The dip that buys the dwell

The flattest dwell is not the longest built a six-bar on the vertex of a coupler curve that is its own mirror image. Symmetry removes every odd term from the curve’s distance to its osculating circle, and one angle at the rocker pin — $\gamma^* = 120.8024°$ — removes the fourth-order term as well, so a dwell link pinned at the centre of curvature holds its far end still to sixth order with no search anywhere in the design.

Then it asked the question a designer would ask of that machine, and got an answer that did not fit the premise. For a dwell that has to stay within a stated band — one per cent of the output’s swing, three tenths, one tenth — the longest dwell does not come at $\gamma^*$. It comes a few degrees below: 4.7°, 3.2° and 2.2° below for those three bands. The flattest machine is not the one to build.

Three numbers invite a law, and that essay proposed one from them. The gaps fell by about 2.1 for each factor of ten in the band, which is close to $10^{1/3} = 2.15$; the dwells grew by 1.54 per decade where $10^{1/6}$ is 1.47. The proposal was a two-term picture — a fourth-order term proportional to how far $\gamma$ is from $\gamma^*$, plus a fixed sixth-order term — and it left the test to a measurement across four or five decades of band, with the question of where the picture breaks down.

That test is made here, across three and a half decades, and it turns out the two-term picture is better than a law of exponents. Solved exactly, it predicts the exponents, the constants in front of them, and one pure number that has no machine in it at all.

The model, solved exactly #

Near the vertex, measure the crank angle $s$ from the vertex and the output from its value there. By symmetry the output’s departure has only even terms, and at angles near $\gamma^*$ the first two are all that matter:

$$f(s) = a\,s^4 + b\,s^6.$$

$b$ is fixed by the machine. $a$ is nought at $\gamma^*$ and changes in proportion to $\gamma^* - \gamma$, with its sign opposite to $b$’s on the side the best angles were found. A band of width $\varepsilon$ means the output’s largest and smallest values over the dwell may differ by at most $\varepsilon$, and the dwell is the widest interval of $s$ around the vertex for which that holds.

At $\gamma^*$, $a = 0$, the output leaves its vertex value monotonically on each side, and it leaves the band when $b s^6 = \varepsilon$:

$$s_0 = (\varepsilon / b)^{1/6}.$$

Below $\gamma^*$, $a$ has the opposite sign to $b$. Take $b$ positive and $a$ negative. The output first goes down, as $a s^4$, then turns as the sixth-order term catches up, and comes back up through its vertex value. Its lowest point is at $s^2 = -2a/3b$, where $f = 4a^3/27b^2$, and it returns to nought at $s^2 = -a/b$.

Everything between the vertex and that return lies between the dip and nought, so the band used there is $4|a|^3/27b^2$. A dwell can be no longer than the return, since past it the output climbs above its vertex value while its minimum is still the dip’s. So the longest dwell comes from making the dip exactly as deep as the band allows:

$$\frac{4|a|^3}{27b^2} = \varepsilon \;\Rightarrow\; a^* = -\left(\tfrac{27}{4}\right)^{1/3} b^{2/3} \varepsilon^{1/3}, \qquad s^* = \sqrt{-a^*/b} = \left(\tfrac{27}{4}\right)^{1/6} \left(\frac{\varepsilon}{b}\right)^{1/6}.$$

That the optimum sits there, and not at some compromise in which the output overshoots its vertex value a little, can be checked by letting the dip be shallower or deeper and asking how long the dwell becomes; a numerical optimisation of the same two-term model over the fourth-order coefficient lands on $a^*$ and $s^*$ to five figures.

Three predictions, and one of them has no machine in it #

The solution predicts three things.

The best angle’s distance from $\gamma^*$ grows as the cube root of the band, because $a^*$ does and $a$ is proportional to that distance.

Both dwells grow as the sixth root, since $s_0$ and $s^*$ both do.

Their ratio is a constant, $s^*/s_0 = (27/4)^{1/6} = 1.37473$, and nothing about the six-bar, its link lengths, its output arm or its tracing point enters it. Any dwell whose error is dominated by a fourth- and a sixth-order term, with the fourth free to be tuned, gains the same factor by being detuned optimally, at every tolerance.

The first two are what the earlier essay guessed from three numbers. The third is new, and it is the one that makes a sharp test, because a pure number is harder to fit by accident than an exponent.

