A cam that holds its follower both ways

The first cam essay sets out the choice every cam has to make about the return stroke. A cam can only push. Something has to bring the follower back, and either a spring does it — force closure, which fails at speed when the spring cannot supply the deceleration and the follower leaves the cam — or the geometry does it, which that essay calls form closure and prices as “a second cam surface, a second follower and an adjustment that has to be maintained”.

That price is quoted rather than computed. This essay computes the simplest form-closed follower there is: two flat faces, parallel, a fixed distance apart, joined into one rigid yoke that closes round the cam. The upper face is pushed up by the cam. The lower face is pushed down by it. There is no spring, and the follower cannot leave the cam because there is nowhere for it to go.

The result is a condition, and the condition is sharper than “a second surface”. It is a statement about the motion itself.

Two faces, one distance #

Put the cam’s centre at the origin and the follower’s line vertical. A single flat face above the cam rests at height $R_0 + s(\theta)$, where $R_0$ is the base circle and $s$ the programme’s lift at cam angle $\theta$. That is simply what a flat-faced cam is: the height of the face is the programme, and the cam is the envelope of every such face.

A second face below the cam touches the part of the cam that is pointing straight down. At cam angle $\theta$ that is the part that was under the upper face half a turn earlier, or will be half a turn later. So its distance below the centre is the programme read half a turn away, $R_0 + s(\theta + 180°)$.

The distance between the two faces is therefore

$$W(\theta) = 2R_0 + s(\theta) + s(\theta + 180°).$$

A yoke is rigid. Its two faces are the same distance apart at every instant, so a cam can drive it exactly — touching both faces at every angle, with nothing to spare and nothing forced — only if

$$s(\theta) + s(\theta + 180°) = \text{constant}.$$

That is the whole condition. It says nothing about the base circle, nothing about the motion law as such and nothing about the lift. It is a relation between the programme and itself, half a turn apart.

The distance measured across the drawn cam #

The identity is a statement about the programme. What a yoke actually meets is a piece of metal, so the breadth is also measured on the cam the figures draw.

The second route builds the cam’s profile as the envelope of a single upper face, sampled at 2,880 angles, turns that drawn curve to each of 360 cam angles, and measures how far it extends along the follower’s line from its highest point to its lowest. It never evaluates $s(\theta + 180°)$ and never uses the identity.

On the programme in the first figure — a cycloidal 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 30 — both routes say the breadth is 70 at every angle, and both say it varies by $2.8 \times 10^{-14}$, which is double precision reporting that nothing varies at all.

There is one condition on that agreement and it came up in the measuring. The extent of a drawn curve along a line equals the sum of its two support heights only if the curve is convex. The quick-rise programme below needs a base circle of 32.85 before a flat face’s cam is convex, as the essay before this one computes. On a base of 30 its drawn extent disagrees with the identity by 0.04, which is the bridged hollow of that essay turning up in this one’s measurement. So the comparison figure below is drawn on a base of 40, where every cam in it is convex and the two routes agree to 0.002.

Three programmes and one flat line #

The figure plots $s(\theta) + s(\theta + 180°)$ through a whole turn for three programmes this field has used.

The matched programme — 10 over 120°, dwell 60°, back over 120°, dwell 60° — is a flat line at 10. Its return begins at 180°, exactly half a turn after its rise began at 0°, under the same law and over the same span. At every angle, however far the rise has got, the return is exactly that far from finishing, and the two displacements sum to the lift.

The standing programme — a rise and a return of 20 over 120° with dwells of 30° and 90° — runs from 10.498 to 20.000, a variation of 9.502, and the drawn cam varies by the same 9.502. Its return begins at 150°, thirty degrees early. At 45° the rise has reached 5.249 and the point half a turn later, at 225°, is 75° into a return that has come down to exactly 5.249 too, so the sum is its least. At 120° the rise has just finished at 20 and the point half a turn later is on the low dwell at 0, so the sum is its greatest.

The quick-rise programme — 20 over 90°, a dwell of 40°, back over 170°, a dwell of 60° — varies by 6.857 by the identity and 6.856 across the drawn cam. It fails for a different reason. Its return is slower than its rise, so no half-turn shift can line them up at all.

So of three reasonable programmes, one can be held by a yoke and two cannot, and nothing about their displacement diagrams looked different in kind until they were compared with themselves half a turn apart.

The second half-turn is the first, reflected #

The identity can be turned round. If $s(\theta) + s(\theta + 180°) = C$ for every $\theta$, then

$$s(\theta + 180°) = C - s(\theta),$$

so whatever the follower does over the first half-turn, it must do upside down over the second. A constant-breadth cam has exactly one free choice, which is its first half-turn. The second is a consequence.

