A lift is a size and a law is a shape

The cam field’s central table ranks four motion laws by what they do to the follower: uniform, parabolic, harmonic, cycloidal, with peak velocity and acceleration coefficients and a verdict about jerk.

That table is printed once and used for every cam anybody builds, and it is worth asking why it can be.

A law has no dimensions in it #

A motion law is a function from a fraction of the rise span to a fraction of the lift. Both arguments are dimensionless and so is the value.

Cycloidal motion is s/h = τ − sin(2πτ)/2π, where τ is the fraction of the way through the rise. There is no length in that expression and no time. It describes a shape of motion and nothing else.

That is not how a law is usually written down, and the difference matters. Written as a displacement in millimetres against a shaft angle in degrees, a law looks like a curve about one cam; written normalised, it is visibly one curve for all cams. The field’s own library stores the second form and multiplies by the lift on the way out, which is the right decision and is worth noticing as one.

The lift h and the span β are what turn it into a displacement: s = h · f(θ/β). The law is the f; the lift is a length and the span is an angle.

So everything the field says about laws is a statement about f, and f is dimensionless.

What that makes transferable #

The peak velocity coefficient, the peak acceleration coefficient, whether the jerk is finite or impulsive, and the ranking of the four laws against each other are all properties of f.

None of them changes when the lift changes, when the base circle changes, or when the whole cam is scaled. That is why the table is printable once, and it is why the field quotes coefficients rather than accelerations: the coefficient is the transferable number and the acceleration is what it becomes on a particular cam at a particular speed.

The site’s own table does this correctly and has always done. Constant-acceleration motion has a peak acceleration coefficient of 4 and cycloidal has 2π; those are pure numbers, they are the same on every cam, and multiplying by h/β² gives the actual acceleration on one.

It is worth noticing how much rests on that. A designer choosing a law is making a decision that will be right for every cam they ever cut, and the four-row table they consult was computed once by somebody who did not know what machine it would be used on. That transferability is exactly what a dimensionless quantity buys, and the same table written in millimetres per second squared would be a table about one cam and would be useless.

The field already separated the shape from the size and put the separation in its units, without ever saying that is what the separation is.

The span is an angle and behaves like one #

A third parameter sits with the law rather than with the lift and it is worth putting there explicitly.

The rise span β is an angle: how much of a cam rotation the rise occupies. It is dimensionless, so a scaling leaves it alone, and it belongs in the shape column.

That gives the law’s arguments a clean split. A programme is a list of segments, each with a law, a span and a rise. The laws and the spans are shapes; the rises are lengths.

So a cam programme is a dimensionless pattern — which laws, over which fractions of a turn — multiplied by one length. Two cams with the same pattern and different lifts have followers whose motions are proportional at every instant, and everything the field says about the pattern applies to both.

That is the sense in which a cam is a shape and a lift, and it is why a programme’s periodicity check — that the rises cancel the returns — is a condition on the pattern with no size in it.

What is not transferable #

The other column, and it is where the design decisions are.

The lift is a length, exponent one. The base circle radius is a length. The profile’s radii are lengths. The face width a flat follower needs is a length — 19.10 for a cycloidal rise and return of lift 10, which is a number about a cam of that lift.

The profile’s curvature is a reciprocal length, exponent −1, like everything in the curvature field. A bigger cam has gentler profile curvatures everywhere.

And the actual acceleration is a length per time squared, so its exponent depends on what is held constant — a cam scaled up and run at the same shaft speed has proportionally larger accelerations, and one run at the same surface speed does not.

The pressure angle is the interesting one #

Between the two columns sits the quantity the field cares most about, and it belongs in the shape column for a reason worth working out.

The pressure angle is the angle between the follower’s motion and the contact normal. It is an arctangent of ds/dθ divided by the base radius plus the displacement — a ratio of a length to a length, so dimensionless.

Scale the cam and both scale, and the pressure angle is unchanged. So the pressure angle curve through a rotation is a shape, transferable between cams of any size.