Measured on the machine #

The measurements use none of the model. For each band, the output of the six-bar is followed through a whole turn in closed form at 14,400 crank angles, the longest interval inside the band is found by a sliding window, and the angle at the rocker pin is scanned in steps of a tenth of a degree and then refined in steps of a hundredth to find the longest dwell. Eight bands are measured: 3%, 1%, 0.3% and 0.1% of the swing, then 300, 100, 30 and 10 parts per million.

The predictions use only two numbers from the machine. The output’s even part is read at crank angles 3° and 6° either side of the vertex, which separates $a$ and $b$; $a$ is read that way at half a degree either side of $\gamma^*$ to give its rate of change with $\gamma$. Nothing about any band goes into them.

Over the four tightest bands the fitted exponents are 0.333 for the best angle’s offset, 0.172 for the dwell at $\gamma^*$ and 0.173 for the longest dwell, against a third, a sixth and a sixth.

The ratio is the cleaner test. Measured at the eight bands, from loosest to tightest, it is 1.3450, 1.3647, 1.3731, 1.3755, 1.3778, 1.3744, 1.3723 and 1.3728. From a tenth of a per cent down, every value is within 0.2% of 1.37473, which is the resolution of a dwell measured in steps of a fortieth of a degree on a dwell of thirty to eighty degrees. The two loosest bands fall short, and the reason is the same one that makes the exponents come out at 0.172 rather than 0.167: a wide band reaches far enough from the vertex that the eighth-order term, which the model leaves out, is no longer small.

The numbers, and where the model stops #

The exponents and the ratio could all be right with the constants wrong. The prediction table says they are not.

At a tenth of a per cent the best angle is measured 2.260° below $\gamma^*$ and predicted 2.264° below it. At ten parts per million it is measured 0.490° and predicted 0.488°. The dwell at $\gamma^*$ is predicted 3.8% short at a tenth of a per cent and 0.73% short at ten parts per million, and the error falls by about a factor of 1.5 for every factor of three in the band.

That answers the question of where the picture breaks down. At three per cent the dwell at $\gamma^*$ is 113° measured against 100° predicted, a 14% miss, and the best angle’s offset of 6.8° is predicted within 4%. So the two-term model is a design tool to about one per cent of the swing and a quantitative theory below a tenth of a per cent. Above one per cent a dwell of more than a hundred degrees of crank reaches so far round the curve that the model’s premise — that two terms describe it — is simply false, and the measured dwells are longer than predicted because the next term happens to help.

What a sixth root costs a designer #

Exponents are abstract until they are turned into the question a specification actually poses, and the sixth root turns into a harsh one.

A dwell that grows as the sixth root of the band doubles only when the band grows by a factor of $2^6 = 64$. Read the other way, a machine that dwells 80° of crank inside a tenth of a per cent of its swing dwells 40° inside sixteen parts per million, and 20° inside a quarter of a part per million. Tightening a dwell’s tolerance by a factor of ten costs a third of its length at every tolerance; tightening it by a factor of a thousand costs two thirds. There is no band at which precision becomes cheap, because the law has no scale in it.

The ordinary fourth-order vertex — a point whose path curvature is momentarily stationary and no more — is worse on the same arithmetic — its dwell halves for every factor of sixteen — and that is the whole quantitative case for the symmetric construction. The detuning cannot change either exponent. What it does change is the constant, and at 1.37 it is worth the same as loosening the band by a factor of $1.37^6 = 6.75$. A designer who has the choice between a more expensive output measurement and a machine detuned by 2.26° gets the same dwell either way, and the detuned machine costs nothing.

In time rather than angle, the numbers are small. At 600 rpm a crank turns 3.6° per millisecond, so the best dwell inside a tenth of a per cent — 80.8° — lasts 22 ms, and the sixth-order machine’s 58.7° lasts 16 ms. A dwell six-bar in a packaging machine is asked, as a Geneva wheel is, to hold a part still for a stated time while something else acts on it, and the extra 6 ms is the difference between an operation that fits and one that does not.

How the dwell is measured, and why it does not need the model #

The dwell is measured so that no assumption about the output’s shape can leak into it. The output angle is followed continuously through a whole turn of the crank — each position found in closed form as two circle intersections, one for the four-bar and one for the dwell dyad, with the branch chosen by nearness to the previous position and the angle unwrapped so a full turn of the output is not read as a jump. The longest run of crank angle inside the band is then found by a sliding window with two monotone queues holding the running maximum and minimum, over the turn written out twice so that a dwell straddling the starting angle is not cut in half.

