A seat with friction stays where it is pushed

A contact turns and slides at once found that a stop between a four-bar’s coupler and its frame removes the four-bar’s one freedom at every normal direction but one. That direction is the line through the coupler’s pole, where the crank’s and rocker’s lines meet. Aimed there, the stop forbids nothing at first order. A stop aimed at its pole holds by curvature settled what it does at second order. A round nose in a hollow seat holds both ways if the seat is tighter than the parallel of the nose centre’s path, and lets the crank turn if it is looser. The threshold on the four-bar of those essays is a seat of radius 1.193 round a nose of 0.3.

Both essays had frictionless contacts, where the force between nose and seat acts along the normal. Real contacts have friction, which lets the force lean from the normal by up to the friction angle, $\arctan\mu$. The normal of a pole-aimed stop passes through the pole and so has no lever about it, but a force leaning off the normal does. The earlier essay ended by expecting friction to let such a stop answer a load at first order, and asking whether the seat’s two-way hold survives it.

It survives, and friction does something other than what was expected.

Friction is a share of a force that curvature decides #

A contact with friction transmits a normal force $N$ and a tangential force of at most $\mu N$ opposing sliding. Everything follows from the second half of that sentence. Friction is a share of the normal force, so it exists only where the contact is pressed. Where the nose leaves its seat as the crank turns, $N$ is nought and so is every force friction could supply, whatever the coefficient.

That is enough to settle the first question before measuring anything. Which stops hold is decided by whether the gap between nose and seat closes or opens as the crank turns, and that is a question about the gap’s derivatives, set by geometry. Friction cannot make a stop hold whose gap opens, and it cannot release one whose gap closes. The threshold stays where curvature put it.

The measurement agrees. Pushed back by a moment of $10^{-3}$ through a contact of stiffness $10^6$, every seat tighter than the threshold holds with and without friction, and friction makes it up to 26 times stiffer. Neither version holds past the threshold of 1.193. Pushed the other way, a seat past the threshold catches only after its gap’s cubic term closes it, a large turn, and there friction changes nothing: at a seat of 1.27 the crank turns 0.173 radians either way. The earlier essay’s hope that friction would turn a stop that releases into one that holds, or a flat face on the pole’s side into a one-way stop, fails for the plainest reason. The contact carries nothing for friction to take a share of.

How the crank’s moment is shared #

To say what friction does to a stop that holds, the crank’s moment has to be split between the contact’s two components. The cleanest route is virtual work. As the crank turns through a small angle $d\theta$, the nose presses into its seat at the rate $g'$, the gap’s derivative in the crank angle, and slides along it at a rate $s'$. The crank’s moment $Q$ does work $Q\,d\theta$, the normal force absorbs $N|g'|\,d\theta$ and friction, opposing the sliding at its limit, absorbs $\mu N |s'|\,d\theta$. So

$$Q = N\,(|g'| + \mu|s'|).$$

That was checked by a route that uses no work at all. The coupler was put in equilibrium under three forces: the crank’s pin force at one end, the rocker’s force along the rocker at the other, and the contact force $N$ along the normal with $\mu N$ along the face against the sliding. The crank’s moment was then read from the pin force. On the seat and on a stop aimed off its pole, at every friction and at turns from $2 \times 10^{-4}$ to $5 \times 10^{-3}$, loading and unloading, the two routes agree to $10^{-5}$ of the moment.

The formula shows exactly where friction matters. For a stop aimed through its pole, $g'$ vanishes at the rest pose and grows as $g''\theta$, while the sliding rate $s'$ does not vanish at all. The coupler turns about the pole, and the contact point, off the pole along the normal, moves square to the normal, straight along the face. On this four-bar it slides at 0.884 per radian of crank. So near the pole the friction arm $\mu|s'|$ is the larger arm, not a correction to the normal one.

