A V pressed onto its nose has a window

Return and stiffness come from the same aim settled how a stop on a four-bar’s coupler should be aimed. A flat face whose normal misses the coupler’s pole by an angle $\psi$ holds the crank one way at first order. It is stiffer the larger $\psi$ is, at every friction, and it springs back when released exactly when $\tan\psi$ exceeds the friction coefficient $\mu$. A stop that must hold both ways and return therefore needs two faces, a V aimed $\pm\psi$ off the pole, each face holding one way.

Every measurement there had one face loaded and the other just touching. A real V rarely sits that way. It is cut a little tight or a little loose, it warms and grows, or it is pressed onto the nose deliberately so that it holds with no play at all. Then both faces press at once, and the nose sits in a wedge. The essay ended by asking whether that wedge can lock, as a roller clutch’s does, and whether a large aim, which the earlier essay found good in every respect, therefore has an upper limit as well as a lower one.

It has, but only for one of the two ways a V can come to press on both faces, and the reason is a matter of direction.

Two ways for both faces to press #

The first way is a rigid V on the frame cut slightly tight. Each face then starts with an interference $c$, a small overlap with the nose that the contact’s stiffness $k$ turns into a normal force $kc$. Nothing moves to make room, so both faces press for as long as the crank stays near home.

The second way is a V that can make room. It rides on a slide along its own axis, and a spring of force $F$ presses it onto the nose. This is a detent, the commonest way of holding a mechanism at a position without play. The V never loses contact with either face at home, and when the crank is pushed hard enough the V rides back along its slide and lets the nose past.

The two look alike in a drawing and behave very differently. The difference is which way the nose can move relative to the V. The crank moves the nose along its path, which at the V runs square to the line from the nose to the pole. So the crank carries the nose across the V and never into it. A rigid V sees only motion across itself. A sprung V also moves along its own axis, which is the direction into and out of the wedge, and that is the direction in which a wedge can lock.

A tight V comes back only to the edge of a dead band #

Push the crank of a tight V and nothing happens at first. Both faces are pressed with $kc$, and the nose must slide on both to move, so the crank needs a breakaway moment before it turns at all. By virtual work it is

$$M_0 = 2\mu k c\,D\cos\psi,$$

where $D$ is the speed of the coupler’s material point at the contact per radian of crank and $D\cos\psi$ is how fast it slides along each face. For the V in the figure, cut $10^{-5}$ tight through contacts of stiffness $10^6$, that is 1.65, measured on the exact four-bar to a part in a million.

Past the breakaway the near face presses harder and the far face eases off, and the crank stiffens at $2kD^2\sin^2\psi$, 208 thousand per radian here. Friction adds nothing to that stiffness. The near face’s friction grows with its normal force as fast as the far face’s shrinks, and their two contributions cancel exactly. The far face lifts when the crank has turned $c/(D\sin\psi)$, 31.0 millionths of a radian, and beyond that the near face carries the load alone, as in the earlier essay.

Let back, the crank does not go home. The far face pushes it towards home with the force it has regained, and both faces’ friction now opens a gap between that push and what it can actually do. The crank stops where the two balance, a dead band of half-width

$$\delta_r = \frac{\mu c\cos\psi}{D\sin^2\psi},$$

7.96 millionths of a radian for this V. That is $\mu/\tan\psi$ of the lift-off turn, 0.26 of it here. The breakaway, the stiffness, the lift-off and the dead band were all measured on the exact gaps and contacts at six aims, and each matched its formula to a part in a thousand or better.

This is the first thing the earlier essay’s rule did not see. A V aimed outside its friction cone returns cleanly from either side when only one face is loaded. Squeezed, it returns to within a band and stops, and the band is set by the squeeze. The V is still reliable, in the sense that no load moves it further than its own dead band once the load is gone. But its home is a small region rather than a point, and a designer who tightened a V to remove its play has replaced the play with something that behaves very like it.

Loose or tight, a larger aim wanders less #

The fair comparison is with the play the same misfit leaves when it goes the other way. Cut a V loose by $c$ at each face and the nose rattles between them. The crank turns $c/(D\sin\psi)$ either way before a face takes it, $2c/(D\sin\psi)$ from one face to the other. That is the earlier essay’s play law for one face, applied twice. Cut tight by the same amount, the crank can rest anywhere within $\pm\delta_r$, and $\delta_r$ is $\mu/\tan\psi$ of the loose V’s half-play.

So a tight fit is the better of the two at every aim outside the cone, and better by more the larger the aim. At 12.8° and a friction of 0.1 the tight band is 0.44 of the loose half-play. At 47.6° it is 0.09, and at 78.5° 0.02. Both shrink as the aim grows. The loose play falls as $1/\sin\psi$, from 10.2 per unit of misfit at 12.8° to 2.2 at 78.5°. The tight band falls much faster, as $\cos\psi/\sin^2\psi$, from 2.25 to 0.022.

