The transmission angle has no size

The transmission angle is the angle between the coupler and the rocker: the geometric part of how much of an applied force becomes useful output torque and how much goes into the bearings.

It is an angle in a triangle whose three sides are link lengths, so scaling the machine leaves it exactly where it was.

What the field is for, restated #

Before the measurement, a sentence about what is being claimed, because the transmission angle is a shape sounds like a small technical remark and it decides what a whole field is worth.

This field exists to say how well a mechanism transmits — where it is efficient, where it approaches jamming, how much of its travel is usable. Every one of those judgements is made from angles and every one is quoted in the literature without reference to a size.

If the underlying quantities were sizes, that practice would be wrong: a rule quoted as a bare number would be a rule about one machine, and the field’s results would not transfer between designs of different scale.

They do transfer, everybody knows they do, and the reason has never been stated on this site. It is that the quantities are dimensionless functions of the proportions, which is a one-line argument and is worth having because the same argument fails for the neighbouring fields — a curvature does not transfer, a tolerance band does not, and a bare number quoted about either is a number about one size.

The measurement #

At a crank angle of one radian the site’s four-bar has a transmission angle of 65.9075°. Scale the machine by 0.8, by 1.25, by 1.6 and it is 65.9075° at each — the fitted exponent under a scaling is exactly zero and the largest departure over six scale factors is zero to the last bit.

Its minimum over a full turn is 54.3147°, at a crank angle of zero. Same at every size.

The control matters and it passes: change the ground length by five per cent and the transmission angle at that configuration moves by 6.4%. So the quantity varies, and its invariance under scaling is a statement about the machine rather than about a constant.

Why it has to be #

Three sides of a triangle, all of them link lengths, and an angle between two of them. Scale all three and the triangle is similar, so every angle in it is unchanged.

There is a version of that which is worth having because it says how far the argument reaches. The transmission angle is computed from the positions of the joints, which the solver returns, and scaling the parameters scales every joint position about the origin. A scaling about a point is a similarity of the plane; a similarity preserves angles; the transmission angle is an angle. Three steps and none of them mentions a four-bar.

So the same conclusion holds for any mechanism this site carries whose transmission behaviour is defined as an angle between two links — a six-bar’s, a slider-crank’s, a Watt chain’s second loop. The argument is about how the quantity is defined rather than about which machine it is defined on.

That is the whole of it, and it means the transmission angle is a shape in the strict sense — a dimensionless function of the lengths that is invariant under the scaling and responsive to the proportions.

Everything the field computes about it inherits the property. Its minimum over a turn is a shape. The crank angle at which the minimum occurs is a shape. The fraction of the turn spent below 40° is a shape. The 222° between a toggle and the worst transmission angle is a shape.

Not one number in this field has a size in it.

Where the worst point falls #

The minimum’s location is worth its own look, because it is a second shape and it is the one a designer needs.

On this four-bar the transmission angle bottoms out at 54.31° at a crank angle of exactly zero — the configuration where the crank points along the frame bar away from the output pivot. That location is a crank angle, which is dimensionless, so it too is unchanged by scaling.

It is also not where a reader would guess, and the field’s own first essay makes the point at length: the worst transmission angle and the toggle are 222° apart on this machine, and both get called jamming. They are opposite situations — one is the mechanism unable to push usefully, the other is the mechanism unable to be pushed at all — and confusing them is the field’s standing hazard.

Both locations are shapes. Both are recoverable from a protractor. Neither depends on the machine’s size, and a design rule about either transfers between scales.

Two configurations, both scale-free, both recoverable, and routinely conflated — which says the difficulty in this field is conceptual rather than metrological, and that is unusual on this site.

What that means for measuring one #

A protractor on the output link recovers three of a four-bar’s four parameters and never the fourth. Those three are the shape, and the transmission angle is a function of the shape.

So a calibration with the cheapest possible instrument — two angle readings, no fixture, no datum, no coordinate machine — recovers the whole transmission-angle curve exactly. Its value at every crank position, its minimum, where the minimum falls, whether the machine ever goes below whatever design rule is in force.

That is an unusually complete answer for an unusually cheap measurement, and it is worth stating plainly because the instinct is the other way. A question about force feels like it needs a real instrument, and the geometric half of it needs a protractor.

