A flat face asks for a convex cam

Two essays in the cams field say the same sentence about a flat-faced follower and neither computes it. The first cam essay says the flat face “removes the pressure angle entirely” and is “paid for with a different constraint — the profile must be convex everywhere, or the flat face bridges a concavity and never touches the bottom of it”. The essay on the cam that cannot be cut closes by noting that its whole analysis is of the roller’s pitch curve, that a flat face “has its own condition”, and that “that is a different test on the same curve”.

This essay runs the test. It turns out to be a single line of arithmetic with a clean derivation, a second route that can check it, and a consequence that the sentence in those two essays does not suggest: the flat face and the roller rank the motion laws by different derivatives of the same motion, so a law chosen for one follower is not automatically the right law for the other.

Why the pressure angle is zero #

A flat face touches a convex cam at one point, and the cam’s surface there is tangent to the face. The contact normal is therefore perpendicular to the face, and if the face is perpendicular to the follower’s guide — the ordinary arrangement — the normal points straight along the guide at every angle of the cam.

That is the whole of the flat face’s appeal. A roller follower’s pressure angle is the angle between the push and the guide, and it grows with the follower’s velocity and shrinks with the cam’s size, so the roller’s cam is sized first by keeping that angle under about 30°. A flat face has no such angle to keep under anything. Its cam could, as far as the push is concerned, be as small as the lift allows.

It cannot, and the reason is the other thing a flat face cannot do. A roller can roll into a hollow in the cam’s surface, as long as the hollow is wider than the roller. A straight edge cannot enter a hollow at all. It rests on the two high points either side and bridges the gap, and the follower sits wherever those high points put it rather than where the programme asked.

So a flat-faced cam must be convex everywhere, and the question is how that condition depends on the programme and on the base circle.

The radius at the contact is one line #

A flat face’s cam is an envelope: the curve that touches every position of the face as the cam turns. Put the cam’s centre at the origin and the face above it, at height $R_0 + s$ at cam angle $\theta$, where $R_0$ is the base circle and $s$ the follower’s lift. The contact sits at a distance $s'$ from the follower’s axis, where $s'$ is the lift’s rate per radian — the result that sized the face in the essay before this one.

So in the fixed frame the contact is at $(s',\; R_0 + s)$, and in the cam’s frame it is that point turned back by $\theta$. Differentiate the turned point with respect to $\theta$ and two things happen at once. The contact moves along the face, because $s'$ changes at the rate $s''$; and the whole picture turns under the face, which moves the contact sideways by its height $R_0 + s$ and upward by its offset $s'$. The upward parts cancel exactly, because the face stays tangent. The sideways parts add.

What is left is that the contact slides along the face at $R_0 + s + s''$ per radian while the face turns one radian relative to the cam. A point moving along a curve at speed $v$ while its tangent turns at one radian per radian is moving on a curve of radius $v$. So

$$\rho = R_0 + s + s''$$

with $s''$ per radian squared. That is the radius of curvature of the cam’s profile at the contact, and the convexity condition is that it stays positive.

The derivation says something the formula alone hides. When $\rho$ goes negative the contact does not merely find a tight bend; it slides backwards along the face. The envelope doubles back on itself in a small swallowtail, which is a curve no solid can have, and the part of it the face would need to touch is inside the metal it is supposed to be the surface of.

The same radius off the drawn curve #

The formula is short enough that a mistake in it would be short too, and the site’s cam library has a documented history of exactly this: a unit slip that put $\rho$ out by a factor of 3,283, and an offset whose sign was wrong for six months while every check looked at a different curve. So the radius is measured as well.

The second route draws the profile — the envelope of the face, sampled at 14,400 cam angles — and fits a circle through three neighbouring drawn points at each angle of interest. It uses no $s''$ and no formula for curvature. On a base circle of 30 the two agree to $6.6 \times 10^{-8}$ relative at every angle checked, and on a base of 8 to $7.2 \times 10^{-7}$.

And the refusal. Hand the formula $s''$ per degree squared rather than per radian squared — the unit slip the library once made — and it disagrees with the drawn curve by 57% of the radius. That makes every cam look far more convex than it is, which is the dangerous direction for an error about convexity to take.

There is a third check, and it is the one that connects this essay to the roller. A flat face is what a roller becomes as its radius grows without limit, and the roller’s rule is that the cut profile’s radius is the pitch curve’s less the roller’s. Take a roller of radius 100 on a prime circle of 130 and that difference is within 1.6 of $R_0 + s + s''$ for a base of 30; at a radius of 1,000 it is within 0.19, and at 10,000 within 0.019. The gap shrinks tenfold for every tenfold growth in the roller, which is what the one term the flat formula drops — the square of the pitch curve’s radial rate, divided by twice its radius — says it should. The roller’s condition and the flat face’s are one condition, and the flat face is its limit.

Where each law bends the cam tightest #

The programme throughout is the one the face-width essay used: a rise of 10 over 120°, a dwell of 60°, a return of 10 over 120° and a dwell of 60°. On a base circle of 8 the figure shows $R_0 + s + s''$ for three laws, and each has its least value in a different place for a different reason.

