Six things a cam drive is not

A cam gives the motion it was asked for, which prescribing motion called a trade rather than an improvement. The displacement becomes free and the derivatives stop being. Cam drives are old and common, from the valve train of every piston engine to the indexing tables and packaging machines of a factory floor, and they have gathered their share of things people say. Each of the six below is what a careful person would say on the evidence of one part of the drive, and each has been measured in the cam essays and found false, incomplete or misapplied.

What makes them worth putting side by side is that they fail the same way. That comes at the end.

One: an offset follower has a lower pressure angle #

The pressure angle is the angle between the follower’s line of travel and the direction the cam pushes it. Past about 30° a sliding follower starts to jam in its guide, since more of the push goes into the guide and less into the lift. The textbook remedy for a steep cam is to move the follower’s line off the cam’s centre. The offset appears in the formula for the pressure angle and can be seen to reduce it.

It reduces it on one stroke. An offset trades the rise for the return found that the offset is subtracted from the follower’s velocity inside the pressure angle, so the stroke whose velocity has the offset’s sign gains exactly what the other loses. On the standing programme, a rise of 20 over 120° and a return over 120° on a prime circle of 40, both strokes peak at 21.22° with the follower centred. Offset by 8, the rise falls to 12.84° and the return rises to 29.44°. Offset the other way and they swap. On any programme whose rise and return mirror each other, the best offset is nought to $6 \times 10^{-12}$.

The offset is worth something only when the strokes are unequal, which is the cam the figure draws. It rises in 90° and returns over 170°, peaks at 27.37° on its rise when centred, and an offset of 5.88 brings both strokes to 21.81°, by handing the gentle return 6.48° of pressure angle it did not need. Turned through the rise with the slider, the contact normal leans furthest from the follower’s line at 42.3° of the cam, and the lean there is the stroke’s peak. An offset is a way of sharing the pressure angle between two strokes, not a way of reducing it. The rule a designer can carry is to centre the follower unless the strokes differ, and then to offset it towards the stroke that needs help by whatever makes the two peaks equal, since the worse of the two is what jams the guide. It also costs something the formula does not show. On the dwells a centred follower has no pressure angle at all, while an offset one leans against its guide for the whole of every dwell.

Two: a flat face has no pressure angle, so its cam can be any size #

A flat-faced follower is pushed at right angles to its face, because the contact normal of a flat face against any curve is square to the face. So its pressure angle is nought at every angle, the constraint that sized a roller’s cam disappears, and the cam can seemingly be as small as the lift allows.

The size limit moves rather than vanishing. A flat face asks for a convex cam found what replaces the pressure angle. A face can touch only the parts of a cam that bulge towards it, and the cam’s radius of curvature at the contact is $R_0 + s + s''$. Where that would go negative the profile needs a hollow, and a flat face bridges any hollow and rides above it.

For a cycloidal rise and return of 10 over 120° the smallest base circle is 5.332, found by the formula and by bisecting on the drawn profile to five decimal places. The comparison with the roller is the interesting part. The roller’s size is set by a velocity, the largest $s'$, which the parabolic and cycloidal laws nearly share, so cycloidal needs only 1.053 times the parabolic law’s prime circle. The flat face’s size is set by an acceleration, the most negative $s''$, which the two laws do not share, so cycloidal needs 1.297 times the parabolic law’s convex base. The follower type changed which derivative sizes the cam.

Below that base the cam still turns and the follower still moves, which is what makes the mistake easy to miss. Cut on a base of 3, the face is held high wherever the profile would need a hollow, over 145° of the turn, by up to 0.218. It is never held low, because a flat face cannot be pushed into a hollow, only across one. So the error shows up as a follower that does not come all the way down between strokes. That is easy to mistake for a spring that is too weak, and it is not a spring problem.

Three: a swinging arm improves any follower #

Carry a cam’s roller on a swinging arm instead of a slide and the pressure angle obeys the offset follower’s formula, with the offset replaced by the distance of the roller’s line of motion from the cam’s centre, and that distance turns with the arm. An arm is an offset that grows with the lift found that a long arm can beat any fixed offset on a cam whose strokes are unequal, because it rebalances the pressure angle as it swings. A designer who has learned that arms are better for rollers will expect them to be better for flat faces too.

