The other pole

The instantaneous centre is the one object in this field that a reader is likely to have met before, and it is usually met in a form that makes it seem larger than it is. The body is rotating about this point. Said that way it sounds like a complete description of what the body is doing, and it is a description of one derivative.

The cleanest way to see the limit is to ask the same question one derivative up. Which point of the moving plane is not accelerating?

There is one, at nearly every position, and it is not the pole.

Both poles come out of the same expression #

A point of the moving plane at fixed-frame offset $w$ from the moving origin has

$$z' = A' + i\varphi'\,w, \qquad z'' = A'' + (i\varphi'' - \varphi'^2)\,w.$$

Setting the first to zero gives the velocity pole:

$$w_P = \frac{i A'}{\varphi'}.$$

Setting the second to zero gives the acceleration pole:

$$w_Q = \frac{-A''}{i\varphi'' - \varphi'^2}.$$

They are different expressions of different quantities. The first needs $\varphi' \neq 0$; the second needs $\varphi'$ and $\varphi''$ not both zero, which is a weaker condition — so there are positions where the plane is instantaneously translating, has no velocity pole at all, and still has a perfectly ordinary acceleration pole.

The two coincide only if $A'/\varphi'$ and $A''/(i\varphi''-\varphi'^2)$ happen to name the same point, which is a coincidence rather than a rule. It does happen — a body in steady rotation about a fixed pivot has both poles at the pivot, and that is the case everyone’s intuition is built from. A crank is exactly that body. So is a gear. The trouble starts when the intuition is carried to a coupler, which is in steady rotation about nothing.

Measuring the gap #

Over 117 positions of the site’s four-bar where both poles exist and both are within a reasonable distance of the mechanism, the distance between them, in coupler lengths:

• closest: 1.49
• furthest: 22.2

They are never the same point. The closest approach is not a near miss — one and a half coupler lengths on a linkage whose coupler is 3.5 units is about five units of separation, which is larger than the whole mechanism.

The measurement is worth having in that form rather than as a statement, because the statement “they are different points” is compatible with them being a hair apart, and a hair apart would make the confusion between them harmless. Five units apart on a four-unit machine does not.

The acceleration pole depends on how fast the crank turns, and the velocity pole does not #

This is where the parameterisation question stops being an aside.

Run the crank at a constant rate and everything in this field is as described. Run it at a rate that is itself changing — accelerate the crank — and the point of the moving plane with zero acceleration moves, because part of a point’s acceleration comes from the crank speeding up rather than from the mechanism’s geometry.

The velocity pole does not move. Scale the input rate by any factor and every velocity scales by the same factor; the point where they all vanish is unchanged. The pole is a property of the mechanism’s configuration, full stop.

So the two poles are not two members of one family. The velocity pole is geometry; the acceleration pole is geometry plus a statement about how the input is being driven. Everything in this field is computed per radian of crank, which amounts to assuming the crank turns at a steady rate, and the acceleration pole reported here is the one that assumption produces. A real machine with a real motor has a different one, and the difference is a dynamics question rather than a kinematics one.

That is the honest reason this field spends one essay on the acceleration pole and no more. The velocity pole earns its place because it is parameterisation-free and organises the entire subject. The acceleration pole earns one essay because the confusion between them is common, and then it stops being geometry.

What the pole is, precisely #

Worth stating in the tightest form available, because the errors are all errors of over-reading.

At this instant, the velocity field of the moving plane is exactly the velocity field of a rotation about the pole. That is the whole content. Every velocity is $\varphi'$ times the perpendicular distance from the pole, directed perpendicular to the ray, and this is exact rather than approximate — it is a rearrangement of $z' = A' + i\varphi' w$.

What does not follow:

That the plane is turning about the pole for any length of time. It is not. The pole is a different point of the fixed plane and a different point of the moving plane an instant later, and the two curves it traces are the centrodes.

That the point at the pole has no acceleration. It generally has a large one, directed along the pole normal, and that is what makes a coupler point’s path curve at all.

That a mechanism pivoted at the pole would do the same thing. It would agree in velocity and disagree immediately afterwards. How immediately is measurable: replace a coupler point by a crank at the centre of its path’s curvature and the two separate like the cube of the crank step. Replace it by a crank at the pole instead and the separation is first order, because the radius is wrong.

