The cam is not the valve

A cam is the mechanism that gives the motion it was asked for. Its whole point is that the follower’s displacement is a designed function of the shaft angle, chosen for its accelerations and its jerk and cut into steel accordingly.

In an engine, that carefully designed motion then passes through a linkage before it reaches the thing that is supposed to do it. The lifter follows the cam, a pushrod goes up, a rocker arm turns, and a pad on the far end of the rocker pushes the valve down.

The rocker is sold as a number: a 1.5, a 1.6, a 1.65. And the number is the ratio of two moment arms at one position of a linkage that swings twelve degrees.

What the ratio is a ratio of #

The rocker turns about its shaft. On the pushrod side, the cup sits at radius r₁; on the valve side, the pad’s centre at radius r₂. The nominal ratio is r₂/r₁ — 61/38 = 1.605 for the geometry here.

That is exactly right at one position, and both arms change as the rocker turns:

• the pushrod leans as the rocker’s cup swings on its arc, so the component of the cup’s motion along the pushrod is no longer the whole of it;
• the pad rolls across the valve tip, and the perpendicular distance from the shaft to the contact point changes as it does.

Neither effect is large and both are one-signed over the lift, which is why the ratio does not oscillate — it falls.

Where the lift goes #

One per cent on a ratio does not sound like much until it is turned into the quantity anybody measures.

The cam here lifts 8.00 mm. Multiplying by 1.605 promises 12.84 mm at the valve. Solving the linkage gives 12.78 mm — 0.51% short, or about 65 microns.

Sixty-five microns is real in an engine — it is a measurable fraction of the valve-to-piston clearance and it is why a builder checks lift at the valve rather than computing it — but the number is not the point. The point is that it is not a property of the cam. Change the rocker’s setup and the shortfall changes; the next essay measures a setup where it falls to 0.03%.

A number that is not a number, one field down #

The cams field’s own version of this argument was about a ratio inside the cam: the pressure angle, which decides how much of the follower’s force goes sideways into its bore, varies through the rise and is quoted as a maximum rather than as a value. The field measured the base circle that a stated maximum implies, and found a second constraint that is usually invisible — a large roller can make a cam unmanufacturable at a pressure angle nobody would worry about.

The rocker ratio is the same kind of object one link further along. Both are properties of a mechanism at a position, both are quoted as single numbers, and both have an established convention about which position — a maximum for the pressure angle, zero lift for the rocker.

What is worth noticing is that the two conventions are different kinds. A maximum is a property of the whole motion and is safe: quoting the worst case cannot mislead about any other position. An operating point is a property of one position and is only safe if the reader knows which one. Of the field’s fourteen rows, every honest one is a maximum, a mean or a stated point, and every dishonest one is a point whose position went unstated.

What the linkage is #

The train is solved as a mechanism rather than as an arithmetic, and the mechanism is one the site already has.

The lifter is a joint on a vertical guide, driven by the cam’s motion law — two slide constraints, one for the bore and one for the lift, which between them place the point. The pushrod is a bar from the lifter to the rocker’s cup, and the cup is a bar from the rocker shaft. That is a slider-crank, with the cam’s lift as the input and the rocker’s angle as the output.

The pad is attached rigidly to the rocker, and the valve’s lift is the drop of the lowest point of the pad — which, for a radiused pad on a flat valve tip, is the drop of the pad’s centre. So the valve’s motion is a coupler point’s motion, and everything the site knows about coupler points applies.

Why the ratio falls rather than wandering #

It is worth asking why the curve is monotonic, because a ratio built from two swinging arms could as easily have a maximum in the middle.

The answer is that both effects have the same sign over this event. As the rocker turns to open the valve, the cup arm swings away from perpendicular to the pushrod, so a given rocker rotation takes more pushrod travel; and the pad rolls in the direction that shortens its effective arm. Each of them reduces the output per unit of input, and they add.

Move the setup — start the rocker half its swing to the other side of square — and the curve keeps its shape and slides: it runs from 1.6134 down to 1.5971, straddling the nominal 1.6053 instead of starting at it. The fall is a property of the linkage; where the fall sits relative to the quoted number is a property of the setup, and the next essay is about what that is worth.