For a flat face the opposite happens, at every arm length. A flat face on an arm is worse found that the cam sees the sine of the follower’s rotation rather than the rotation, and differentiating a sine twice brings in a square of the follower’s speed that costs convexity. The smallest convex base circle is 10.66 on a slide. A pivot 20,000 away needs 10.71, one at 1,000 needs 11.61, one at 150 needs 18.46, and one at 90 needs 28.68. A pivot at 60 has no convex cam at any base circle at all. The slide is the arm’s limit as the pivot recedes, and it is also its best case. Standing the face off its pivot moves the swing down the sine and recovers part of the loss, from 28.68 to 26.14 on an arm of 90, but never all of it.

The claim fails because it generalises a result about a first derivative to a second. The arm helps a roller because a roller’s limit is its pressure angle, a slope. It hurts a flat face because a flat face’s limit is convexity, a curvature, and the arm’s geometry adds curvature.

Four: a groove holds its roller on one wall #

A spring-loaded follower can leave its cam when the follower’s deceleration asks for more force than the spring gives. A grooved cam holds the roller between two walls so that cannot happen, and it is natural to picture the roller running against the inner wall all the way round, with the outer wall as a guard.

A roller in a groove changes walls with the speed found that the groove chooses its wall by the sign of one force, the follower’s mass times its acceleration plus the load pressing it in, and the acceleration grows as the square of the speed. With 0.5 kg and 20 N, the quick-rise programme has no crossovers below 267.6 rpm, two up to 505.5 rpm, and four above. The standing programme begins crossing at 356.8 rpm. The claim describes a slow cam and a fast cam as if they were the same machine, and the groove holds its roller in the sense that the roller cannot leave the groove. It does not hold it on one wall.

The crossovers matter because a groove must have clearance, and the roller flies across it. The time a crossover takes measured how hard it lands: with an exponent in the clearance that the motion law decides, two-thirds where the force passes smoothly through nought and one half where it steps. Under a constant-acceleration law the crossovers are pinned where the law’s acceleration steps and arrive as steps of 3,200 N and 1,600 N at 3,000 rpm. The groove is quiet only below its first threshold, and the threshold is a speed, not a property of the cam.

The threshold also says what a designer can do about it. The first crossover comes where the follower’s largest deceleration, times its mass and the speed squared, first equals the load, so the threshold speed goes as the square root of the load over the mass. A groove whose follower is also spring-loaded towards the cam, with twice the load, crosses first at $\sqrt{2}$ times the speed, 378 rpm instead of 268 on the quick-rise cam. A lighter follower does the same. Neither removes the crossovers; each moves them out of the running range, which is the only way a groove is quiet.

Five: the smoothest law is the gentlest #

Cam designers are taught to prefer laws with continuous acceleration, and cycloidal motion is the standard example: its acceleration rises from nought and returns to it, so its jerk is finite everywhere. From there it is a short step to thinking of it as the gentle law, and of a law with steps in its acceleration as the harsh one.

It is the gentlest in one measure and the harshest in another. The law that costs least differentiated the drawn displacements and found constant acceleration peaking lowest, at $5.556 \times 10^{-3}$ per degree squared for this rise, with steps at its junctions. Cycloidal peaks highest, at $8.727 \times 10^{-3}$, 57% more, and arrives at the dwell with none. The displacement curves give no hint of either, and the choice between them is a choice of what to be harsh about. A cycloidal law is kind to anything that rings, since its acceleration never steps. It is unkind to anything whose force or stress goes with the peak acceleration: a spring that must hold the follower on, a bearing, the groove’s walls. On a light, stiff follower with a strong spring, constant acceleration may be the gentler cam.

Six: a camshaft on a timing chain runs the law it was cut to #

A cam is cut in cam angle and runs in time. The two are the same only when the camshaft turns steadily, and a camshaft behind a roller chain does not. A cam on a chain steps at every tooth found that the chain leaves its sprocket at a radius that shrinks and recovers as each pin seats, so the camshaft speeds up through every tooth and its angular acceleration jumps back at every seating.

