Machine Kinematics
solved, then drawn
Concept

Transmission angle — where it appears

The angle between a four-bar's coupler and its output link, which decides how much of the coupler's push turns the rocker. It is the linkage's version of a pressure angle, and a design keeps it away from zero and from a straight line.

Named by 25 essays across 9 fields — each of them below, with the objects they name alongside it.

The output is a band, not an angle. The rocker's angle through one turn of the crank, for a four-bar whose four lengths are each specified to ±0.01. The line is the nominal mechanism; the band is where the output of an actual one lies, found by building all sixteen extreme combinations of the four lengths at every crank angle and solving each. The band is not a constant width: it is 0.73° at its widest, near 30°, and 0.36° at its narrowest — a factor of 2.0. Which of those a designer is told depends entirely on where the mechanism was measured.

A length is a range

Every figure on this site so far has been drawn from four numbers. No four numbers were ever cut. Give each of them a tolerance of ±0.01 and the rocker's output stops being an angle and becomes a band 0.73° wide at one part of the turn and 0.36° wide at another — and which of those a designer is told depends only on where somebody measured.

practice · Tolerance A four-bar at 60°, solved. Ground 4, crank 1, coupler 3.5, rocker 3. Every joint position here is the output of a Newton–Raphson solve on the loop-closure equations, converged to 0.0e+0 — not a placement that looked right. Grashof's condition classifies these lengths as a crank rocker, and sweeping the crank through 360° confirms it: 120 of 120 positions assemble. The transmission angle at this instant is 66.9°.

Four bars and four pins

The smallest interesting machine there is. Four lengths decide everything about it — which link can turn all the way round, how hard it pushes, where it stops and whether it can be assembled at all — and every one of those is a number that falls out of a solve rather than a judgement about a drawing.

linkages · Fourbar A 12-tooth ratchet, holding. A ratchet wheel with a 8° tooth face and its pawl. The long line is the face extended; the short one through the contact is the normal the tooth pushes along, and both belong to the wheel and turn with it. The pawl holds, and the reason is which side of those two lines its pivot is on: 1.080 wheel radii from the face's line and 0.495 from the normal's, and on opposite sides of them. Nothing about how hard anything pushes enters the question: a normal is a direction and a moment arm is a length with a sign. Dragging turns the wheel the way it may go, and the pawl's tip is solved onto the wheel's surface at every frame: it is lifted up a tooth's back and dropped into the next notch, once per click.

A joint that works one way

A pawl either holds a ratchet or is levered out of it, and which one happens is decided by two lines through the contact. One of them is the rule a workshop quotes. The other is the boundary that rule leaves out, and a check written to confirm the quoted rule turned out to be incapable of failing.

timing · Intermittent 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 Two things that are not the same configuration. The mechanical advantage of a four-bar, from the velocity solution: for a lossless mechanism it is the reciprocal of the output-to-input speed ratio, so wherever the output momentarily stops the force ratio diverges. It reaches 1673 at 222° — a toggle, where crank and coupler line up, and the reason a knee-joint clamp holds with almost no effort. The transmission angle is at its worst somewhere else entirely: 46.5° at 0°, 222° away, where coupler and rocker line up instead. Both get called "the mechanism jamming" and they are opposite situations — one is enormous force output, the other is force going into the bearings.

Two things called jamming

A four-bar's mechanical advantage peaks at 1,673 in one configuration and its transmission angle collapses in another, 222° away. Both get described as the mechanism jamming. One is enormous force output and the other is force disappearing into the bearings, and they are opposite situations.

wrong · Misconception The four lengths do not matter equally. Each length's average contribution to the output band, for a tolerance of ±0.01 on all four, averaged over the 48 crank positions the mechanism reaches. The coupler contributes 38% of the total and the rocker 11% — a factor of 3.4 between the ends of the ranking. A tolerance specified equally on all four therefore spends most of its cost buying accuracy the mechanism does not notice, which is what a sensitivity ranking is for.

The four lengths do not matter equally

Averaged over a whole turn, the coupler contributes 38% of a four-bar's output band and the rocker 11% — a factor of 3.4 between the ends of the ranking. A tolerance specified equally on all four therefore spends most of its money buying accuracy the mechanism cannot use, and the ranking that says so costs four linear solves.

practice · Tolerance The transmission angle through one turn. μ is the angle at B between coupler and rocker, computed from each solved position rather than from a formula. It runs from 54.3° to 100.3° for these lengths. The shaded band is the usual design rule — keep μ between 40° and 140° — and this linkage stays inside it throughout. The rule is about geometry alone: nothing here knows about friction, and a mechanism with a comfortable μ can still be a poor machine.

