Machine Kinematics
solved, then drawn
More than one input

Two inputs and one output

Every mechanism in this collection so far has had one input, and its output has been a function of that input. A differential does not. Its cage turns at the mean of two wheels, so knowing one of them tells you nothing at all about where the third shaft is going — and that is not a complication of the mechanism, it is a different kind of object.

Assumes Counting and measuring mobility and Epicyclic ratios, two ways.

16 min read 9 figures Solved, not drawnTwo routes to the same numberThe crank is the argument

Turn the crank of a four-bar and the coupler goes somewhere. Turn it again to the same angle and the coupler goes to the same place. That is so ordinary a property of a mechanism that it takes an effort to see it as a property at all: the configuration is a function of the input, the input is one number, and every essay on this site up to here has been about what that function does — where it is steep, where it is degenerate, where it stops having a solution, how many branches it has.

A car’s differential is not like that. Its cage — the housing the crown wheel is bolted to — turns at the average of its two wheels, and that is the whole of its kinematic content. Hold the cage still and the two wheels turn in opposite directions at whatever speed they like. Turn the cage and the wheels can do anything whose average is right. There is no function from the cage’s angle to a wheel’s angle, and there never was one to find.

A bevel differential, turning. cage in, hold left, with every member's speed taken from the train's null space and every angular position that speed integrated. The teeth are marked at the pitch points rather than cut as involutes — the flank is the teeth field's subject — but the count is the tooth count and the positions are the solved ones, so what turns and how fast is real. Drag it and watch which way each member goes: left 0.000 · right 2.000 · cage 1.000.
Fig. 1 Two side gears with a pinion between them, and the pinion’s axis carried by the cage. The pinion is an idler: whatever it does to one side gear it undoes to the other, so it drops out of the relation entirely and what is left is one equation saying the cage turns at the mean. Drag it and watch — the two wheels are going opposite ways, and the cage is not moving.

This field is about mechanisms of that kind, and the reason it is a field rather than an appendix is that the change is not one of degree. A mechanism with two freedoms and one output is a different object, it needs a different question asked of it, and the answers turn out to be the numbers on the side of every gearbox anybody has driven.

What “one input” was doing, unnoticed

Count the mechanisms this site has solved. A four-bar has one crank. A slider-crank has one crank. A cam has one shaft. A Geneva drive has one driver. An escape wheel has one arbor, and even the state machine that field ended in was a machine over one input’s travel. A Gough platform has six legs, and six freedoms to match. An arm has six joints and a tool with six coordinates.

In every one of those the count of inputs equals the count of freedoms, and the consequence is that specifying the inputs specifies the mechanism. It is why “solve the mechanism” has meant, throughout, name the crank angle and everything else follows. The mobility count has been a check on that arrangement rather than a subject in itself: mobility 1 with one crank is a mechanism that works, mobility 0 is a structure, mobility 2 with one crank was — until now — a mistake, a constraint left out.

Here it is the design. A differential has mobility 2 and one shaft anybody drives. So does every epicyclic gear train, which is a fact the gears field already stated and then walked away from: three shafts and one equation relating them, so two must be specified before the third is determined. That essay used the freedom to make a point about quoting ratios. This field takes it as the object.

The relation, written down

Take the epicyclic. Sun, ring, carrier, planets. Willis’s equation says

ωS−ωCωR−ωC=−zRzS\frac{\omega_S - \omega_C}{\omega_R - \omega_C} = -\frac{z_R}{z_S}

and clearing the fractions and collecting terms, what it says is

zS ωS+zR ωR−(zS+zR) ωC=0.z_S\,\omega_S + z_R\,\omega_R - (z_S + z_R)\,\omega_C = 0.

One linear equation in three unknowns. Not a formula for anything — an equation. It is satisfied by a two-dimensional space of triples, and the mechanism permits exactly the motions in that space and no others.

That is worth sitting with, because it is the shape of every claim in this field. The mechanism is not a machine that computes an output from an input; it is a constraint, and what it does is rule things out. Sun at 1 and ring at 0 leaves the carrier no choice but zS/(zS+zR)z_S/(z_S+z_R). Sun at 1 and carrier at 1 leaves the ring no choice but 1. Sun at 1 and nothing else said leaves the ring and carrier free along a whole line of possibilities, and the mechanism has no opinion about which.

