The law that costs least is not the smoothest

Four ways to lift a follower by 20 units over 120° of cam rotation, prescribed rather than derived from a linkage’s geometry. All four do the same job and their displacement curves are hard to tell apart.

Their derivatives are not.

The four laws #

Uniform velocity rises linearly. The velocity steps from zero at the start, so the acceleration is an impulse — infinite in the ideal case, and in a real follower whatever the stiffness of the train permits. Used only at very low speed.

Parabolic, or constant acceleration, accelerates for half the rise and decelerates for the other half. It has the lowest possible peak acceleration for a given lift and time, which is a genuine optimality result. Its acceleration steps at the start, the midpoint and the end, so its jerk is impulsive three times.

Simple harmonic is a half cosine — the motion a slider-crank is wrongly said to produce. Its acceleration is a cosine — smooth in the middle, and stepping at both ends.

Cycloidal is a straight line minus a sine. Its acceleration is a full sine, going to zero at both ends.

The ordering, measured #

Peak acceleration for the same rise, differentiated from the generated profiles:

The measured and closed-form columns agree to eight digits for parabolic and cycloidal, and to four for harmonic — two independent routes to the same quantity, one from the sampled curve and one from the formula.

And the ordering is the point: the smoothest law has the largest peak acceleration. Cycloidal peaks 57% above parabolic. Choosing the smooth law costs 57% more inertia force, and choosing the gentle one costs an impulsive jerk.

Reading the derivatives #

The four laws differ in which derivative is allowed to be discontinuous, and saying it that way makes the hierarchy obvious rather than a list to memorise.

Uniform velocity has a step in velocity. Its acceleration is therefore an impulse: infinite in the ideal case and, in a real follower, whatever the stiffness of the train permits before something yields.

Parabolic has a step in acceleration, at the start, the midpoint and the end. Its jerk is impulsive three times per rise.

Simple harmonic has a step in acceleration at the two ends only. Its middle is smooth, which is why its displacement curve looks like the gentlest of the four and why it is the one most often misapplied.

Cycloidal has no step in acceleration anywhere. Its acceleration is a full sine that reaches zero at both ends, so its jerk is finite.

Each step up that ladder buys one more continuous derivative and costs peak acceleration, and the whole of cam design is choosing where on it to stop.

Why the peak has to rise #

The trade is not a coincidence of these four particular functions, and the reason is a constraint on the area under the acceleration curve.

The follower starts at rest and ends at rest, so the acceleration’s integral over the rise is zero — it must decelerate as much as it accelerates. And the displacement is fixed, which fixes the double integral.

Given those two constraints, the profile that minimises the peak is the one that holds the maximum for as long as possible: constant acceleration, then constant deceleration. That is the parabolic law, and its optimality is exact rather than approximate.

Any law that softens the corners — and softening the corners is precisely what buys continuity of the higher derivatives — must therefore spend less time at the peak, and so must have a higher peak to cover the same area. The smoothest law has the largest peak because it has to.

Which one is right #

It depends on what the rise connects to, and the deciding case is arriving at a dwell — the intermittent motion that a Geneva mechanism produces by geometry instead.

A dwell is stationary, so acceleration there is zero. A law that arrives with acceleration still non-zero produces a step, and a step in acceleration is an impulse in jerk — a shock the whole mechanism feels, which excites every resonance in the follower train.

So: for a rise that meets a dwell, cycloidal, and pay the 57%. For a rise that meets another rise, or for a low-speed mechanism where inertia is not the concern, parabolic or harmonic, and take the lower peak.

Simple harmonic is the one most often misapplied, because it is smooth in the middle and stepped at the ends, and its displacement curve looks like the smoothest of the three.

Differentiating a sampled curve #

The accelerations above are central differences on the generated displacement, which introduces a choice: the step size.

A central difference has truncation error falling as $h^2$ until round-off takes over and the error rises as $1/h^2$. Picking $h$ by eye is how people end up differentiating noise, so both ends are checked: halving the step must improve the answer, at every step size used. Measured, the error falls from 3.7 × 10⁻⁵ at $h = 0.4$ to 9.1 × 10⁻⁸ at $h = 0.02$, quartering with each halving — which is second-order convergence, and confirms the differencing is still in the regime where it is measuring the curve rather than the noise a badly posed measurement returns.

The laws not covered here #

Real cam design uses more than these four, and the extras exist to break the trade rather than to accept it.

Modified trapezoidal rounds the corners of the parabolic acceleration profile, keeping most of the low peak while removing the impulses. Polynomial laws specify boundary conditions on displacement, velocity, acceleration and jerk and solve for the coefficients — a 3-4-5 polynomial matches acceleration at both ends, a 4-5-6-7 matches jerk as well.

The pattern is that each additional derivative made continuous costs peak acceleration, and the design question is how many to buy.

What is not modelled, and why it decides the answer #

Everything on this page is the geometry of a prescribed motion. Nothing here has mass, and mass is what makes the peak acceleration matter.

The force needed to accelerate the follower is its mass times the acceleration, so the peak acceleration sets the peak force — and the peak force decides whether the follower stays in contact with the cam. A spring holds it down; if the required deceleration exceeds what the spring can supply, the follower lifts off and lands again, which is an impact the whole train feels and which is the dominant failure mode of a high-speed cam.

That is why the 57% penalty for cycloidal motion is a real cost rather than a bookkeeping one. It means a 57% stronger spring, or a 57% heavier preload on every bearing in the train, for the same lift and speed.