Cycloidal acceleration is a sine, so $s''$ is smooth and the radius dips smoothly. Its least value is 2.668, and it falls at 212.4° — 32.4° into the return, where the follower has only dropped from 10 to 8.88 and is already decelerating hard. That combination of a high follower and a large negative $s''$ is what makes the curve dip, and the dip is interior to the stroke.

Constant acceleration has $s''$ that is constant and positive for the first half of each stroke and constant and negative for the second, with a step in the middle. The radius therefore jumps down at the step and then recovers as the follower rises. Its least value, 3.889, is right at the step, at 60.0°, where the follower is exactly halfway up.

Simple harmonic motion has its largest deceleration at the very end of the rise, where the follower has reached full lift, and then drops to zero on the dwell. The radius is least just before the dwell, at 6.750, and it then jumps up to $R_0 + 10$ the moment the dwell begins.

Those two steps deserve a sentence of their own. A step in a law’s acceleration is a step in the profile’s radius of curvature, and a cam surface whose radius jumps from 28.75 to 40 on a base of 30 — as the harmonic cam’s does — is convex and perfectly cuttable, and it is also a surface along which the contact stress jumps. That is a wear question and not a kinematic one. The geometry puts it on the drawing where it can be seen.

The smallest convex base, two ways #

Since $\rho = R_0 + s + s''$ must stay positive, the smallest base circle is the negative of the least value of $s + s''$ anywhere on the turn. For the programme above that gives 1.250 for harmonic motion, 4.111 for constant acceleration and 5.332 for cycloidal motion.

The second route does not use the formula. It bisects on the base circle, draws the envelope at each trial, and asks one question of the drawn polyline: does any step of it run backwards along the face it is touching? A convex cam’s contact slides one way along the face as the cam turns; a swallowtail makes it double back. It returns 1.250, 4.103 and 5.332.

The cycloidal and harmonic values agree to five decimal places. The constant-acceleration value is 0.008 low, and the gap has a definite cause: that law’s least radius is at a step, so just below the true threshold the stretch where the radius is negative is a sliver whose width depends on the follower’s velocity at the step, and 5,760 samples per turn can land either side of it. The route finds the threshold to within a sample, which at that step is 0.008.

The first version of the second route did not ask about direction, and its failure is worth the paragraph because the conjugate-pair essay found the same class of error in the roller’s offset: a construction that looks right in every picture and is wrong in one number. It asked whether the drawn polyline ever turned the wrong way, and it reported 5.05 for the cycloidal law against the formula’s 5.332. Just below the threshold a swallowtail is a few hundredths of a degree across and its two cusps turn sharply in the right direction, so the turning test saw two large turns of the correct sign and missed the short backward step between them. A test on direction of travel cannot miss that step, because it is first order in the radius itself.

The roller reads velocity and the flat face reads acceleration #

Now set the flat face’s demand beside the roller’s. The table adds two columns: the width of face each law needs, which is the range of $s'$, and the prime circle a centred roller needs to keep its pressure angle under 30°, found by bisecting on the pressure angle the cam library has always computed.

Both followers rank the three laws in the same order — harmonic least demanding, cycloidal most — and it would be easy to stop there. The ratios say something else. Cycloidal motion needs 1.053 times the constant-acceleration law’s prime circle for a roller, and 1.297 times its base circle for a flat face. On the roller the two laws are nearly tied. On the flat face the cycloidal law costs almost a third more.

The reason is which derivative each condition reads. A roller’s pressure angle is $\arctan(s'/(R_0+s))$, so what sizes its cam is the peak velocity, and the cycloidal and constant-acceleration laws have exactly the same peak velocity: twice the average, $2h/\beta$. That is also why they need the same face width, 19.10. A flat face’s convexity is $R_0 + s + s''$, so what sizes its cam is the peak deceleration where the follower is high, and there the two laws differ by the famous factor: cycloidal motion peaks at $2\pi$ against constant acceleration’s 4, which is the 57% the motion-law essay measures.

So the ledger that essay keeps gains a line. Choosing the cycloidal law for its finite jerk costs 57% in peak acceleration, and on a roller cam that cost is invisible in the cam’s size because the roller does not read acceleration. On a flat-face cam the same choice is paid for in metal.

A faster rise widens the gap #

The separation is not an accident of one programme. Take the quick-rise programme from the offset essay — a rise of 20 over 90°, a dwell of 40°, a return over 170° and a dwell of 60° — and read the same columns.

The roller needs a prime circle of 34.08 under constant acceleration and 35.02 under cycloidal motion: a ratio of 1.027. The flat face needs a base circle of 22.401 against 32.853: a ratio of 1.467. A faster rise puts more of the programme’s difficulty into its acceleration, which scales as the lift divided by the span squared, while the velocity scales only as the lift divided by the span. Both demands scale with the lift itself, so as the scaling essay would predict the ratios between laws are shapes and survive any change of size. The flat face feels the square and the roller feels the first power.

The harmonic law is the cheapest for both followers on both programmes, at 26.06 for the roller and 20.000 for the flat face on the quick rise. It gets there by accepting a step in acceleration at the start and end of every stroke, which is why the motion-law essay does not recommend it for a rise that ends in a dwell. A designer who wants a small flat-face cam and smooth motion is being asked for two things that pull against each other more sharply than they do on a roller.

