Machine Kinematics
solved, then drawn
Drawn wrongly

The ratio that is not a number

A four-bar's output-to-input speed ratio runs from −0.29 to 0.51 through one turn and changes sign on the way. Quoting a single figure for it quotes the average of that curve, which the mechanism never exhibits. A gear pair is the case where the same phrase is honest, and it is honest by construction.
14 min read 7 figures The constant that is not constantSolved, not drawn

A statement that a mechanism’s ratio is three to one is a sentence with a hidden claim in it. It claims the ratio does not change.

For a gear pair that is true, and it is true because the involute was invented to make it true. For almost everything else it is false.

A ratio that is not a number. The output-to-input angular velocity of a four-bar, computed from the velocity solution at every position. It runs from -0.398 to 0.333 — it changes sign, because the rocker turns back — with a mean of 0.000 that no instant of the cycle actually exhibits. Quoting a single figure for a linkage's ratio is quoting the average of that curve. A gear pair is the case where the same phrase is honest: its ratio is 0.5000 and stays there, which is not a coincidence but the property the involute was invented to guarantee.
Fig. 1 The output-to-input angular velocity of a four-bar, computed from the velocity solution at every position. It changes sign, because the rocker turns back. The dashed line is the mean, which no instant of the cycle exhibits.
A ratio that is not a number. The output-to-input angular velocity of a four-bar, computed from the velocity solution at every position. It runs from -0.506 to 0.393 — it changes sign, because the rocker turns back — with a mean of 0.000 that no instant of the cycle actually exhibits. Quoting a single figure for a linkage's ratio is quoting the average of that curve. A gear pair is the case where the same phrase is honest: its ratio is 0.5000 and stays there, which is not a coincidence but the property the involute was invented to guarantee.
Fig. 2 A second set of lengths, the same instrument. The ratio varies over the turn again, by a different amount and with its extremes in different places — which is what makes it a function of the mechanism rather than a number attached to the type.

Where the number comes from

Every position on this site is solved, and differentiating the constraint equations gives the velocities from the same Jacobian the position solve already formed. So the ratio at each instant is available for any mechanism the solver can position, without needing a closed form for it.

That matters here because the closed form for a four-bar’s velocity ratio is not memorable and the curve is the point rather than the formula.

Why a linkage cannot have a constant ratio

A four-bar with a rocker output cannot possibly have one: the rocker reverses, so its velocity passes through zero and changes sign twice per turn. A ratio that changes sign is not a number.

Even a drag-link four-bar, where both attached links rotate continuously, has a ratio that varies — considerably. That variation is sometimes the point: quick-return mechanisms exist precisely because the output is faster in one direction than the other, which is what is wanted when the working stroke should be slow and the return fast.

So the variation is not a defect to be minimised. It is a design variable, and a mechanism quoted by a single ratio has had its most interesting property averaged away.

The gear case, and why it is special

A gear pair’s ratio is constant, and that is a strong statement: constant at every instant of every tooth’s engagement, not merely on average over a turn.

It holds because the common normal at the contact is a fixed line — tangent to both base circles — so the moment arms about the two centres never change. The whole tooth form exists to produce that property, and it is why the sentence “the ratio is 2:1” is honest for gears and misleading for nearly everything else.

Worth noting what it is not honest about even so. The ratio is constant kinematically; the transmitted torque varies with the pressure angle and with which pair of teeth is carrying, and real gears have transmission error from manufacturing tolerance and tooth deflection. The constancy is a property of ideal geometry.

Where the number comes from here

Every position on this site is solved, and the velocities come from differentiating the same constraint equations the position solve already used — so the ratio at any instant is a by-product rather than a separate calculation.

That matters more than it sounds. The closed form for a four-bar’s velocity ratio exists, is standard, and is not memorable; and it is specific to the four-bar. The Jacobian route works for any mechanism the solver can position, which is why the Peaucellier cell and the redundant parallelogram can be given velocities without anybody deriving anything.

It also gives an independent check. Differentiating the constraints symbolically and differencing two nearby solved positions are genuinely different calculations, and requiring them to agree is what caught this site’s worst bug: the velocity solve negated its right-hand side twice, every velocity came out reversed, every mechanism still drew perfectly, and the finite-difference check reported a relative error of exactly 2 — which is the signature of a sign flip rather than an inaccuracy.

Quick return, and the variation as a design variable

The variation is not always a defect to be minimised. Sometimes it is the point.

