One wheel on ice

Everybody who has driven in winter knows the picture: one wheel spinning uselessly on ice, the other standing still on tarmac, the car going nowhere. It is the standard demonstration that an open differential is a poor device, and it is usually described as the differential sending all the drive to the wheel with no grip, as though the mechanism were making a choice and choosing badly.

Half of that is outside this site and half of it is a kinematic statement that is worth getting exactly right.

What the mechanism is doing on the ice #

The relation is

$$\omega_{\text{left}} + \omega_{\text{right}} = 2\,\omega_{\text{cage}}$$

for equal side gears. Now put the left wheel on tarmac, where it does not turn, and ask the null space what the mechanism permits. It returns $\omega_{\text{right}} = 2\,\omega_{\text{cage}}$, exactly — the fraction $2/1$, with nothing rounded.

So the spinning wheel turns at precisely twice the propshaft’s crown wheel, and the differential’s condition is satisfied the entire time. Nothing has broken. The mechanism is not confused, has not chosen anything, and is doing exactly what it does at every other moment of its life: fixing the mean of two speeds and leaving their difference free.

That is the whole of the kinematics. The question why does the wheel with grip not turn is a question about what is pushing, and it has an answer — an open differential applies the same torque to both side gears, so the torque available at either wheel is capped by whichever has less grip — and that answer is a statement about forces and is not in this library. The site has kept that boundary since its foundation and it is worth keeping here rather than gesturing across it, because the gesture is where the misattribution comes from. The mechanism is blamed for something the mechanism does not decide.

What is worth saying precisely, because it is the useful half:

• The kinematic relation permits the one-wheel-spinning motion. It also permits both wheels turning together, and every split in between. A mechanism that permits many motions is not thereby responsible for which one occurs.
• The relation is necessary rather than optional. It is not that the differential allows the wheels to differ as a convenience; it is that a car cannot turn a corner without them differing, which is the next section.
• Removing the freedom is a kinematic change and can be discussed here in full. That is what a differential lock is, and it is the third section.

The mechanism exists because the constraint is otherwise unsatisfiable #

Take the axle out of the car and look at the two wheels rolling on the ground. Rolling without sliding is a constraint on velocities: a wheel of radius $r$ rolling on a path travels $r\omega$ per unit time along that path.

Now put the car on a circle of radius $R$ measured to the middle of the axle, with the wheels a track $t$ apart. The inner wheel’s path has radius $R - t/2$ and the outer’s $R + t/2$, and both wheels go round the circle in the same time, so their speeds must be in the ratio

$$\frac{\omega_{\text{outer}}}{\omega_{\text{inner}}} \;=\; \frac{R + t/2}{R - t/2}.$$

At $R = 12$ m with a track of 1.55 m that is 1.138 — the outer wheel must turn 13.8% faster or one of them is sliding. Nothing about grip has been mentioned. This is a statement about arc lengths, and it would be equally true of two wheels rolling on a sheet of paper.

A solid axle imposes a second condition, $\omega_{\text{left}} = \omega_{\text{right}}$, and the two conditions have no common solution except $R = \infty$. Two equations, one unknown ratio, no answer. That is the whole reason the differential exists, and it is a mobility argument of exactly the kind the constraint field is built on: the mechanism has to be given a freedom, or a constraint has to be broken by force.

The differential gives it the freedom in the cleanest possible way. Its condition is on the mean of the two wheel speeds; the turn’s condition is on their difference; and two conditions on two independent quantities never collide. Whatever radius the car is on, the difference the turn wants is available and the mean the drive wants is enforced, and neither knows about the other.

The locked axle, measured #

Lock the differential and the mechanism changes in a way that is entirely within this site’s competence. A diff lock is a clutch — the same object as the shift elements of the enumeration, one linear condition $\omega_{\text{left}} = \omega_{\text{right}}$ — and it takes the axle’s mobility from two to one.

The axle is then a mechanism with one freedom being asked to satisfy two rolling constraints, which is one too many. It is overconstrained, in exactly the sense this site uses the word: the constraints are not independent of each other in the way the count assumes, and something has to give. In a spatial linkage the surplus constraint is either absorbed by a geometric coincidence or it stops the mechanism moving. Here it is absorbed by a tyre sliding.