Why an angle can still depend on the geometry #

There is a trap in the phrase an angle is dimensionless and it is worth clearing before the essay leans on it.

Being dimensionless does not make a quantity a shape. The number seven is dimensionless. What makes the transmission angle a shape is that it is dimensionless and it responds to the proportions — 6.4% under a five per cent change in the ground length, which is the control the probe applies to every row.

Conversely, being a length does not stop a quantity from having a scale-free version. The Grashof margin is a length; the same margin divided by the shortest link is a shape, and the two say different things about the same machine.

So the classification is not is this quantity an angle. It is what happens to its value when every length is multiplied and nothing else is, and the answer has to be measured because the reasoning about it can be wrong. One row of the site’s own survey was predicted to be a shape and came back at exponent −1.

That row is a tolerance band computed from a fixed absolute tolerance, and it is a useful thing to have beside this field’s result: an angle computed from lengths that all scale is a shape, and an angle computed from lengths and a fixed absolute number is not.

And the pressure angle likewise #

A cam’s pressure angle is the angle between the follower’s direction of motion and the normal at the contact, and it is decided by the ratio of the follower’s velocity to the base radius plus the lift.

Every one of those is a length, and the pressure angle is an arctangent of their ratio. Scale the cam — base circle, lift, everything — and the ratio is unchanged and so is the angle.

So the pressure angle is a shape too, and the same conclusion follows: a measurement of a cam’s follower motion in angle recovers the pressure-angle curve without recovering the cam’s size.

The two angles this site uses to describe force transmission are both shapes, which is not a coincidence: both are defined as angles between directions, and directions are what a similarity preserves.

The velocity ratio is a shape too #

Worth running the whole field through the probe rather than one quantity, since the result is the same each time and the accumulation is the point.

The velocity ratio — how fast the output turns per unit of input — is a ratio of two angular rates, dimensionless, exponent zero. Scale the machine and it does not move.

The time ratio between forward and return strokes is a ratio of crank angles. Exponent zero.

The mechanical advantage, in its geometric form, is the reciprocal of the velocity ratio. Exponent zero.

And the instant centre’s position, expressed as a fraction along the line between two pivots, is a ratio. Exponent zero — though the instant centre’s distance from a pivot is a length and is not.

That last pair is the field’s one place where a reader has to be careful. The instant centre is a point in the plane, so its coordinates are lengths; every use the transmission field makes of it is a ratio of distances along a line, which is a shape.

Two quantities called by one name, one with an exponent and one without, which is the same hazard the Grashof margin has and is worth flagging wherever a field’s central object is a point.

What is not recovered, and it is the important half #

The field’s own boundary says the transmission angle is the geometric part of force transmission and that whether a mechanism actually moves under load depends on friction, which is outside the site entirely.

The scaling result sharpens that. The part that is geometry is a shape; the part that is outside is not.

A friction coefficient is dimensionless, so it too would be unchanged by scaling. But the force required, the torque delivered and the bearing loads are not: force scales with whatever produces it, torque scales with force times length, and none of that is in this field.

So the boundary between what this site computes and what it does not runs exactly along a line the scaling argument can see. Everything on the inside is a shape; everything on the outside has either a force or a material property in it, and forces have sizes.

That is a pleasing coincidence and it is not one. The site’s boundary is drawn at the point where inputs stop being geometry, and geometry is what a similarity acts on.

The one place a size does enter this field #

For completeness, since not one number in this field has a size in it is a strong claim and there is a qualification.

The transmission angle is defined at a configuration, and a configuration is a crank angle, which is a shape. So the curve is a function from shapes to shapes and nothing anywhere has a length in it.

What does have a length is the machine the curve is about. Quoting a transmission angle without saying which four-bar is quoting a number about a family — which is fine, and is what makes the number transferable — and quoting it without saying which crank angle is quoting a number about nothing.

So the field’s own reporting convention is already right in one respect and worth noticing: every transmission angle on this site is quoted with the configuration it was computed at, because the field understood from the start that the quantity varies through the turn.

A field that quotes the configuration and not the size has already made the distinction this essay is about, without stating it. The size was never quoted because it was never needed, and it was never needed because every quantity was a shape.

