A rotor nobody drew

A rotary engine has two moving parts of consequence: a shaft with an eccentric on it, and a three-cornered rotor riding on that eccentric inside a housing shaped like a fat figure of eight. The rotor turns at a third of the shaft’s rate and its three corners sweep the wall.

The housing’s shape has a name and a formula. It is an epitrochoid — the path of a point at radius $R$ on a circle that rolls inside a larger one — and it is the one curve in the machine that somebody wrote down:

$$P(t) = \big(e\cos 3t + R\cos t,\; e\sin 3t + R\sin t\big).$$

The rotor’s shape has no such formula in any useful sense, and it does not need one. Given the housing and the motion, the rotor is the envelope, and the same routine that generates a gear tooth from a straight rack generates it — corners included.

The motion, which is the whole input #

Two numbers describe a rotary engine’s geometry: the generating radius $R$ and the eccentricity $e$. Their ratio $K = R/e$ is the only shape parameter it has, and everything the geometry decides is a function of it. The engine drawn here has $R = 100$ and $e = 14$, so $K = 7.14$, which is in the range real engines use.

The motion is as simple as the shape is not. With the shaft at angle $\varphi$:

• the rotor’s centre sits at $e(\cos\varphi, \sin\varphi)$ — it orbits, at the eccentricity, once per shaft turn;
• the rotor itself has turned through $\varphi/3$.

That is it. Three shaft turns to one rotor turn, an orbit and a rotation, and the housing standing still. Hand that pair of motions and the housing’s curve to the meshing equation and ask what shape stays in contact.

What comes back has corners #

The generation returns contacts of two kinds, and telling them apart is the whole reading of the figure.

Most of them lie between radii $R - 2e$ and $R$ from the rotor’s own centre, and they move as the shaft turns: those trace the flanks. Three of them sit at radius exactly $R$, at $0°$ and $\pm120°$ in the rotor’s frame, and they are contacts at every shaft angle: those are the apexes.

Measured: the apex contacts come out at radius $100.000000$ against a generating radius of $100$, and the flank’s closest approach to the rotor’s centre is $72.0000$ against $R - 2e = 72$. Neither number was put in. The generation was given a curve and two motions, and it returned a shape whose three corners are at the generating radius and whose waist is two eccentricities inside it.

A corner of an envelope is not a corner anybody drew. It is where two branches of the same envelope meet, and it is the geometric reason a rotary engine has an apex seal problem: the seal is riding on a point of the rotor where the surface is not smooth, which is the one place a conforming seal cannot conform.

That the apexes are contacts at every position is the other half of the same statement. The rotor touches its housing at three points, always, and nowhere else — the flanks are clear of the wall by design, and the three chambers between them are sealed by those three corners and nothing else. A piston engine seals a chamber with a ring running round a bore; this one seals three chambers with three lines running along a wall that is not round.

Two branches, and only one of them is a rotor #

The generation returns more contacts than the rotor has boundary, and the extra ones are worth understanding rather than discarding quietly.

At most shaft angles the meshing equation has roots at radii larger than $R$ as well as smaller. Those are perfectly good solutions of the equation: they are the places where the housing curve, seen from the rotor, is tangent to a curve on the outside. What they trace is the shape a housing would have to be if the rotor were the given body — the other member of the pair, generated in the other direction, which the round trip in this field’s second essay is about.

So the filter is not a numerical convenience. It is the statement that the rotor is the body inside the housing: keep the contacts nearer the rotor’s centre than the housing is, and the shape that comes back is the one that fits inside. Keep the others and the shape that comes back is the housing, rediscovered.

That is also why the apex contacts are the ones that never move. An apex is where the inner branch and its neighbour meet, and at that meeting the two solutions coincide — the corner is a double root of the meshing equation, which is exactly what a cusp of an envelope is.

Two lobes and three corners is one member of a family #

Nothing in the construction is specific to two lobes. Put $N$ lobes on the housing — trace it with $e\cos(N{+}1)t + R\cos t$ — and drive the rotor at $1/(N{+}1)$ of the shaft rate, and the generation returns a rotor with $N + 1$ corners.

Measured across the family at $R = 100$, $e = 14$:

Every member puts its corners at the generating radius and its waist at $R - 2e$, evenly spaced, without being told to. The one-lobe case — a two-cornered rotor in a housing shaped like a fat oval — is the arrangement of the first working machines; the two-lobe case is every rotary engine anybody has driven; four corners and three lobes has been built and is worse, because the chambers get thin.

