The wheel is the coupler

Lift the bonnet of a car with double wishbones and what is in there is a four-bar. Two arms pivot on the chassis, an upright joins their outer ends, and the wheel is bolted to that upright. The chassis is the frame, the arms are the two cranks, and the upright is the coupler.

That is not an analogy. It is the same object the site’s second field opened with, rotated ninety degrees so that the ground link stands on end, and everything the field established applies unchanged: two branches, a transmission angle, an instantaneous centre, a coupler point tracing a curve of degree six.

What changes is which quantities anybody cares about. Nobody asks a four-bar for its camber. A suspension is asked for nothing else.

Camber is the coupler’s rotation #

Camber is how far the wheel leans from vertical. The wheel is rigid with the upright and the upright is the coupler, so the camber curve is the coupler’s angular displacement, and nothing else.

For the geometry drawn here — a lower arm of 426 mm, an upper of 282, both angled so their lines meet inboard — the wheel goes from +1.54° at eighty millimetres of droop to −2.71° at eighty of bump.

The curvature is the interesting part. “Camber gain” is quoted as a rate — degrees per millimetre, or degrees per degree of body roll — and it is a derivative of a curve that bends. A rate measured at ride height and applied over eighty millimetres of bump under-predicts the camber by about a degree, which is roughly the whole camber budget of a road car’s alignment.

That is the ratio that is not a number in its politest form: not a quantity the mechanism lacks, but a derivative quoted where a function was needed. The field’s ledger files it under point rather than quoted, because the operating point is at least conventional.

Why the arms are different lengths #

The upper arm here is two-thirds the length of the lower, which is deliberate and is the reason the camber curve exists at all.

Make both arms the same length and parallel and the mechanism becomes a parallelogram: the coupler translates, camber never changes relative to the body, and therefore changes exactly as much as the body rolls. A car in a corner leans, and the outer wheel — the one doing the work — leans with it, onto its outer edge.

Shorten the upper arm and the coupler rotates as it rises, leaning the wheel in as it comes up. Get the ratio right and the wheel stays roughly upright relative to the road while the body leans. That is the whole reason a short-long-arm suspension is shaped the way it is, and it is a coupler-rotation argument from beginning to end.

Where the tyre touches, and two answers #

Scrub is how far the contact patch slides sideways across the road as the wheel moves. It matters because that sliding is what makes a car tramline over ridges and what wears a tyre in a straight line.

Here the site found two answers to what ought to be one question, and they disagree by more than the quantity itself.

The textbook answer. Take the point of the upright that is at the contact patch at ride height, and track it. It is a coupler point, its path is a coupler curve, and the site has been drawing those since the foundation.

The disc answer. A wheel is a rigid disc, so the point of it that touches the road is the lowest point of the disc. When the wheel cambers, the lowest point moves across the tread — by roughly the tyre radius times the sine of the camber angle — and that motion is part of where the tyre meets the road.

Neither is a mistake in arithmetic. They are answers to two different questions: where has this piece of the upright gone and where does the wheel touch the road. For tyre wear the second is the relevant one; for the geometry of a steering axis the first sometimes is. What is not defensible is computing one and calling it the other, and the ratio of 1.2 between the disagreement and the quantity says how far that gets anybody.

This is exactly the kind of thing the site’s own habit is supposed to catch, and the habit is the reason it was caught: two routes to one number, and when they disagree the disagreement is the finding rather than a bug to be tidied.

The strut answers differently #

A MacPherson strut replaces the upper arm with a sliding rod. The upright is rigid with a rod that passes through a mount on the body; the rod slides and turns in that mount.

That is a prismatic pair between two moving links — the mount is fixed but the rod’s axis leans as the wheel moves — and until this phase the site could not write one. slide pins a joint to a line given by fixed numbers. The constraint a strut needs is that the fixed top mount lies on the line through two joints of the rod, which is one equation, and rail is that equation.