What it depends on is the ratio of the lift to the base circle. Double the lift and hold the base circle and the pressure angle rises; double both and it does not.

So the design variable is a ratio and the field has been treating it as two numbers. A cam with a lift of 10 on a base circle of 20 and one with a lift of 5 on a base circle of 10 have identical pressure-angle behaviour, and the site’s figures quote both lift and base circle where one ratio would do.

What jerk is, dimensionally #

The field’s sharpest result deserves the treatment because it is where a reader might expect the argument to break.

Constant-acceleration motion has an impulsive jerk and cycloidal does not. Jerk is the third derivative of displacement with respect to time, so it has a length and a time cubed in it and is plainly not dimensionless.

The claim about it is, though. Whether the third derivative of f is bounded or contains a delta is a property of f, which is dimensionless, and it is unchanged by multiplying f by any lift and by rescaling its argument by any span.

So the field’s verdict — this law is smooth and that one is not — transfers between cams of any size and any speed, and the magnitude of the jerk does not.

A qualitative claim about a dimensionless function is scale-free even when the quantity it is about is not. That distinction runs through the whole scaling survey and this is its clearest instance: an unbounded quantity, a bounded quantity, and a statement about which is which that has no units in it.

The same applies to every other verdict in the table. Which law has the lowest peak acceleration coefficient, which has the lowest peak velocity, which is continuous in its second derivative — all statements about f, all transferable, all quoted as coefficients rather than as accelerations.

What a measurement of the follower recovers #

Apply the field’s own question to a cam that exists.

Measure the follower’s displacement as the cam turns. That reading is a length, so the size is visible and the lift comes back.

Measure the follower’s displacement as a fraction of its total travel — which is what a normalised measurement gives — and the reading is dimensionless. The law comes back exactly; the lift does not; and neither does the base circle, because the base circle affects the profile and not the follower’s displacement at all.

That last is worth pausing on. A translating follower’s motion does not depend on the base circle. Given a law and a lift, the follower does the same thing whatever radius the cam is cut on; the base circle decides the profile’s shape, the pressure angle and whether the profile undercuts, and none of that reaches the follower’s displacement.

So a measurement of the follower recovers the law and the lift and says nothing whatever about the base circle. That is a genuine unidentifiability of a different kind from a scaling: not a direction the readings cannot span, but a parameter the observable does not contain.

A third kind of unidentifiability #

The base-circle result deserves its own section because it is a kind of unrecoverability the survey has not met before.

The two kinds so far are a null direction — a continuous family the readings cannot distinguish, found by a rank — and a discrete ambiguity — several isolated answers, found only by a search.

This is neither. The base circle is not in a null direction of the follower-displacement model, because it is not in that model at all: the follower’s displacement is h·f(θ/β) and no base circle appears. Its column of the identification Jacobian is identically zero, not small.

A zero column is technically a null direction, and calling it one loses what is interesting. A null direction is a combination of parameters that cancels; a zero column is a parameter the observable does not mention. The first is discovered by a decomposition and the second by reading the model.

A parameter that does not appear in the observable is a modelling error rather than an identifiability finding, and the right response is to remove it from the list rather than to report it as unrecoverable. The parameter-list essay’s rule covers exactly this: a column at the noise floor should not be in the model.

What makes it worth an essay is that the base circle is unmistakably a property of the cam, so leaving it out of a model of the cam feels wrong — and it is right, for a measurement of the follower. A measurement of the profile is a different observable and the base circle is in it.

The undercut condition is a shape #

One more quantity, and it lands where the pattern predicts.

A roller follower cannot track a concave stretch of profile whose radius is smaller than the roller’s. That is a comparison of two lengths, so the condition is a comparison of a ratio against one and is dimensionless.

Scale the cam and the roller together and the condition is unchanged. Scale the cam and hold the roller — which is what happens when a standard roller is used on cams of different sizes — and it changes, because the ratio changes.