That the pole is where a mechanism’s mechanical advantage is decided. That one is nearly true and is the source of most of the useful applications — the ratio of two links’ angular rates is a ratio of distances from a common instantaneous centre, and the site uses that in earnest. It is a velocity statement, it is correct, and it says nothing about forces without a virtual-work argument on top of it.

The centrodes are the honest version of “rotating about it” #

There is a sense in which a body really is rotating about its instantaneous centre, and it is a stronger and stranger statement than the loose one.

The pole traces a curve in the fixed plane and another in the moving plane. The motion is the second curve rolling on the first without slipping. Not approximately — exactly, for the whole motion, not just near an instant. Every planar motion is the rolling of one curve on another, and the two curves are the centrodes.

So “the body is rotating about the instantaneous centre” is right if it is heard as “the body is at each moment rotating about the current point of contact between two rolling curves”, which is a description of a wheel rather than of a pivot. The point of contact moves along both curves. A wheel rolling on a road has its instantaneous centre at the contact patch, and nobody thinks the wheel is pivoting there.

That reading also explains why the acceleration pole is somewhere else. The rolling picture makes the velocity field obvious and says nothing about accelerations, because the contact point is itself moving and the rate at which it moves — the pole velocity — enters every second-order quantity. It is exactly the $\delta$ of Euler and Savary’s relation, and the fact that the pole moves is the whole reason coupler curves are curved.

The three-body version, which is where the confusion usually starts #

The place most readers meet an instantaneous centre is not a coupler at all. It is a pair of gears, or a wheel on a road, or a lever — bodies whose instantaneous centre is a physical point that stays put, and where the loose reading is harmless because the strict one happens to agree.

A gear turning on a fixed shaft has its instantaneous centre at the shaft, at every instant, forever. Both centrodes degenerate to points. The velocity pole and the acceleration pole coincide, sit at the shaft, and stay there. Every intuition built on that case is correct about that case.

The trouble arrives with the third body. Kennedy’s theorem says that for any three bodies in planar motion the three pairwise instantaneous centres are collinear, and the useful case is exactly the one where two of the three centres are physical pins and the third is not. That third centre — the coupler’s, relative to the frame — is the one that wanders, and it inherits the word “centre” from the two that do not.

The site uses that construction in earnest and it is correct. A gearset’s lever diagram is an instantaneous-centre argument; so is the epicyclic ratio taken two ways; so is the roll centre of a suspension, whose whole essay is about the fact that it moves and is quoted as though it does not. In every one of those the statement being made is about velocities, the answer is right, and the word “centre” carries an implication about permanence that the arithmetic does not.

Two poles, two circles, and a question the second one answers #

There is a second reason to have the acceleration pole in view, and it is that it makes the inflection circle less arbitrary.

A point’s path curvature is zero when its acceleration has no component across its velocity. Since the acceleration field is a fixed rotation-and-scaling about the acceleration pole, and the velocity field is a rotation about the velocity pole, the condition “acceleration parallel to velocity” is a relation between the two poles and the point — and working it out gives a conic, which turns out to be the circle of points going straight.

So the inflection circle can be described as the locus of points whose offset from one pole is parallel, after a fixed rotation, to its offset from the other. That is a purely projective-looking statement about two points and a fixed angle, and it produces a circle for the same reason two angles subtended at the ends of a chord produce one.

It is not the derivation this site uses, because the algebraic route gives the coefficients directly and the coefficients are what the conic fit is checked against. It is worth recording as the geometric reading, because it makes the circle’s dependence on both poles visible where the algebra hides it inside $\varphi'$ and $\varphi''$.

Where the acceleration pole is genuinely useful #

Having said what it is not, it is worth saying what the acceleration pole does buy, because it is not nothing.

The acceleration field of the moving plane, like the velocity field, has a simple form once its zero is known: every point’s acceleration is a fixed linear map applied to its offset from the acceleration pole. The map is $i\varphi'' - \varphi'^2$, which is a rotation by a fixed angle combined with a fixed scaling — so every point’s acceleration makes the same angle with the line joining it to the acceleration pole, and its magnitude is proportional to that distance.