That jump enters the follower’s acceleration as the law’s velocity times the shaft’s angular acceleration, so every law gets a step once per tooth, whatever its own smoothness. On eleven teeth the cycloidal and simple harmonic laws step by 0.392 of their peak, constant acceleration by 0.615. A cycloidal rise, chosen precisely because its acceleration never steps, gets four steps inside its 120° rise. The step falls as one over the tooth count, not faster, because the chain’s speed ripple shrinks as $1/n^2$ while its rate grows as $n$. A cam behind a bent joint found the same kind of term from a Hooke’s joint, where it moves the same cycloidal rise’s peak 35% up or 17% down at a 30° bend, depending only on where in the joint’s half-turn the rise is placed.

The harm is shared unequally among laws, and the difference is instructive. The step at a seating is the law’s velocity times the shaft’s jump, so it matters most where the law’s velocity and acceleration are both large at once. Measured on a 120° rise, the worst peak’s excess over a steady shaft falls from 38% at eleven teeth to 8.7% at forty for constant acceleration, from 16.5% to 3.1% for cycloidal, and from 7.7% to 0.5% for simple harmonic motion, whose peak acceleration sits where its velocity is nought. The designer’s levers are the tooth count, and a law whose peaks fall where it is moving slowly.

The law a cam is cut to is a function of cam angle. The follower receives it as a function of time, through whatever turns the camshaft, and any unsteadiness in the drive adds its own acceleration to the law’s.

And one thing a cam drive is #

The six share a shape, and it is a shape specific to cams. Each claim reads the drive at first order and the drive is decided at second.

The offset was read from the pressure angle, a slope, and its cost appeared as a slope on the other stroke. The flat face was judged by its pressure angle, and its limit turned out to be a curvature, $R_0 + s + s''$. The arm was judged by what it does to a roller’s slope and failed on a face’s curvature. The groove was pictured by where the programme’s displacement pushes, and its walls are chosen by the sign of an acceleration. The smooth law was judged by its jerk, and the stresses in the drive go with its peak acceleration. The chain was assumed to deliver the displacement in time, and it adds an acceleration of its own.

That suggests a working rule, stated as a rule rather than a result. A cam drive is designed from its displacement and must be checked in its second derivatives: the profile’s curvature against the follower, the follower’s acceleration against its spring, groove or load, and the camshaft’s own acceleration against the law’s velocity. A profile is an envelope found how long a broken curvature can go unnoticed when every check looks at the pitch curve and none at the surface that gets manufactured. The second derivative is where cam drives go wrong, and it is also where they are least often looked at.

What the six do not cover #

Dynamics of the follower train. Every number here is kinematic or quasi-static: the follower as a rigid mass on a rigid train. A real valve train has stiffness and rings, and the steps above excite it; how much depends on the train’s own frequencies, which are not measured.

Wear and lubrication. Pressure angle and curvature decide the contact’s geometry. The stresses and film thickness at the contact decide the cam’s life, and nothing here computes them.

Manufacture. A cam cut to a profile is a cam cut with a cutter of some radius, and a cutter, like a roller, cannot reach a hollow tighter than itself. The cam that cannot be cut found a combination where the pressure angle is comfortable and the cam still cannot be made; it belongs beside claim two and is not repeated here.

Still open: six things an indexer is not #

Cam drives now have their six. Intermittent drives have a field of their own measurements, from ratchets and escapements to Geneva wheels and their locks, and a set of things said about them that nobody has put to the test together: that a Geneva with more pins indexes more smoothly, that a finer ratchet is a more precise one, that a locked wheel is a still wheel, that a curved slot gives a smooth start.

The distinct argument would be six such claims about indexing mechanisms, each answered from the field’s own measurements. The curved slot’s step, which a curved slot moves the step found conserved between entry and exit, and a ratchet’s resolution, which is its pitch whatever the workmanship, are two already measured. The work would be finding the other four and stating each claim in the words a designer would use, and then asking whether indexers too fail in one characteristic way, as cam drives fail at second order.