A square root instead of a cube root #

Without friction the seat’s normal force grows as $k|g''|\theta^2/2$ with the turn, and its lever grows as $|g''|\theta$, so the moment grows as the cube of the turn and the turn as the cube root of the load. That was the earlier essay’s result. With friction the lever has a part that is there from the start, $\mu|s'|$, so the moment grows as the square of the turn:

$$Q = k\,\frac{|g''|\theta^2}{2}\,\big(|g''|\theta + \mu|s'|\big).$$

Measured by pushing the crank until the contact’s moment matches the load, the turn follows that cubic to 2% at every load and friction swept. With friction it goes as the square root of the load until the turn reaches $\mu|s'|/|g''|$, where the two arms are equal. On this seat that is 0.057 radians at a coefficient of 0.05, far beyond any load in the sweep. At a moment of $10^{-4}$ the seat turns $6.9 \times 10^{-4}$ radians with no friction and $3.1 \times 10^{-5}$ at 0.3, twenty-two times less. At a moment of 1 it turns $1.5 \times 10^{-2}$ and $3.1 \times 10^{-3}$.

For a designer this is good news with a qualification. The earlier essay found the pole-aimed seat soft under light loads, because its lever has to be grown by the turn. Friction supplies a lever from the start, so a seat with ordinary friction is far stiffer at small loads than its frictionless idealisation. The qualification is the next section.

Released, the seat does not come back #

Take the load off and the contact has to slide back the way it came, so friction reverses. The crank’s moment on the way back is

$$Q = N\,(|g'| - \mu|s'|).$$

While $|g'|$ is smaller than $\mu|s'|$ that is negative. The normal force the contact carries cannot push the crank back against friction, and the stop stays where the load left it. For the seat at the pole $|g'| = |g''|\theta$, so a seat pushed less than $\mu|s'|/|g''|$ does not move at all when released. On this seat that is 0.023 radians at a coefficient of only 0.02, and 0.34 at 0.3. Every load that leaves the four-bar still recognisably at its stopped pose leaves it stuck where it was pushed. At a middling coefficient of 0.1 the seat stays wherever it is left by any turn up to 0.11 radians, more than six degrees of crank, and the largest load in the sweep turns it 0.0052, twenty times less. Within that range, nothing a load does to this seat is undone by taking the load away, and the range is far larger than the turn the contact’s stiffness allows at every load in the sweep, from a ten-thousandth up to one.

The cycle figure shows it. The seat at the pole with friction 0.1 is pushed to a turn of 0.004 and released, and it stays at 0.004. The unloading curve falls straight to nought moment without any turn coming back. A flat stop aimed off its pole behaves differently. Its normal misses the pole by 5.33° at the contact, and at friction 0.05 it springs back to its stopped pose. At 0.1 it too stays where it was pushed. In each case the area between the loading and unloading curves is the work friction took.

The friction cone decides #

Why 0.05 lets the off-pole stop come back and 0.1 does not is one line of geometry. The coupler turns about its pole, so the material point of the coupler at the contact moves square to the line from the pole. If the normal misses that line by an angle $\psi$ at the contact, the contact presses in at $|g'| \propto \sin\psi$ and slides at $|s'| \propto \cos\psi$, and their ratio is exactly $\tan\psi$. Measured on seven stops, the ratio equals the tangent of the angle between normal and pole line at the contact to $10^{-3}$. So the stop stays put when released exactly when $\tan\psi < \mu$. That is when the line from the contact to the pole lies inside the friction cone.

The table bears it out on every stop and friction where $\tan\psi$ and $\mu$ are not within a hundredth of each other. A stop missing its pole by 1.07° locks from a coefficient of 0.05; one missing by 5.33° from 0.1, since tan 5.33° is 0.093; one missing by 12.8° only at 0.3; and one missing by 21.3° at none of them. The table’s second number is the other side of the same geometry. While the stop is being pushed, friction adds $\mu|s'|$ to the lever $|g'|$ and so multiplies its stiffness by $1 + \mu/\tan\psi$. That is seventeen times for the 1.07° stop at friction 0.3 and 1.8 times for the 21.3° one. Aim a stop near its pole and friction makes it both stiff and sticky. Aim it well off and friction changes little.

A stiff pin buys grip with release found the same cone deciding a cam cleat. The cleat holds a rope while the line from its contact to its pivot leans no more than the friction angle, and a cleat leaning too little locks against a backward pull. A stop and a cleat are the same contact read from opposite sides. For the cleat, the cone is what makes it grip. For the stop, it is what keeps it from letting go.