That is more restrictive than it sounds, and it can be read off as rules:

• The return must use the same law as the rise. A cycloidal rise of 10 over 120° with a harmonic return over the same span and the same dwells is otherwise a perfectly good programme. Its breadth varies by 1.154.
• The return must take exactly as long as the rise. The quick-rise programme above fails on this alone.
• The two dwells must be equal. A rise of 10 over 120°, a dwell of 50°, a return over 120° and a dwell of 70° misses by 1.657, although only one dwell moved by ten degrees.
• Beyond that, anything. A rise over 150° with two dwells of 30° has constant breadth, and so does the matched programme under the harmonic law. The law itself is free, as long as the return repeats it.

The consequence a designer feels is the second rule. A great many cams exist precisely because the working stroke and the return should take different times: a fast feed and a slow retract, or a quick return. None of those can be a constant-breadth cam, and the reason is not that the yoke is badly designed. It is that the yoke measures across the cam, and across the cam the two strokes are the same stroke.

Lost motion or interference, and the yoke only chooses which #

Suppose a yoke is closed round the standing programme’s cam anyway. There is still a width to choose, and the choice turns out not to matter in the way anyone would hope.

Make the yoke as wide as the cam’s greatest breadth, 80, so that it fits at every angle. At 120° both faces touch. At 45° the cam is only 70.498 across, so the lower face stands 9.502 clear of it, and over that distance the follower is driven by nothing. It is held up by the upper contact and held down by gravity, inertia or luck. That is lost motion, and it is a follower that is form-closed at some angles and not closed at all at others.

Make the yoke as narrow as the cam’s least breadth, 70.498, so that it never has lost motion. Then at 120° the cam is 80 across and the yoke is 9.502 too narrow. It cannot be assembled on the cam at that angle, and if forced on it will jam there.

Any width between the two splits the same 9.502 between the two failures. The yoke’s width decides where the programme’s asymmetry is paid, and never whether. The variation of the breadth is a property of the programme, and the only thing that removes it is a different programme.

A mechanism the count says is a structure #

There is a second way to see why the condition has to be exact, and it is one met before, in a different mechanism.

Count the mechanism. A frame, a cam and a follower make three links. The cam’s pivot and the follower’s slide are two lower pairs, each removing two freedoms, and the contact between the cam and a single face is a higher pair that removes one. Grübler’s count gives $3 \times 2 - 2 \times 2 - 1 = 1$: one freedom, which is the cam turning and the follower going where it is told.

Add the second face’s contact and the count is zero. By the count, a cam in a yoke is a structure and cannot turn at all.

It turns, on the matched programme, because the second contact is redundant. It removes no freedom the first did not already remove, and it can only be redundant if the geometry makes it so at every angle — which is the constant-breadth condition, again, arrived at from the other side. That is the shape of the mechanism Grübler says cannot move: an overconstrained chain that moves only because a dimension is exactly right.

It is also the shape of the seventh contact, which adds no rank to a part already held and is therefore the only contact in the set that can fail to touch. On the standing programme the yoke’s second face is exactly that contact. It is the one that stands 9.502 clear at 45°.

Opposite radii that share the breadth #

A constant-breadth cam has a property that is visible in its outline once it is pointed out, and it follows in one line from the flat face’s curvature.

The radius of curvature at a flat face’s contact is $\rho = R_0 + s + s''$. The radii at the two contacts of a yoke are that expression at $\theta$ and at $\theta + 180°$, and their sum is

$$2R_0 + \bigl[s(\theta) + s(\theta+180°)\bigr] + \bigl[s''(\theta) + s''(\theta+180°)\bigr].$$

On a constant-breadth cam the first bracket is constant, so its second derivative, which is the second bracket, is zero. The two radii sum to the breadth, exactly, at every angle.

On the matched cam the sum is 70.00 to $1.0 \times 10^{-7}$ through the whole turn, while each radius on its own swings between 24.67 and 45.33. Where the cam bends tightest on one side it is flattest directly opposite, by exactly the amount that keeps the two faces parallel. Every curve of constant width has this property, from the circle, whose opposite radii are equal, to shapes with corners. A constant-breadth cam is one of those curves built to order from a programme.

Convex, twice over #

The flat face’s own condition still applies, and the yoke doubles it.

A flat face needs the cam convex where it touches, so both contacts need $\rho > 0$. Because the two radii sum to the breadth, a positive radius on one side is the same thing as a radius less than the whole breadth on the other. So a constant-breadth cam is convex exactly when every radius lies between zero and the breadth.

For the matched cycloidal programme that holds on any base circle above 5.332, the same threshold the flat face alone needs. The yoke adds nothing to it. What it does add is a reading of the threshold. Below 5.332 the cam has a hollow somewhere, the face over it bridges the hollow and is held high, and the opposite face — rigidly attached — is dragged high with it and forced into the cam on the other side. On a single face a hollow is lost lift. On a yoke it is interference.

