A stop aimed at its pole holds by curvature

A contact turns and slides at once found the one direction in which a stop stops nothing. A four-bar has one freedom, and its coupler, relative to the frame, turns about a pole: the point where the crank’s line and the rocker’s line meet. A stop between the coupler and the frame takes that freedom away at every normal direction but one. Aimed through the pole, the stop’s normal is square to the contact point’s velocity, the point slides along the stop’s face, and the rank of the constraints does not rise. The count, which sees four links, four pins and one contact, says the result is a structure at every angle alike.

That essay left the aimed stop at first order and said what the next order would ask. A contact point that starts out moving along a face does not stay on it: it moves along a curve, the face is a curve, and one of them bends away from the other. If the path bends into the face, any motion at all runs the point into the material and the stop holds after all. If it bends away, the four-bar moves and the point leaves the face. The question is which curvature decides, and the natural answer is the curvature of the contact point’s own path, its osculating circle.

That answer is right for a stop with a sharp point and wrong for every real one. A stop has a nose, and a nose has a radius, and the measurements below say that the curve which decides is not the one the nose touches.

The gap is flat at the pole, and its curvature decides #

The four-bar is the one from the essay before: frame 4, crank 1.2, coupler 3.4, rocker 2.8, the crank at 70°. Its coupler’s pole is at (2.505, 6.883), well above the linkage. On the coupler, at (1.593, 2.160), sits the centre of a round nose of radius 0.3. The line from that centre to the pole is the stop’s normal, and the nose touches the frame where that line leaves it, at (1.650, 2.454). The frame’s side of the contact is a seat: a circle through the contact point, centred on the normal line, of any radius, hollowed round the nose or bulging towards it or flat.

With the four-bar turned by a small angle from rest, the nose either stands off the seat or would sink into it, and the gap between them is one number. At rest the gap is nought, since the nose touches. Its first derivative is nought as well, to $8 \times 10^{-13}$, because the normal passes through the pole and the contact is moving along the face. What is left is the second derivative, and its sign is the whole verdict.

The five seats are aimed identically and differ only in curvature. A hollow of radius 0.6 is in the way at once, in both directions, and so is a hollow of 0.8. A hollow of 1.6 opens both ways, and so does a flat face. The seat is either in the way on both sides of rest or on neither, which is what a quantity with a zero slope and a nonzero curvature does. At the first order the stop was blind; at the second it is decisive, and the deciding number lies between 0.8 and 1.6.

This is the situation It moves to first order and not at all found in a framework of two bars in line, where the rank left a freedom and the second order took it back. The difference is that bars pull and push, while a contact only pushes. A negative second derivative here means interference both ways, which a one-sided contact can refuse; a positive one means separation both ways, which it cannot. So the curvature decides between a stop that holds in both directions and a stop that holds in neither. There is no second-order state in which it holds one way only.

The nose is a disc about its centre #

The measurement that decides where the threshold lies is simple enough to state before it is taken. The nose is a disc. Where it is depends only on where its centre is, since a disc turned about its own centre is the same disc. So the gap between the nose and a seat of radius $R$ is the distance from the nose’s centre to the seat’s centre, compared with $R - r$ for a hollow or $R + r$ for a bump. The seat acts on the nose’s centre, and it acts through a circle: the circle of radius $R - r$ about the seat’s centre, inside which the nose’s centre may go.

The four-bar moves the nose’s centre along a path of its own, a coupler curve like any other. Near rest that path has an osculating circle, and every point has a centre of curvature on the line to the pole. For the nose’s centre it is at (1.423, 1.283), a distance 0.893 below it. If the circle the seat allows is tighter than 0.893, the path leaves it on both sides and the nose runs into the seat. If it is looser, the path stays inside and the four-bar turns.

The figure shows exactly that picture. The dashed circle is where the nose’s centre may go; the solid curve is where the four-bar sends it. With a seat of 0.8 the allowed circle has radius 0.5, the path is flatter than it, and the path leaves the circle at once in both directions. With a seat of 1.4 the allowed circle has radius 1.1, the path curves more than it does, and the path runs inside.

