Six things a centre is not

The instantaneous centre is the first object most readers meet in this subject and the one they are most likely to have been told something wrong about. It is not a hard idea; it is a narrow one, and the errors are all errors of reading more into it than it says.

What it says, in full: at this instant, the velocity field of a moving plane is exactly the velocity field of a rotation about one point. That is a statement about velocities and it is exact.

Here are six things that do not follow. Each comes with a number, because a false claim that is nearly true is a different problem from one that is wildly wrong, and the two need different warnings.

One. It is not the point that is not accelerating #

Wrong, and by a lot.

A body in steady rotation about a fixed pivot has zero velocity and zero acceleration at the pivot, and that is the case everybody’s intuition is built from — a crank, a gear, a wheel on a fixed axle. On any of those, the two questions have the same answer.

On a coupler they do not. The point with zero velocity satisfies $A' + i\varphi' w = 0$ and the point with zero acceleration satisfies $A'' + (i\varphi'' - \varphi'^2)w = 0$, and those are different equations in different quantities.

Over 117 positions of the site’s running four-bar, the two points are never nearer than 1.49 coupler lengths and get as far apart as 22.2. On a linkage whose coupler is 3.5 units and whose frame is 4, the closest approach is about five units — larger than the machine.

The point at the instantaneous centre is generally being accelerated hard, along the pole normal, and that acceleration is what makes every coupler point’s path curve at all. The essay on the second pole is about the gap and about what the acceleration pole is genuinely good for.

Two. It is not a fixed point of anything #

Wrong, and this is the one the word “centre” causes.

The instantaneous centre is a different point of the fixed plane at every instant, and a different point of the moving plane at every instant. It traces one curve in each: the two centrodes.

The honest version of “the body is rotating about it” is the rolling statement: the motion is the moving centrode rolling on the fixed centrode without slipping, and the instantaneous centre is the contact point. That is a description of a wheel, not of a pivot, and nobody thinks a rolling wheel is pivoting at its contact patch.

The site has a whole essay about what this costs in practice. A suspension’s roll centre is an instantaneous centre found by Kennedy’s construction, it is quoted as a height in millimetres in catalogues and in setup sheets, and it moves through the suspension travel — by metres, on a linkage whose parts move by tens of millimetres.

Three. A body pivoted there would not do the same thing #

Wrong, and measurably so, at first order.

The tempting substitution is: since the plane is turning about this point, replace the mechanism by a crank pivoted there. That agrees in velocity and disagrees immediately afterwards, because the radius is wrong — the distance from the point to the instantaneous centre is not the radius of curvature of the point’s path.

The correct substitution is a crank at the point’s own centre of curvature, which is a different point for every point of the plane and is given by Euler and Savary’s relation. Done that way the substitution is right to third order and the two paths separate like $\Delta^3$; done at the pole it is right to first order and they separate like $\Delta$.

That is a difference of two whole powers, and it is measured directly rather than argued.

Four. It is not the centre of curvature of anything’s path #

Wrong, and this one is the same mistake as three, stated the other way round.

The centre of curvature of a point’s path lies on the ray from the instantaneous centre through the point, which is a real and useful fact and the reason the two get conflated. Where on the ray is settled by Euler–Savary and is almost never at the pole itself.

Being at the pole would require $1/r_0 = 1/r - 1/(\delta\sin\psi)$ to give $r_0 = 0$, which needs $r = 0$: the point would have to be the pole. So the only point whose centre of curvature is at the instantaneous centre is the instantaneous centre, whose curvature is undefined because it is not moving.

The site’s machinery refuses to report a curvature there rather than returning a very large number, and the refusal is a check the gate runs. A figure that quietly reported an enormous curvature at the pole would carry a caption about an extremely tight bend at the one place in the plane where there is no curve at all.

Five. It does not settle a force #

True of something else.

The claim in circulation is that mechanical advantage is decided at the instantaneous centre, and there is a correct statement nearby: the ratio of two links’ angular rates is a ratio of distances from a common instantaneous centre, by Kennedy’s theorem.

That is a velocity statement and it is exactly right. Turning it into a statement about forces needs a virtual-work argument on top of it, and the virtual-work argument needs the mechanism to be ideal — no friction, no inertia, no elasticity. The site uses the velocity statement in earnest in the gearset lever diagram and in the epicyclic ratio taken two ways, and is careful in both to say that what is computed is a relation between rates.

The distinction has bite at a toggle position, where the velocity ratio goes to infinity and the mechanical advantage is quoted as infinite. What actually happens there is that a small input motion produces no output motion, so an ideal mechanism transmits unbounded force and a real one deflects, binds, or breaks. The geometry is right about the ratio and silent about everything that decides which.

Six. It is not a property of one body #

Nearly right, and worth the pedantry.

An instantaneous centre belongs to a pair of bodies. The coupler has one instantaneous centre relative to the frame, another relative to the crank, another relative to the rocker, and they are three different points.