Nothing in that procedure knows where the vertex is or that the output is even about it. If the detuned output did not dip and return, or if the dwell were not centred on the vertex, the window would report whatever the machine actually does. That independence is what makes the agreement between a two-coefficient model and a sliding window at 14,400 samples a test rather than a restatement.

What the optimum looks like #

The solution also says what the best machine’s output does, and it is easier to see than to derive.

At $\gamma^*$ the output sits at its vertex value and leaves it as the sixth power on both sides, crossing out of the band at ±29.4°. It never uses the lower half of the band at all: every value it takes during the dwell is on one side of its starting point.

At the best angle, 2.26° below, the output first moves the other way, reaches exactly one band’s depth at two symmetric points, turns, and comes back through its vertex value at ±40.4°. The dwell ends there, at the moment the output has used the whole band twice over and returned to where it started. Detuning spends the half of the band that the sixth-order machine leaves empty.

Dragging the band through three decades shows the same picture narrowing: in units of the band the dip is always exactly one band deep and the return is always at the vertex value, and in crank angle the whole figure shrinks as the sixth root of the band. That is the ratio seen directly — the same shape, rescaled, at every tolerance.

It is also a familiar move in a different costume. A Chebyshev approximation spreads its error evenly between its extremes instead of concentrating the fit at one point, and the error curve of the best approximation touches the band’s edge alternately. The detuned dwell does the same with the only two coefficients it has: it touches the band’s lower edge twice and its vertex value three times.

What sixth order is worth, and from which tolerance #

The earlier essay’s case for the symmetric curve was that sixth-order contact is better than fourth. The law says how much better and says that “better” depends on the band.

An ordinary vertex, where the curve’s error is fourth order, dwells until $a s^4 = \varepsilon$, so its dwell grows as the fourth root of the band rather than the sixth. Measured on a six-bar built the same way on a vertex of a curve whose rocker has been made 2% long, the fitted exponent is 0.263.

A smaller exponent means a dwell that shrinks more slowly as the band tightens, which is the side that wins at tight tolerances. At three per cent the ordinary vertex dwells 113.6° and the symmetric machine at $\gamma^*$ 113.3°: level. At ten parts per million the ordinary vertex dwells 14.1°, the sixth-order machine 26.4°, and the detuned symmetric machine 36.3° — two and a half times the ordinary one.

So the symmetric construction buys nothing at a loose band and buys a factor that grows without limit as the band tightens, and the detuning multiplies it by 1.37 at every band. A designer working to a few per cent should not pay for symmetry at all.

What this does not settle #

It is one machine. The ratio $(27/4)^{1/6}$ has no machine in it, but the two coefficients that set the offset and the dwells do. A different symmetric curve has different $a$, $b$ and $\partial a/\partial\gamma$; the laws transfer and the constants must be read off again, which takes two evaluations of the output.

The band is on the output angle. A dwell specified as a band on the dwell link’s far pin position, or on a follower driven from the output, is a band on a different function of the crank angle, with its own $a$ and $b$ and the same exponents.

Tolerances on the links are not modelled. Every length here is exact. A real six-bar’s lengths carry manufacturing error, and the offset from $\gamma^*$ that the best dwell needs at a hundredth of a per cent of the swing is half a degree — a tolerance on an angle at a rocker pin that a real part would struggle to hold. The design is then a question about which band the manufacturing can deliver, and the right angle as a tolerance is the nearest measurement of that kind.

The asymmetric case is untouched. An ordinary vertex has odd terms as well, and a fifth-order term tilts the dip; the optimal detuning of a curve without symmetry is a two-coefficient problem with a different answer.

Still open: the dwell whose error has an odd term #

Every result here rests on symmetry killing the odd terms. The searched dwell six-bar of a dwell made from a curve was found on a curve with no symmetry, and its error near the best stretch has a third- or fifth-order part that tips the output one way.

Its distinct argument would be the same optimisation with $f(s) = a s^4 + c\,s^5 + b s^6$ and the dwell no longer centred on the vertex. Two things would come out of it. Whether the best dwell still spends its band as one dip returning to its starting value, or whether the odd term makes the optimum lopsided, with the dwell starting at the band’s lower edge and ending at its upper one; and whether there is a closed-form ratio in that case too, depending on one dimensionless combination $c/\sqrt{ab}$, which would let a designer decide from three coefficients whether chasing symmetry is worth more than detuning a curve that has none.