Every gear is a line in one plane. The axes are the speeds of two members; every motion the gearset permits is a point of this plane, and there are two dimensions of them because the gearset has two freedoms. A brake or a clutch is one more linear condition, so it is a line through the origin — and the ratio it produces is that line's slope. This is what it means to say a transmission has no ratio until something is engaged: the mechanism is the plane, and a gear is a direction in it. The diagonal is direct drive, where everything turns together.
Fig. 2 The plane of motions, with two of the members’ speeds on the axes. Every point is something the gearset permits. There is no ratio anywhere in this picture until a line is chosen, and each line is one more constraint: a brake, or a clutch, or a decision about which shaft is driven.

A gearbox is a set of lines

Now the mechanical fact that makes this a subject rather than a curiosity. Because the permitted motions form a plane, one more linear condition cuts it to a line, and a line through the origin in the space of speeds is exactly a fixed proportion between the members: a ratio.

There are only two kinds of extra condition available in hardware, and every automatic transmission ever built is made of them:

  • a brake, which holds one member to the case: ωm=0\omega_m = 0;
  • a clutch, which locks two members together: ωa=ωb\omega_a = \omega_b.

Both are linear, both are homogeneous, and both are cheap. That is the whole engineering idea. The gearset is a plane; the shift elements are lines in it; a gear is a line; and shifting is choosing a different one.

What a Ravigneaux gearset can be made to do. Every distinct ratio the gearset offers, as a reduction. Each one is the same mechanism with one more constraint imposed — a brake holding a member to the case or a clutch locking two members together — and the input is a choice as well, which is why a four-speed needs an input clutch rather than four brakes. The shaded rows are the 5 a real transmission on this gearset is sold with; the rest are ratios the mechanism has and the gearbox does not buy the elements to reach. Every value is exact: the reductions are ratios of integers and are printed as such.
Fig. 3 Every distinct ratio a Ravigneaux gearset offers, computed by adding one row to the constraint matrix and reading off what is left. Seven of them, from three tooth counts. The shaded rows are the five a real four-speed transmission on this gearset is sold with; the other two are ratios the mechanism has and the gearbox does not buy the elements to reach.

Seven, and five sold. The two that go unused are not rounding or approximation — they are exact ratios of integers, available from the same casting, and they are left on the table because reaching them would need an extra clutch and because the ladder they would make is not one anybody wants. A transmission is a gearset plus a choice of which lines to build the hardware for, and separating those two things is most of what this field has to say.

Where the arithmetic is easier than usual, and where it is not

This site’s habit is that a count and a measurement have to agree, and where they disagree the count is wrong. Mobility by Grübler’s formula against mobility by the rank of the constraint Jacobian is the standing example, and the spatial field is largely the story of the formula losing.

A gear train is the one place on this site where that fight does not happen, and the reason is worth stating because it is unusual. Everywhere else the constraint Jacobian is evaluated at a configuration: its entries are sines and cosines of the current pose, and its rank can change as the mechanism moves — which is exactly what a singularity is. A gear train’s constraint has no configuration in it at all. The entries are tooth counts. They are the same at every position, so the rank is the same at every position, and a gear train has no singularities of that kind whatever.

The constraint matrix of a simple planetary. One row per mesh, one column per member, and every entry a tooth count. That is the whole of the kinematics: what the train permits is the null space of this matrix, and its dimension — 2 here — is the mobility. Nothing in it depends on where anything has got to, which is the one respect in which this field is easier than the rest of the site: elsewhere a Jacobian is evaluated at a configuration and its rank can change as the mechanism moves. A gear train's cannot. The elimination is done in exact rationals, so the ratio that comes out is a fraction rather than a number near one.
Fig. 4 The constraint matrix of a simple planetary. One row per mesh, one column per member, and every entry an integer count of teeth. The mobility is the dimension of its null space, and because nothing in the matrix depends on where anything has got to, that dimension is the same at every position the train can reach.

What follows is more than convenience. Because the matrix is integral, its null space can be computed in exact rational arithmetic, and a gear ratio comes out as a fraction rather than as a double that is nearly right. The Simpson gearset’s second gear on this site is 37/5437/54, not 1.45951.4595. That matters twice: once because the applied field already argued that a gear ratio is the one exact number in a catalogue, and once because several results in this field are counting results — how many trains have a given ratio, whether two ratios are the same — and asking whether two doubles are equal is not the same question.