The number of corners is not a design choice either. It follows from the ratio, which follows from the lobe count, which follows from which trochoid was drawn. There is one free decision in the whole engine — $K$ — and everything else is a consequence.

Where the shape stops existing #

The ratio $K$ cannot be anything. The housing is a trochoid, and a trochoid has a cusp when its tracing point stops moving.

Differentiate the housing’s equation and the speed of the tracing point comes out as a sum of two terms — one of size $3e$, one of size $R$ — which cancel when $R = 3e$. Measured over a fine sweep at a series of $K$, the slowest the tracing point ever goes is exactly $|R - 3e|$, at $t = 90°$, and it reaches zero at $K = 3$ to fourteen places.

So $K = 3$ is a hard floor. Below it the housing has loops rather than a boundary, and there is no engine. Above it the housing is smooth and the question becomes one of proportion: as $K$ rises the two lobes become shallower, the chambers become thinner, and the swept volume falls away. Real engines sit between about 6 and 8 — the Mazda units are near 7 — which is the usual sort of compromise between a shape that works and a shape that is worth building.

It is not a Reuleaux triangle #

The rotor is drawn, described and sold as a Reuleaux triangle: the curve of constant width made of three circular arcs, each centred on the opposite corner. It is not, and the difference is not a subtlety of draughtsmanship — a Reuleaux flank would foul the housing.

Two measurements, and the second is the one that cannot be argued with.

The flank against the arc. Take the Reuleaux triangle through the same three apexes — arcs of radius $R\sqrt3$, centred on the opposite corners — and compare it with the generated flank. The largest departure is 1.21 mm on a rotor whose generating radius is 100, and the true flank is the shallower of the two.

The width. A Reuleaux triangle has the same width in every direction; that is what it is for, and it is why it turns inside a square. The generated rotor’s width runs from 150.0 to 174.1 over the directions — a spread of sixteen per cent. It is not a curve of constant width, and no amount of drawing convention makes it one.

The mistake is a natural one. Both shapes are three-cornered, roughly triangular, and symmetric under a third of a turn; the Reuleaux triangle is the famous three-cornered curve and it has a rolling story attached to it. But the rotor’s flank is an envelope of an epitrochoid, and there is no reason a curve produced that way would be an arc of anything.

There is a real relationship and it is worth having, because it is the reason the drawing convention persists: for large $K$ the rotor’s flank approaches the Reuleaux arc, since the eccentricity that distinguishes them is what makes the housing’s lobes at all. At $K = 7$ the difference is a millimetre; the engine with no difference at all would have no chambers.

What K decides #

Since $K = R/e$ is the only shape parameter, it is worth setting out what moves when it does.

Small K — near the floor of 3. The housing’s lobes are deep, the waist between them is narrow, and the rotor’s flanks are correspondingly deep: the waist sits at $R - 2e$, which at $K = 3$ is a third of the generating radius. The chambers are large, so the engine is voluminous for its size. The trouble is that everything is extreme: the housing’s curvature is severe near the waist, the apex’s path is violently non-circular, and at $K = 3$ exactly the wall has a cusp and there is no engine at all.

Large K — say 12 or more. The housing is nearly circular, the rotor is nearly a Reuleaux triangle, and the chambers are thin crescents. The geometry is gentle and there is almost no swept volume: at large $K$ the machine approaches a round rotor in a round hole, which does nothing.

The middle. Production engines sit near $K = 7$, and the drawings here use $7.14$. It is the usual sort of compromise — enough chamber to be worth building, enough smoothness in the wall for a seal to follow it — and it is a single number that fixes the entire geometry of the machine, which is unusual and is worth noticing on its own.

What the machine is doing to itself #

Two consequences of the shape are geometric and are worth stating here because they are usually explained with forces.

The rotor’s flanks are shallow, and that is not an aesthetic choice. The waist of each flank sits at $R - 2e$, so the deeper the flank, the larger the eccentricity, the more violent the housing’s lobes, and the closer $K$ gets to the floor at 3. The whole engine is a compromise on one number.

