Return and stiffness come from the same aim

A seat with friction stays where it is pushed put friction into the contact between a four-bar’s coupler and a stop on its frame, and found two things that seemed to pull against each other. Friction makes a stop stiffer, because the force at a pressed contact may lean off the normal and so gains a lever about the coupler’s pole. The gain is largest for a stop aimed near its pole, seventeen times at a coefficient of 0.3 for a stop whose normal misses the pole by just over a degree. And friction makes a stop sticky: released, a stop whose pole lies inside its friction cone stays where the load left it, while one aimed outside the cone springs back.

That essay ended with what looked like the natural design question. Aim a stop just outside the cone and it returns and still gains $1 + \mu/\tan\psi$ in stiffness. Since real coefficients wander by a factor of two between clean and dirty surfaces, how much stiffness does a guaranteed return cost?

The answer is that it costs none, and the question was a product of how the gain was counted. What return costs is something else, and it is the other side of the stop.

A large factor on a vanishing number #

Everything about a stop on a mechanism with one freedom comes down to two rates at the contact. As the crank turns, the coupler turns about its pole, so the coupler’s material point at the contact moves square to the line from the pole, at some rate $D$ per radian of crank. If the stop’s normal misses that line by an angle $\psi$, the point presses into the face at the rate $|g'| = D\sin\psi$ and slides along it at $|s'| = D\cos\psi$. The earlier essay measured both, and their ratio is $\tan\psi$ to a thousandth on seven stops.

A stop of contact stiffness $k$ pushed through a small turn $\delta$ presses in by $|g'|\delta$ and carries a normal force $k|g'|\delta$. By virtual work its moment on the crank is that force times $|g'| + \mu|s'|$, the normal lever plus the friction lever. So its moment per radian is

$$K = k\,|g'|\,\big(|g'| + \mu|s'|\big) = k D^2 \sin\psi\,(\sin\psi + \mu\cos\psi).$$

Friction’s gain is the second bracket over $\sin\psi$, which is $1 + \mu/\tan\psi$ and grows without limit as the stop is aimed at its pole. That is the earlier essay’s seventeen. But the whole expression has a $\sin\psi$ in front, and that vanishes at the pole. Friction multiplies most exactly where there is least to multiply. Written as $\tfrac12(1-\cos2\psi) + \tfrac{\mu}{2}\sin2\psi$, the bracket rises from nought at the pole to its largest value at $\psi = 90° - \tfrac12\arctan\mu$. That is 87.1° at a coefficient of 0.1 and 81.7° at 0.3, a face very nearly square to the path of the coupler’s point, tilted towards the pole by half the friction angle. Everywhere between the pole and that face the stop gets stiffer as it is aimed further off, at every coefficient. A designer reading the formula without the earlier essay’s framing would aim a stop as far from its pole as the mechanism allows, and would never think of the pole as the stiff direction at all.

Measured by deflecting each stop under a moment of $10^{-6}$ and dividing, the stiffness follows the closed form to a hundredth at every aim and friction. A stop missing the pole by 1.07° gives 271 per radian with no friction and 4,631 at 0.3, which is the seventeen-fold gain. A stop missing by 21.3° gives 104,000 with no friction and 184,000 at 0.3, a gain of only 1.77, and it is forty times stiffer than the near-pole stop at the same friction. At 42.4° the figures are 367,000 and 488,000. The order never reverses: at every coefficient tried, each stop in the sweep is stiffer than every stop aimed nearer its pole.

The seat at the pole sits below all of them, and in a different way. It has no first-order stiffness, since its lever $|g'|$ is itself grown by the turn. A stop aimed at its pole holds by curvature found it giving way as the cube root of the load without friction, and the earlier friction essay found the square root with it. Its secant stiffness under a moment of $10^{-3}$ is 5.9 per radian at a coefficient of 0.1 and 10.2 at 0.3, small even beside the flat stop that misses its pole by a degree.

The earlier essay’s gain was a true number about a true comparison: friction against no friction, for one stop. It was the wrong comparison for choosing a stop’s aim, which compares stops with each other at the friction they will actually have.

Return is decided by the same angle #

The other property was settled by the earlier essay and is only re-measured here, across a wider sweep of aims, because it is now one half of a single statement rather than the other end of a trade.