Three shafts, and which one is driven

One more consequence of the relation being symmetric. Willis’s equation does not name an input. It relates three speeds, and calling one of them the input is something a designer does afterwards, with a coupling. So the same gearset gives a different ratio for every assignment of the roles — and this is where the “one gearset, several ratios” fact everybody knows actually comes from.

For a 24-tooth sun in a 72-tooth ring:

held driven driven output reduction
ring sun carrier 4
sun ring carrier 4/3
carrier sun ring −3
carrier ring sun −1/3
ring carrier sun 1/4
sun carrier ring 3/4

Six numbers, one set of gears, and every one of them exact. The three with the carrier held run backwards, and they run backwards for a reason that is visible in the mechanism rather than in the sign of a formula: with the carrier held the planet is an idler between the sun and the ring, an idler reverses the sense, and nothing else in the train reverses it back.

A simple planetary, turning. sun in, hold ring, with every member's speed taken from the train's null space and every angular position that speed integrated. The teeth are marked at the pitch points rather than cut as involutes — the flank is the teeth field's subject — but the count is the tooth count and the positions are the solved ones, so what turns and how fast is real. Drag it and watch which way each member goes: sun 1.000 · ring 0.000 · carrier 0.250.
Fig. 5 The classic case, drawn: ring held, sun driven, carrier out, reduction 4. Every member’s speed came out of the null space and every angular position in the drag is that speed integrated, so the picture cannot show a combination the teeth refuse.

The mechanism that makes a turn possible

The differential is worth one more paragraph here because it is the clearest case of a two-freedom mechanism existing to solve a problem that has no one-freedom solution.

A car going round a corner has its two driven wheels on circles of different radii. They are a track apart — call it 1.55 m — so on a 12 m circle the outer wheel travels a path 1.55 m longer per revolution of the car than the inner one, and rolling without sliding is a constraint on their speeds: they must be in the ratio of those radii, which is 1.138, or one of them is scrubbing.

An axle without a differential imposes a second condition — the two wheels turn together — and the two conditions are incompatible except when the car is going straight. That is not a matter of grip or of how much torque anything can take. It is two equations that have no common solution, and the mechanism has to be given a freedom or the constraint has to be broken.

What a turn asks of an axle. Two wheels a track apart on a circle must roll at different speeds or slide, and the difference is a pure fact about arc lengths. The bar is that difference as a percentage of the mean — which is what a locked axle has to make up in sliding, whatever the surface and whatever the torques. It is largest where the turn is tightest, which is the manoeuvre a locked differential is least usable in: at an 8 m radius the two wheels differ by 19.4% of their mean speed. The total sliding per circle is the same at every radius — it is 2π times the track, 9.74 m, because a full circle turns the vehicle through the same angle whatever its size — so what a tight turn changes is how little distance that scrub is spread over. Nothing here is about grip. The differential exists because the constraint is otherwise unsatisfiable.
Fig. 6 What a turn asks of an axle, as a pure statement about arc lengths. The bar is the difference between the two wheels’ rolling speeds as a fraction of their mean — 19.4% at an 8 m radius. The total sliding per full circle is the same at every radius, because a circle is a circle: it is 2π times the track, 9.74 m, and what a tight turn changes is how little distance that has to be spread over.

The differential gives the axle the freedom, and gives it exactly: the mean of the two wheel speeds is fixed by the propshaft and their difference is left alone, so the turn’s condition, which is a condition on the difference, never meets the drive’s condition, which is a condition on the mean. Two constraints on two independent quantities. They do not collide because they cannot.

