Holding an axle still

A live axle has to be held sideways. It is a beam across the car with a wheel at each end, it moves up and down, and something has to stop it moving across while it does.

The two answers are a hundred and fifty years old and one of them is a linkage this site opened a whole field with.

The rod, and why its error is one-signed #

A Panhard rod is one link: body at one end, axle at the other. The axle end has to stay at a fixed distance from the body end, so as the axle rises the rod swings, and the axle is pulled towards the body’s pivot.

The arithmetic is a right-angled triangle. For a rod of length $P$ with the axle displaced vertically by $y$, the lateral movement is $x = \sqrt{P^2 - y^2} - P \approx -y^2/2P$, which is quadratic in the travel and — this is the part that matters — has the same sign whichever way the axle moves. Bump and droop both pull the axle towards the pivot.

So on a rough road the car is not shaken from side to side; it is shifted, twice per bounce, in one direction. That is a qualitatively different nuisance from a symmetric error and it is a consequence of the geometry rather than of the numbers: any single link produces a one-signed lateral error, because a circle is convex.

For a 900 mm rod and ±80 mm of travel, the number is 3.56 mm.

The straight-line linkage, turned inside out #

The other answer is Watt’s linkage, which the curves field met as one of the eighteenth century’s attempts at drawing a straight line without a straight edge. Two links pinned to the frame, a coupler between their free ends, and a point on the coupler that traces a path with a long straight stretch.

Under a car it is used inverted. The central link’s middle is bolted to the axle and the two side links run to the body — so the point that traced the straight line is now the part that is held still, and the frame the linkage was pinned to is now the thing that moves relative to it.

That is a kinematic inversion in the exact sense the linkages field defined: the same chain with a different link held fixed. The straight line is the same curve read from the other end.

The order, measured rather than asserted #

Saying the Watt is better is easy. Saying how it is better needs the order of each error, and the order is the useful part.

An error that goes as the square of the travel and one that goes as the fifth power are not the same kind of claim. Halve the travel and the first falls by four; the second falls by thirty-two. So the comparison between the two linkages is not a number — it depends on how much the suspension moves.

The Panhard rod’s fitted order is 2.001, which is the exact result the triangle gives and a good check that the fitting is honest.

The Watt linkage’s is 5.011, and that is a surprise worth pausing on. A straight-line linkage’s tracing point is usually described as having an inflection — a vanishing second derivative — which would make the error cubic. Fifth order means the third-derivative term vanishes too, and it does so because of the symmetry of this particular design: equal side links, the tracer at the middle, both links perpendicular to the central one at ride height.

Detune that symmetry and the order drops. With one side link twenty per cent longer than the other, and the tracer moved to the position the classical rule gives, the fitted order falls to about 2.7 — the error is smaller at some travels and grows faster. The fifth order is a property of the symmetric design, not of Watt linkages, and the site found the classical tracer rule by scanning rather than by looking it up: sweep the tracer’s position along the central link and the error collapses at l₂/(l₁+l₂), measured at 0.602 against a predicted 0.600.

How the tracer rule was found #

The classical rule for a Watt linkage with unequal side links is that the tracing point divides the central link in the inverse ratio — nearer the end whose link is shorter. This site did not look it up; it scanned for it, which is worth describing because the scan is three lines and the result is unambiguous.

Fix the geometry, put the tracer at a fraction u along the central link, and measure the lateral error at full travel. Sweep u from 0.3 to 0.7. The error falls, collapses, and rises again — 8.5 mm at u = 0.30, 1.4 mm at 0.50, and 0.7 µm at the bottom of the trough, which the scan puts at u = 0.602.

The prediction is l₂/(l₁+l₂) = 0.600. The scan’s resolution is 0.001 in u, and the difference between 0.602 and 0.600 is within the width of the trough at that resolution.

Two things about that are worth keeping. The first is that a scan over one parameter is often cheaper than finding the rule, and it produces a number rather than a citation. The second is that a scan gives the shape as well as the position: this trough is narrow — moving the tracer one per cent of the link’s length from the optimum multiplies the error by twenty — which is a manufacturing tolerance on the pin’s position, and no textbook statement of the rule mentions it.