The transmission angle

The angle at which the coupler meets the rocker decides how much of an applied force becomes useful output torque and how much goes into the bearings. It is pure geometry, it is computed here from every solved position rather than from a formula, and it is the number a linkage is judged by after Grashof has said it turns.

linkages · Fourbar Three machines a protractor cannot tell apart. The same four-bar at 0.60×, 1.00×, 1.50×, drawn one inside another at the same crank angle. Every one of them puts its output link at 102.914064°, and the three readings differ by 2.8e-14° — which is the solver's floor rather than a difference. A protractor on the output link is reading a function of the ratios of the lengths, so it is the same function for every member of this family, at every crank angle, exactly. Whatever such an instrument recovers, it is not the size of the machine.

The direction no protractor can see

A four-bar's output angle depends only on the ratios of its lengths. That is a sentence anybody would agree to, and it has a consequence with a number attached: the vector of the four lengths is annihilated by every row of the machine's own identification Jacobian, to 7.6 × 10⁻¹⁵, at every pose, for ever.

metrology · Parameter Where the coupler is pivoting, at 70°. At any instant the coupler is turning about one point — not a pin, and usually not on the mechanism at all. Kennedy's theorem finds it: the crank and coupler share the pin at A, the coupler and rocker share B, so the coupler's centre relative to the frame must lie on both O₂A extended and O₄B extended, and it is where they cross. The dashed lines are that construction. The cross is a completely different route to the same point — the place where the coupler's solved velocity field is zero, computed from the Jacobian and knowing nothing about Kennedy. Across 119 positions the two agree to 2.7e-15. At this instant the centre lies outside the frame — the two construction lines are nearly parallel, the coupler is close to translating, and the pivot has run off rather than gone missing.

Where the coupler is turning

At every instant the coupler of a four-bar is rotating about a single point — not a pin, and usually not on the mechanism at all. Track that point in two different frames and you get two curves which, rolled on each other without slipping, reproduce the coupler's motion exactly. The bars are one way of producing it and not the motion itself.

curves · Coupler 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 The wheel is the coupler — double wishbone. The suspension solved at 0 mm of bump, with the whole travel ghosted behind it. The two arms are the cranks and the upright between them is the coupler; the wheel is bolted to that coupler, so camber is the coupler's rotation and nothing else. Camber here is 0.00° and the contact patch has moved 0.0 mm across the road. The cross is the instantaneous centre of the upright, found from the solved velocity field; the roll centre is where the line from it to the contact patch crosses the car's centreline, and it is at 73 mm here.

The wheel is the coupler

A double wishbone is a four-bar standing on end whose coupler carries a wheel, so camber is coupler rotation and scrub is a coupler point's path. Both are computable, and the second one comes out with the opposite sign from the model every suspension book uses — by more than the whole scrub.

applied · 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 How wrong the trapezoid is, arm angle 15.6°. The difference between the outer wheel's angle and the angle that would put all four wheels on one circle. It is zero at straight ahead by construction — both wheels point forwards — and it reaches 2.06° at 35° of lock. "One hundred per cent Ackermann" names a condition this linkage meets at 1 angle and nowhere else, and no four-bar can do better than a handful: the condition is not a rational function of the crank angle, and the linkage is.

The steering that is never right

For four wheels to roll without scrubbing, the two front wheels must point at different angles, and the relation between them is a cotangent condition no four-bar can satisfy. The trapezoid under every car meets it at straight ahead and, if the arm angle is chosen well, at exactly one other angle — 0.34° out at worst instead of 2.06°.

applied · Fourbar The slider turns round at the dead centre. The slider's position against the crank angle, through the dead centre. The curve has a maximum there — that is what a dead centre is — so the slider retreats on both sides of it, and the latch works because a mechanism resting past the top of this curve has to be pushed back up it before anything can move. Set 4° past, the slider is 71 µm below the peak, and because the curve is quadratic there, doubling the setting quadruples the depth.

Locked on purpose

A toggle clamp, a landing-gear downlock and the catch on a folding table are all the same mechanism parked a few degrees past its dead centre, where the slider's motion is second order in the crank's. Seventy-one microns of slider travel undoes a latch set four degrees over — and setting it eight degrees over does not double that, it quadruples it.

applied · Fourbar One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything.

The angle that holds the lock

A locking face cut exactly concentric with the pallet arbor has no tendency to hold itself: the tooth's push aims straight at the pivot and its moment is zero, so the smallest disturbance opens the lock. Tilt the face and the moment becomes ρ sin δ — and the same tilt, by virtual work, is exactly the rate at which the wheel is driven backwards. Draw and recoil are one angle read twice.

timing · Intermittent 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 One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything.