A design rule that transfers #

One practical consequence, and it is the reason a design rule can exist at all.

The rule that a four-bar’s transmission angle should stay above 40° is quoted without reference to size, in every textbook, and it works. Now the reason is available: the quantity is a shape, so a rule about it is a rule about proportions, and proportions transfer.

Compare with a rule about a length — a minimum bearing diameter, a minimum wall thickness. Those are size-dependent and every one of them is quoted with a qualifying condition or as a fraction of something.

A design rule can be a bare number exactly when the quantity it bounds is a shape, and a bare number quoted about a quantity with an exponent is a rule that is silently about one size. The 40° rule passes the test; a rule about a minimum radius of curvature would not, and the curvature field’s quantities are the case where it fails.

The margin again #

Everything the field computes is a shape, and there is a margin worth reporting for the same reason the Grashof class needs one.

The transmission angle’s own margin is how far its worst value sits above the design rule: 54.31° against 40° is 14.31° of margin on this machine, and against a rule of 50° it would be 4.31°.

That margin is itself a shape, since both quantities are angles, so it transfers between sizes and it is recoverable from a protractor. It is also the number a designer actually wants, and the site quotes the minimum without the rule and therefore without the margin.

One number, computed already, and the difference between the worst transmission angle is 54.31° and the worst transmission angle clears the usual rule by fourteen degrees.

A tolerance on the transmission angle is not a shape #

The exception that proves the rule, and it is the one place this field’s results acquire an exponent.

Ask how much the worst transmission angle moves when the four lengths are held to ±0.01. That is a sensitivity times a tolerance, the sensitivity is a reciprocal length and the tolerance is a length, so the product is dimensionless — and it depends on the ratio of the tolerance to the lengths.

Scale the machine and hold the tolerance at ±0.01 and the band on the transmission angle halves, exponent −1. Express the tolerance as a percentage instead and the band is a shape.

So the transmission angle itself is size-free and its uncertainty is not, unless the tolerance is written proportionally. A bigger machine held to the same absolute tolerance has a more certain transmission angle, from geometry alone.

That is the same finding the tolerance field’s band produces, arriving at this field’s own quantity, and it means a statement like the worst transmission angle is 54.31° ± 0.4° is a statement about a machine of a particular size even though the 54.31° is not.

A caution against the slogan this nearly is #

Everything in this field is a shape, measured — exponent zero with a control that moves it by 6.4% — and the field is therefore the only one on the site whose entire output survives a scaling untouched. Its objects are angles by definition, so the result is not a discovery so much as a consequence, and the useful part is what it lets a reader stop worrying about.

Nothing in a transmission-angle result is about the machine it was computed on. The worst point of the sweep, the value there, the design rule that says keep it above forty degrees, the velocity ratio, the pressure angle: all of them transfer between a model and a press unchanged, and none of them wants a length quoted beside it.

The boundary between what this site computes and what it declines to compute runs along nearly the same line, which is where the temptation starts. What the field computes is dimensionless; what it hands to somebody else has a force in it, and a force has a size. That is a sharper statement of the site’s own limit than kinematics not dynamics, and it agrees with it everywhere in this field.

It is not a rule about the site, and it would be a bad one. A coupler point’s position is a length and is squarely inside. So is a swept area, a clearance, a stack-up, a strand length, a base circle. An entire field here is nothing but sizes and it is as much a part of the subject as this one. The boundary is not dimensionless is in, dimensional is out; it is that this particular field’s outputs happen all to be dimensionless because its objects are angles, which is a fact about the transmission angle and not about kinematics.

The near-miss is worth spelling out because slogans of this shape are how a boundary gets misremembered. A reader who took dimensionless is in away from here would conclude that the tolerance field and the clearance field are outside the subject, which is precisely backwards: those are the fields where the machine stops being a diagram and starts being an object, and they are the fields where the site’s results have to be most careful about which machine they are about.

What is true, and worth carrying, is narrower and more useful. When a field’s outputs are all dimensionless, its results transfer and its measurements are cheap. That is why this field’s design rules have survived a century of machines of every size, why a protractor is a sufficient instrument for the whole of it, and why the one number here that does have a size — a tolerance on the transmission angle, expressed as a band on the lengths — is the one that has to be quoted with the machine it belongs to.