Machine Kinematics
solved, then drawn
Concept

Limit position — where it appears

A configuration at which a mechanism's output has reached the end of its travel and reverses. It is where the transmission angle is often worst and where the timing of a quick-return is decided, and it is found as a dead centre of the linkage.

Named by 10 essays across 6 fields — each of them below, with the objects they name alongside it.

The gap is a function, and it has corners. The smallest gap over every tested pair, at each of 360 solved positions of the crank. Two things are visible that a check at the ends could not report. The minimum, 0.1041, is in the middle of the travel and not at either end. And the curve has 4 corners, each one a change in which pair is closest — the colours — so the function is piecewise smooth rather than smooth, and its minimum is not where a derivative vanishes. Refining off the sample grid by golden section moves the answer by 1.4e-5, which is what a corner rather than a smooth minimum looks like.

A gap with corners in it

The clearance between two parts is a function of the crank angle, and it is not a smooth one. It has a corner wherever the closest pair of features changes hands, so its minimum is not where a derivative vanishes and is not at either end of the travel.

bodies · Body Where the output stops, and why the return is quicker. A crank-rocker's output reaches an extreme exactly when the crank and coupler line up — stretched out, so O₂ to B is 4.50, or folded back, so it is 2.50. Nothing about the rocker enters the condition, which is why the limits can be written down rather than searched for. Those two crank angles are 40.8° and 228.5°, so the crank spends 187.7° going one way and 172.3° coming back while the rocker covers the same 40.0° both times. The ratio is 1.0894 predicted and 1.0894 measured over 7200 swept positions — a shaper cuts on the slow stroke and returns on the fast one, and this number is what the proportions are chosen to get.

The return stroke is quicker

A crank-rocker's output stops at two definite places, and the crank angles at which it does are calculable without touching a solver. The interesting number is not where they are but how far apart — because the crank turns at a constant speed and the output covers the same swing twice in unequal times.

linkages · Fourbar How far the crank turns before the rocker does. With a clearance of 0.01 at each pin, the crank must be turned this far on reversal before the rocker moves at all. Through most of the cycle it is about 2.8°, and at its best 1.44°. At the two positions where the rocker reverses — marked — it is unbounded: the output velocity passes through zero there, so no amount of crank rotation moves the rocker out of its clearance band. The peak in a plot like this is therefore a property of the sampling and not of the mechanism; it reads 402° at 96 samples and grows without limit as the sampling is refined. The number worth quoting is the plateau.

How far the crank turns first

Reverse the input of a four-bar with a hundredth of clearance at each pin and the output does not move for about 2.8° of crank rotation. At the two positions where the rocker reverses it does not move at all, however far the crank is turned — the lost motion is unbounded there, and the peak in any plot of it is a property of the sampling rather than of the mechanism.

practice · Clearance 20,000 four-bars, and not one with three circuits. 20,000 four-bars with the ground at one and the other three lengths drawn from 0.05 to 3, each counted exactly. crank-rocker: 3,031, of which 3,031 have two circuits, 0 have one and 0 cannot be assembled; double crank: 3,086, of which 3,086 have two circuits, 0 have one and 0 cannot be assembled; Grashof double rocker: 1,497, of which 1,497 have two circuits, 0 have one and 0 cannot be assembled; triple rocker (non-Grashof): 12,386, of which 0 have two circuits, 9,528 have one and 2,858 cannot be assembled. The most circuits any linkage has is 2.

Never three circuits

A four-bar has one circuit or two, and twenty thousand random four-bars counted exactly contain no exception. The reason is a count of four points where the two assemblies merge, which makes the configuration curve a curve of genus one, and Harnack's theorem allows a real curve of genus one two pieces and no more. A six-bar's curve has genus five or seven, and its circuits go to four and six.

algebra · Algebra Where a calibration measures. The four-bar drawn at the 6 poses a selection chose, one over another, with the tracing point marked at each. The largest gap between consecutive chosen poses is 150°. They are spread because rows of the identification Jacobian at nearby poses are nearly the same row, which is a statement about the matrix and reads here as a picture of a machine in visibly different configurations.

The pose the machine cannot reach

A measurement plan is drawn against the nominal machine and executed on the real one, and the real one does not go quite where the drawing says. A pose that falls outside the travel returns no reading at all — which is not an error, not a failure of the instrument, and not nothing: it is a measurement of the limit position.

metrology · Parameter Every pivot on these arcs gives a 60° swing at a time ratio of 1.2. The rocker's pivot O₄ is fixed and its two limit positions B₁ and B₂ are 60° apart. At a limit the crank and coupler are in line, so the crank pivot sees the chord B₁B₂ at the angle δ = 180°(Q − 1)/(Q + 1) = 16.36°, and the points that see a chord at a fixed angle are arcs of two circles. The thick stretches are the 828 sampled pivots that give a crank-rocker with exactly this swing and ratio; the rest of each circle gives a linkage of another kind. The linkage drawn is the member whose worst transmission angle is largest, at both limits: ground 1.222, crank 0.478, coupler 1.130 and rocker 1, with its worst transmission angle 40.32°.

