Where the pad touches

The end of a rocker arm is not a pin. It is a curved pad bearing on the flat end of a valve stem, and the point where they touch moves as the rocker swings.

That movement has a name in the trade — the wipe, or the sweep — and a rule attached to it: set the geometry so the pad is square to the valve at mid-lift, not with the valve shut. Engine builders shim rocker studs to achieve it and argue about it in forums.

It is geometry, and it is computable, and the answer is a factor of four.

Where the contact point is #

A pad of radius ρ, with its centre carried on the rocker at some radius from the shaft, bearing on a flat valve tip. The tip is horizontal; the pad is a circle; so they touch at the pad’s lowest point, which is directly below the pad’s centre.

Two consequences, both worth stating because both are used.

The valve’s lift is the drop of the pad’s centre. The radius cancels: a bigger pad sits higher and drops by the same amount. That is why the previous essay could compute the lift from the centre’s motion without knowing ρ at all.

The contact point’s horizontal position is the pad centre’s horizontal position, and the pad centre is on an arc about the rocker shaft. So the contact wanders across the valve tip by exactly as much as the pad’s centre moves sideways, which is the arc’s horizontal extent over the swing.

Why the symmetry is the whole answer #

A point on a circular arc, moving through an angle, sweeps horizontally by an amount that depends on where on the arc it starts.

Start at the bottom of the arc — with the arm perpendicular to the direction of travel, which is what “square” means — and the horizontal motion is second order in the angle: the point comes back towards where it started. Start off to one side and the horizontal motion is first order and much larger.

So the geometry is symmetric about wherever it is squared up, and the swing is twelve degrees. Square it at the valve-shut position and all twelve degrees are on one side of the symmetry. Square it at mid-lift and there are six degrees each side, and the sweep is what a six-degree half-swing costs rather than a twelve-degree full one.

The measurement: 1.35 mm becomes 0.34 mm, a factor of exactly about four, which is the ratio a second-order quantity gives when its argument is halved.

And the lift comes back too #

The same shim does something to the ratio, and this is the part that is not folklore because nobody quotes it.

The rocker’s instantaneous ratio falls monotonically through the event — 1.605 down to 1.588 with the geometry squared at rest. Move the setup to mid-lift and the same curve slides: it now runs from 1.6134 down to 1.5971, straddling the nominal 1.6053 rather than starting at it.

A ratio that is above nominal for the first half of the lift and below it for the second integrates to very nearly the nominal. And it does:

The mid-lift setup delivers the promised lift to three parts in ten million. That is not because the ratio has become constant — it still varies by one per cent — but because the variation has been centred, and a centred variation cancels when it is integrated.

Two effects, one adjustment, and a reason to be careful #

It would be easy to present this as one finding. It is two, and they happen to be improved by the same shim on this geometry:

• the wipe is smaller because the arc’s horizontal excursion is second order about the square position;
• the lift is closer to nominal because the ratio’s variation is centred.

Both follow from centring the swing, and on this geometry both improve together. They are not the same statement, and it would be a mistake to expect them always to move together — a rocker whose ratio variation is dominated by the pushrod rather than by the pad could have its wipe minimised at one setup and its lift error at another.

The site’s habit here is to compute both rather than to compute one and infer the other, and the gate requires both to improve: the wipe to more than halve, and the peak lift to move closer to the arm ratio’s promise. An assertion on one alone would pass on a geometry where the other had got worse.

Getting the factor right, and the assertion that would have hidden it #

The factor of four is a second-order argument and second-order arguments are easy to state carelessly, so it is worth saying exactly what was measured.

The wipe is not halved by halving the swing; it is quartered, because a point near the bottom of an arc moves horizontally as the square of the angle from the bottom. Twelve degrees on one side gives a sweep proportional to 12²; six degrees each side gives two sweeps proportional to 6² in opposite directions, which is a range of 36 rather than 144 — a quarter.