The apexes travel further than the rotor turns. Each apex is at radius $R$ from an orbiting centre, so its path in the housing frame is the epitrochoid itself — the corner traces the wall it seals against, exactly. That is the defining property of the design: the housing is the apex path, and the rotor is what has to be there for the apexes to be where they are.

What it took to get the corners right #

Two things went wrong in producing the figures above, and both are worth recording because they are the failure modes of any envelope computation.

The first is the branch problem described above: an early version filtered the contacts by a fixed radius — anything below $R - r$ is the rotor — which is a number that looks like the tip radius and is not one. It truncated the corners, leaving a rotor whose flanks were right and whose apexes were flat, and every measurement that looked at the flank agreed with the geometry. The test against the material, sweeping the housing over the candidate shape and asking what survived, is what caught it.

The second is that a corner is where the sampling is worst. The envelope’s parameter runs slowly along a flank and then turns through the corner quickly, so a uniform sample in shaft angle crowds points on the flanks and leaves the corner represented by a handful. The apexes are therefore placed rather than sampled — at the radius and the angles the generation says they are at, both of which are measured — and the flanks are drawn between them.

That is a general shape for this kind of drawing. A curve with a singularity in it cannot be represented by uniform sampling of its own parameter, and the useful move is usually to compute where the singularity is exactly and hand it to the drawing separately.

Why this is the same picture as a gear tooth #

It is worth being explicit, because the two mechanisms have nothing in common at the level of what they are for.

A rack cutting a gear: a straight edge, a rolling motion, an envelope, an involute. A rotary engine: an epitrochoid, an orbit-and-rotation, an envelope, a three-cornered rotor. The routine is the same one — the same equation, solved the same way, with the contacts mapped into the second body’s frame — and the two mechanisms differ only in which curve is handed in and what the two bodies are doing.

That is why the ledger in this field’s first essay has six rows rather than one, and it is why a rotary engine is a conjugate pair. Nobody calls the housing and the rotor a gear pair, and geometrically that is what they are: two bodies in continuous contact whose shapes determine each other.

Solve for the corner, do not sample it #

The corner being where the sampling is worst is recorded above as a thing that went wrong, and it generalises into a rule about drawing envelopes that is worth stating, because the naive repair does not work.

An envelope’s corner is where two branches of the same solution meet. Away from it the boundary is smooth and a sampled polyline approximates it to the square of the step; at it the two branches arrive with different tangents, and no polyline through samples on either side passes through the corner unless a sample happens to land there.

The naive repair is to sample more finely, and it fails in a specific way. Halving the step halves the distance from the nearest sample to the corner, so the corner is cut off by half as much — an error falling as the first power of the step while the rest of the curve improves as the second. Refine uniformly and the corner becomes, relatively, worse and worse: it is the only place where the drawing’s error is not improving at the rate everything else is.

The repair that works is to solve for it. The corner is at the shaft angle where the two branches coincide, which is an equation with a root, and finding the root costs a bisection. Put a sample exactly there and the polyline has the corner as a vertex, exactly, with no cutting off at any step size.

That is the same habit the intermittent field applies to a switch, arriving in a completely different subject. In both cases the interesting configuration is the boundary between two regimes, in both cases it is the root of a condition rather than a point of a sweep, and in both cases sampling the interval finely is the wrong instrument for a quantity that lives at its end.

Which gives a short rule for any figure of this kind. A curve with a corner has to be drawn in pieces: solve for the corners, and sample each smooth arc between them. A single sweep across the whole parameter range produces a picture that is right in the middle and wrong exactly where the mechanism’s most distinctive feature is — which on this rotor is its three apex seals, the parts the whole engine is famous for having trouble with.

What is left out #

Force, as everywhere. What presses the apex seals against the wall, what the gas does, what the bearing carries: all outside. The rotor here is a shape that stays in contact with another shape, and every number quoted is a length or an angle.

Sealing as an engineering problem. The geometry says the apexes are corners and the corners are the seal; it does not say how a seal is made to follow a corner along a wall that is not a circle, which is the problem that took the design twenty years and is not a kinematic one.

Combustion, in every respect. The three chambers change volume as the shaft turns and their areas add to a constant; what happens inside them is somebody else’s subject entirely.

And the third dimension. Everything here is planar: a real rotor is a prism of this cross-section with side seals and a recess machined into each face, and the recess is a deliberate loss of geometric compression that the next essay measures.