One test, three mechanisms

Whether a pawl holds, whether an escapement's lock draws itself deeper, and how much a four-bar's coupler can do for its rocker are the same question asked three times: on which side of a pivot does a contact normal pass? All three are one cross product, none evaluates a force, and the three answers are used for completely different things.

timing · Intermittent The transmission angle through one turn. μ is the angle at B between coupler and rocker, computed from each solved position rather than from a formula. It runs from 54.3° to 100.3° for these lengths. The shaded band is the usual design rule — keep μ between 40° and 140° — and this linkage stays inside it throughout. The rule is about geometry alone: nothing here knows about friction, and a mechanism with a comfortable μ can still be a poor machine.

What is still outside

Eight essays here end by saying that clearance, backlash, friction or wear are not modelled. This field took two of those four, because a tolerance is a set of geometries and a clearance is a short link, and both are questions about where a mechanism can be. The other two are not, and this is the page that says exactly where the line falls and why it is where it is.

practice · Practice The transmission angle through one turn. μ is the angle at B between coupler and rocker, computed from each solved position rather than from a formula. It runs from 54.3° to 100.3° for these lengths. The shaded band is the usual design rule — keep μ between 40° and 140° — and this linkage stays inside it throughout. The rule is about geometry alone: nothing here knows about friction, and a mechanism with a comfortable μ can still be a poor machine.

The transmission angle has no size

The geometric half of force transmission is an angle in a triangle whose three sides scale together, so it is the same at every size — 54.31° at its worst on this machine, whatever units the drawing is in. Which means a measurement that cannot recover a single length recovers the whole of what this field computes.

transmission · Transmission 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 Which pin's play costs the most. Each pin's clearance taken one at a time, at 0.01 on links of 1 to 4, with the direction swept rather than assumed. The ranking runs A 32%, B 25%, O₂ 22%, O₄ 21% — a spread of 1.52 against the 3.43 the four lengths spread over. Clearances are more evenly weighted than length tolerances because each pin joins two links and so enters two of the four sensitivities, which is why the best bearing buys less than the best-held length does — and why it still goes somewhere the load path does not suggest.

Which pin to buy

The four lengths of a four-bar contribute 38, 26, 25 and 11 per cent of its output error — a spread of 3.4. Its four pins contribute 32, 25, 22 and 21 — a spread of 1.5. Clearances are more evenly shared than length tolerances, because every pin joins two links and so appears in two of the four sensitivities, and that changes what a better bearing is worth.

practice · Clearance How far the driven link turns is a fact about the lengths, not the chain. Each of the sixteen eight-link chains, given the arbitrary placement its own layout produces, driven from its first available choice of frame and input, and swept until a frame stops closing. 10 of the sixteen reach every angle and the rest rock through between 107° and 244°. Nothing in this chart is a property of the chains. Change the placement and the bars change; the census above them does not. It is here because it is the sharpest way to say what this field does and does not decide, and because the temptation to read a topology census as a catalogue of machines is exactly the mistake it prevents.

The chain has no lengths

Every number in this field survives multiplying every link by a different scale factor, because there are no lengths to scale. Which is also the statement of what a census cannot decide — and the sixteen eight-link chains, each given one arbitrary set of dimensions and driven, produce a chart in which nothing belongs to the chains.

topology · Topology 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 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 Which numbers have a size. Twelve quantities from eight fields of mechanism kinematics, each scaled by taking every length in its mechanism up and down together and fitting the power its value follows. A shape lands on zero, a length on one, a curvature on minus one, an area on two, and every row lands on an integer to 3.3e-11. The sorting is not a units argument — it is measured from each field's own library, by asking that library the same question. What it says is that most of what kinematics computes is a shape: a mobility, a Grashof class, a transmission angle, a contact ratio and a velocity ratio are all unchanged by making the machine bigger, so all of them are recoverable from a measurement that cannot recover a single length. The row worth stopping at is the tolerance band: held to a fixed ±0.01 it lands on minus one, because ±0.01 is a length and a bigger machine held to the same absolute tolerance is a better machine. Written as a percentage the same band lands on zero. One quantity, two drawing conventions, two different kinds of number.

Which numbers have a size

Take every length in a mechanism up and down together and fit the power each computed quantity follows. A transmission angle lands on zero, a coupler point's speed on one, a path curvature on minus one, an enclosed area on two — and a tolerance band held to a fixed ±0.01 lands on minus one, which nobody would guess.

metrology · Parameter

Named alongside it

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

Grashof's conditionFour-barDead centreConstraintToggleToleranceCrank-rockerStrokeVelocity ratioJacobianIdentifiableLimit position

All concepts