This is the same observation the linkages field makes about a slider-crank: the piston’s motion is not harmonic, and how far from harmonic depends on the rod-to-crank ratio, which is a design number rather than a fact about slider-cranks. A rocker is a slider-crank with an extra arm and it inherits the whole family of behaviours.

Two routes to the valve’s motion #

The valve’s lift is computed here from the solved linkage. There is a second route — write the rocker’s angle as a closed form in the cam’s lift and differentiate — and it is instructive that it is harder, not easier.

The closed form needs the pushrod’s obliquity, which is an arcsine; the pad’s contact, which is a rolling condition; and the composition of the two, which is where the algebra becomes an exercise. The solve needs neither: assemble the constraints, drive the input, read the output.

That is the site’s standing preference stated in the one case where it is unarguable. The closed form is available and nobody would enjoy it, and — the part that matters — it would be a second implementation of the same geometry, which is exactly the thing the fleet’s consolidation spent three thousand deleted lines undoing. What the site does instead is check the solve against something structurally different: the ratio at zero lift must be exactly r₂/r₁, which is arithmetic, and it is, to eleven figures.

The cam’s law survives, mostly #

A more searching question than the peak lift: does the rocker change the shape of the motion?

The cam’s law here is cycloidal, chosen in the cams field as the one that costs least in acceleration among the simple laws. Its virtue is that its acceleration starts and ends at zero, so the follower is not slapped at the start of the rise.

Passing it through a ratio that varies by one per cent scales the whole motion by a factor that is itself a function of position. The valve’s displacement is therefore not the cam’s law times a constant, and its second derivative — the valve’s acceleration — is not the cam’s acceleration times a constant either.

The effect is small and it is not zero, and the honest summary is that a rocker preserves the qualitative properties of a motion law and perturbs the quantitative ones. A cycloidal cam still gives a valve zero acceleration at the ends of its rise, because a ratio that is finite and smooth cannot turn a zero into a non-zero. What it does is put a small position-dependent scaling on everything in between.

By the top of the lift the arms have swung again, and the ratio has moved with them: the rocker is a linkage rather than a lever, so the number stamped on it is one instant of a function rather than a property of the part.

The number is an operating point, and the convention is real #

The field’s ledger files the rocker ratio as point rather than quoted, and the distinction is worth defending because it is the friendliest verdict in the table.

The quantity exists at every position. It is a ratio of two real moment arms and it can be measured with a dial gauge. What makes the quoted figure honest is that everybody in the trade knows a rocker ratio is quoted at zero lift, and everybody who needs the peak lift measures it rather than multiplying.

That is a working convention doing its job. It is also exactly the mechanism by which a roll centre and a camber gain go wrong: a convention that is common knowledge inside a trade is invisible to everyone outside it, and the number gets used as though it were a property.

The difference between the two cases is not the arithmetic. It is that the rocker’s convention comes with an established practice of checking — cam cards specify lift at the valve, and builders measure it — and the suspension’s does not.

What the shortfall is, and what it is not #

It is worth separating two things that both look like “the valve does not lift as far as it should”.

The first is what this essay measures: the ratio is not constant, so the product of a constant ratio and the cam’s lift is not the valve’s lift. That is a geometric shortfall, it is 65 µm here, and it is entirely predictable from the linkage.

The second is everything a real engine adds — lash taken up, pushrod compression, rocker flex, lifter bleed-down — all of which also reduce the valve’s lift, all of which depend on speed and temperature, and none of which is here. Those are usually larger.

The reason to compute the first anyway is that it is the part that does not go away. A stiffer pushrod removes some of the second; nothing removes the first except changing the geometry. And when a builder measures lift at the valve and finds it short of the multiplication, this is the part of the discrepancy that will be there again tomorrow.

That is the general reason to draw a boundary and compute up to it rather than throwing the whole problem at a dynamic simulation: the geometric part is the part that is a property of the design.

What a designer does about it #

Three things, and the site can say something about each.

Quote lift at the valve. Which is what cam cards do: they give the lobe lift and the valve lift at a stated rocker ratio, and the second is the one that matters. The multiplication is done once, carefully, by somebody with the geometry.