Its camber curve is a different animal: −0.65° at full bump against the wishbone’s −2.71°, over the same travel. A strut leans the wheel much less as it rises, which is the geometric half of why struts are cheaper and why cars that use them are set up differently. The reason is visible in the mechanism: the upright’s angle is the angle of the line from the lower ball joint to the top mount, and that line hardly turns when the lower ball swings on a long arm.

What is drawn versus what is solved #

Every position in every figure here comes from a Newton solve of the closure equations, and there is a specific reason to be strict about that in a suspension rather than lazy.

The input is the lower arm’s angle and the output everybody wants is wheel travel, and those are not proportional. Sampling the arm angle evenly samples the travel unevenly, densely at one end and sparsely at the other, and a camber curve plotted that way misreports its own curvature. So the sweep is done the other way round: pick the travel, find the arm angle that produces it by bisection, and solve there.

That bisection is where the strut caught the site out. It refuses to assemble past about a fifth of a radian either side of ride height — the rod would have to pass through its own mount — and a bisection started on a wide bracket lands on an unreachable angle at its first step, gets a refusal, and reports the whole travel as unreachable. The wishbone swept fine and the strut returned nothing at all.

The fix is the one the site uses everywhere else and had not used here: continuation. Walk the arm angle outward from ride height in small steps, seeding each solve with the previous solution, and keep the range that assembles. A refusal that survives being seeded is real; one that does not is the solver’s. With seeding, the strut sweeps the full ±80 mm.

The centre the wheel is turning about #

Everything in this essay is the coupler’s motion, and a coupler’s motion at any instant is a rotation about one point. Find that point and the camber rate, the scrub rate and the roll centre all fall out of it, because they are all the same rotation read at different places.

The point is the instantaneous centre, and for a wishbone there are two ways to it.

Kennedy’s construction. The upright shares a pin with the lower arm and another with the upper, so its centre relative to the frame must lie on both arm lines extended. It is where they cross — at ride height here, 1494 mm inboard and 230 mm up.

The velocity field. Solve the mechanism, ask the Jacobian for the velocity of two points of the upright, and find where the perpendiculars to those velocities meet. This route knows nothing about arms; it works for any mechanism whose velocity field can be solved.

They agree to 1.1 × 10⁻¹⁵ over the travel, which is arithmetic noise, and the agreement is asserted rather than assumed. That matters because the second route is the only one available for the strut — a strut’s upright is not carried by two bars, so there is no second line to intersect — and a route used where it cannot be checked has to be checked somewhere it can.

Once the centre is known, the camber rate is one over the distance from it to the wheel, and the scrub rate is the height of the contact patch above it times the same angular rate. A suspension’s whole behaviour at a position is three numbers: where the centre is, and how fast the coupler turns about it.

Sampling, motion ratio and the thing a spring sees #

There is a second quantity that comes free from the same sweep and is worth naming, because it is the one a spring engineer wants: the motion ratio, the wheel’s travel per unit of movement at the spring’s mounting point.

It is the same kind of object as the scissor lift’s rise-per-ram — a ratio of two solved displacements — and it has the same property: it is not a constant. A spring mounted at two-thirds of the way out on the lower arm sees a motion ratio that changes through the travel, because the arm swings and the spring’s line of action swings with it.

This essay does not draw it, and the omission is deliberate rather than an oversight: the interesting version of that curve needs the spring’s mounting geometry, which is a fourth set of coordinates nobody has agreed on. What the essay establishes is the machinery — every such ratio here is a derivative of two solved positions, and every one is a curve.

Where the geometry came from #

The suspension drawn here is not a real car’s. Its numbers were chosen so that the mechanism does what a front suspension does — an instantaneous centre inboard rather than outboard, a roll centre at a plausible height, a camber curve of the right sign and roughly the right size — and then everything else was measured rather than chosen.

That order matters and the first attempt got it wrong. The arms were first given as lengths, with the ball joints placed wherever the lengths reached, and the resulting mechanism had its arm lines converging on the wrong side of the car: an instantaneous centre 3 metres outboard, a roll centre a couple of millimetres below the ground, and a camber curve that gained camber the wrong way in bump.

