A twist steadies what it cannot tighten

The clearance that is the seal measured a section. Two conjugate rotors cut back by the same amount leave exactly twice it between them at every angle of the mesh; two rotors that are not each other’s conjugates leave a clearance that swings from material to a hole and back within one lobe pitch, and no undersize and no centre distance repairs it. The second shape is not a choice, so a rotor run against the wrong one is run against a shape that was computed for something else. Both statements are about a plane through the machine, and a real blower’s rotors are not in a plane. They are twisted, so that no section along the shaft is at the same phase of the mesh as its neighbour.

That is usually explained as quietness, and it is. The question it leaves is arithmetic rather than acoustic: a twisted pair’s clearance is not one number but a whole family of them at once, and it is not obvious what the family does. The optimistic reading is that a twist averages a bad profile into a good one — that a pair whose section seals unevenly could be made to seal evenly by never letting its worst angle happen at more than one place along the rotor. The pessimistic reading is the opposite, and it turns out to be the true one.

Nothing new has to be meshed #

The reduction is the whole of why this is computable at all, and it is worth stating before any measurement rests on it.

A helical rotor is the planar rotor swept along the shaft while turning: the section at axial position $z$ is the planar profile turned by $\kappa z$, where $\kappa$ is the twist rate. Its mate is the planar mate turned by $-\kappa z$, because the pair runs at one to minus one and a helix of one hand demands a helix of the other. So at shaft angle $\varphi$ the section at $z$ is a pair of bodies in exactly the relative position the flat pair occupies at input angle $\varphi + \kappa z$.

The mesh therefore does not have to be solved again. Every clearance a twisted pair can present is a value the flat pair’s profile already takes, and the only thing the twist decides is which stretch of that profile is present at one instant. A rotor whose ends differ by $\tau$ lobe pitches shows a window $\tau$ periods wide, slid along the profile by the shaft angle. Nothing outside the flat profile’s own range can appear, and nothing inside it can be removed.

It is worth being clear about what that does and does not assume. It assumes the twist is uniform — a constant lead, which is what a hobbed or milled rotor has — and it assumes the section is the same shape at every $z$, which is what a swept profile means. It does not assume the profile is conjugate, symmetric, small, or anything else: the reduction is a statement about rigid motions and holds for a pair of arbitrary shapes. And it makes $\tau$, not the helix angle, the quantity the argument is about. A long rotor at a shallow helix and a short one at a steep helix with the same wrap behave identically here, which is the same insensitivity the gear field’s overlap ratio has, arrived at from the clearance instead of from the teeth.

The open area is a moving average, and it goes flat #

The seal between the two sides of a blower is a line running the length of the rotors, and the geometric quantity attached to it is the area open across it — the gap integrated along the shaft. Divided by the rotor’s length that is a mean clearance, which is directly comparable with the flat profile and is what the measurements below report.

Written out, the mean clearance at shaft angle $\varphi$ is

$$\bar g(\varphi) \;=\; \frac{1}{\tau P}\int_0^{\tau P} g(\varphi + s)\,\mathrm{d}s ,$$

with $P = 2\pi/n$ the lobe pitch and $g$ the flat pair’s profile. That is a moving average of a periodic function, and moving averages have a property worth the two lines it takes to say. Decompose $g$ into harmonics of the pitch. Averaging the $m$-th harmonic over a window $\tau$ periods wide multiplies it by $\sin(\pi m \tau)/(\pi m \tau)$ and shifts it by half the window. At $\tau = 1$ that factor is $\sin(\pi m)/(\pi m)$, which is nought for every whole number $m$ — not for the largest harmonic, not for the first few, but for all of them at once. What survives is the term with no $m$ in it, which is the mean.

So a wrap of exactly one lobe pitch makes the open area of the seal constant through the turn, whatever the profile is. The measurement agrees and it is not close: on the mismatched pair the mean clearance swings 4.757 through a lobe pitch when the rotors are flat, and 2.4 × 10⁻⁴ when they are wrapped one pitch. Two wraps give 3.8 × 10⁻⁴. Those residues are the profile’s own sampling and fall with it; the argument says they are nought.

The second route is the one worth having, because the first is an integral and an integral is easy to get right for the wrong reason. Taking the flat profile’s Fourier coefficients, attenuating each by its own window gain and summing them back up never mentions a window or a rotor at all, and it reproduces the measured curve — swing for swing, shape for shape — at every wrap tried. The two agree to 2.4 × 10⁻⁴ at one pitch and to within four hundredths of the profile’s peak-to-peak everywhere else.

Every whole pitch is a zero, and the fractions are not a ramp #

The integer wraps are the clean statement and they are not the useful one, because a rotor is as long as it is and a designer short of axial length is buying a fraction.