Set the rocker up properly. The shortfall is a function of where the rocker sits at zero lift, and it can be reduced by an order of magnitude with a shim. That is the next essay.

Measure. A dial gauge on the valve retainer costs nothing and settles it. Every quantity in this essay was computed from a model; the model is a rigid linkage with no clearance, no lash and no deflection, and an engine has all three.

Where the rocker puts an acceleration the cam did not #

The ratio is one per cent off across the event, and the essay has cashed that out as sixty-five microns of lift. There is a second consequence that is not a per-cent effect at all, and it lands on the quantity the cam’s law was chosen for.

Write the valve’s lift as a function of the lifter’s: $v = f(s)$, with $f'(s)$ the instantaneous ratio $\rho$. Differentiate twice with respect to the cam angle and

$$v'' = \rho\, s'' + \rho'\, (s')^2 .$$

The first term is the cam’s own acceleration scaled by the ratio, which is the one-per-cent effect. The second term is new: it exists only because the ratio varies, it is proportional to the square of the lifter’s velocity, and it has nothing to do with the cam’s law at all.

Now look at where that term is largest. $(s')^2$ peaks at mid-lift, and a cycloidal law is chosen precisely because its acceleration passes through zero at mid-lift — that is the smooth reversal the law exists to provide. So the rocker inserts an acceleration at exactly the point of the event where the cam was designed to have none, and it inserts it in proportion to the square of the fastest the valve train ever moves.

That is a sharper statement than the ratio varies by one per cent, and it is a different kind of statement. The lift shortfall is a small correction to a large number. The mid-lift acceleration is a small number added where the design had zero, so the relative change there is not one per cent — it is unbounded, in the sense that any nonzero addition to zero is a complete change of the value.

Whether it matters is a dynamics question and this field cannot answer it. What the geometry establishes is that the cam’s law is preserved in shape and not in its second derivative, that the departure is concentrated at mid-lift rather than spread over the event, and that its size is governed by $\rho'$ — the rate at which the ratio changes — rather than by the ratio’s total variation.

Which gives the shimming argument a third justification alongside the two already made. Centring the swing makes $\rho'$ smallest through the middle of the event, where $(s')^2$ is largest, so the two factors of the new term are minimised where they multiply hardest. An uncentred setup does the reverse: it puts the steepest part of the ratio curve at the fastest part of the motion.

And it explains why a cam profile designed for a direct-acting follower is not simply transferable to a rocker train at the nominal ratio. The law that costs least through a rocker is not the law that costs least at the follower, because the rocker adds a term the cam designer did not put there — and correcting for it means designing the profile against $f$ rather than against the lift.

What the model leaves out #

Valve lash. A solid-lifter train has a deliberate clearance in it — a few thousandths of an inch — so the first part of the cam’s rise takes up the gap and the valve does not move at all. The site’s practice field has exactly the machinery for that: a clearance is a short link with a free direction, and the lost motion it produces is computable. Adding it here would move every number in this essay by an amount comparable to the shortfall being measured, and it is not added.

Compliance. A pushrod is a long thin steel rod under load and it shortens. At high engine speed the whole train deflects enough that the valve’s motion differs from the geometric one by more than everything measured here — which is why serious valve-train analysis is dynamic and this essay is not.

The cam-to-lifter contact. The lifter is treated as a point following a prescribed displacement; a real one is a flat or a roller with its own contact geometry, and the flat follower’s contact offset is the subject of the essay after this one. That one is geometry, and it is computed.

One more omission is worth naming because the site has the machinery and chose not to use it here. The valve train has a spring, and a spring is why the follower stays on the cam at all — the cams field’s whole treatment of pressure angle and base circle is about a follower being pushed, and a valve is pulled shut by a spring whose force is the only thing keeping the train in contact. Whether it does at a given speed is the classical valve-float question, and it is a dynamics problem: the geometry says where the valve can be and the spring says whether it is there.

What is here is the linkage, solved: the ratio a rocker actually delivers at each position, the lift the valve actually reaches, and the fact that both are properties of a mechanism with a setup rather than of a number on a box.