What a Simpson gearset can be made to do. Every distinct ratio the gearset offers, as a reduction. Each one is the same mechanism with one more constraint imposed — a brake holding a member to the case or a clutch locking two members together — and the input is a choice as well, which is why a four-speed needs an input clutch rather than four brakes. The shaded rows are the 4 a real transmission on this gearset is sold with; the rest are ratios the mechanism has and the gearbox does not buy the elements to reach. Every value is exact: the reductions are ratios of integers and are printed as such.
Fig. 7 A Simpson gearset’s five distinct ratios, which are what a three-speed automatic’s shift pattern is made of. One casting, one sun, two rings, and the whole gearbox is that matrix with one extra row added five different ways.
The lever of a simple planetary. Each member sits at a position on the lever fixed by the tooth counts alone, and its speed is the height of one straight line over that position. The line here is drawn through sun and ring; every other member is plotted where the train's null space puts it, and lands on the line exactly — the residual is zero in rationals, and 2.2e-16 once the coordinates have been rounded to doubles for the drawing. Where the line crosses the axis is the member that is standing still, and that is what a brake does: it pins the line to the axis at one position and leaves it free to pivot there. The planets are on the lever too, off the end of it, which is where they belong — they are members of the train and are not shafts anybody can reach.
Fig. 8 The same plane drawn as a line, which is how the trade draws it. Each member sits at a position fixed by the tooth counts and its speed is the height of one straight line over that position — a diagram that is usually offered as a mnemonic and is exact, for a reason the third essay in this field works out.

What this field will not say

The differential’s reputation is not about speeds at all. It is about torque: the open differential’s habit of sending the drive to whichever wheel has least grip, the limited-slip devices sold to stop it, the whole argument about what happens when one wheel is on ice.

None of that is here, and it is not a gap left to be filled later — it is the same boundary this site has kept since its foundation. Kinematics answers what motions a mechanism permits. It does not answer which of the permitted motions actually happens, because that is decided by what is pushing, and nothing in this library knows what a force is.

What can be said, and is worth saying because it is nearly always said wrongly, is that the kinematic relation is satisfied in the one-wheel-on-ice case. The mechanism is not confused and nothing has failed. The wheel on ice turns at twice the cage and the wheel with grip turns at nothing, and the mean is right, and the constraint the differential imposes is met exactly. One wheel on ice is about that, and about what a locking differential does — which is a kinematic change, because it is a constraint, and it takes the axle’s mobility from two back to one.

Two more exclusions, stated once for the field:

  • Efficiency. A compound epicyclic can be given a reduction of two hundred to one, and nobody builds one at that ratio because it stops moving. That is a force argument, the ratio is exactly what the tooth counts say it is, and the essay on those drives says so rather than pretending the geometry knows.
  • The shift itself. A clutch here is a constraint that is either present or absent. What happens while it is engaging — both constraints partly applied, the members’ inertia, the speeds being dragged together — is dynamics and is outside.

Why this field’s figures look different

There is a visible consequence of two freedoms that runs through every picture in this field, and naming it explains a difference in style that would otherwise read as a change of taste.

A one-freedom mechanism has an output curve. Turn the input, plot what comes out, and the whole behaviour of the mechanism is a line on a graph — a coupler curve, a lift profile, a transmission angle against crank angle. Every field on this site before this one draws that line, and the drawing is complete: there is nothing about the mechanism that is not somewhere on it.

A two-freedom mechanism has no such curve. Its behaviour is a surface, and a picture of it is either a surface — hard to read, and hard to draw honestly at this size — or a family of slices, each of which is a choice somebody made. Neither is the complete object that a coupler curve is.

So this field draws something else instead, and the something else is the relation rather than the motion. A lever diagram is a picture of the constraint: one line, showing the whole two-dimensional family at once by making a member’s position the horizontal coordinate. A null space is the same object written as algebra. A ratio table is the family sampled at the places hardware can hold it.

That is why the field’s figures are diagrams where the rest of the site’s are traces, and it is not an editorial decision. A mechanism with more freedoms than inputs has a relation and not a function, and a relation is drawn by drawing the relation. Attempting to draw a motion would mean fixing one freedom arbitrarily and then showing a curve that is an artefact of the choice — which is exactly what a slice is.

It also explains why the shift elements are so central here. Every one of them is a way of turning the relation back into a function: add a linear condition, get a line, get a ratio. A gearbox is a mechanism that spends its life being converted from a two-freedom object into a one-freedom one, several times a minute, and the ratios this field enumerates are the results of those conversions rather than properties the gearset has on its own.

Which is the sharpest way to say what the field is about. The rest of the site studies mechanisms that do one thing; this one studies a mechanism that does a family of things, and the engineering is in choosing which.

The ledger of the field

Twelve essays on this ladder and three on older ones. What holds them together is the claim in the title: the input does not determine the output, and everything a transmission is follows from what has to be added before it does.