The fractions do not behave the way the word averaging suggests. Half a pitch leaves 54% of the swing, three quarters leaves 19%, and a quarter leaves 57% — so the first quarter of a pitch buys almost nothing and the third buys more than the first two together. Worse, the curve is not even monotone: between three tenths and four tenths of a pitch the swing rises, from 2.775 to 2.924. More twist is not uniformly better, and the interval over which it is worse is wide enough to fall into.

The window gain explains all of it, including the parts that look like noise. At half a pitch the fundamental is multiplied by 0.637 and the second harmonic by exactly nought; at three quarters the fundamental keeps 0.300 and the second harmonic returns with its sign reversed at 0.212. A profile with a sharp tightest place carries a great deal of its swing in harmonics above the first, and those are the ones a narrow window kills first — which is why the early fractions flatten the profile’s corners while leaving its overall rise and fall nearly untouched, and why the measured share at a quarter pitch, 57%, sits so far below the fundamental’s own 90%.

And the zeros repeat. Two whole pitches is a zero, three is a zero, and between them the swing humps back up to 0.866 at about one and a half. A pair wrapped one and a half pitches is worse than one wrapped a single pitch and is using half as much rotor again to be so. The design rule this produces is therefore sharper than more wrap is better: wrap to a whole pitch, and prefer one whole pitch to any fraction above it that is not also whole.

What the wrap moves is one statistic and not the seal #

Here is the half that reverses the optimistic reading, and it follows from the same reduction with no arithmetic at all.

A moving average moves no average. The mean of $\bar g$ over a turn is the mean of $g$ over a turn, exactly, at every wrap — and the measurements say so: the mismatched pair’s average clearance is −0.4085 flat, −0.4085 at a quarter pitch, −0.4085 at one pitch and −0.4085 at two. Negative, because this pair has more material than clearance over the pitch. The twist has not removed any of it. It has spread it evenly along the rotor and held it there.

The extremes are worse than untouched. For any wrap of a whole pitch or more the window covers the entire period, so at every shaft angle the tightest section anywhere along the rotor is the profile’s own tightest, −2.420, and the widest is its own widest, 2.337. A flat pair reaches its worst angle once a pitch and is clear of itself the rest of the time — its tightest section at its loosest instant is 2.337, a genuine gap. A pair wrapped three quarters of a pitch has already lost that: its best instant reads −2.417. The twist has not moved the worst place; it has made the worst place permanent.

For a pair whose profile crosses zero that is a change of kind and not of degree. Flat, the rotors touch once a lobe pitch and are apart otherwise, which is a knock — intermittent, at a known angle, audible, and survivable by a machine whose rotors are timed — which they must be, since the pair cannot drive itself and a Roots blower’s rotors are carried round by gears on their shafts rather than by each other. Wrapped, they are touching somewhere at every instant, which is a rub. The quantity a seal is specified at is its tightest place, and the tightest place has not improved by so much as the sampling.

The pair that had nothing to average #

The comparison that says what the twist is actually for is the one against a pair that was already right.

Two conjugate rotors cut back by the same amount have a clearance of exactly twice the undersize at every angle of the mesh — a flat profile, established before any of this, and specified where a tolerance belongs, because the tool is the definition and an undersize is one setting of one operation. A flat profile has no harmonics above the mean, so there is nothing for a window to attenuate, and the wrap does precisely nothing: the conjugate pair’s swing is 5.2 × 10⁻⁴ before the twist and 1.1 × 10⁻⁷ after, both of them the sampling rather than a quantity. The mismatched pair’s swing falls from 4.757 to 2.4 × 10⁻⁴ over the same change, which looks like an enormous improvement and is an improvement in one statistic only.

So the two rotor pairs end up at the same steadiness and nowhere near the same seal. The conjugate pair is steady at twice its undersize, which is a number a drawing can specify and a machine can hold. The mismatched pair is steady at −0.4085, which is interference. A designer reading only the swing would conclude that the twist had made the second pair as good as the first, and the reason that conclusion is available is that the swing is the statistic the twist was chosen to improve.

This is the same lesson the centre-distance comparison reached from the other side. There, a designer buying clearance by opening the shafts spends the whole of the opening and gets a twenty-fourth of it as seal, because the gap is a component of a translation along a normal that turns. Here, a designer buying uniformity by twisting spends axial length and gets uniformity — genuinely, exactly — around an average that was never a seal to begin with. Neither purchase is fraudulent and neither is what it looks like, and in both cases the thing that separates the appearance from the fact is asking which number is a seal. The seal is the tightest place. It is the only one of these quantities that the twist leaves entirely alone.

One column moves #

Laid out across the wraps, the arithmetic is four columns of which three are constants.

The fourth column is the one a designer should read. Flat, this pair is clear of itself by 2.337 at its best instant. At a quarter pitch that has fallen to exactly nought — the rotors are in contact somewhere for the whole turn from that wrap onward. At half a pitch it is −0.942 and by three quarters it is −2.417, which is the profile’s own deepest interference, now present at every shaft angle without exception.

