Locked on purpose

Every other essay on this site treats a dead centre as a hazard. It is where a mechanism’s output stops responding to its input, where the transmission of motion collapses, where a sweep loses frames and a designer loses control of the machine.

This one is about a product.

A toggle clamp, the catch that holds a folding table flat, the downlock on an aircraft’s landing gear and the over-centre fastener on a flight case are all the same mechanism: a slider-crank taken past its dead centre and rested against a stop. The singularity is not avoided. It is bought deliberately, and what makes one latch better than another is a property of the geometry a first-order reading cannot see.

What a dead centre is, restated #

The linkages field met dead centres as limit positions and as one of the two things called jamming, and it is worth restating precisely which of them this is.

In a slider-crank, the slider reaches the end of its stroke when the crank and the coupler are in line. At that configuration:

• the slider’s velocity is zero for any crank velocity — the output has stalled;
• the crank’s velocity is unbounded for any slider velocity — the input is infinitely sensitive to the output;
• the mechanism is at a limit of its travel in the slider’s coordinate, and continuing to turn the crank brings the slider back.

The third of those is what a latch uses. Past the dead centre, pushing the slider inward — the direction a load would push it — drives the crank further onto its stop rather than back through the dead point. The mechanism is not constrained from moving. It is constrained from moving in the direction it would have to move first.

The security is a length, and it is small #

How locked is it? The honest question is: how far would the slider have to travel to bring the mechanism back to the dead centre, past which it can open?

For the geometry here — a 40 mm crank, a 150 mm coupler — set four degrees past, the answer is 71 µm.

That is not much. It is a thousandth of the stroke. A latch is not held shut by a large geometric barrier; it is held by a very small one that is very hard to cross.

Why it is hard to cross, in one number #

The reason is the same singularity, read as a ratio. Near the dead centre the crank turns a great deal per unit of slider movement, and away from it, not much.

At the stop, the ratio is about eight times the mid-stroke value. So the 71 µm the slider must travel corresponds to four degrees of crank rotation, and four degrees of crank rotation is a movement of the handle a person can see. A load pushing on the slider has to supply that whole rotation before anything gives.

The design consequence is the reverse: a latch is easy to release deliberately — the handle has plenty of leverage over that 71 µm — and hard to release accidentally, because nothing that pushes on the output can do it without turning the input.

The second order is where the design lives #

Here is the property that makes this essay worth writing.

The slider’s retreat from the dead centre is quadratic in the crank’s rotation past it. Near the maximum of a smooth curve the first derivative is zero, so the leading term is the second, and the depth of lock goes as the square of the over-centre angle:

Doubling the setting quadruples the depth. That is measured — each row is a separate sweep of the mechanism — and asserted: the check requires the ratio of the four-degree depth to the two-degree depth to be 4.00 within a quarter, and it comes out at 4.000.

A first-order analysis of this mechanism sees nothing at all. At the dead centre the derivative of the slider with respect to the crank is zero, so a linearised model says the slider does not move as the crank turns, which is exactly true and exactly useless: it says the latch has infinite security. All of the behaviour is in the second derivative, which is the same statement as the toggle essay’s observation that a toggle and a bad transmission angle are opposite situations 222° apart on the same linkage.

Setting the latch: two costs pulling opposite ways #

The over-centre angle is the one number a designer of a latch chooses, and the two consequences of choosing it pull in opposite directions.

Set it further over and the depth of lock grows as the square, so the latch is harder to shake loose. It also means the handle has to travel further to release it, and — the part that matters in practice — the mechanism has to deflect further to get over the peak when it is being closed, because a real latch is closed by forcing it through the dead centre against a preload.

Set it barely over and the depth is microns. A latch with a one-degree setting has 4.5 µm of lock, which is within the elastic deflection of almost anything, and it will be shaken open.

The compromise real hardware settles on is a few degrees, which is exactly where the quadratic makes the depth tens of microns — comparable to the clearance in the joints and small compared with any deliberate movement. It is a nice example of a design number that is neither arbitrary nor derived from a rule: it is where two curves of the same mechanism cross.

The quadratic decides the setting, not the two costs #

The over-centre angle was presented as a compromise between depth and release effort, and the quadratic settles it more sharply than a compromise, because it also governs how well the setting can be held.

The depth goes as the square of the angle past centre, so a setting error $\delta$ moves the depth by roughly twice the fractional error: a latch set one degree over, assembled with half a degree of variation between units, has a depth varying by a full factor of two either way. The same half degree on a latch set eight degrees over moves the depth by about an eighth. The relative uncertainty in the lock is worst exactly where the lock is shallowest, and the two effects compound rather than trading off.

That is a stronger argument for setting further over than the depth alone gives, and it is the argument a manufacturer would actually make. A latch whose depth is 71 µm nominal and might be 20 µm or 160 µm depending on where the stop landed is not a latch with a small margin; it is a latch with an unknown margin, and the two are different problems. Pushing the setting out until the depth is well clear of the assembly variation converts an unknown into a small known, which is what a tolerance is for.

The limit on pushing it out is the release effort, and that grows only linearly — the handle’s mechanical advantage over the slider degrades as the mechanism moves away from the singularity, roughly in proportion to the angle. So the two quantities scale differently: depth as the square, effort as the first power. A latch set twice as far over is four times as deep and about twice as hard to release, which is a favourable exchange and is the reason real hardware sits at a few degrees rather than at a fraction of one.

There is a ceiling, and it is not in this arithmetic. Far enough past centre the mechanism is no longer near its singularity at all, the quadratic stops describing it, and the latch becomes an ordinary linkage holding by geometry rather than by a stationary point — at which stage the handle has to be actively restrained rather than merely parked. Where that transition sits depends on the crank-to-coupler proportion, and the useful reading of the whole family is that the good settings are the ones where the quadratic still holds and the depth has cleared the clearances, which on this geometry is a band a few degrees wide.

