Machine Kinematics
solved, then drawn
Concept

Crank-rocker — where it appears

A four-bar in which the input turns fully and the output swings between limits. Which of the four possible behaviours a linkage has is decided by its four lengths through Grashof's condition, and it can be checked by sweeping the crank.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

Grashof's classification, predicted and then swept. Four sets of link lengths. For each, Grashof's condition predicts from the lengths alone whether the input can rotate a full turn, and the solver then attempts all 180 positions and reports how many assembled. The prediction and the measurement agree in every case, which is what licenses quoting the classification for a mechanism nobody has swept.

Grashof, predicted and then swept

Add the shortest link to the longest. If the total does not exceed the other two, some link can turn a full revolution. It is a sentence about four numbers, it was published in 1883, and it is the kind of claim this site refuses to print without measuring — so every linkage here is also asked for all 360 positions and required to agree.

linkages · Fourbar 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 Grashof's classification, predicted and then swept. Four sets of link lengths. For each, Grashof's condition predicts from the lengths alone whether the input can rotate a full turn, and the solver then attempts all 180 positions and reports how many assembled. The prediction and the measurement agree in every case, which is what licenses quoting the classification for a mechanism nobody has swept.

Grashof is a shape test

The oldest classification in the subject compares sums of lengths, so it is unchanged by making the machine bigger — which means a protractor recovers it exactly without recovering a single length. What it does not recover is the margin, and the margin is what says whether the classification is safe.

linkages · Fourbar Crank angle θ and crank angle −θ, joined. The coupler curve alone, with 12 chords, each joining the point the tracing point reaches at a crank angle θ between 0° and 180° to the point it reaches at −θ on the same assembly. Every chord is square to the dashed axis, to 7.5 × 10⁻¹⁶ in the cosine, and every midpoint lies on it. Measured at 48 pairs, the reflection of one point misses the other by at most 4.2 × 10⁻¹⁵. The crank's own angle is the pairing: the two places the curve crosses its axis are crank angles 0° and 180°, the only angles equal to their own negatives.

A symmetric curve from a lopsided machine

A four-bar with a crank of 1, a ground of 3 and a rocker of 2.5 has no symmetry anywhere in it. Make the rocker, the coupler and the arm from the rocker pin to the tracing point one length, and the curve it draws is its own mirror image to 4 × 10⁻¹⁵, about a line through the rocker pivot turned from the ground line by exactly half the coupler's angle at that pin.

curves · Coupler Turning all the way round, against being made of something. Seven four-bars, classified by Grashof's inequality on their four lengths and then asked a question Grashof cannot answer: with a bearing pedestal at each ground pivot, how wide may the links be? The two instruments have nothing in common — one is an inequality on four numbers, the other counts sign changes of (B − A) × (G − A) over a sweep — and they agree about something Grashof was not for. Every four-bar that turns all the way round sweeps a link straight over a ground pivot, so its closest approach is exactly zero and no positive width is admissible; not one of the rockers does, and they take widths up to 0.20 of their shortest link.

The crank that cannot turn all the way

Grashof's inequality says which four-bars turn fully. Ask instead how wide their links may be with a bearing at each ground pivot, and the same inequality answers the opposite question: every four-bar that turns all the way round sweeps a link straight over one of its own pivots.

bodies · Body Eight four-bars, one from each region the three signed sums cut. One linkage from each of the eight sign patterns of T₁ = g + c − a − b, T₂ = g + b − a − c and T₃ = b + c − a − g, each solved at a crank angle in the middle of its range. The thick arc round the left pivot is where the input's pin can go and the arc round the right pivot is where the output's can: crank-rocker + + +, input 100% of a turn and output 22%; double crank − − +, input 100% of a turn and output 100%; rocker-crank − + −, input 22% of a turn and output 100%; double rocker + − −, input 23% of a turn and output 21%; 0–π rocker + + −, input 76% of a turn and output 72%; π–π rocker + − +, input 84% of a turn and output 51%; π–0 rocker − + +, input 84% of a turn and output 85%; 0–0 rocker − − −, input 67% of a turn and output 91%. Grashof's condition names four of these and calls the other four one thing; the arcs show that the four triple rockers differ in which way along the ground line each rocker swings through.

Eight kinds of four-bar

Grashof's condition gives a four-bar one of four names and calls every linkage that fails it a triple rocker. Three signed sums of the lengths give eight, and a census of four thousand random linkages finds every one moving exactly as its signs say — because the planes where those sums vanish are the only places a four-bar's motion can change its kind.

linkages · Fourbar One crank, four machines, one area. Four crank-rockers sharing only a crank of length 1, with grounds, couplers and rockers all different, each tracing the point 30% of the way from the crank pin to the rocker pin. The curves have different shapes and sizes and different places in the plane, and the area each one encloses is 2.199115 = (1 − 0.3)·π·1² — on both assemblies of every one, with a worst difference of 3.2 × 10⁻¹⁴.

The area a coupler point encloses

Trace a point on the line through a crank-rocker's two moving pins and the region its curve encloses has area (1 − u)πa²: the crank pin's own circle, scaled by how far along the line the point sits. No ground, coupler or rocker length appears in it. Four machines that share only a crank enclose 2.199115 each, to 3 × 10⁻¹⁴.

curves · Coupler 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 Three machines drawing one oval, and where each keeps its area. The crank-rocker with ground 4, crank 1, coupler 3.5 and rocker 3, its tracing point at (0.45, 0.5) of the coupler, and the two other four-bars Roberts's construction gives for the same curve, each drawn holding the same point of one oval and each with its input pin's path dashed in the input colour and its output pin's in the output colour. The shaded oval encloses 2.116354 for all three. In the crank-rocker the crank pin goes round and carries 1.727876 of it; in the double rocker neither pin goes round and the coupler's turn carries 1.727876; in the rocker-crank the output pin goes round and carries 1.727876. The remaining 0.388478 is the same in all three.