A quick-return mechanism is one whose output moves slower in one direction than the other, and the asymmetry is deliberate: on a shaping machine the cutting stroke should be slow and powerful and the return should be fast, because the return does no work. The Whitworth mechanism and the offset slider-crank are both ways of getting it, and both work by making the crank sweep unequal angles for the two halves of the output’s travel.

For an offset slider-crank the asymmetry follows directly from the geometry: with the slide displaced from the crank centre, the two dead positions are no longer 180° apart in crank angle, so one stroke takes more of the revolution than the other. The ratio of the two is the time ratio, and it is a number a designer chooses by choosing the offset.

None of that is visible in a quoted “velocity ratio”. Averaging the curve destroys exactly the property the mechanism exists to provide.

The piston, which is the most repeated case

Every introduction to the slider-crank says the piston moves harmonically.

The piston that is not a sine wave. Every introduction to the slider-crank says the piston moves harmonically, and it does not. Plotted is the difference between the solved piston position and a pure cosine of the same amplitude, for four connecting-rod ratios. At L/r = 2 the departure reaches 0.268 of a crank radius; at L/r = 6 it is 0.084, and only in the limit of an infinitely long rod does it vanish. The shape of that residual is the second harmonic, it is why engine balance shafts run at twice crankshaft speed, and it is a direct consequence of a rod having a length.
Fig. 3 The difference between the solved piston position and a pure cosine, for four rod-to-crank ratios. It reaches a quarter of a crank radius at L/r = 2 and never reaches zero for any real rod.

The residual is dominated by a term at twice crankshaft frequency, and that second harmonic is why engine balance shafts exist and why they run at double speed. A statement that would be true only for an infinitely long connecting rod is taught as though it were true generally, and the error it hides is the reason for a whole category of engine hardware.

What to say instead

Three sentences that are defensible where the single number is not.

The mean ratio over a cycle is 0.11. True, checkable, and useless for anything except gross throughput.

The ratio varies from −0.29 to 0.51. More useful, and the shape of the variation is what a designer actually needs.

The ratio is 2.0000 at every instant. Available only for gears, and only because of the involute.

The habit worth adopting is asking, whenever a ratio is quoted, whether the mechanism is one where the word means an instantaneous value or an average — and the same question applies to any single number describing something that varies.

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.
Fig. 4 Another quantity that varies through the turn and is often quoted as one number. The transmission angle of the same linkage runs from 54° to 100°; the figure that matters for design is the worst of them, not the mean.
A ratio that is not a number. The output-to-input angular velocity of a four-bar, computed from the velocity solution at every position. It runs from -0.756 to 0.598 — it changes sign, because the rocker turns back — with a mean of 0.000 that no instant of the cycle actually exhibits. Quoting a single figure for a linkage's ratio is quoting the average of that curve. A gear pair is the case where the same phrase is honest: its ratio is 0.5000 and stays there, which is not a coincidence but the property the involute was invented to guarantee.
Fig. 5 And a third, chosen so the crank is a larger fraction of the frame. The variation grows with it: the closer the linkage comes to its own limits, the further the instantaneous ratio strays from whatever mean somebody quotes.

What makes the gear case genuinely different

It is worth being precise about why one mechanism gets to quote a number and the others do not, because “gears are constant and linkages are not” is a fact rather than an explanation.

The reason is that a gear pair’s contact normal is a fixed line in space. The two bodies touch somewhere along it, that somewhere moves, and the line does not — so the perpendicular distances from the two centres to it never change, and those distances are what set the velocity ratio.

A linkage has no such invariant. Its instantaneous relationship between input and output is set by the current geometry of the whole loop, and the loop’s geometry changes continuously as it moves. There is nothing to be constant.

That is also why the involute is not one tooth form among several. It is the answer to the requirement that the common normal be fixed, and essentially the only answer for arbitrary centre distances. The constancy is manufactured deliberately, at some cost, because it is not available for free.

The instantaneous centre, and why the ratio has a geometric form

There is a construction that makes the varying ratio of a four-bar visible without any calculation, and it is worth having because it explains the shape of the curve rather than only its values.

Any rigid body in plane motion is, at each instant, rotating about some point — its instantaneous centre. For the coupler of a four-bar that point is found by extending the crank and the rocker until they meet: both pins on the coupler are moving perpendicular to their own links, so the centre of rotation lies on both lines.