Two ways to be stuck, and only one of them is this #

There is a second mechanism people call a lock and it is a different thing, which the wrong field separated with a number: a toggle and a bad transmission angle are 222° apart on the same linkage, and both get called jamming.

A toggle — this essay’s subject — is a limit of the output’s travel. The mechanism is at the end of what it can do, the output has stalled, and the input has all the authority.

A bad transmission angle is a configuration where the coupler is nearly perpendicular to the direction it needs to push. The mechanism is nowhere near a limit; it moves perfectly well, and it does so with the force in the coupler going mostly into the bearings rather than into the output. Nothing is locked. Something is being wasted.

The reason they are confused is that both feel like a mechanism that has stopped being useful. The reason it matters here is that only the first can be built into a fastening. A latch parked at a poor transmission angle is a latch that still opens under load and loses most of the closing effort to friction — the worst of both.

Telling them apart in a real mechanism is easy once the distinction is stated: at a toggle, two links are in line; at a bad transmission angle, two links are at right angles. The site’s own toggle figure draws both configurations on the same linkage, and the 222° between them is the measure of how far apart two things called by one name can be.

One further consequence of the same square is worth naming because it runs the other way and is easy to miss. The depth of lock is quadratic in the angle past centre, so it is also quadratic in any geometric error that shifts where the dead centre falls — a coupler a fraction long, a pin hole out of position, a crank not quite at its nominal radius. Those move the dead centre itself rather than the setting, and a latch set close to centre is again the sensitive one, because a small shift in where the singularity sits is a large fraction of a small offset. So the case for a generous setting is made three times over by the same exponent: against assembly variation, against dimensional error, and against the clearances the mechanism runs with.

What the site can and cannot say about it #

Everything above is a displacement. Not one sentence of it is a force, and the boundary is worth drawing sharply here because a latch is usually described in force terms.

The usual description is that at the dead centre the mechanical advantage is infinite, so no load on the slider can turn the crank. That is a statically correct statement about an idealised mechanism, and it is not what this site computes. What it computes is the displacement relation: the slider cannot move inward without the crank rotating backwards through a peak. The force statement follows from that by virtual work — if the output cannot move without the input moving a great deal, then a force at the output produces a small torque at the input — but the derivation is the force field’s, and the geometry is the whole of what is here.

The reason this matters rather than being pedantry is that the geometric statement survives friction, compliance and preload, and the force statement does not. A real latch has a spring pushing it over centre and a stop with some give in it; its mechanical advantage is not infinite because nothing in a real machine is. What stays true is that the output has a local maximum and the mechanism is parked past it, and that is a fact about the linkage’s dimensions.

Building the model, and why the sweep is fine and the closed form is not #

The latch is a sliderCrank — the mechanism the linkages field has drawn since the foundation, with no modification whatever. What this essay adds is a sweep that is dense enough near one position to resolve a quadratic.

That sounds trivial and is the whole implementation. The depth of lock is 71 µm on a mechanism whose stroke is 80 mm, so a sweep with a hundred and sixty points over ±40° puts about four points inside the region where the quantity being measured lives. The figures here sweep at that density and then interpolate, and the depth is measured by finding the position nearest the dead centre and the position nearest the stop and subtracting, which is a difference of two solved positions rather than a difference of two samples of a fitted curve.

A closed form exists — the slider’s position is a cosine plus a square root, and the second derivative at the dead centre can be written down — and it was not used, for the reason the scissor lift’s gain was not: an expression plotted is a plot of an expression. A wrong link length, a wrong branch, an assembly that does not close, and the closed form will keep producing a smooth curve about a mechanism the model does not have.

There is a specific version of that risk here. The dead centre is exactly where a Newton solve is worst conditioned: the Jacobian is singular there, and the site’s solver reaches it by damping, taking a small step, and reporting a residual of 10⁻¹³ rather than 10⁻¹⁶. Every figure in this essay is drawn through that region, and the reason it works is the damped least-squares escalation the solver has carried since the foundation — the same machinery that lets a redundant parallelogram be solved at all.

Where else this shape turns up #

The latch is not the only mechanism on this site that lives on a singularity deliberately.

A parallel platform’s direct singularity is the same phenomenon with the roles reversed: there the platform can move with every actuator locked, which is a loss of control and the reason those singularities are mapped and avoided. A latch is the inverse-kinematics singularity — the input can move without the output moving — and that one is safe to sit on.

The pattern is worth naming because the two are constantly confused. A singularity where the output cannot move is a lock. A singularity where the output can move without the input is a runaway. Both are rank deficiencies of a Jacobian; they are deficiencies of different Jacobians, which is precisely the distinction the parallel field spent an essay establishing.

What the ledger says #

The row is filed as point rather than quoted, and the distinction is worth defending.

“Locked” is not a quantity the mechanism lacks. It is a statement that is true at one position — the position the latch is set at — and it is accompanied by a definite number, the 71 µm, that says exactly how true. A specification that said “locked, 4° over centre” would be entirely honest, and a great deal more useful than “locked”, because the second number is what a reader would need to know whether the latch survives being dropped.

The four-bar ladder ends here, four rungs above the four-bar itself. It has gone from a mechanism whose crank turns to a suspension whose crank rocks, to a linkage asked to generate a function it cannot, to a mechanism deliberately parked where the function generation fails completely. What has stayed constant is the method: solve the closure, sweep the input, read the output, and never quote a value without saying which position it came from.