Nothing in that column is an accident of the profile chosen. It is the window again: the best instant is the largest value the tightest-section-in-the-window can take, and widening the window can only lower it, monotonically, until the window covers the period and it is pinned at the profile’s minimum. The wrap that flattens the average is at least as wide as the wrap that makes the worst case permanent, necessarily, because it is the same window. A pair cannot have the first without the second. That is not a limitation of helical rotors; it is what a moving average is, and it is the reason a twist has to be spent on a profile that is already good rather than on one that is not.

What this does not settle #

Nothing here is flow. The area open across a seal is geometry. A leakage rate is a pressure, a viscosity, a path length and a temperature, and none of those appears anywhere in this collection. The claim made is that a flat area profile has no worst instant, not that a machine with one leaks less than a machine without.

The seal measured is rotor against rotor. A real blower also seals rotor against housing and rotor end against end plate, and a twist changes the second of those in ways nothing here computes. On a machine whose casing clearance is the larger of the two, everything above is a correction to the smaller term.

The twist is uniform and the section is one shape. A constant lead and a swept profile are what the reduction needs. A rotor with a varying lead, or one whose profile is relieved towards its ends, is a different family of sections and the window argument does not describe it.

The profile is sampled. The flat pair’s clearance is computed at seventy-two angles round a lobe pitch on outlines of a hundred and fifty points a tip, and the window is read from that by interpolation. Every residue quoted as nought above — 2.4 × 10⁻⁴, 3.8 × 10⁻⁴, 1.1 × 10⁻⁷ — is that sampling and falls when it is refined. The arguments are exact; the numbers are not, and the difference is stated rather than hidden.

One mismatched pair, one conjugate pair, two lobes. The mismatched pair is a cycloidal rotor against a circular-tipped one of the same height, which is a specific and fairly violent mismatch. A pair that is nearly conjugate has a small swing and the whole comparison shrinks with it, and a partner exists for any shape at all, so the space of pairs that are merely close to conjugate is enormous; whether the fractional-wrap behaviour has the same shape when the harmonics are differently distributed is not measured here.

The same arithmetic, met once before and stopped sooner #

The integer result is not new to this collection, and the place it appeared first is worth naming because the two arrivals are at different quantities.

Contact that runs along the tooth found that a helical gear pair whose overlap ratio is an exact integer has a total length of contact that does not vary at all through a base pitch — measured at 2.1 × 10⁻¹⁴ mm of variation, which is the arithmetic’s floor. That is the same statement as this one with the seal’s open area replaced by the contact length, and the same proof works: both are integrals of a periodic quantity over a window of exactly one period.

What separates the two is what happens either side of the integer, and what the constancy is worth. In a gear pair the quantity being held constant is the thing that matters — a varying contact length is a varying stiffness and that is where the noise comes from, so flattening it is the purchase. In a rotor pair the quantity being held constant is an average, and the thing that matters is a minimum. The gear field’s advice — aim for an overlap ratio of at least one — transfers to rotors in form and not in force, because a rotor pair that needs it is a rotor pair whose profile is wrong.

There is a third reading, and it is the one that makes the window argument worth having beyond either machine. A wrap is a way of turning a quantity that varies with the input into one that does not, by spending a dimension the mechanism was not using. A cam’s follower cannot do it, because a cam has no second dimension to spend, and its contact is a point that leaves the profile at both ends rather than a line running the length of anything. A gear train’s ratio error cannot be spent that way either, because the error is not periodic in anything the train is free in. Rotors and helical teeth can, and what they get for it is the same in both cases: the mean, exactly, and nothing else.

Still open: the wrap that is not uniform #

Everything above reads one profile through a rectangular window, because a uniform twist weights every section equally. A rotor relieved towards its ends — cut back a little at the tips of its length, which is what a real machine does to keep a thermally grown rotor off its casing — weights the sections unevenly, and the seal’s open area becomes a weighted moving average of the same profile.

Its distinct argument would be that weighting chosen rather than suffered. A rectangular window has the gains $\sin(\pi m\tau)/(\pi m\tau)$, which is why its zeros are sharp, why they need a whole pitch, and why the swing humps back up between them. A relief profile is a different window with different gains, and the question is whether a shaped relief can put a zero at a wrap shorter than one pitch — trading a little of the mean, which a relief costs by definition, for the constancy that a full pitch of rotor currently has to buy. Two things would come out of it. The shortest wrap at which any weighting can flatten the area, which is a statement about what a rotor’s length is worth; and whether the weighting that does it leaves the extremes alone, as the rectangular window does, or whether a window that tapers can keep the profile’s worst angle off the rotor for part of the turn — which is the one thing a uniform twist has been shown not to do.