So the threshold seat is the one whose allowed circle is the path’s own osculating circle: a seat centred on the nose centre’s centre of curvature, of radius $0.893 + 0.3 = 1.193$.

The natural guess uses a different point. The point of the coupler that sits at the contact, W in the figure, moves on its own coupler curve, and that curve’s centre of curvature is at (1.503, 1.693), with a radius of 0.776. Both centres lie on the normal through the pole, as they must, and they are 0.417 apart. A seat cut to the contact point’s osculating circle would be a hollow of 0.776; the seat that just holds is a hollow of 1.193. The two are not a small correction of each other. Between them is every seat from 0.776 to 1.193, and every one of those holds while the guess says it will not.

The seat that just holds is cut to a path it never touches #

The measurement is a bisection on the sign of the gap’s second derivative, with the seat’s curvature as the unknown, taken for noses of seven radii from nought to 0.8. It is set against two predictions: the path of the nose’s centre moved out by the nose, and the contact point’s osculating circle.

With the frame between the nose and the pole, the threshold seats are 0.893, 0.943, 0.993, 1.093, 1.193, 1.393 and 1.693: the path radius plus the nose, at every radius, to better than $10^{-8}$. The contact point’s circle agrees at nought and nowhere else. For a nose of 0.05 it says 0.873 against a true 0.943. For 0.8 it says 0.602 against 1.693, and it has gone the wrong way. As the nose grows, the contact point moves along the normal towards the pole, where the coupler’s points bend more sharply, so the guess shrinks while the truth grows.

It is not a coincidence that the rule has that form. A profile is an envelope derived the same law for cams: the surface a roller follower rides on has the curvature of the roller centre’s pitch curve less the roller’s radius, because the surface is the envelope of the roller as its centre runs round. A seat at its threshold is that envelope, taken over the four-bar’s motion rather than the cam’s. What the contact point does is not part of it. The contact point is a point of the coupler that happens to be at the contact now; a moment later the contact is somewhere else on the nose, and the point that was there has gone its own way.

Which side the frame is on #

The frame can reach the nose from the other side: from below, with the contact on the nose’s far side from the pole. Then the seat is a bump rather than a hollow, and the nose’s centre may go anywhere outside a circle of radius $R + r$ about the bump’s centre. The nose centre’s path curves towards that bump, since its centre of curvature lies below it.

Here the threshold bumps are 0.893, 0.843, 0.793, 0.693, 0.593, 0.393 and 0.093: the path radius less the nose, again to better than $10^{-8}$. And the verdicts turn over. On this side a flat face holds: the path curves into it, so any motion runs the nose’s centre towards the face. Every bump sharper than the threshold lets go, because a sharp enough bump curves away faster than the path approaches. The contact point’s circle is wrong here too, and wrong the other way — 1.019 for a nose of 0.3, against a true 0.593.

So the side of the contact on which the frame lies decides whether a flat stop aimed at the pole holds, before any curvature is chosen. With the frame between the nose and the pole, the stop needs a hollow seat tighter than the path’s radius plus the nose’s. With the frame on the far side, it holds with a flat face or any bump of radius above the path’s radius less the nose. The pole tells a designer where to aim; the centre of curvature tells them which side to stand on.

Across the crank’s travel #

One pose is one measurement. The same nose, fixed to the same point of the coupler, was re-aimed through the pole at six crank angles from 40° to 115°. The pole travels a long way over that range, from (3.630, 3.046) to (−18.147, 38.917), and the nose centre’s path radius changes with it.

At 40° the path radius is 0.310 and the threshold seat 0.610; at 115° they are 1.597 and 1.897. At every angle the seat is the path radius plus 0.3, to better than $10^{-9}$, and at every angle the contact point’s circle is something else — 0.219 at 40°, where a designer following it would cut a hollow nearly three times too tight. The rule is not a property of one pose. It is the envelope law, and it holds wherever the four-bar stands.

