The cam that cannot be cut

A cam is a shape that produces a prescribed motion, and the usual reason a designer is told to make it bigger is the pressure angle: the steeper the angle between the follower’s direction of travel and the normal to the cam’s surface, the more of the driving force goes into pushing the follower sideways in its guide rather than along it, and past about 30° the follower jams — the cam equivalent of a linkage’s transmission angle, and decided by the same question of how much of an applied force turns into useful output.

That is one constraint. There is a second, it is less often mentioned, and it can bite first.

Two curves, one of which is the part #

A roller-follower cam has two curves and it matters which is which.

The pitch curve is where the roller’s centre travels. It is the base radius plus the follower’s displacement, plotted in polar coordinates against the cam angle, and it is the curve the motion specification directly produces.

The cut profile is the actual surface of the metal. It is the pitch curve offset inward by the roller’s radius, along the pitch curve’s own normal at every point.

An offset curve is a well-behaved object as long as the offset distance is smaller than the radius of curvature. When it is not, the offset curve turns itself inside out: neighbouring normals cross before they have gone the offset distance, the “inner” curve loops back on itself, and the region it encloses has been visited twice.

Physically, that means the cutter has removed material at one part of the cycle that the profile needs at another. The cam has a cusp or a loop where it should have a smooth flank, the follower does not do what the specification asked, and no amount of care in manufacture recovers it. The condition is

$$\rho > r_{\text{roller}}$$

on every convex stretch of the pitch curve, and it is hard rather than advisory.

Why the failure is invisible in the specification #

A designer working on a cam sees a displacement diagram: lift against cam angle, with the rises and dwells laid out. Everything about the specification lives there, and the site’s motion-law essay is entirely about what that diagram implies for the derivatives.

The curvature failure is not in that diagram at all.

The displacement diagram is a plot in Cartesian coordinates of lift against angle. The pitch curve is the same data plotted in polar coordinates about the cam’s axis, and the transformation from one to the other brings in the base radius, which is nowhere in the specification. Two cams with identical displacement diagrams and different base circles have different pitch curves with different curvatures, and one may be cuttable and the other not.

So the check cannot be done on the specification. It has to be done on the part, after the base circle has been chosen, and it has to be redone every time the base circle changes. That is an ordinary engineering fact and it is worth stating because the pressure angle has exactly the same property and is checked routinely, while the curvature is not.

Curvature in polar form, with a factor of 3,283 in it #

The pitch curve is given as r(θ) — radius against cam angle — so the radius of curvature comes from the polar formula

$$\rho = \frac{(r^2 + \dot{r}^2)^{3/2}}{r^2 + 2\dot{r}^2 - r\,\ddot{r}}$$

with the derivatives taken with respect to the cam angle in radians.

That last clause is not pedantry. The cam library reports its derivatives per degree, because a cam programme is specified in degrees and every figure’s axis is in degrees, and substituting per-degree derivatives into the formula above puts ρ out by a factor of 57 in the first-derivative terms and 3,283 in the second-derivative one. The result is a number, it is smooth, it varies plausibly with the base radius, and it is wrong by two orders of magnitude.

The sign is kept rather than taken as an absolute value, and that is deliberate too. A negative ρ is a concave stretch of the pitch curve, where the roller sits on the outside of the bend and can be as large as it likes; only the convex stretches limit the roller. Reporting |ρ| would lose exactly the distinction that decides whether there is a problem.

Checking it against the drawn curve #

The formula is one route. The other fits a circle through three neighbouring points of the pitch curve that the profile generator actually produces, and compares the fitted radius with the formula’s.

They agree to about 10⁻⁵ relative. What is interesting is what happened when the check was asked to demonstrate that it converges.

The three-point fit does not get better with more points. Sampling twice as finely halves the truncation error and simultaneously makes the three points more nearly collinear, so the circumcentre is computed from a difference of numbers that agree to more digits. Measured, the disagreement went the wrong way — 8.3 × 10⁻⁵ at 1,440 samples and 3.7 × 10⁻⁴ at 2,880 — and a check written to require convergence failed on a curvature that was right.