  • The relation as a null space, and what its dimension is.
  • The lever diagram — the exact linear picture of a gearset, which turns out to exist precisely when the frame carries no teeth, and to be exact rather than approximate for the same reason.
  • Holding a member as a choice of line, and the enumeration of every brake and clutch a gearset offers.
  • Four speeds from two tooth counts, matched against a transmission somebody actually sold.
  • Why the steps in a gearbox cannot all be what the design rule asks for.
  • The wheel on ice, and the locked axle as an overconstrained mechanism.
  • The continuously variable ratio, where the second pulley’s radius is a root of an equation rather than a choice, and the rule of thumb that says the two radii add to a constant turns out to be wrong by 2.6%.
  • The reductions built out of a difference of two nearly equal quantities — an epicyclic, a screw and a chain hoist — which share one conditioning number and fail in one way.
  • The reductions a single planetary cannot give, which are bounded not by anything about ratios but by the arithmetic of assembling planets.
  • Reverted trains, where asking for a ratio and for the shafts to be in line is asking for a solution of two Diophantine equations, and there is no reason for one to exist.
Which trains turn as a block. Give every member the same speed and ask each mesh whether it is satisfied. A train whose frame carries no teeth says yes — nothing is straining, because every mesh relation is about differences of speeds — and that one motion is what makes the lever diagram exact. A train with a gear on the frame says no, and has no lever at all. The bar is the largest mesh residual, in teeth per unit of speed; the first four are exactly zero and the last two are not near it.
Fig. 9 The property the next few essays turn on, tested where it can fail. Give every member of a train the same speed and ask each mesh whether it is satisfied. A train whose frame carries no teeth says yes — nothing is straining, because every mesh relation is about differences — and that one motion is what makes the lever diagram exact. A train with a gear on the frame says no, and has no lever at all.

The habit is unchanged from every field before it: two routes to each number, and a check that can lose. Here the two routes are the graph’s null space, which knows nothing about which gear is a sun, and the closed forms every reference prints, which know nothing about matrices. They agree exactly, in rationals, and the next essay is about why “exactly” is available here and what it buys.

There is a small practical corollary about reading this field’s figures, and it is worth stating because it inverts the usual instruction. Elsewhere on this site a figure’s slider explores the mechanism and the reader is invited to move it; here a slider explores a choice the hardware makes, and what it shows is which line out of the family a shift element has selected. So the thing to watch is not how the mechanism moves as the control changes but which relation is in force, and the two are different questions asked of similar-looking controls.

Where this goes next

Mobility, counted and measured. Grübler's criterion counts links and joints and knows nothing about the dimensions; the rank of the constraint Jacobian measures the dimensions and knows nothing about the topology. They agree for four of these five. The parallelogram with a redundant third bar is the exception: the formula declares it a structure with zero degrees of freedom, and it moves. The formula is the one that is wrong, because it cannot see that the third bar's constraint equations are already implied by the other two. Counting and measuring mobility Grübler's criterion counts links and joints and never asks how long anything is. The rank of the constraint Jacobian measures the lengths and never asks what a joint is. Two calculations with no inputs in common, producing one number — which is the only arrangement under which agreement is evidence. Sun 24, ring 72, planet 24. An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. With the ring held, a sun input turns the carrier at 1 + 72/24 = 4.000 to one. Willis's equation and the tabular superposition method are independent derivations and both are computed here; the site requires them to agree, because planetary ratios are the most commonly mis-stated numbers in mechanism work and the sign errors are notorious. Epicyclic ratios, two ways An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. The sign errors are notorious, so every ratio here is computed by Willis's equation and by the tabular method, and the two are required to agree. A simple planetary, as a graph. Members are vertices and meshes are edges. Each edge carries the two tooth counts and the body the two axes are stationary in — its carrier — and that third label is the whole of what makes an epicyclic different from an ordinary train. Write the mesh relation relative to the carrier and one formula covers both: an ordinary train is the case where every carrier is the frame. This train has 2 meshes across 5 members and 2 freedoms. A ratio is a null space Write a gear train as a graph — bodies for vertices, meshes for edges, and on every edge the body the two axes are stationary in — and one formula covers a countershaft gearbox, a planetary, a harmonic drive and a car's differential. The ratio is the null space of a matrix whose entries are tooth counts, so it comes out as a fraction and not as a number that is nearly one.

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 11 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.

Degrees of freedomDifferentialEpicyclicMobilityNull spaceShaftTransmission relationTwo-degree of freedomVelocity ratioWillis equation