The measurement is 1.353 mm against 0.339 mm, which is 3.99. The prediction is 4. It is close enough that the assertion could have been written as “the wipe is quartered”, and it is written as “the wipe is more than halved” instead.

That is deliberate. A threshold that encodes the exact factor is a threshold that fails when the geometry changes, and the geometry here is one plausible rocker among many: a longer pad radius, a different pushrod angle or a different swing changes the number and not the argument. The assertion is written at the strength of the claim the essay actually makes — that centring the swing substantially reduces the wipe — and the exact 3.99 is reported rather than required.

The site has been caught by the opposite choice. A check written at the number that came out first, with a threshold chosen to fit it, is a check that says the computation has not changed rather than that it is right; the spatial field’s constraint-drift measurement made exactly that mistake and was saved only because the answer had been derived on paper before the code ran.

What the wipe costs, and why anybody cares #

The sweep is a sliding contact under load, so it is where a valve tip and a rocker pad wear. That is a tribological question and this site does not do wear.

What it can say is the geometric half, and the geometric half is the part a designer controls. The practice field made the same separation about pins: which pin to buy ranks a four-bar’s joints by how much their clearance costs the output, and then says plainly that wear is a different ranking, computed from the relative rotation at each pin rather than from the error each contributes. It named the sliding-distance ranking as a third quantity and did not compute it.

This essay computes exactly that quantity for one contact: the sliding distance per cam event, 1.35 mm or 0.34 mm depending on the setup. It is the first sliding distance the site has computed anywhere, and it is a small down payment on a question the practice field left open.

What the sweep would be on a different rocker #

The numbers here belong to one geometry, and it is fair to ask how much of the finding travels.

The wipe scales with the swing, squared, and with the pad’s distance from the shaft. A rocker with a bigger valve lift swings further and wipes more; a rocker with a longer valve-side arm swings less for the same lift and wipes less. Nothing about the argument changes: the sweep is still second order about the square position, and centring it is still worth a factor of about four.

What does not travel is the lift result. The near-perfect cancellation at mid-lift depends on the ratio’s variation being close to linear in the rocker’s angle, which it is here because both contributions are small and smooth. A geometry with a more sharply curved ratio would centre less cleanly, and the essay’s table would show a smaller improvement.

So the transferable statement is the weaker one, and it is the one to keep: centring the swing on the square position reduces both the wipe and the lift error, by amounts that depend on the geometry, and the direction of the improvement does not. That is exactly the shape of claim a site like this should be making — a sign and a mechanism that transfer, with the magnitudes attached to a stated example.

What the model has and has not #

Has: the pad’s radius, the valve tip’s flatness, the rocker’s arc, the pushrod’s obliquity, and a real cam motion law with the geometry solved at every step of it.

Has not: valve lash, which shifts the whole event; the valve guide’s clearance, which lets the stem tilt and moves the contact; and any load at all, so nothing here says which way the sliding happens under force or what it does to the surfaces.

There is also a modelling choice worth flagging. The valve tip is treated as flat and the pad as a circular arc, which is the usual arrangement. Some rockers use a roller on the valve end instead, which replaces sliding with rolling and makes the whole question different — the contact point still moves across the valve tip, but the pad’s surface no longer slides against it. The geometry of that is a rolling-contact problem and is not built here.

Why the pad has a radius at all #

A question the geometry answers in passing: why is the rocker’s contact face curved rather than flat?

If both surfaces were flat, they would have to stay parallel, and a rocker’s face turns twelve degrees while the valve’s does not turn at all. The contact would be at an edge, and the whole load would go through a line at the corner of the pad.

Give the pad a radius and the contact is at the lowest point of a circle, which exists at every angle. The pad can turn as much as the rocker turns and the contact stays a proper contact. The radius does not need to be any particular value for that — it needs to be finite.

What the radius does decide is how far the contact moves for a given rotation, which is a second-order effect on the wipe, and how concentrated the contact stress is, which is outside this site. So the radius is chosen for the stress and the wipe follows, which is the usual order of things: the geometry is decided by something else and its kinematics is inherited.