Conditioning wins over truncation here. So the check requires both densities to agree with the formula and reports which agreed better, rather than assuming which will. The negative case carries the test instead: the same comparison against a curve drawn on a base circle 20% larger disagrees by 0.2, which is four orders of magnitude worse, so the comparison is capable of noticing a difference.

That is a general point about convergence tests and it is worth carrying. A check that asserts a specific error behaviour is asserting something about the numerics, not about the thing being measured, and it can fail on a correct answer. The centrode rolling check on this site does assert convergence and is right to, because there the truncation genuinely dominates; the difference is that it was measured rather than assumed in both cases.

Where the curvature is worst #

The formula gives ρ at every cam angle, and where its minimum falls is worth knowing because it is not where intuition puts it.

On a dwell the pitch curve is a circular arc of radius equal to the base radius plus whatever the lift is there, so ρ is exactly that radius — large, and never the problem.

On a rise the pitch curve is spiralling outward, and its curvature depends on how hard the follower is accelerating. The tightest bend is where the follower’s acceleration is most negative relative to the radius: at the top of a rise, where the follower is decelerating to a stop, the curve is bending most sharply toward the centre.

That is the counter-intuitive part. The place a cam is most likely to be uncuttable is not where it is steepest but where it is levelling off, and a designer looking at a displacement diagram for the dangerous region will look at the wrong end of the rise.

It also means the choice of motion law affects the curvature limit directly. A law with a large peak deceleration — cycloidal, which peaks highest of the standard three — bends the pitch curve more tightly at the top of its rise than one with a gentler peak. So the law that is chosen for smoothness costs something in manufacturability, which is one more entry in the ledger that essay keeps.

Which constraint binds first #

Now the two limits together.

At the smaller roller the two floors change places. The curvature stops binding, the pressure angle becomes the constraint that sets the size, and which of the two is doing the work is a fact about the design rather than about cams.

Both constraints are cured by the same medicine — a bigger base circle — and for an ordinary roller they are never in competition. Finding a case where the curvature binds first took searching, and what the search found is worth the essay.

For a twenty-millimetre lift and a roller of eight or twelve millimetres, the pressure angle is always the tighter of the two. Every base radius that undercuts has already failed on pressure angle by a wide margin, so the curvature limit never decides anything and a designer who checks only the pressure angle is, in practice, safe.

It is the large roller that changes which constraint binds. And large rollers are chosen for a reason that has nothing to do with kinematics: the contact between roller and cam is a higher pair, so the load is carried on a line rather than a surface, the contact stress is high, and the standard way to reduce it is to make the roller bigger.

So the situation in the figure — a modest lift, a roller two-thirds the size of the base circle — is not contrived. It is what happens when a stress calculation and a kinematic one are done by different people. The first row has a peak pressure angle of 21.2°, comfortably inside the 30° rule, and a minimum radius of curvature of 19.4 against a roller of 34. The cam cannot be cut, and nothing in the pressure-angle check says so.

The parallel with gears #

The word “undercutting” is doing double duty on this site and the two uses are worth putting side by side, because they are the same phenomenon.

A gear tooth is undercut when the generating rack reaches below the point where the involute begins, and removes material the flank needs. A cam is undercut when the generating cutter, offset by the roller radius, reaches across a tight bend and removes material the profile needs.

In both cases the failure is in the generating process rather than in the shape, and in both cases the desired shape is perfectly reasonable — it is the sweeping of a tool that destroys part of it. The distinction shows up in what happens if the part is made another way: a gear form-milled with a cutter shaped like the tooth space is not undercut whatever its tooth count, and a cam ground with a wheel much smaller than the design roller has no self-intersection to worry about. What breaks in each case is the generation, and the rule is a rule about hobbing and about milling rather than about geometry. The desired shape exists; it is the sweeping of a tool that destroys part of it. In both cases the condition is a comparison between a radius of curvature and a tool dimension. And in both cases the fix is to make the thing bigger — more teeth or more shift on a gear, more base circle on a cam.

The difference is what “bigger” costs. A gear can be shifted without changing its ratio, so the fix is nearly free. A cam’s base circle cannot be changed without changing the cam’s size, its inertia, its surface speed and the space it occupies, so the fix is paid for directly.

The cutter is not the roller #

One more distinction that catches people, and it is the reason the constraint is sometimes described as avoidable.