The path radius itself is where the design work is. By the Euler–Savary relation it is set by the nose centre’s distance from the pole and by the direction of the normal through it. The inflection circle marks the coupler points whose paths are momentarily straight. A nose centre placed on that circle is moving straight for the moment, and its threshold is a flat face: any hollow at all then holds. Placed off it, the nose centre’s path bends one way or the other, and the threshold becomes a hollow or a bump accordingly.

At the threshold the stop holds one way #

The gap figure has a sixth curve, dashed, for the seat of 1.193 itself. Its second derivative is nought by construction, and the curve is not flat for that. The gap’s third derivative there is −1.56. So a crank turned forward, towards larger angles, drives the nose into the seat, and a crank turned back draws it off: at a fifth of a radian the gap is −0.0020 one way and +0.0021 the other.

That is the one exception to the two-way rule, and it sits exactly on the boundary. At the threshold the stop holds in one direction only, as an ordinary stop does, but without the first-order lever an ordinary stop has; it resists as the square of the motion rather than in proportion. A seat cut to the threshold is the worst of both. It is a one-way stop that gives like a two-way one, and a manufacturing error of either sign in the seat’s radius turns it into one of the other two cases.

Play enters as a square root #

A stop is never cut touching. There is a clearance, and the question a designer asks of it is how far the mechanism moves before the stop takes up. For an ordinary stop the answer is familiar: the gap closes at the rate $g'$, so the crank turns $c/|g'|$ before the nose touches, and it does so in one direction only. A ratchet with no teeth used the same quotient for a roller clutch.

For a stop aimed at its pole there is no rate. The gap closes as the square of the turn, so a clearance $c$ is taken up after a turn of $\sqrt{2c/|g''|}$, and the same distance in either direction.

For the seat of 0.8, whose gap has a second derivative of −0.782, a clearance of $10^{-4}$ lets the crank turn 0.0159 rad forward and 0.0162 back, against a law of 0.0160. At $10^{-6}$ the two directions agree to two parts in a thousand. At the larger clearances they part, by 6% at $10^{-3}$ and 19% at $10^{-2}$: that is the third derivative again, making the seat tighter forward than back.

The flat stop aimed 5° off the pole has a gap that closes at 0.082 per radian, and it takes up a clearance of $10^{-4}$ after a turn of 0.00122 — thirteen times less than the pole-aimed seat. At $10^{-6}$ the ratio is 131. The square root makes the pole-aimed stop relatively looser the tighter it is made: every factor of a hundred in the clearance buys a factor of ten in its play, and a factor of a hundred in the other stop’s. On a frame of 400 mm, a clearance of a hundredth of a millimetre lets the seat’s crank rock 0.9° each way, and the ordinary stop’s 0.07° one way.

The ordinary stop has a limit the seat does not. Its gap curves away too — its second derivative is +0.945 — and for clearances above $g'^2/2g'' = 0.0036$ the nose’s path bends away from the face before it ever reaches it. At a clearance of $10^{-2}$ it never touches in either direction. The seat is the only one of the two that is certain to engage at every clearance, because its curvature and not its aim is what brings the nose and the face together.

Loaded, the lever has to be grown #

Clearance is play with no load on the mechanism. With a load, the question is stiffness: a moment on the crank must be answered by the contact’s force times its lever about the crank, and a real contact deflects under its force. Give the contact a normal stiffness $k$, so that its force is $k$ times the depth the nose would sink into the seat, and ask how far a moment $Q$ turns the crank.

For an ordinary stop the lever is there from the start. The force the contact exerts on the crank is its normal force times $g'$, and the penetration is $g'$ times the turn, so the turn is $Q/(k\,g'^2)$: linear in the load. For the stop aimed at its pole the lever is nought at rest. It has to be grown by turning the crank, the penetration grows as the square of the turn, and the moment the contact returns grows as its cube. The turn is $(2Q/(k\,g''^2))^{1/3}$.