Once that point is located, the velocity ratio follows from similar triangles. The output-to-input angular velocity ratio equals the ratio of the distances from the instantaneous centre to the two fixed pivots. No trigonometry, no derivatives.

Three facts fall straight out. When the crank and rocker are parallel, their extensions meet at infinity and the ratio is one — the two links turn at the same rate for that instant. When the crank lies along the coupler — the dead centre — the instantaneous centre lands on the fixed pivot itself and the output velocity is zero. And the ratio is unbounded in between, which is the quantitative version of “not a number”.

The construction is also what makes the varying ratio designable. Moving the ground pivots moves where the instantaneous centre travels, which moves the whole ratio curve. That is how a quick-return mechanism is proportioned, and it is why the transmission angle and the ratio are related without being the same thing: both are read off the same geometry, one as a force question and one as a velocity question.

Where the varying ratio is the point

It is easy to read this essay as a list of reasons a linkage is worse than a gear at producing a ratio. In most machines that use a linkage, the varying ratio is what the linkage was chosen for.

Quick return. A shaper, a slotting machine or a crank-and-slotted-lever drive cuts on one stroke and returns on the other. Cutting slowly and returning fast is free time, and it comes entirely from the ratio’s asymmetry through the cycle.

Force multiplication where it is needed. A press linkage has enormous mechanical advantage near the bottom of the stroke and very little at the top, which is exactly the distribution the work requires — fast approach, slow squeeze.

Dwell. Some four-bar proportions hold the output nearly stationary through a substantial part of the input rotation, because the coupler curve has a segment of near constant radius. That is a dwell produced by a linkage rather than by a cam, and it is cheaper and quieter, at the cost of being approximate.

Motion generation. When the requirement is a path rather than a ratio, the varying ratio is not a side effect at all — it is one description of the path being right.

So the honest summary is not that a linkage has a bad ratio. It is that a linkage does not have a ratio, and the design question is what shape the variation should be. A gear train has a number and no shape; a gear train’s number is exact and unchangeable, which is a strength when a number is what is wanted and a limitation when it is not.

The phrase to be suspicious of

Almost every misuse in this area starts from the same construction: the ratio of the mechanism. It is worth collecting the cases where the phrase is safe and where it is not.

Safe. A gear pair, a gear train, a belt or chain drive on round pulleys, an epicyclic with one member held. In each of these the ratio is a constant determined by counts or diameters, exact for as long as the teeth mesh, and independent of position. It is a number.

Not safe. Any linkage. A four-bar, a slider-crank, a six-bar, a coupler point driving anything. The output-to-input velocity ratio of a linkage is a function of configuration, it varies through every revolution, and at a dead centre it is zero or unbounded depending on which way the ratio was written.

Deceptive. A linkage quoted with an average ratio, which is a real quantity — total output travel over total input travel — and which describes the instantaneous behaviour nowhere. A quick-return mechanism has an average ratio of one and forward and return strokes that differ by a factor of two, which is the entire point of it.

The practical consequence is a sizing error. Choosing a motor from a linkage’s average ratio underestimates the peak torque, because peak torque occurs where the ratio is least favourable and that configuration is not the average one. The correct calculation sweeps the mechanism, computes the ratio at every position, and sizes from the worst — which requires solving the mechanism through its travel rather than characterising it with a number.

That is the shape of nearly every problem this site addresses. A quantity that is constant in one family gets carried over to a family where it varies, the vocabulary comes with it, and the variation becomes invisible because the language has no place to put it. The transmission angle has the same problem in reverse: it is a function of configuration, everyone knows it, and it still gets quoted as a single number for a mechanism.

Sizing from the worst case

The practical consequence of a varying ratio is a sizing calculation, and it is worth writing out because the wrong version of it is common.

The torque a motor must supply at any instant is the load torque divided by the instantaneous velocity ratio, ignoring losses. Since the ratio varies through the cycle, so does the required torque, and the motor must supply the maximum rather than the mean.

The wrong version divides the load by the average ratio and specifies a motor from that. For a mechanism whose ratio varies by a factor of two through the cycle — which is unremarkable — the result underestimates the peak demand by a factor of two, and the machine stalls at a particular crank angle rather than everywhere, which makes the fault look intermittent and mysterious.

The right version sweeps. Solve the mechanism at enough configurations, compute the ratio at each from the velocity solution, divide the load torque by it, and take the maximum over the sweep. That is a few dozen solves, and it produces the actual number.