And it is why the jerk matters separately. Jerk is the rate of change of acceleration, so it is the rate of change of force — which is what excites the resonances of a follower train that is, unavoidably, elastic. A finite jerk does not remove the vibration; it removes the impulse that would excite every mode at once.

Neither of those quantities appears in any figure here. This site computes what the profile demands; whether the mechanism can supply it is a dynamics question and the one that actually decides whether a cam works.

Laws beyond the four #

Real cam design uses more than these four, and the additions exist to break the trade rather than to accept it.

Modified trapezoidal rounds the corners of the parabolic acceleration profile with sinusoidal segments, keeping most of the low peak while removing the impulses. It is the usual industrial default and is a compromise designed by inspection of exactly the plot above.

Polynomial laws specify boundary conditions on displacement, velocity, acceleration and jerk and solve for the coefficients. A 3-4-5 polynomial matches acceleration at both ends; a 4-5-6-7 matches jerk as well, at a further cost in peak.

The pattern is that every additional derivative made continuous costs peak acceleration, and the design question is how many to buy — which is exactly the question the choice of mechanism poses one level up.

Polynomial laws, and why the exponents are odd #

Beyond the four classical laws, the standard modern approach is to write the displacement as a polynomial in the normalised cam angle and choose its coefficients to satisfy boundary conditions. It is worth seeing why this always produces the same family.

At each end of a rise there are conditions to satisfy: displacement, velocity, and acceleration at minimum, jerk if the follower train is stiff enough to care. Two ends, three conditions each, gives six constraints, which needs six coefficients — a fifth-degree polynomial, the 3-4-5 polynomial, whose name comes from the exponents that survive.

Extend to jerk at both ends and there are eight conditions and a seventh-degree polynomial: the 4-5-6-7. Each additional derivative matched at the boundaries costs two degrees and raises the peak acceleration, which is the same trade the four classical laws show in a cruder form.

The reason to prefer a polynomial over a cycloidal law is not the peak acceleration — cycloidal is competitive — but the freedom. A polynomial’s coefficients can be chosen against any set of boundary conditions, including asymmetric ones, and the classical laws cannot: they are what they are, and a rise that must start with zero acceleration and end with a specified non-zero one is outside their vocabulary.

The reason to be careful with them is that a high-degree polynomial fitted to many conditions can behave badly between the boundaries. Nothing in the constraints controls the interior, and a 4-5-6-7 with poorly chosen normalisation can have an interior acceleration excursion larger than the boundary values suggest. That is the polynomial-interpolation problem in a mechanical costume, and the defence is the same: plot it and look, or better, measure it and assert.

Why the comparison had to be made periodic #

The measurement behind this essay was wrong the first time, and the way it was wrong is instructive enough to record.

The four laws were first compared on a rise-and-dwell programme: rise through some angle, dwell through the rest. The peak accelerations came out at 5.0 × 10⁴ per degree² — for all four, identically, which is not a result any theory predicts.

The number was the seam. A programme that rises and then dwells has a discontinuity in displacement or in one of its derivatives at the join unless the join is constructed carefully, and the difference quotient across that join dominated everything the laws themselves were doing. Four assertions failed at once on the same artefact, which was the clue: independent checks do not usually fail together unless what they share is the problem.

The fix was to require the programme to be periodic — total displacement over a full cam revolution must be zero — and to construct the test programmes as rise-and-fall pairs that satisfy it. With that in place the four laws separated in exactly the predicted order, and the peak accelerations became the numbers this essay quotes.

A second, smaller version of the same error lived in the dwell-join check, which measured the maximum of a quantity over a window near the join instead of its value at the join. That reported a ratio of 1.6 where the correct measurement gives 33 — a check that was technically measuring something, and not the thing whose name it carried. Both corrections are the same lesson, and it is the one the gear-mesh test learned as well: a number appearing where a number was expected is not evidence that it is the right number.

The ordering as a general result #

The four laws rank in a fixed order — uniform worst, then parabolic, then harmonic, then cycloidal — and the ranking is not specific to cams. It is an instance of a trade that appears wherever a motion has to be specified.

The quantity being traded is smoothness against peak magnitude. A motion that gets from rest to rest in a given time and distance has an average acceleration that is fixed by those numbers alone. Whatever the law does, the area under the acceleration curve is determined; all the law chooses is the shape.

Making the acceleration continuous at the ends — which is what buys finite jerk — forces it to start at zero, which means it must be larger somewhere in the middle to enclose the same area. That is the whole mechanism of the ordering, and it says the peak must rise as more derivatives are made continuous. It is not a property of the particular functions chosen.

The same trade governs polynomial motion laws, where each extra boundary condition costs two polynomial degrees and raises the interior peak. It governs motion profiles in servo control, where S-curve profiles buy finite jerk with higher peak acceleration than trapezoidal ones. It governs the Geneva mechanism’s compromise, which achieves continuous velocity at entry and pays with a step in acceleration.

So the design question is never “which law is smoothest” — cycloidal, always — but which discontinuity the machine can afford. A slow, heavily damped, lightly loaded mechanism can take a jerk step and save the peak acceleration. A fast one with a spring-returned follower cannot, because the peak negative acceleration is what decides whether the follower stays on the cam at all.

Both ends of that answer are numbers this site computes rather than describes, which is the only reason the comparison is worth making twice.