What a face does on a cam that is not convex #

Below the smallest base the programme still defines an envelope, but the envelope is not the boundary of any solid. The question for a real follower is what it does instead, and it has a definite answer: the face rests on the highest point of whatever was cut in its direction, which is the convex hull of the envelope.

On the cycloidal programme with a base circle of 2, the drawn envelope doubles back into a swallowtail near the top of the rise and again near the start of the return. At 58.5° of cam rotation the programme asks for a lift of 4.750. The face cannot reach the contact the programme specifies, because that contact is inside a hollow, and it rests instead on the hull’s bridging edge at 5.196. The follower is held 0.446 high.

The magnified panel is not a flourish. On a cam 24 across, a hollow 0.45 deep is a few pixels at the scale the whole cam is drawn, which is exactly the size of defect that looks like nothing on a drawing and is a quarter of a millimetre on a part. The window on the right is placed round the specified contact and magnified five times, and the edge that bridges the hollow is drawn clipped to that window rather than left out because its ends are outside it.

Held high, never low #

The realised lift is computed at every one of 1,440 cam angles by turning the drawn profile under the face and taking its highest point. It departs from the programme over 177.5° of the turn, by up to 0.446, and on a base of 3 by up to 0.218 over 145°.

It is never below the programme, and that is a geometric certainty rather than a feature of these numbers. The point the programme specifies is always on the envelope, so the highest point of the envelope can only be at or above it. A flat face on a non-convex cam cannot be pushed into a hollow, only across one. The error is one-sided, and it is always an overshoot of the lift in the stretches where the programme is decelerating.

That makes the failure unusually easy to misread in a test. A follower that rises slightly too far near the top of every stroke and returns late at the start of every return looks like a follower with a stiff spring or a sticky guide. It looks like dynamics. It is geometry, it was decided when the base circle was chosen, and no spring or lubricant will change it.

And on a convex cam the same computation returns the programme exactly: on a base of 6, just above the cycloidal limit, the worst departure is $7 \times 10^{-15}$. The check that the face delivers its programme is not a check calibrated to fail; it passes precisely where the formula says the cam is convex and fails precisely where it says it is not.

Reconciling this with the face #

The face-width essay found that the face must span the range of $s'$, and that the base circle does not enter that width at all. It was careful to say that the independence is conditional on the profile existing, and this essay supplies the condition.

The two results are the two derivatives of one motion. The face’s width reads the velocity: 19.10 for the cycloidal programme, whatever the base circle. The cam’s existence reads the acceleration, together with the lift at the point where the acceleration bites: a base circle above 5.332. A flat-face design has to satisfy both. It needs a face wide enough for the fastest part of the motion and a cam large enough for the hardest deceleration, and neither number tells the designer anything about the other.

On the base of 30 the face-width essay drew, the cycloidal cam’s least radius is 24.67, far from zero. That is why the question never arose there, and why it would arise at once on any attempt to make that cam compact.

What a zero pressure angle leaves out #

The claim that a flat face has no pressure angle is exact and slightly misleading, and the misleading part should be stated.

The push is along the guide, but it does not act on the guide’s axis. It acts at the contact, which sits $s'$ to one side — 5.37 at the angle drawn in the first figure, and up to 9.55 during the stroke. A force along the guide applied off its axis is a force plus a moment, and the moment presses the follower’s stem against one side of its bearing and the other side at the opposite end. A flat-face follower can jam in a short guide by exactly that moment, with a pressure angle of zero throughout.

So the flat face does not eliminate the roller’s problem; it converts an angle into a lever arm, and the lever arm is the same $s'$ that sizes the face. A long guide tolerates it and a short one does not. None of that is modelled here, any more than friction is modelled in the pressure-angle essays. The geometry says where the lever arm comes from and how large it gets. The guide’s length decides whether that matters.

Nor is sliding. A flat face slides across the cam at every instant, as a roller does not, and it wears in proportion to that sliding and to the contact force. Both are outside a kinematic model, and both are the usual reasons a designer who could use a flat face chooses a roller anyway.

What comes next: a face that holds both ways #

A flat face introduces something a roller does not have: a face can have a twin. Two parallel flat faces, one above the cam and one below, joined into a rigid yoke, touch the cam on opposite sides at once. That drives the follower up with one face and down with the other, with no spring and no possibility of the follower leaving the cam.

The price of that is a condition on the programme rather than on the base circle, and it is the next question: a cam that holds its follower both ways must have the same breadth at every angle, which forces its second half-turn to be the mirror of its first. The convexity this essay computes is part of that condition too. A constant-breadth cam’s opposite radii add up to its breadth, and each has to stay positive.

Two directions are left open behind it. The oscillating flat follower, whose face swings about a pivot, has a contact offset and a convexity condition with the pivot’s distance in them, and neither has been derived here. And the moment the offset contact puts on a follower’s stem is a quantity this essay names and does not compute. It is the flat face’s counterpart of the pressure angle, and it deserves the same two-route treatment.