Getting the configuration wrong, and how it showed #

The Watt linkage in this essay was built wrongly first, and the way it announced itself is worth recording because it is the argument for measuring an order at all.

The first version had the central link horizontal with the side links running diagonally to the body — a perfectly good five-bar, assembling and moving smoothly at every position of the travel. Its lateral error was 39 mm at 80 mm of travel, eleven times worse than the Panhard rod it is supposed to beat, and its fitted order came out at 0.88.

Nothing in the solve complained. The mechanism closed to 10⁻¹⁴ at every position; the figure drew; the path was a smooth curve. What said something was wrong was the order: 0.88 is not a power law any linkage produces, and an error growing more slowly than linearly is the signature of a quantity that is not what it is supposed to be.

The correct configuration has the central link standing across two parallel side links, and it gives 34 µm and order 5. The lesson is the site’s standing one: a smooth wrong answer is the dangerous kind, and the thing that catches it is a quantity with a known shape — an order, a limit, a symmetry — rather than a plausible number.

Why not use an exact one #

Peaucellier’s cell draws an exact straight line, measured on this site at 9.8 × 10⁻¹⁶ of its span — fourteen orders of magnitude better than Watt’s. It has never been used to locate an axle.

The reason is not that nobody thought of it. It is that the cell has eight links and six pins to Watt’s four and four, every one of those pins is a bearing that wears and rattles, and the exactness is destroyed by clearance long before the geometry runs out. The practice field measured exactly this trade: an ideal mechanism’s exactness is worth nothing if its clearances contribute more error than the geometry does.

Thirty-four microns of geometric error, on an axle whose bushes deflect by millimetres under load, is already far past the point where more exactness buys anything. That is the honest engineering reading of the ledger’s bounded verdict: the claim “it locates the axle” is false, the falsity is 34 µm, and no application cares.

Two mechanisms, one question, five verdicts apart #

The two rows this essay produces sit at opposite ends of the ledger’s vocabulary, and they make the same claim.

The Panhard rod’s “it locates the axle” is filed quoted: the mechanism has no such property, and the arc is not a small deviation from the claim but the mechanism’s whole behaviour.

The Watt linkage’s identical sentence is filed bounded: false, by 34 microns, with an order that says how that scales.

The distinction is the field’s most practically useful one. Both claims are false as stated. One is false in a way that means the mechanism is doing something else entirely, and the other is false in a way that means the mechanism is doing its job and the sentence was written by somebody in a hurry.

Telling them apart takes exactly one number — and the number is not the error but the order, because an error is a measurement at one travel and an order is a claim about all of them.

What the inversion changes and what it does not #

It is worth being precise about the inversion, because “the same linkage upside down” can sound like hand-waving and is not.

A kinematic chain is a set of links and joints. A mechanism is a chain with one link declared to be the frame. Watt’s engine linkage and a car’s axle-locating linkage are the same chain with different links declared — in the engine, the link carrying the two fixed pivots is the frame; under the car, that link is the body and it is the axle whose motion is described.

What is preserved by an inversion is every relative motion in the chain: the same joint angles occur in the same order, the same coupler curve is traced relative to the same link. What changes is which motion anybody calls the output.

So the straightness measured here is exactly the straightness the curves field measured, and it had to be — the two figures compute it from different code and the agreement is a check on both. What the inversion changes is the question: there, how straight is the line this point draws; here, how still is this point held while everything else moves.

That is the fourth time this site has used an inversion to turn one mechanism into another — the slider-crank into the quick-return, the four-bar into its four mechanisms, the straight-line linkage into an axle locator — and each time the useful thing was that no new analysis was needed.

What the two orders do over a range of travel #

An order is a claim about how a number changes, so the pair of orders can be cashed out into a statement about vehicles rather than about mechanisms — and the statement runs opposite to the obvious one.