Two refinements matter. The load itself usually varies with configuration too — a press does most of its work near the bottom of the stroke — so the worst case is the maximum of the product, not the product of the maxima, and those can occur at different crank angles. And near a dead centre the required torque diverges, so the sweep must be read against the mechanism’s working range rather than over the full revolution.

That is the whole of what this essay recommends in place of a ratio: a curve, a working range, and a maximum taken over the two together. It is more work than a number and it is what solving the mechanism was for.

A universal joint's output speed, through one turn. The output shaft of a universal joint does not turn at the input's speed. It runs fast, then slow, twice per revolution, and the further the shafts are from being in line the worse it is: at 40° the output is running between 0.766 and 1.305 times the input speed. The curves are the closed form cos β / (1 − sin²β cos²θ); the dots are the ratio measured by solving the loop at θ ± 10⁻⁵ and differencing the solved output angle, which is a route the formula plays no part in. The worst disagreement across all three is 8.3e-11.
Fig. 6 The variation, measured. Two routes to the same curve: the closed form as a line, and the ratio differenced from solved positions as dots. The mean over a full turn is exactly 1 at every shaft angle.
The piston that is not a sine wave. Every introduction to the slider-crank says the piston moves harmonically, and it does not. Plotted is the difference between the solved piston position and a pure cosine of the same amplitude, for four connecting-rod ratios. At L/r = 2 the departure reaches 0.268 of a crank radius; at L/r = 8 it is 0.063, and only in the limit of an infinitely long rod does it vanish. The shape of that residual is the second harmonic, it is why engine balance shafts run at twice crankshaft speed, and it is a direct consequence of a rod having a length.
Fig. 7 The same departure carried out to a rod eight times the crank. It falls as the rod lengthens and does not reach zero at any finite length, which is the sense in which the sine wave is a limit and not an approximation somebody could tighten.

The mechanism everybody has met

The clearest case of this essay’s argument is one that was not available to it until the site left the plane, and it is worth adding because it is in every car ever built.

A universal joint’s velocity ratio is not 1. It runs between cos β and 1/cos β twice per revolution, where β is the angle between the two shafts, so at 30° the output shaft is running a third faster at some points of the turn than at others. Nothing about that is a manufacturing tolerance and no amount of precision reduces it — it is what four axes through a point do.

What makes it the exemplary case is the average. Over a full turn the input and output make exactly the same number of revolutions, at any shaft angle. Averaged, the joint is a perfect coupling; the mean ratio is 1 to the last decimal place and always has been. What the average hides is the entire engineering problem, because the excursion produces a torque variation at twice shaft frequency that fatigues everything either side of it.

So the universal joint is this essay’s argument with the summary statistic being not merely uninformative but exactly correct and exactly useless. A number that is right and answers a question nobody asked is harder to argue with than one that is wrong.

There is a compact test for whether a quoted ratio is honest, and it needs no measurement at all. Ask what the ratio is a ratio of. If both quantities are counts — teeth, revolutions, pins, contacts — the ratio is exact and a single number is the right report. If either is a length, an angle or a position, the ratio is a function of configuration and a single number is the value of that function somewhere. That sorts every mechanism this site has measured without computing anything: a gear pair’s ratio is teeth over teeth and is a number; a four-bar’s is a distance over a distance and is a curve; a variator’s is a radius over a radius and is a curve; a chain drive’s is teeth over teeth on average and a curve within a tooth. The test is worth having because it can be applied to a specification sheet by a reader with no access to the mechanism, and because it is right in both directions — every exact ratio on this site is a quotient of counts, and every quotient of counts on this site is exact.

Where this goes next

The involute, unwound. Hold a string taut against a circle and unwind it: the end traces this curve. Two properties follow immediately and between them they are the whole of gear geometry. The string is always tangent to the base circle, and it is always perpendicular to the curve. So the normal to an involute at any point is a tangent to its base circle — measured on the generated points here, to 2.3e-5 — and when two involutes touch, the common normal is a line tangent to both base circles, which does not move as the gears turn. Why a tooth is an involute A gear tooth is not a shape anybody chose for its looks. It is what one requirement forces — that the ratio of the two shafts' speeds stays exactly constant while the contact point slides along the flank. Impose that and the curve is essentially determined. 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. 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.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

The 8 of 35 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

AverageConstraintDead centreGear ratioHarmonicInstant centreInvoluteQuick-returnRatioStrokeVelocity ratio