A design that undercuts at one size may not at another, and which is a question about the ratio of the roller to the cam rather than about either. The field computes an undercut margin at one size, and whether that margin transfers depends on whether the roller scales with the cam.

That is the same structure as the transmission angle’s tolerance: a dimensionless quantity compared against something chosen independently, and the comparison moving with size while neither the geometry nor the choice does.

Reverse-engineering a cam #

The measurement question has a practical form and the answer is more useful than it sounds.

Somebody has a cam and wants to know what it is: what law, what lift, what base circle, what spans. The available measurements are the follower’s motion and the profile itself.

From the follower’s motion: the law exactly, as a normalised curve; the lift, if the displacement is measured with a rule; the spans, from the crank angles at which the segments begin and end. Not the base circle.

From the profile: everything, because the profile is a curve in space and contains the base circle as its smallest radius on the dwell.

So the cheap measurement — watch the follower — recovers the design intent and not the manufacture, and the expensive one recovers both. That is a clean division and it is the reverse of the usual expectation, since the follower’s motion is the thing the cam exists to produce.

A cam’s purpose is fully measurable and its geometry is not, from the observation anybody would make first.

A conjugate pair behaves differently #

A cam and a conjugate follower — where one profile is given and the other computed — is the field’s other object, and the scaling question has a different answer there.

The conjugate profile is derived from the given one and the centre distance. Scale both profiles and the centre distance and the conjugacy still holds, so the construction is a similarity like the curvature field’s.

What is not scale-free is the relation between the two profiles at a fixed centre distance. Enlarge one profile and hold the centres and the pair no longer meshes; the centre distance is a length that has to scale with the profiles.

So the conjugate construction takes three lengths — two profile scales and a centre distance — and the condition relates them, exactly as Bennett’s condition relates lengths and twists. A ratio, in other words, and the free parameter is one common size.

A parameter the observable does not contain #

Two ways a parameter can be unrecoverable have appeared on this site already and this field has a third, which is worth separating from them because the three fail differently and are diagnosed differently.

The scaling kind is a direction in parameter space along which nothing moves: the parameters are all there in the model, the observable depends on all of them, and one combination happens to cancel. It shows up as an exact zero in a singular value, it is found by looking at a spectrum, and it is repaired by measuring one length.

The discrete kind is several separated parameter vectors that fit equally well — three cognate linkages, or the wrong assembly. No spectrum shows it, because every one of the answers is a perfectly ordinary well-conditioned minimum, and it is found only by starting the fit somewhere else and seeing where it lands.

The third kind is the base circle of a cam with a translating flat-faced follower, and it is neither. The observable does not contain the parameter at all. The follower’s displacement is the lift curve and nothing else; the base circle is added to it to place the contact, and it cancels out of the displacement exactly. There is no direction in which the parameters trade off, because the parameter never enters. A column of the identification Jacobian is not small — it is zero, at every pose, for every cam.

The diagnosis is different in a way that matters. A scaling shows up in a rank computed from data. A discrete ambiguity shows up under a multi-start. A parameter absent from the observable shows up in neither, because a fit that includes it simply reports whatever it started with and a fit that omits it converges beautifully. The only thing that finds it is reading the model, and the reason it is easy to miss is that the parameter is unquestionably real: it is on the drawing, the cam cannot be cut without it, and a modeller listing the machine’s parameters would list it without hesitating.

That suggests a habit worth having before any calibration is planned. Take each parameter in turn and ask whether the observable depends on it at all — not how strongly, but whether the symbol appears once the model is written out. It costs a few minutes, it needs no data, and it catches the one failure that no amount of data catches. A pose plan optimised over a parameter the readings cannot see is optimising over a coordinate the problem does not have.

And a roller follower does contain it, which is the useful contrast: swap the flat face for a roller and the base circle enters the contact geometry, the column stops being zero, and the parameter becomes ordinary. The unrecoverability was never a property of the cam. It was a property of what was being watched.