That is worth noticing because it is the reverse of most of this field. A steering arm angle is chosen for its kinematics; a suspension’s arm lengths are chosen for their kinematics; a rocker pad’s radius is chosen for its contact stress and its kinematics is a consequence. Both kinds of decision exist and it is useful to know which one is being made.

The wipe is a band, and the band has to fit #

There is a third consequence of centring the swing that neither the trade’s rule nor the two effects above mention, and it is the most purely geometric of the three.

The contact point does not merely move; it moves across a band on the valve tip, and the band has a position as well as a width. Squared up with the valve shut, the swing runs from the square position outward in one direction only, so the band lies entirely to one side of where the pad sits at rest. Squared up at mid-lift it runs both ways, so the band straddles that point. Same pad, same valve, and two quite different regions of the tip in use.

That matters because a valve tip is a small flat of a few millimetres and it has an edge. A band of 1.35 mm sitting off to one side needs the tip to extend 1.35 mm in that direction beyond the rest position, plus whatever margin the assembly’s own variation demands; a band of 0.34 mm straddling the rest position needs 0.17 mm either side. The centred setup therefore asks for roughly a fifth of the tip width the uncentred one does, and asks for it symmetrically, which is the arrangement a round tip is actually shaped for.

So the failure mode of a badly shimmed rocker is not only faster wear and a lift that is short. It is that the contact can run off the edge of the tip near full lift, where the load is highest and the pad is furthest from square — and a contact at an edge is a line load on a corner rather than a Hertzian patch, which is a different and much worse kind of contact than the one the parts were designed for.

None of that needs any force to establish. The band’s width and position are geometry, the tip’s diameter is a dimension, and whether one fits inside the other is a comparison. What forces decide is how bad it is when it does not fit, and the geometry has already said whether the question arises.

It also explains a detail of the folklore that is otherwise unmotivated. Engine builders check the wipe pattern by marking the valve tip and running the engine over, and what they look at is not only how wide the mark is but where it sits — centred is right, offset is wrong, and a mark running to the edge condemns the setup. That is exactly the band and its position, read directly off the part, and the shim is the adjustment that moves it.

The rule, restated #

The trade’s rule is “square at mid-lift”. What the geometry says is:

Centre the swing on the square position, because both the sweep and the ratio’s deviation are symmetric about it, and a symmetric error over a symmetric range is a smaller error.

That is more general than the rule, and it is checkable, and it applies to a great many mechanisms with an adjustable rest position. It is the same argument that puts the precision points of an approximate synthesis at Chebyshev’s spacing rather than uniformly, and the same argument that makes an optimised steering arm angle equioscillate: an error that is allowed to be signed and centred is smaller than an error that is one-signed over the same range.

Three fields have now arrived at that from three directions — synthesis, steering, and a shim under a rocker stud — which is about as good a sign as this site gets that an idea is worth having.

It also closes the cam ladder in a way worth noticing. The cams field began by prescribing a motion and cutting a profile to deliver it; it then found that the profile cannot always be cut, that the base circle is decided by two constraints rather than one, and that the follower’s contact is a geometry of its own. This essay ends it at the far end of the same chain: the motion has been designed, the profile cut, the follower followed — and the last two components, an arm and a pad, still have a per cent and a millimetre to say about what the valve does. A prescribed motion is prescribed at one point of a machine, and everything after that point is a mechanism.

A last observation about the band that generalises past valve gear. Any contact between a curved face on a swinging arm and a flat on something else has this structure: the contact point migrates across the flat, the migration is symmetric about the position where the arm is square, and the flat has to be large enough to contain the whole excursion with the arm’s actual working range on it. Cam followers, brake pad carriers, clamp pads and the seat of a rocking bearing are all the same geometry, and in every one of them the two design questions are how wide the band is and where it sits. The trade’s rule about shimming is a special case of an arrangement this site has drawn several times without naming the common part.