Released, the contact must slide back against friction, and the moment that drives it back is $N(|g'| - \mu|s'|)$, which is positive exactly when $\tan\psi > \mu$. So a stop either returns the whole way or keeps the whole turn. At a coefficient of 0.1, every stop missing the pole by 4.3° or less keeps all of its turn and every stop missing by 6.4° or more keeps none of it; the cone’s half-angle is 5.7°. At 0.2 the switch is between 8.5° and 11.7°, around a cone of 11.3°. At 0.3 it is between 16.0° and 21.3°, around 16.7°. No stop in the sweep keeps part of its turn, and none needs to be tested for it: the sign of one bracket decides.

Put next to the stiffness, the two say one thing. The angle that sends a stop back when released is the angle that makes it stiff while loaded, and more of it does more of both. A designer who must have return across a friction band takes the aim just past the band’s upper cone, and finds the stop stiffer at the band’s lower coefficient than any stop that does not return. There is no stiffness left on the table, because there was never any on the near side.

What the aim does cost: the other side #

Why aim a stop near its pole at all, then? The two earlier essays chose the pole for a reason, and the reason survives. A stop aimed off its pole holds one way. Its gap closes when the crank turns towards it and opens when the crank turns away, and it is the pole, where the gap neither closes nor opens at first order, that allows a single curved face to close it both ways at second order.

The cost is measured by turning the crank the way an aimed seat does not hold and scanning the exact gap for where it closes again. A flat face never closes: once its gap opens it opens for good, at every aim. A curved seat does close, after the curvature has caught up with the first-order opening. The gap’s quadratic puts that at $2\tan\psi\,|s'|/|g''|$, and the measurement agrees at the smallest aims, 0.044 radians of free turn at 1.1° against 0.042 estimated. At 2.1° it is 0.092 and at 4.3° 0.214. From 6.4° the gap’s higher terms take over and the seat has not closed within 1.2 radians, nearly seventy degrees of crank. By then it is no longer a seat in any useful sense on that side.

That is the price a single face pays for return. With friction anywhere near the ordinary range, the aim that guarantees return is already past the point where the seat’s curvature can catch the other side. The seat at the pole is the only aim at which one face holds both ways, and it holds both ways only by curvature and only with the stickiness the earlier essay found.

Two faces for two sides #

A stop that must hold both ways and return both ways therefore needs two faces, one for each side, each aimed off the pole by more than the friction angle. Put both on the frame, each touching the coupler’s round nose, and they form a V. One face’s normal is turned off the line to the pole one way, the other’s the other way. Turned towards either face the crank presses that face at first order and lifts off the other.

A V holds the wheel to first order found the same shape doing the same job for a Geneva wheel’s lock. There the flat chord, whose normal passes through the wheel’s centre, held only to second order, and tilting two faces a few degrees off it gave a first-order hold with a small disc. The pole plays the part the wheel’s centre played. A face whose normal passes through it holds at second order or not at all, and two faces tilted each way off it hold at first order.

The V drawn here misses the pole by 12.3° each way, a degree outside the cone for a coefficient of 0.2. Pushed at friction 0.1, it turns in proportion to the load both ways, $1.9\times10^{-8}$ radians under a moment of $10^{-3}$. The seat at the pole turns $1.7\times10^{-4}$ under the same moment, 8,800 times as far, and grows as the square root of the load where the V grows in proportion to it. Released, the V returns from every load either way. The seat keeps exactly the turn its load gave it, to a millionth, every time.

That last measurement says something about the seat that the earlier essay stated only as a failure. A seat that stays where it is pushed never stays further than it was pushed. Its lost return is bounded by its own deflection under the largest load it has carried, and its deflection is already the thing that limits it as a stop. Where the seat is stiff enough for the job, its stickiness is a hysteresis of the same small size. The seat is not a bad stop because it fails to return; it is a soft one, and its failure to return is its softness remembered. Under a moment of $10^{-3}$ at friction 0.1 it keeps $1.7\times10^{-4}$ radians, and under a moment of 1 it keeps $5.2\times10^{-3}$, both exactly the turns it took while loaded.