A second disc buys back the programme #

The way out of the reflection rule is the one the first cam essay names and prices in words: a second cam surface. Put two discs on the same shaft, side by side. The front disc drives the upper face and is the ordinary flat-face cam for the programme. The rear disc drives the lower face, and nothing requires it to be the same shape as the front one.

What shape it must be follows directly. The lower face is always $W$ below the upper one, and the upper one is at $R_0 + s(\theta)$. So the rear disc’s height in the downward direction must be $W - R_0 - s(\theta)$ at every angle. That is a support function, the rear disc is its envelope, and it exists for any programme at all. The standing programme’s asymmetry, which no single disc could carry, is carried by the difference between two discs.

It still has to be convex, and that is where the price appears. The rear disc’s radius at its contact is $W - R_0 - s - s''$, so the yoke must be wider than $R_0$ plus the greatest value of $s + s''$ anywhere on the turn. For the standing programme on a base circle of 30 that greatest value is 30.66, reached at 237.6° where the follower is low and accelerating hard upward. The yoke must be at least 60.66 wide.

At a width of 80, as drawn, the rear disc’s tightest radius is 19.34, and the span from the top of the front disc to the bottom of the rear one, measured across both drawn discs at every whole degree, is 80.000 with a variation of $4.3 \times 10^{-14}$. The yoke is held exactly, on a programme that cannot be held by one disc, by a shape nobody chose.

This is the principle of desmodromic valve gear in its plainest geometric form. An engine’s desmodromic mechanism uses rockers rather than a yoke, but the claim is the same: one surface for each direction, and a second surface computed from the first rather than designed.

Any programme, at a price in size #

The quick-rise programme is the harder test, because it fails the reflection rule on every count: a fast rise, a slow return, unequal dwells. On a base circle of 40 its front disc is convex. The rear disc needs a yoke wider than 92.85, and at 100 its tightest radius is 7.15 — much closer to the limit than the standing programme’s 19.34 at 80. The span across the two drawn discs is 100.000 at every whole degree.

The rear disc in that figure is visibly a different object from the front disc, lopsided in the other direction. That is the programme’s asymmetry made into metal. A single constant-breadth cam carries its asymmetry nowhere, because it is not allowed any. A twin carries all of it in the difference between two shapes, and the price is that the second shape has to be large enough to stay convex where the first one’s follower accelerates hardest.

So the two-disc cam is not free form closure. It is form closure bought with a second surface, a wider yoke and a larger rear disc. The size is set by the programme’s peak acceleration where the follower is low, the same derivative the flat-face essay found sizing a single flat face’s cam. Choosing a gentler law, as the motion-law comparison prices it, shrinks both discs at once.

What the spring was also doing #

A return spring is usually described as the cheap alternative to form closure, and the description leaves out two jobs a spring does that a yoke cannot.

It takes up clearance. A yoke on a constant-breadth cam must be exactly as wide as the cam’s breadth. Any working clearance is lost motion at every reversal of the contact force, and the follower crosses the gap and strikes the other face each time. A spring holds the follower against one surface and the clearance never opens.

It takes up wear. A cam and its follower wear, and a spring-held follower simply rides a fraction of a millimetre lower. A yoke’s breadth is cut into the metal and the yoke’s width into the frame, and wear on either face opens a gap that nothing closes. The two-disc cam has the same problem twice, with the added need to keep the two discs’ angular positions matched on their shaft.

None of that is modelled here, and neither is the elasticity of the yoke, the inertia of the follower, or the twisting a two-disc yoke feels when its two faces are pushed in opposite directions in two different planes. What the geometry settles is the condition that decides whether form closure is available at all, and its exact price in shape. A constant-breadth cam needs a programme that is its own reflection; a twin needs a yoke wide enough for the programme’s hardest acceleration; and neither tolerates the clearance a spring would have hidden.

What comes next: a roller held in a groove #

The third form-closed follower is the one the first cam essay mentions first: a roller running in a groove cut into the cam’s face. Its two walls are the pitch curve offset outward and inward by the roller’s radius, so they are the same distance apart along every normal whatever the programme. A groove needs no reflection rule and no second disc, and it is the form-closed cam most often actually built.

What it has instead is a crossover. A roller touches one wall or the other, never both, and it changes walls whenever the force the follower needs changes direction. At that moment the roller must reverse the direction it spins in and cross whatever clearance the groove has. Where those crossovers fall is decided by where the programme’s acceleration changes sign against the load, which is a kinematic question with a definite answer for each motion law. How many a cycloidal programme has, where they fall, and how the constant-acceleration law’s step changes them is the next question. It has not been computed.

Behind it is the roller version of this essay’s yoke. A yoke with two rollers instead of two faces also measures across the cam, but along the line of the rollers’ centres rather than along the follower’s line, and its constant-breadth condition includes the pressure angle. Whether it forces the same reflection rule is a question the offset essay’s formula could answer. The conjugate-pair construction would supply the second route.