At a stiffness of $10^6$, a moment of $10^{-4}$ turns the pole-aimed seat by $6.9 \times 10^{-4}$ rad and the ordinary stop by $1.5 \times 10^{-8}$: forty-six thousand times as much. At a moment of 1 the seat turns $1.5 \times 10^{-2}$ and the stop $1.5 \times 10^{-4}$, a ratio of 99. The cube root is kindest to heavy loads and cruellest to light ones. A pole-aimed seat is soft exactly where a latch is expected to be firm, at the small disturbing loads a mechanism sees all the time.

The measurement is a bisection on the balance of moments, with the gap measured on the four-bar at every trial turn; both laws agree with it to 3% over four decades of load. The rigid-body picture that went before is the limit of this one. With $k$ infinite, the seat holds at any load with no turn at all, but it does so with a contact force that is infinite at rest — a normal through the pole has no moment about anything the four-bar can turn, so no finite force along it can answer a load.

Where latches and detents sit #

A detent is a stop aimed at its pole on purpose. A spring-loaded ball dropping into a hollow on a shaft pushes along a normal through the shaft’s axis, which is the shaft’s pole on the frame, and it holds the shaft both ways with a single contact. It holds softly, at second order, which is what a detent is for. The hollow is sized so that a hand can push through it. The measurements above are the same thing on a coupler, whose pole is not at any pivot and moves as the mechanism does, which is why it is easy to aim at without meaning to.

A latch is the opposite design intent. It should hold hard in one direction, and it holds at first order only if its normal misses the pole. How much a stop holds was the sine of that miss. What these measurements add is what happens as the miss goes to nought. The latch does not go on weakening in proportion; it changes kind. It becomes a detent if its face curves the right way and nothing if it curves the wrong way. A latch built within a degree or two of its pole should be checked for which it is becoming, and the check is the seat against the nose centre’s path, not against the contact.

Exact constraint makes the same demand in three dimensions. Six hold nothing found that a coupling’s contacts leave a part a direction in which it is held only by the load, and the seventh contact found a contact that constrains nothing at first order. Wherever a contact’s normal passes through the instantaneous axis of the motion it is meant to stop, what it does next is set by curvature. And the curvature that counts is always that of the path of the centre of the contact’s own rounding.

What the curvature does not settle #

Friction. Every contact here is smooth. A pole-aimed contact slides along its face as the four-bar moves, and friction resists sliding: with a coefficient $\mu$, a normal within $\arctan\mu$ of the pole meets a friction force that can stand in for the missing lever. Whether friction makes the seat’s curvature irrelevant, or only moves the threshold, is not measured.

One four-bar, one nose position. The nose sits at one point of one four-bar’s coupler, re-aimed at six poses. The envelope rule is a statement about any coupler point and any motion, and it held at every one tried, but no census over nose positions or linkages was taken.

A contact stiffness that is a single number. The load measurement treats the contact as a linear spring along its normal. A Hertzian contact stiffens as it is pressed, which would change the cube root to a different power. The ordering of the two stops would not change, since the lever argument is independent of the contact law, but the exponent would.

Circles only. Every seat is a circle and every nose is a circle. A seat with a curvature that changes along it matches the path to third order where its curvature’s derivative matches the path’s, which is the cubic at the threshold made adjustable, and no seat of that kind is measured.

Still open: a seat with friction #

The seat at its pole holds by curvature because it has no lever, and friction is the other way a contact can resist a motion it has no lever against. A pole-aimed nose moves along its seat by sliding. Friction opposes that sliding, and the friction force has a lever about the crank’s pivot even when the normal force has none.

The distinct argument there would be the threshold with friction. For a coefficient $\mu$, a stop is in its friction cone whenever the contact’s resultant can lie anywhere within $\arctan\mu$ of the normal. That means a stop aimed within that angle of its pole can answer a load at first order, from one side, as a stiff pin buys grip with release found for a pivot. The measurement would take the seat of 0.8 and the flat face and add a coefficient from nought to 0.3. It would ask whether the seat’s two-way hold and square-root play survive, whether friction turns the flat face on the pole’s side into a one-way stop, and whether the envelope rule still marks the boundary once the contact’s force is a cone rather than a line.