Where three machines keep one area

Roberts's theorem gives every four-bar two others that draw the same coupler curve, and so enclose the same areas. Measured, they do, to 10⁻¹² — but each keeps the area in a different place: the crank-rocker in its crank pin's circle, the double rocker in its coupler's turn, the third machine in its output pin's circle. What is left over has no closed form, and it is one number in all three.

curves · Coupler Four offset slider-cranks, one from each region the limit leaves. One slider-crank from each region of T₂ = b − a + e and T₃ = b − a − e, with the slide the dashed vertical line a distance e from the crank's pivot and the ground line dashed across. The thick arc round the pivot is where the crank's pin can go and the thick stretch of the slide is where the slider can: crank-rocker + +, crank 1, rod 3.5, offset 0.8, the crank reaching 100% of a turn and the slider between 2.37 and 4.43 on each side of it; double rocker − −, crank 3.5, rod 1.2, offset 0.6, the crank reaching 22% of a turn and the slider between 2.22 and 4.66 on each side of it; 0–π rocker + −, crank 2, rod 2.5, offset 1.6, the crank reaching 65% of a turn and the slider running from −4.21 to 4.21 through the ground line; π–π rocker − +, crank 2, rod 2.5, offset −1.6, the crank reaching 65% of a turn and the slider running from −4.21 to 4.21 through the ground line. These are the four of the eight four-bar kinds in which T₁ is positive.

Four kinds of slider-crank

An offset slider-crank is a four-bar whose output bar and ground have grown without bound, and in that limit the three signed sums that sort four-bars into eight kinds lose one of their signs. Four kinds survive. A census of four thousand finds every slider-crank moving as its region predicts, the textbook condition for a full crank turn turns out to be one region exactly, and each of the four kinds that vanish is carried, at a length that can be written down, into the survivor that shares its other two signs.

linkages · Fourbar A parallelogram at its flat position, exact and built slightly wrong. Left, a parallelogram with ground 3, cranks 2 and coupler 3 lying flat, where its two assemblies meet; faintly, the two ways it can go on, drawn at 30°. Middle, the same linkage with its coupler 0.05 too long, drawn at the closest it can come to the flat position: 7.49° away on either side, so the input's circle is thick where the input can go and red across the 15.0° it can never enter. Right, the input crank 0.05 too short, at the flat position: its two assemblies put the output crank 43.76° apart, and they do not meet at any input angle.

A parallelogram a micron wrong

A parallelogram linkage sits exactly where two kinds of four-bar meet, so a parallelogram that has actually been made is always one of four other machines. Make one bar a micron wrong on a 300 mm frame and the input stops a tenth of a degree short of lying flat, or the output turns round there with an acceleration that grows as one over the square root of the error. A third bar turns the square root back into a misfit of one micron.

constraint · Mobility 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 What region each four-bar's two cognates land in. One row per region of length space. Against each, the regions of the two four-bars Roberts's construction gives for the same coupler curve, and how many of the 24000 chains in the census landed in that region. Every row has one entry: across the whole census, and at each of 5 tracing points, the original's region decides its cognates' regions with nothing left over. The four Grashof regions are above the rule and the four triple rockers below it, and no row crosses it — a crank-rocker has a double rocker and a rocker-crank, a double crank has a double crank and a double crank, a rocker-crank has a rocker-crank and a double rocker, a double rocker has a crank-rocker and a crank-rocker, a 0–π rocker has a 0–π rocker and a 0–π rocker, a π–π rocker has a π–π rocker and a π–0 rocker, a π–0 rocker has a 0–0 rocker and a 0–0 rocker, a 0–0 rocker has a π–0 rocker and a π–π rocker.

The kind is decided before the lengths are

Roberts's construction hands a four-bar two others that draw its curve, and which of the eight kinds those two are is settled by the kind of the first — not by its lengths within that kind, and not by where the tracing point sits. Twenty-four thousand chains at five tracing points produce no exception, and the reason is one line: the tracing point enters the construction only as a scale, and a region is scale-blind.

curves · Coupler How evenly the rocker is driven through the working stroke. The rocker's speed through the working stroke, divided by its mean over the stroke, against the fraction of the stroke's duration, for a crank-rocker with a 60° swing and a time ratio of 1.2 of its own, ground 1.2223, crank 0.4783, coupler 1.1298, output 1. Every curve must start and end at nought, because the rocker stops to reverse; what differs is the middle. The number after each name is the fastest speed over the slowest while the rocker covers the central 80% of its swing. Driven directly at constant speed the crank-rocker gives 1.91 at a time ratio of 1.20. The drag link that gives the highest time ratio, 2.71, gives 2.77: a hump in the middle of the cut. The most even design that still reaches 2, a drag link of ground 1, crank 5, coupler 4.5, output 2.25, gives 1.21 at a ratio of 2.03 — more even than the crank-rocker alone. Dragging moves that design's phase.

A quick return that cuts evenly

A drag link ahead of a crank-rocker buys a shaper its time ratio of 2 with both stages at 40°. The drag link that buys the most ratio drives the cut unevenly — its fastest speed through the middle of the stroke is 2.77 times its slowest — and a different drag link at a different phase reaches 2.03 with a ratio of 1.21, which is more even than the crank-rocker driven alone at constant speed. Up to a ratio of about 2.4 the second stage can improve both specifications at once.

linkages · Fourbar

Named alongside it

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

Grashof's conditionTransmission angleDouble rockerChange pointClassificationCoupler curveQuick-returnAssembly branchLimit positionSigned areaTime ratioTriple rocker

All concepts