The cam is used with a roller of some radius. It may be cut with a tool of a different radius — a milling cutter, a grinding wheel — and the two need not match.

If the cutter is smaller than the design roller, the cut profile can be produced without the cutter itself self-intersecting. What cannot be produced is a profile that the design roller will follow correctly, because the offset relationship between pitch curve and profile is a statement about the roller, not about the cutter. The metal comes out looking like the intended profile; the follower still does not do what the specification asked, because at the tight bend the roller cannot reach into the profile far enough.

So the condition ρ > r_roller is about the follower, not the manufacturing tool, and no choice of cutter relaxes it. The word “undercutting” is misleading here in a way it is not for gears: the gear failure really is the cutter’s doing, and this one would happen even if the cam were carved by hand to the exact intended shape.

What the smallest base circle is #

Since the constraint is a hard one, the useful quantity is the smallest base radius at which the cam can be cut, and it is found by bisection on the base radius using the curvature test itself — so the answer is a property of the curve that gets drawn rather than of a rule of thumb.

For the figure’s programme and a 34-unit roller it is 35.95. Below that, no cam. Above it, a cam that can be manufactured and whose pressure angle was never in question.

A designer given that number has the information the two-constraint picture is for: the base circle is bounded below by the manufacturing limit and, separately, by whatever the pressure angle requires, and the binding one is whichever is larger. Quoting only the pressure-angle bound is the common practice and it is safe for small rollers and wrong for large ones.

Fixing it without growing the cam #

Since a larger base circle is expensive, it is worth listing what else can be done, because a designer whose only lever is size will accept a cam twice as large as it needs to be.

Use a smaller roller. The constraint is ρ > r_roller, and the roller is the term the designer usually has most freedom over. What it costs is contact stress, which rises as the roller shrinks, so this trades a kinematic failure for a wear one.

Spread the rise over more degrees. The curvature at the top of a rise depends on the follower’s acceleration there, and acceleration scales as lift divided by span squared. Doubling the span quarters the acceleration and slackens the bend considerably. What it costs is cycle time, or dwell, and often there is none to give.

Change the motion law. A law with a lower peak deceleration bends the pitch curve less. This is genuinely free in the sense that it costs no material and no time — and it is not free at all in the sense that it changes the accelerations the follower and everything attached to it experience, which is what the law was chosen for in the first place.

Offset the follower. Moving the follower’s line of travel off the cam’s centre changes the pressure angle asymmetrically, improving it on the rise at the expense of the return. It affects curvature less directly but it can buy enough pressure-angle margin to allow a smaller base circle, which then has to be re-checked against the curvature.

The general shape of the problem is that all four levers move both constraints, in different proportions and sometimes in opposite directions — which is why the two limits are worth computing separately and plotting together rather than folded into a single rule.

What this does not compute #

Three honest limits, in keeping with the rest of the site.

The curvature test says whether the offset curve is self-intersecting. It does not say whether the cam is strong — the contact stress at the roller depends on the curvatures of both bodies and on the material, and that is not kinematics.

It assumes the roller is a circle rolling on the profile without slipping in the plane, which is the standard idealisation. A real roller has a bearing with clearance in it, and the clearance shows up as lost motion at the follower.

And the whole analysis is of the pitch curve’s curvature, which is the right thing for a roller follower and the wrong thing for a flat-faced one. A flat-faced follower has its own condition — the cam must be convex everywhere, or the flat face bridges a concavity and never touches it — and that is a different test on the same curve. This essay’s figures use roller followers throughout, and the captions say so.

The two size constraints being different reasons is the rung’s point and the combination it names is the useful part. A cam can pass the pressure-angle check comfortably and fail the curvature one, so a design validated on the first alone is validated on the wrong criterion — and the failure is invisible in the specification, because a profile with a curvature radius below the roller’s is drawn perfectly happily and cut into something else. Two independent conditions on one dimension, with different causes and different consequences, and the binding one changes with the motion programme. That is why the base radius cannot be chosen by a rule of thumb: the rule of thumb encodes whichever constraint bound on whatever mechanism it came from, and it is silent on the other. The honest procedure is the one this field runs — compute both margins across the whole programme, take the smaller, and report which one it was — because the answer and the reason for it are both design information.