The Panhard’s error goes as the square of the travel and the Watt’s as the fifth power, so the ratio between them goes as the cube. Halve the travel and the Panhard improves fourfold while the Watt improves thirty-twofold, and the Watt’s relative advantage grows by a factor of eight. Double the travel and the advantage shrinks by eight.

Put the measured numbers in and the range comes out. At ±80 mm the two errors are 34 µm and 3.56 mm, a ratio of about one to a hundred and five. At ±160 mm the Watt is at roughly 1.1 mm and the Panhard at 14.2 mm, a ratio of one to thirteen. Follow the cube law to where the ratio reaches one and it arrives at about ±380 mm of travel — four or five times what a road car has, and beyond what any wheel-located axle does.

So the Watt linkage wins across every travel that exists, and it wins by wildly different margins at the two ends of the range. That is a more useful statement than the Watt is better, and it explains a pattern in what gets built that the single comparison at 80 mm does not.

A short-travel vehicle is where the geometry pays most. A stiffly sprung saloon with 50 mm of travel gets a Watt error under 5 µm against a Panhard’s 1.4 mm — a factor of three hundred, and comfortably below every other error in the axle. The linkage is worth its four extra joints there because the thing it buys is enormous.

A long-travel vehicle is where it pays least. An off-road axle with 200 mm of travel is in the region where the cube law has eroded most of the advantage, and it is simultaneously the vehicle with the least room for a Watt linkage’s central bellcrank and the most exposure to what a mechanism collects underneath it. Both arguments point the same way, and a Panhard rod on a vehicle like that is not a compromise made in ignorance of the geometry — it is the case where the geometry is worth the least.

That is the transferable half of this essay applied to itself. The 34 µm is a fact about one linkage at one travel and would have to be re-measured for any other; the pair of orders is a fact about the two mechanisms, and it answers the question at every travel at once, including the ones nobody has built.

What is left out #

The rod’s compliance, first of all. A Panhard rod is a steel bar with a rubber bush at each end, and under a cornering load the bushes deflect by an amount comparable to the 3.56 mm the geometry gives. A real car’s axle moves sideways for two reasons and this essay computes one of them.

The roll centre, second. A live axle’s roll centre is where the axle-locating linkage says it is — the Panhard rod’s centre is on the rod, and the Watt linkage’s is at its central pivot — and it moves with the travel exactly as the independent suspension’s does. It is not computed here because the construction for a beam axle is a different one and would need its own model, but the same objection applies to quoting it as a height.

And the third dimension: everything here is a front view, and a real axle also has to be located fore and aft by trailing arms or a torque tube, whose arcs interact with the lateral linkage’s. That is a spatial problem, and the site has the machinery for it in the spatial field — a Watt linkage and a pair of trailing arms is a closed spatial loop whose mobility is exactly the sort of thing Kutzbach’s count gets wrong. It is named here and not built.

What the essay establishes #

Three statements, each of which is a measurement rather than a preference.

A single link cannot hold anything still. Its error is quadratic, one-signed and set entirely by its length: a longer rod is proportionally better and there is a limit to how long a rod fits across a car. Nothing about the pivot positions changes the shape of the answer.

A four-bar can, to a stated order. Not exactly — the curves field settled that in the eighteenth century’s terms — but to fifth order in the travel for the symmetric design, which over a suspension’s stroke is thirty-four microns.

The order is the transferable part. An error at one travel is a fact about a car; an order is a fact about a mechanism, and it is what lets somebody with a different suspension work out their own number. Measuring it costs seven solves and a straight-line fit, and it is the measurement that caught this essay’s own wrong configuration when nothing else did.

One qualification on the cube law, since it is being pushed a long way outside where it was measured. An order fitted over ±80 mm describes the leading term of an expansion, and an expansion’s leading term is a good description only while the terms after it are small. At ±380 mm a Watt linkage is nowhere near its designed operating region — the side links have swung far enough that the symmetry the fifth order depends on is long gone — so the crossing point computed above is not a prediction about a mechanism anybody could build. It is the statement that the crossing lies far outside the range where either mechanism is used, which is all the argument needs and is as much as the fit can support.