Nothing was wrong with the solver or the arithmetic. It was a mechanism nobody would build, drawn correctly. The fix was to state the geometry as ball joint positions and let the arm lengths follow, because the thing that decides everything — whether the two arm lines converge inboard or outboard — is visible in the positions and invisible in the lengths.

The numbers this suspension is sold with #

Three of the ledger’s fourteen rows come from this mechanism, and they are of three different kinds — which is why it is the field’s opening machine rather than its most surprising one.

One point decides both #

Camber gain and scrub are treated above as two outputs, and the instantaneous centre says they are two readings of one thing — which is the most useful consequence of taking the suspension as a four-bar rather than as a mechanism with its own vocabulary.

The coupler’s motion at any instant is a rotation about a single point. Every quantity anybody wants from the suspension is a property of that rotation: the wheel leans at a rate set by how far the wheel is from the centre, and the contact patch moves sideways at a rate set by how far the patch is from the centre in the other direction. Camber gain and scrub are the two components of one rotation about one point, and once the point is located both follow without any further solving.

That reframes the arm-length ratio as a single knob with two outputs rather than as a design variable with a preferred value. Making the arms equal and parallel sends the centre to infinity: the coupler translates, the wheel does not lean, and the contact patch is carried sideways by the full arc of the arms — zero camber gain and the worst scrub available. Shortening the upper arm brings the centre in from infinity, and every millimetre it moves trades some scrub for some camber gain. There is no setting that improves both, because there is only one point and the two quantities are its two coordinates seen from the wheel.

So a double wishbone’s fundamental compromise is not a matter of taste or of tyre data; it is that a one-degree-of-freedom mechanism has one instant centre and two things are wanted of it. Every published argument about suspension geometry is an argument about where to put that point, and the disagreements are about which of the two coordinates matters more on a given car.

It also says what a designer gains by adding parts. A mechanism with more links can put the instant centre somewhere a four-bar cannot, or move it along a different path through the travel, which is precisely what a five-link or a multi-link rear suspension is for. The extra links do not buy new quantities; they buy control over where one point goes and how it moves — which is what the extra dimensional parameters of a longer chain buy, stated in the vocabulary of one mechanism.

What this leaves out #

Everything about the tyre. A real contact patch is not a point; it is a patch, with a pressure distribution that moves under load and slip, and the “scrub” a tyre actually does is a distributed slip rather than a rigid point sliding. The 12.2 mm here is the geometric displacement of an idealised contact point, which is the right input to a tyre model and is not a tyre model.

Also everything about compliance. Every bush in a suspension deflects under load, and the resulting motion — compliance steer, compliance camber — is comparable in size to the geometric motion measured here. A car is set up on both together. The boundary drawn in what-is-still-outside is exactly where it always was: this site computes where a mechanism can be, and what a loaded rubber bush does about it is a different subject.

And nothing here is about steering. The upright’s third job is to carry a steering axis, and turning the wheel about that axis while the suspension moves produces bump steer — a coupling between two mechanisms that the steering essay names and does not compute, because it needs a spatial model and every mechanism in this essay is planar.

What the essay does establish is the frame the next three are built in: a suspension is a four-bar, camber is its coupler’s rotation, and the interesting quantities are all derivatives of a solved sweep — which means every one of them is a curve, and every number quoted for one is a point on that curve chosen by somebody.

One qualification on treating the instant centre as the design object, and it is the same one that makes the roll centre a quoted number rather than an exact one. The centre is an instantaneous property, so it is a different point at every position of the travel, and a suspension is not designed at one position — it is designed over a range. So the real object is the path the centre traces through the travel, and the two quantities a designer reads are the path’s position and how fast it moves. A geometry with a well-placed centre at ride height and a centre that runs away at full bump is a geometry that behaves well in the showroom and badly on a road, and nothing measured at a single position would say so.