A swing and a time ratio

A shaper's specification gives the rocker's swing and how much quicker the return must be than the cut, and those two numbers do not fix a linkage. They leave a one-parameter family of crank-rockers on the arcs of two circles, every member exactly right, and the transmission angle chooses between them — which is also what decides that a 60° swing cannot return more than 1.207 times as fast and keep 40°.

linkages · Fourbar A drag link driving a crank-rocker at a phase of 284.8°. A drag link, ground 1, crank 2.5, coupler 2.25, output 1.5, whose output crank carries the crank of a crank-rocker with a 60° swing, ground 1.2223, crank 0.4783, coupler 1.1298, output 1, turned 284.8° ahead of it, drawn at an input angle of 40°. The two cranks on the middle pivot are one rigid part. The thick arc at the right is the rocker's swing. At this phase the whole machine returns 2.71 times as fast as it works; the crank-rocker alone, driven at constant speed, returns 1.2 times as fast.

A drag link ahead of a crank-rocker

A crank-rocker with a 60° swing keeps a transmission angle of 40° only up to a time ratio of 1.207, and at a ratio of 2 no crank-rocker keeps even 20°. Drive its crank from the output of a drag link, whose cranks both turn but not at the same speed, and a pair in which each stage keeps 40° returns 2.71 times as fast as it works. The phase between the two stages decides almost all of it: the same two linkages give anything from 1.003 to 2.71.

linkages · Fourbar The configuration space of a crank rocker. Every point of the square is a pair of angles — the crank's and the rocker's — and the curve is where the coupler is exactly the right length to join them. That curve is the mechanism: one equation in two angles leaves one freedom, which is the mobility. It has 2 components, and that is the two assembly branches. A built linkage cannot cross between them, because there is no path in the set to cross by — and each goes all the way round, which is what makes the crank a crank. The square's left and right edges are the same line, and so are its top and bottom.

Branches were components all along

Seven words have been used for one thing. An assembly branch, a circuit, an assembly mode, a working mode, a posture and a branch defect are all statements about the connected components of a mechanism's configuration space — and once that is said, a four-bar's two circles, a platform's six modes and a synthesis defect stop being three subjects.

serial · Algebra A centred slide's time ratio is its drag link's split of half a turn. The output angle of a drag link against its input angle over one turn: ground 1, input crank 4.019, coupler 3.920, output crank 1.339, the double crank that splits half a turn of its output most unevenly while keeping a worst transmission angle of 40.0°. A centred slider-crank driven by its output reaches its two dead centres at output angles exactly half a turn apart, whatever its rod, so its forward and return strokes take the input angles between the two dotted levels and outside them. At a phase of 66.3° between the stages they are 104.5° and 255.5° of input, a time ratio of 2.443. The largest over every phase angle is 2.443, against 2.304 for the best link of the quarter-step grid: no rod can raise it.

A centred ram cannot be timed past its drag link

A shaper's drag link and slider-crank were held to a combined time ratio of 2, and the centred slide's rod turned out to have a best length. Asking for a quicker return was supposed to shorten that rod and narrow the offset slide's lead. The first happens; the second runs the other way. A centred slide's dead centres are half a turn of its crank apart whatever its rod, so its ratio is the drag link's split of half a turn and nothing else: 2.304 on the grid, 2.443 off it, and no rod moves it. Below that ceiling the centred slide's best cut worsens from 1.118 to 1.466 and the offset slide's lead grows from 1% to 28%. Above it only an offset slide with a short rod reaches, and no slide keeping 45° reaches 3.

linkages · Fourbar A ratio of 3 costs both stages forty degrees. The largest quick-return ratio of a shaper's drive against the worst transmission angle it must keep, from 30° to 60°: an offset slider-crank alone, 1.448, 1.360, 1.286, 1.224, 1.172, 1.128, 1.092; a drag link alone ahead of a centred slide, 2.867, 2.632, 2.443, 2.284, 2.130, 1.968, 1.809; and the two in series, both stages keeping the angle, 3.975, 3.427, 3.006, 2.696, 2.432, 2.184, 1.957, against the product of the separate ceilings, 4.153, 3.580, 3.143, 2.796, 2.496, 2.221, 1.976. The train reaches 3 at 40.0°. Marked: 40°, where the train reaches 3.006.

A ratio of three costs forty degrees

A shaper's two-stage drive, a drag link ahead of a slider-crank, could not be given a return three times as quick as its cut while its drag link kept 40° and its slide 45°. Drawn as a frontier, the largest ratio against the worst transmission angle both stages keep, a ratio of 3 costs 40.0° in both. The two stages in series reach 95.7% to 99.0% of the product of their separate ceilings at every angle from 30° to 60°, so two stages relax one stage's bound very nearly by a product. Shared unequally, every degree the drag link gives up buys the slide between one and a half and three: the drag link is the stage to let lean. And the trains on the frontier cut badly, the ram's fastest speed in the middle of its cut 3.2 times its slowest at a ratio of 3. On the way, the drag link's own ceiling at 40° turns out to be 2.443, not the 2.365 a single search had found.

linkages · Fourbar

Named alongside it

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

Transmission angleQuick returnGrashof's conditionTime ratioCrank-rockerDesign ruleAssembly branchCircuitClearanceConfiguration spaceConnected componentDead centre

All concepts