Nothing in the rigid V locks, at any aim. It cannot, because a wedge locks by being driven deeper into itself, and the rigid V’s nose is never driven into it. Its home is fixed by the linkage, and the faces can only press harder or less hard as the crank turns across them. For the rigid V, the earlier essay’s conclusion survives in full: aim as far off the pole as the mechanism allows. The aims that are good for stiffness and return are also good for fit.

The catch is getting the nose in. A rigid V whose faces are turned 80° off its axis is nearly a slot, its walls almost square to the nose’s path. A nose cannot be moved into a slot along the path it then holds. It has to be put there along the V’s own axis, by assembly, or by a V that comes to meet it, and a V that comes to meet the nose and is held against it by a force is a sprung V.

A sprung V can be pushed back, and that is where it locks #

The detent’s V moves along its axis, so carrying the nose past a face means pushing the V back. Two balances settle what that costs. The carrier’s balance along its slide decides the normal force at the pressed face. The spring pushes the V on with $F$. The face’s normal is turned an angle $\beta$ off the axis, and friction at the face, pointing against the sliding, has a component along the axis too. So

$$N = \frac{F}{\cos\beta - \mu\sin\beta}$$

while the crank pushes the V back, and $F/(\cos\beta + \mu\sin\beta)$ while the V comes forward. Virtual work then gives the crank’s moment from the normal force and the friction, as in the earlier essays:

$$M_\text{hold} = F D\,\frac{\sin(\psi + \varphi)}{\cos(\beta + \varphi)}, \qquad M_\text{return} = F D\,\frac{\sin(\psi - \varphi)}{\cos(\beta - \varphi)},$$

with $\varphi = \arctan\mu$ the friction angle. When the V’s axis runs through the pole, $\beta = \psi$ and these are the familiar $FD\tan(\psi \pm \varphi)$ of a block pushed over an inclined face. The moments measured on the exact four-bar, by solving for the carrier’s motion that keeps the pressed face touching, agree with them to a part in a hundred at every aim and friction.

The returning moment is the earlier essay’s result in a new form. It is negative while $\psi < \varphi$, so a detent aimed inside the cone does not bring the crank back: at a friction of 0.2 a V turned 8° off its axis pushes with $-0.04F$, and one turned 12° with $+0.02F$, just enough. The holding moment is the new thing. It grows with the aim, gently at first, $0.56F$ at 20° and $0.81F$ at 30°, then fast. At 60° it is $2.75F$, at 67° $4.50F$, at 73° $9.34F$, and at 79° it is not finite at all.

That is the lock. The carrier’s balance has a denominator, $\cos\beta - \mu\sin\beta$, and it reaches nought when $\beta = 90° - \varphi$. Beyond that no normal force at the face, however large, has enough component along the slide to push the V back against the friction. The V’s face lies inside its own friction cone measured along the axis, and the nose wedges it the way a roller wedges a clutch’s race. A ratchet with no teeth built a one-way clutch on exactly that condition. A detent wants the opposite, and it gets the clutch whether it wants it or not once its faces are turned far enough.

The window has its two edges on different lines #

So a detent’s aim has a window, $\varphi < \psi$ and $\beta < 90° - \varphi$. Bisection on the measured moments finds both edges exactly. At a friction of 0.1 the window runs from 5.71° to 84.29°, at 0.2 from 11.31° to 78.69°, and at 0.3 from 16.70° to 73.30°. It narrows as friction rises and closes at a coefficient of one, where both edges reach 45°.

The two edges are measured from different lines, and the essay’s figures keep them apart for that reason. The lower edge is about return, return is a turn of the coupler about its pole, and the angle that decides it is $\psi$, at the contact, from the line to the pole. The upper edge is about the carrier, the carrier slides along the V’s axis, and the angle that decides it is $\beta$, the face’s turn off that axis. On a point-sized nose the two would coincide. On the nose of radius 0.3 here the contact sits off the axis, and $\psi$ runs ahead of $\beta$ by half a degree at small aims and by 3.6° near the lock. So at a friction of 0.1 the V locks when its faces miss the pole at the contact by 87.86°, not the 84.29° the simple statement would give. A designer who measured both edges from the line to the pole would think the window 3.6° narrower at its top than it is.

Inside the window: the right angle #

Within the window the choice of aim is a choice about how the detent feels. Its holding moment sets how hard it must be pushed. Its returning moment sets how surely it pulls the crank home. Their ratio says how much of the work done pushing the crank out is given back on the way home, and so how nearly reversible the detent is.