The two stops also differ in what they ask of their contacts. The normal force a stop carries for each unit of crank moment is $1/(|g'| + \mu|s'|)$. For a face of the V at 12.3° and friction 0.1 that is about 3.6. For the seat at the pole under a small load, where its lever is almost all friction, it is about 11. So the V carries its load with a third of the seat’s contact force, which matters for wear before it matters for stiffness.

Designing for a band of friction #

The named question can now be answered as it was asked. Suppose the coefficient at the contact may be anywhere from 0.1 to some upper value, and the stop must return at every one of them.

For a band of 1.5, up to 0.15, the V is aimed at 9.5° and gives 34,000 per radian at the low coefficient. For a band of two it is aimed at 12.3° and gives 52,000. For three, 17.7° and 95,000; for four and a half, 25.2° and 174,000. Each returns at both ends of its band. A wider band asks for a larger aim, and the larger aim is stiffer. The guarantee has a price only in the sense that it forbids the small aims, and the small aims were the soft ones.

So the rule a designer can carry is short. Aim each face of a stop well off its pole, by more than the largest friction angle the contact may ever see, and use two faces where the mechanism must be held both ways. Aim at the pole only when a single face must hold both ways and the stop can be soft and sticky — a detent that must not spring a mechanism away from wherever it is left is such a case.

Where this joins the other contact tests #

One test, three mechanisms found that a pawl, an escapement’s lock and a coupler’s grip on its rocker all ask on which side of a pivot a contact normal passes. The earlier friction essay widened the normal into a cone, which gave the test a third answer, inside, and found that answer to be sticking. This essay adds a fourth reading of the same picture: how far outside the cone the pivot lies, which is how stiff the stop is.

A ratchet with no teeth built a one-way clutch by placing a roller’s wedge inside its friction cone, and a joint that works one way found a pawl’s two holding lines to be the cone’s edges. Those devices want the sticking. A stop that must return wants the opposite, and the V is to a stop what a roller clutch is to a ratchet: the same contact read the other way round.

What this does not settle #

A V must fit. Two faces touching one nose at one crank angle is a redundant pair on a mechanism with one freedom, and the seventh contact found redundant contacts constraining nothing until something deforms. Any error in where the V is cut leaves the nose on one face with a gap at the other. That gap becomes play of $c/|g'|$ on that side, first order in the error, by the law the curvature essay measured for a stop off its pole. At a misfit of $10^{-4}$ that is about $5\times10^{-4}$ radians for the 12.3° V, against $\sqrt{2c/|g''|} = 0.016$ for the seat at the pole with the same clearance. The V’s fit is not optimised and its tolerance is not computed.

Friction in the joints. Only the stop’s contacts have friction. A stiff pin buys grip with release found that a pin’s friction circle moves where a force may pass, so the pole itself is blurred by the four-bar’s own joints, and so is every angle measured from it.

Clearance in the joints. Why a hinge works gave every joint a clearance of its own, which turns the pole into a small region. A stop aimed far from it is insensitive to where in the region it lies, which is a further argument for the large aims; how insensitive is not measured.

Contact stress. The normal force per unit of crank moment is estimated above for two stops, but the contacts’ Hertzian stress, which depends on the nose’s and faces’ curvatures as well as the force, is not computed, and nor is the wear that follows from sliding at $|s'|$ under it.

Still open: the nearest a V’s faces may be to each other #

A V’s faces must be aimed off the pole by more than the friction angle, and they must both touch the nose. The further apart their aims, the stiffer each face, and the more the V’s two contacts wedge the nose between them. A V whose two faces are turned far enough off the pole each way approaches a slot whose walls face each other square to the nose’s path, which holds hardest of all and is the least forgiving of misfit.

The distinct argument would be that wedge priced. A V’s two faces both press the nose when a small misfit, a thermal growth or a preload closes the gap at the second face. The measurement would take a misfit of each sign, give the V an aim from the friction angle to sixty degrees, and compute the force the two faces then squeeze the nose with, the crank moment needed to move the nose out of the wedge, and whether the wedge can lock — a V whose half-angle lies inside the friction cone for the squeeze, as a roller clutch does. If it can, there is an upper limit on the aim as well as a lower one, and the band between them is the honest design window for a stop that must hold both ways and return.