How much sliding? Per unit of cage rotation the two wheels want to travel different distances, and the difference over one complete circle of the vehicle is the difference in the two path circumferences:

$$2\pi\left(R + \tfrac{t}{2}\right) - 2\pi\left(R - \tfrac{t}{2}\right) = 2\pi t.$$

The radius cancels. A locked axle scrubs $2\pi t$ per full circle driven, whatever the radius — 9.74 m for a track of 1.55 m, on a car park manoeuvre and on a motorway sweeper alike. What a tight turn changes is not the amount of sliding but how little distance it is spread over: at $R = 8$ m the 9.74 m is spread over 50 m of travel, and at $R = 40$ m over 251 m.

That is a nicer result than the one the question seemed to promise, and it comes out of the geometry with no work at all once the question is asked the right way round. It also matches the experience it describes: a locked axle is unbearable in a car park and merely unpleasant on a long curve, which is a statement about scrub per metre and not about scrub.

What “twice the cage” is a statement about #

The doubling is the one number everybody knows about a differential, and it is worth being clear about what kind of number it is, because the popular version is a statement about torque and the true version is not.

Take the relation $\omega_L + \omega_R = 2\omega_C$ and set $\omega_L = 0$. Then $\omega_R = 2\omega_C$. That is an identity of the constraint, not a prediction about what a car will do: it says if the left wheel is stationary and the cage is turning, then the right wheel is turning at twice the cage. Its content is entirely conditional, and it is true whether the left wheel is stationary because it is on tarmac, because it is in a vice, or because somebody is holding it.

The null space returns it as the exact fraction $2/1$, and the check that guards it does two things rather than one. It asserts the doubling; and it asserts that a differential with unequal side gears — 16 teeth against 20 — does not return 2. That second half is what makes the first a measurement rather than a slogan: a check that reported 2 for every differential would report 2 for a mechanism that was not a differential at all.

For unequal side gears the relation is $z_L\omega_L + z_R\omega_R = (z_L + z_R)\omega_C$, so the cage turns at a weighted mean and a held wheel makes the other turn at $(z_L+z_R)/z_R$ times the cage. Nobody builds one, because there is no reason to want the wheels of a car treated unequally — but the machinery does not assume the symmetric case, and the check would notice if it had been assumed somewhere.

One level up: the centre differential #

The same argument applies between axles rather than between wheels, and it is why part-time four-wheel drive is unusable on tarmac.

A car turning about a centre on the rear axle’s line has its rear axle travelling on a circle of radius $R$ and its front axle on a circle of radius $\sqrt{R^2 + L^2}$, where $L$ is the wheelbase — the front axle is a wheelbase further along the arc, so it is further out. At $R = 12$ m with a 2.7 m wheelbase those are 12.00 and 12.30, a ratio of 1.025.

Two and a half per cent does not sound like much. Per full circle it is $2\pi(12.30 - 12.00) = 1.88$ m of path difference between the two axles, which at a 0.32 m wheel radius is 0.94 of a wheel revolution. At $R = 8$ m it is 1.39 revolutions.

With a centre differential that difference is simply permitted. Without one, the front and rear driveshafts are locked together and the whole driveline has to absorb almost one wheel-turn of wind-up in a single circle — through tyre scrub if there is any to be had, and through twist in the shafts if there is not. This is why a part-time system is engaged only on loose surfaces, where the tyres will slip a little continuously and unwind it, and why doing a full lock-to-lock turn on dry tarmac in four-wheel drive makes the transmission bind.

What a limited-slip differential is, and is not #

Two more devices, and the boundary runs between them.

A locking differential is kinematic, and it is what has just been analysed: a clutch that removes a freedom, with everything about its consequences computable here.

A limited-slip differential is not. Whether it is a clutch pack, a viscous coupling or a gear-based torque-biasing unit, its mechanism is the open differential and its added behaviour is a relation between torques — it resists a speed difference rather than forbidding one. Its state space is the same two-dimensional space; the null space is unchanged; nothing in this library can tell one from an open differential, because kinematically there is nothing to tell.

That is not a limitation being apologised for. It is the useful content of the distinction: a device that changes what motions are possible and a device that changes which possible motion happens are different kinds of thing, and the second kind cannot be drawn.

Three wheels and a mechanism that is genuinely overconstrained #

The locked axle is the simplest case of a shape this site has spent a whole field on, and it is worth putting the two side by side.