One test, with a cone in place of a line #

The friction cone at a contact is a familiar object under other names. One test, three mechanisms found that whether a pawl holds, whether an escapement’s lock draws itself deeper and how much a four-bar’s coupler can do for its rocker are all one question. On which side of a pivot does a contact normal pass? For a frictionless contact the normal is a line, and the test is which side of the line the pivot lies.

With friction, the contact can press along any line inside its cone, so the test has three answers instead of two. The pivot can lie clear of the cone on one side, clear of it on the other, or inside it. Inside it, the contact can press with a force on either side of the pivot, and which one it presses with is decided by which way the mechanism is trying to move. That is the locking this essay measured, and it is the same event as a joint that works one way found at a pawl’s tooth. There, one of two lines through the contact is the rule a workshop quotes and the other the boundary it leaves out. The cone’s two edges are those two lines.

A ratchet with no teeth is the device built entirely on this. A roller clutch holds one way because its roller wedges between two surfaces whose normals meet at an angle inside the friction cone, and it runs free the other way because the wedge opens. The stops here differ only in which question is asked of the cone. A clutch asks whether it will grip when driven. A stop asks whether it will let go when the load is taken off.

What the pole-aimed seat is, with friction #

Put together, friction leaves the seat’s geometry alone and changes its mechanics completely. The seat still holds exactly when it is tighter than the nose centre’s path. Its play in a clearance is unchanged, $\sqrt{2c/|g''|}$ either way, because play is how far the crank turns before the nose touches and friction acts only after it does. What changes is everything that happens once it touches. The seat is much stiffer at small loads, and it keeps whatever turn a load gives it.

That second property is the one a designer must decide about. A stop’s job is usually to bring a mechanism back to a known pose. A frictionless seat at the pole does that, but with a soft cube-root give. A real seat at the pole will not do it at all, since its pole lies inside every friction cone. It holds the mechanism wherever the last disturbance left it, within the seat’s play. The seventh contact found a redundant contact constraining nothing. Here a contact constrains perfectly well and does not restore. Where return matters, the stop should be aimed off its pole by more than the friction angle, at the cost of holding only one way at first order. Where stiffness and damping matter more than return, the seat at the pole with friction is the better stop than either essay’s frictionless analysis suggested.

What friction does not settle #

Stick and slip at the start. The analysis takes friction at its limit throughout, which is right once the contact slides. Before it slides, a real contact sticks with a tangential stiffness of its own. That adds a short elastic stretch before the square-root law begins, and it is not modelled.

Friction in the joints. Only the stop’s contact has friction here. The four-bar’s pins have friction too, and a stiff pin buys grip with release found that a pin’s friction circle moves where a force may pass. The pole itself is then blurred by the joints’ friction circles, and so is the cone test.

Vibration. A stop that stays where it is pushed can be walked away from its pose by small reversing loads, each leaving a residual turn. Whether repeated disturbances accumulate or cancel is a dynamics question. The loops in the cycle figure are the energy each push and release costs, which makes a frictional seat a damper as well as a stop; how much damping, against how much drift, is not computed.

Clearance in the joints. Why a hinge works gave every joint a clearance of its own. The pole is then not a point but a small region, and a stop aimed at it is aimed at a smear. Whether the cone test survives that, or whether a stop near its pole locks only for some positions of the joints within their play, is not measured.

Still open: a seat that returns and stays stiff #

The two good properties parted company. Friction’s stiffness needs the pole inside the cone, and return needs it outside. A stop can be designed with its normal missing the pole by just more than the friction angle, returning when released and still gaining $1 + \mu/\tan\psi$ in stiffness while pushed.

The distinct argument would be that margin measured against an uncertain friction. Real coefficients vary by a factor of two between clean and contaminated surfaces. A stop aimed just outside the cone at the design friction is inside it at a higher one, and it locks. The measurement would find the aim angle that returns for every coefficient in a stated band, and the stiffness it then keeps at the lowest coefficient. It would compare that stop with the seat at the pole, which never returns, and with a stop aimed far off, which returns always and stiffens little. The designer’s question is how much stiffness a guaranteed return costs.