That is the acceleration analogue of the pole’s property, and it is exactly as strong. It is what makes graphical acceleration analysis possible at all: draw the acceleration of one point, find the pole, and every other point’s acceleration is a rotation and a scaling away.

The fixed angle is $\arctan(\varphi''/(-\varphi'^2))$ and it is not $90°$ except by accident, which is where the acceleration picture stops resembling the velocity one. In the velocity field the angle is exactly a right angle at every instant on every motion. That difference — a right angle by structure against an arbitrary angle by circumstance — is a reasonable summary of why the velocity pole is the object this field is organised around.

The trammel, where the arithmetic goes quiet #

One motion in the field’s ledger has $\varphi'' = 0$ exactly: the elliptic trammel, whose rod angle is the input, so the plane’s angular rate is $1$ and every higher derivative of it is zero.

For that motion the acceleration pole expression collapses to $w_Q = A''/\varphi'^2 = A''$, and the map from offset to acceleration becomes $-\varphi'^2$ — a pure negative scaling, no rotation. So every point of the trammel’s rod is accelerating straight towards the acceleration pole, at a rate proportional to its distance from it. That is centripetal acceleration about a point, which is what a body in steady rotation does, and the trammel’s plane is not in steady rotation about anything.

It is a good example of the kind of thing that happens when a general relation is evaluated on a special case: the answer is correct, it looks like a familiar picture, and reading the familiar picture back into the general case would be wrong. The trammel’s poles are one coupler length apart at the sampled position. They are still two different points.

One object in the field depends on the schedule #

The observation that the acceleration pole moves when the crank’s rate changes and the velocity pole does not is worth promoting into a classification of the field’s whole apparatus, because it says which of its objects belong to the mechanism and which belong to the mechanism plus a drive.

A geometric quantity depends on the path the moving plane traces through the space of positions, and not on how fast it is traced. Reparameterise the motion — drive the crank at any rate at all, accelerating, decelerating, stopping and restarting — and a geometric quantity is unchanged. The instantaneous centre is one: it is where the velocity field vanishes, and scaling every velocity by a common factor does not move a zero.

A kinematic quantity in the stricter sense depends on the schedule as well. The acceleration pole is one, and it is the only one in this field: its expression carries $\varphi''$, which is a fact about how the crank is being turned rather than about where the mechanism is.

Running that test over the field’s objects sorts them completely. The centrodes are geometric — they are the loci of a geometric point. The inflection circle is geometric, because a path’s curvature is a property of the path. The cubic of stationary curvature is geometric for the same reason. The acceleration pole is not. Every other essay in this field is about the shape of a motion, and this one is about a motion being executed.

That gives the field a check it can actually run rather than merely a distinction to remember. Reparameterise and re-measure. Drive the crank on a non-uniform law, recompute every quantity, and require that all of them come back identical except the acceleration pole, which must move. A quantity that changes when it should not has a $\varphi''$ in it that nobody intended, which is the commonest way for a geometric derivation to acquire a schedule.

It is also the honest reason the acceleration pole gets one essay and no ladder. A field built on the shape of motions has one object in it that is not about shape, and the right treatment of such an object is to compute it, say what it is, say what it depends on, and not build anything on top of it. Everything downstream would inherit a dependence on the drive law, and the mechanisms this field is about do not have one.

The measurement that would have caught the confusion #

If the two poles were nearly the same, this essay would not be worth writing. So the assertion behind it is not “the two poles differ” — which is easy to satisfy by rounding — but a bracket: the closest approach over a full turn must exceed a stated threshold, and the measured 1.49 coupler lengths clears it by three orders.

That shape of assertion is the one this site has learned to prefer. A check that a difference is nonzero passes on noise. A check that a difference is at least some size fails if the two quantities are secretly one quantity, and it also fails if a later change to the mechanism’s proportions makes the claim untrue — which is the case worth catching, because the essay’s prose quotes the number.

The same shape appears in the essay that lists six things an instantaneous centre is not: each of the six is paired with a measurement of how badly wrong it goes, and the ones that turn out to be nearly right are labelled as nearly right rather than being dropped.