In the spatial field, an overconstrained mechanism is one whose constraint count says it cannot move and which moves anyway, because the constraints are not independent: a Sarrus linkage, a Bennett loop, a universal joint. The count is wrong and the rank is right, and what saves the mechanism is a geometric coincidence built into it on purpose.

A locked axle in a turn is the same arithmetic with the opposite outcome. One freedom, two rolling constraints, and no coincidence available — the two constraints are genuinely independent unless the turn radius is infinite. So the count is right, the rank is right, and the mechanism does not move without something giving way.

What gives way is a tyre, and this is where the analogy stops being exact and starts being informative. In a rigid-body mechanism a surplus constraint either stops the motion or is absorbed as strain. Here it is absorbed as sliding at a contact, which is a third possibility the rigid-body accounting does not have a place for, and which exists only because a rolling constraint is a constraint on velocities that a surface is free to violate at a price.

That is the same structure as the ratchet’s pawl, one field back: a contact is a constraint that holds in one direction and gives way in another, and mechanisms built on contacts do things that mechanisms built on joints cannot.

Per metre rather than per circle #

The sliding coming out at $2\pi$ times the track per circle, with the radius cancelling, is a satisfying result and it is quoted in the wrong units for comparing against anything else. Divided by the distance actually travelled it becomes a rate, and the rate is where the locked axle takes its place in the field’s ledger.

One circle of radius $R$ is $2\pi R$ of travel and costs $2\pi t$ of sliding, so the sliding per metre driven is simply $t/R$ — the track over the turn radius, with the $2\pi$ gone. On the field’s own car, 1.55 m of track at a 12 m radius, that is 12.9 per cent of the distance travelled, shared between the two wheels.

Set that beside the other scrub figures this field has measured at the same radius and the ordering is stark. Two front wheels steered wrongly, both held parallel, scrub 1.71 per cent. A three-axle rig steered as well as it can be scrubs 5.99 per cent at its worst wheel. A locked axle scrubs 12.9 per cent between its two, which is several times either — and it does so on a mechanism with no steering error in it at all and no surplus axle.

The reason for the gap is worth stating because it is structural rather than a matter of degree. A steering error is a misalignment: the axes nearly meet, and the residual is the small amount by which they miss. A locked axle is a contradiction: the two rolling conditions demand different speeds from a shaft that has one, and the disagreement is the full difference between the two wheel paths rather than a residual of anything.

The rate form also says where a locked axle is tolerable and where it is not, which the per-circle form conceals. $t/R$ falls as the turn opens out, so a locked axle on a motorway is doing almost nothing — at a 500 m radius the rate is three parts in a thousand — and on a farm track at full lock it is catastrophic. That is exactly the usage pattern differential locks are built for: engaged at low speed and tight radii on loose ground where the tyre can slide cheaply, and required to be disengaged on the road.

And it puts a number on the standard warning. Driving a locked axle round a car park at a 6 m radius costs $1.55/6$, a quarter of the distance travelled, in sliding — which is why a locked differential on tarmac produces the noise and the wear it does, and why the instruction is always to unlock before the surface gets grippy rather than before the speed rises.

Where this sits beside the steering #

The applied field already owns a piece of the same turn: the steering trapezoid, whose job is to give the two front wheels different steer angles so that their axes meet at the turn centre. This essay is about the two driven wheels having different speeds for the same reason.

They are the same geometry seen twice, and it is worth naming the symmetry. In a turn the four wheels are on four different circles about one centre. The steering linkage is the mechanism that arranges the directions; the differential is the mechanism that arranges the rates. Both exist because a rigid vehicle turning about a point imposes conditions that a naive mechanism — parallel steering arms, a solid axle — cannot satisfy, and both are approximations of exactly the kind this site keeps measuring: an Ackermann trapezoid gets the angles right at one radius and wrong elsewhere, while a differential gets the rates right at every radius, exactly, with no error term at all.

That difference is worth the last paragraph. The steering linkage has to approximate because it is a four-bar being asked to realise a trigonometric relation, and a four-bar’s coupler curve is a sextic that can only be made to pass through so many prescribed points. The differential does not have to approximate, because the condition it enforces — a mean — is linear, and linear conditions are exactly what a gear train’s mesh relations are. A mechanism whose specification happens to lie in the space its constraints span does the job exactly. That is rarer than it sounds, and it is why the differential is one of the few mechanisms this site has met that is doing precisely what it claims with no residual anywhere.