sweeps the drive in 2 equal arcs still flags: the points some arc's sweep of one part shares with the same arc's sweep of the other. With 2 arcs that is 0.859 square units. The parts are drawn at their closest approach.

Four arcs and the false alarm is gone

A swept region forgets when, so two parts whose regions overlap may never meet: Chebyshev's two arms share 4.66 square units of ground and never come within 0.38. Cut the drive into arcs, sweep each part over each arc, and compare regions only arc against the same arc. The test stays sound and costs the same sweep, and it forgets less with every cut. Four arcs clear Chebyshev's arms and eight clear Peaucellier's worst pair. A certificate that needs no grid asks for three times as many, and the same arms made wide enough to touch are refused at every number of arcs.

bodies · Body Every shared cell asks for a cut between two visits; 2 cuts answer all of them. Chebyshev's two arms: each of the 5,510 cells of ground both parts occupy at some point of the drive, as the stretch of drive angle between one part's last visit to it and the other part's first — every such cell is visited by each part in one unbroken stretch, so each gives exactly one interval. Sorted by where they end, one in 21 drawn. A partition of the drive clears the pair exactly when every interval holds a cut. The vertical lines are the fewest cuts that do, placed greedily at the earliest unstabbed end: 1.072 and 1.428, making 3 arcs of 0.360, 0.356, 0.274 radians. The thicker intervals are the ones the rule cut at; they do not overlap, so no partition clears the pair with fewer cuts.

Three arcs, cut in the right places

Equal arcs clear Chebyshev's two arms at four and Peaucellier's long arm and rhombus bar at eight. Each cell of ground the two parts share is visited by each part in one unbroken stretch of the drive, so each asks for exactly one cut, somewhere between the two visits — and the fewest cuts that answer every cell are found greedily from one sweep and proved least by the intervals that forced them. Three arcs clear Chebyshev's arms and five clear Peaucellier's pair, and halving only the arcs that still flag finds four and five at three and four times the sweeping.

bodies · Body 6 pairs, 16 cuts if each is cut alone, 4 for all of them together. Peaucellier's cell's pairs of parts that never touch but whose whole-drive regions share ground, each in its own lane: its intervals between visits to shared cells (thin, one in several drawn) and the fewest cuts that pair needs alone (solid ticks). long arm A / rhombus QB: 5 arcs; long arm A / rhombus PB: 3 arcs; long arm B / rhombus AQ: 5 arcs; long arm B / rhombus AP: 3 arcs; rhombus AQ / rhombus PB: 3 arcs; rhombus QB / rhombus AP: 3 arcs. Cut separately they need 16 cuts. The dashed lines are the fewest cuts for every pair at once, 4, at −1.132, −0.290, 0.658, 1.360 radians — as few as the worst pair alone needs — and the intervals they were placed at do not overlap, so no partition does with fewer.

Four cuts serve the whole cell

A region test between moving parts needs its drive cut where each pair's two visits to shared ground separate, and one pair of Peaucellier's arms needs four cuts. The cell has six such pairs, and cut one at a time they would need sixteen. Cut together they need four, the same four moments, because every pair crosses its shared ground on either side of them. Built from a whole-drive region test the cell would take six planes; the four cuts give back the true four, where equal arcs need eight to do it.

bodies · Body 3 handovers, and 3 cuts where the shared ground stands empty. Four in turn: each tool's lift against the cam angle, from its own cam, with a thick bar under it while its front is inside the shared ground. The dashed lines are the fewest cuts that clear every pair of tools at once, 3, at 114.5°, 204.5°, 294.5°, and the shaded bands are the windows each may move in, 65° to 115°, 155° to 205°, 245° to 295°: the times when no tool is inside the shared ground. The pairs need 1 cut at worst and 6 between them; the station changes hands 3 times after the turn's seam.

A cam machine is cut at its handovers

In one symmetric linkage, every pair of parts crosses its shared ground on either side of the same few moments, so the whole machine needs no more cuts of its drive than its worst pair. Four tools driven by four cams into one work station do not share their moments. Each pair needs one cut, the pairs need six between them, and the machine needs three: one for each time the station changes hands. Each cut can move only within the time the station stands empty, which is the cam diagram's own timing margin, and when a schedule is squeezed until that margin closes, the tools collide.

bodies · Body Two stations, each shared by its own tools, and an arm that visits both. A cell with two work stations 3 apart. Station A is fed from the left and pressed from above; station B is pressed from above and emptied to the right; an arm pivoted below and between them picks a part up at A, swings it to B and parks midway. Every tool is 0.3 wide and reaches 0.15 past its station's centre, so the shaded squares are the only ground any two tools share, each square shared by its own station's tools and the arm. At 230° station A holds nothing and station B nothing.

Two stations share a cut when they hand over together

A cam machine's region test needs one cut of its drive for every time its shared ground changes hands. A cell with two work stations has two pieces of shared ground, each with its own handovers, and a single cut can serve both if it falls when both stations stand empty. So the cell needs somewhere between its busier station's count and the two stations' sum, and its cams decide where. Out of step, a press-and-transfer cell needs four cuts; with the same tools and strokes put in step, two. The transfer arm that carries the part between stations makes one shared cut for free, because while it swings it is at neither.

bodies · Body

Named alongside it

The objects these essays reach for when they reach for this one.

InterferenceClearanceConfiguration spaceLink bodyLayer assignmentApexCompression ratioConjugate-actionConstraintCouplerDimensionless ratioEccentricity

All concepts