For $\beta = \psi$ the ratio is $\tan(\psi - \varphi)/\tan(\psi + \varphi)$, and it is largest at 45°, where it is $\big((1-\mu)/(1+\mu)\big)^2$: 0.669 at a friction of 0.1, 0.444 at 0.2, 0.290 at 0.3. Measured on the four-bar it peaks with the faces turned 44.3° off the axis, at 0.682, 0.461 and 0.307. That is slightly better than the ideal, because the contact’s lead over the axis gives the returning moment’s numerator a few degrees more. It falls to nothing at both ends of the window: at the bottom because the return vanishes, at the top because the hold grows without limit.

The right angle is also the aim least disturbed by friction it was not designed for. Across coefficients from 0.1 to 0.3, a V turned 45° holds with between $1.14F$ and $1.72F$ and returns with between $0.78F$ and $0.53F$. A V at 20° holds with similar steadiness, between $0.45F$ and $0.68F$, but its return falls 3.5 times, to $0.07F$, close to not returning at all. A V at 70° returns steadily and holds with anything from $3.7F$ to $16F$, a factor of four, because it sits close to the lock at the upper coefficient. The window’s centre is the only aim with a wide margin at both edges: at a coefficient of 0.3 it sits 31° above the lower edge and 28° below the upper.

A detent, a clutch and a stop are one contact #

One test, three mechanisms found that a pawl, an escapement’s lock and a coupler’s grip on its rocker all ask which side of a pivot a contact normal passes. A seat with friction stays where it is pushed widened the normal into a cone, which gave the test the answer inside, and found that answer to be sticking. The earlier aim essay read how far outside the cone the pivot lies as how stiff the stop is.

This essay adds the second direction. A contact that is free to move two ways has two cones to be inside, one for each motion. For the coupler’s turn about its pole the cone decides return. For the carrier’s slide along its axis the cone decides whether the contact can be passed at all. A V holds the wheel to first order found a V locking a Geneva wheel to first order. Its V was rigid on the driver and never had to be pushed back, so only the first cone applied. A joint that works one way found a pawl’s holding lines at the edges of its friction cone, which is the second cone’s edge seen from the other side. A detent is the mechanism that needs both, and the window is where it lies outside both at once.

What this does not settle #

The detent’s slide. The carrier is taken to slide without friction. A slide with friction adds a term to the carrier’s balance that resists its motion both ways, raising the holding moment, lowering the returning one, and moving the lock to a smaller aim. Its size depends on the slide’s own geometry and is not computed.

Speed. Every moment here is quasi-static. A crank that arrives at a detent with speed carries the V back on its own inertia and the spring’s, and whether a detent near its lock can be knocked past by a blow that could not push it past is a question about impact, not equilibrium.

Wear and the friction coefficient’s own range. The band from 0.1 to 0.3 stands for clean and dry surfaces against slightly worn ones. A detent that wears its faces changes their aim as well as their friction, and neither drift is modelled.

Redundancy and fit for the sprung V. The seventh contact found that redundant contacts constrain nothing until something deforms. The sprung V has no fit problem at home, since its spring takes up any misfit, which is much of why detents are sprung. But a V carried on a slide whose axis is not where it was meant to be presses its two faces unequally, which shifts home. That shift is not measured here, and nor are the clearances why a hinge works gives every joint.

Joint friction. As in the earlier essays only the V’s contacts have friction. A stiff pin buys grip with release found that a pin’s friction circle blurs where a force may pass, so the four-bar’s own joints blur the pole and every $\psi$ measured from it.

Still open: a rigid V whose spring is the linkage #

The rigid V was found unable to lock because its nose never moves along the V’s axis. That is true of a linkage whose links are rigid. Real links stretch, and a rigid V cut tight, or a V that grows warmer than its mechanism, presses the nose into the linkage’s own compliance. The linkage then plays the part of the detent’s spring, pushing the nose back out along the V’s axis with a stiffness of its own.

The distinct argument would be that compliance included. Give each link of the four-bar a stiffness along its length, so that a force on the nose along the V’s axis moves the nose by a computed amount, and squeeze the nose into a rigid V by a thermal growth. The measurement would follow the squeeze through a heating and cooling cycle and ask whether the nose comes back out when the growth goes away. A V turned past 90° less the friction angle would, by the detent’s balance, keep the nose wedged in after the cooling. The linkage would then be left carrying a locked-in force that no crank moment can release, and the question is whether a rigid V aimed for stiffness can trap its own mechanism in this way. If it can, the upper limit found here for the detent applies to every V, sprung or not, and the earlier essay’s advice to aim as far off the pole as possible has a ceiling it did not know about.