The usual usage — “the coupler’s instantaneous centre” — silently means “relative to the frame”, which is almost always what is wanted and is a reasonable shorthand. It stops being reasonable in a mechanism where more than two bodies move independently, and Kennedy’s theorem is the machinery for keeping the three-body case straight: for any three bodies, the three pairwise centres are collinear.

The site’s transmission field is where this earns its keep. A gearset with two degrees of freedom has no single ratio at all, and the lever diagram that makes it tractable is a set of pairwise instantaneous centres arranged on a line. Reading any one of them as “the” centre of a member gives nonsense.

Where each of the six is actually used, correctly #

Every one of the six errors has a correct neighbour that is in daily use, and the corrections are only useful if the neighbours are visible too.

The acceleration pole is a real object with a real use: the acceleration field of a moving plane is a fixed rotation-and-scaling about it, so graphical acceleration analysis is possible in the same way graphical velocity analysis is. What it is not is the velocity pole.

The centrodes are the honest form of “rotating about it”, and they are more useful than the loose claim, because a motion is completely determined by its two centrodes and two mechanisms sharing centrodes are interchangeable. The trammel’s exact straight line is a centrode argument and nothing else.

The pivot substitution works at the point’s own centre of curvature, and how long it works is measurable. That is the whole of this field’s practical content.

The centre of curvature is on the ray from the pole, which is a real constraint and halves the problem of finding it. Euler and Savary supply the rest.

The velocity-ratio claim is exactly right as a velocity claim and is used throughout the transmission field.

The pairwise reading is what Kennedy’s theorem is for, and it is what makes a three-body mechanism tractable at all.

So the six are not six things to unlearn. They are six places where a correct statement has been widened by one word — from velocities to accelerations, from an instant to a while, from a pair to a body — and the widening is where the trouble is.

The measurement that would settle any of them on a given mechanism #

A reader with a mechanism in front of them can check the whole list with two sweeps, and it is worth saying how, because the errors are the kind that survive being argued about and do not survive being measured.

Sweep one: the two poles. At each position, find the point with zero velocity and the point with zero acceleration and record the distance between them. If that distance is a small fraction of the mechanism, claims one and three are harmless approximations on this machine; if it is comparable to the mechanism or larger, they are not. On the site’s four-bar the minimum is 1.49 coupler lengths and the errors are not harmless.

Sweep two: the pole’s own path. Record where the instantaneous centre is at each position and how far it moves per radian of input. That number is the pole velocity, it is the $\delta$ of Euler and Savary’s relation multiplied by the angular rate, and it is the size of the correction every one of the six claims is missing. A mechanism with a nearly stationary pole is one where the crank intuition nearly works.

Both sweeps are cheap and both are already what this field computes for other reasons. The point of stating them as a procedure is that “the instantaneous centre moves” is a fact about a particular machine as much as about the concept, and how much it matters is a number that machine has.

What is left, and it is not nothing #

Stripping away six over-readings makes the object look diminished. It is not, and the remaining content is worth stating positively.

The velocity field is a rotation about that point, exactly. Every velocity is $\varphi'$ times the distance from it, perpendicular to the ray. That is a complete description of one derivative of the motion of an infinite plane, in one point and one number.

It organises everything else. The circle of points going straight passes through it, with its diameter along the direction the pole is itself travelling. The cubic of stationary curvature has a double point there. Every point’s centre of curvature is on its own ray from it. Both special curves are most cheaply held in polar form about it.

It is constructible in two lines. Kennedy’s construction crosses the crank extended with the rocker extended, and the result agrees with the derivative route to $1.5\times10^{-13}$ over a full turn. That is why the object survived into every engineering curriculum: it is the cheapest useful thing on the drawing.

The pattern the six share #

Five of the six errors have one structure, and naming it is more useful than the individual corrections.

They generalise from a body in steady rotation about a fixed pivot. On that body the instantaneous centre is fixed, it is the acceleration pole, a pivot there reproduces the motion exactly, it is the centre of curvature of every point’s path, and it is a property of the body. Every one of the six claims is true of a crank.

A coupler is not a crank, and the difference is that its instantaneous centre moves. The rate at which it moves is the pole velocity, it is the $\delta$ in Euler and Savary’s relation, and it is the source of every second-order phenomenon in the curvature field. The claims fail exactly to the extent that the pole is moving, which is why they fail hardest on a fast-swinging coupler and hardly at all near a position where the pole is momentarily stationary.

That gives a one-line diagnostic worth carrying: if the argument would be unchanged when the pole is moving quickly, it is probably a velocity argument and correct; if it would not, it is probably one of the six.

A seventh, which is not a mistake but a shorthand #

One more claim is worth separating from the six, because it is not wrong and it is where the wrong ones come from.

“The mechanism is instantaneously equivalent to a four-bar with pivots at the two centres of curvature.” That one is true, it is the classical equivalent linkage, and it is genuinely useful: any mechanism at any position can be replaced by a four-bar whose pivots are the two moving pins’ centres of curvature, and the replacement reproduces the motion to second order.

The equivalent linkage is what a cam-and-follower is replaced by when somebody wants to reason about it as a linkage, and it is why a cam contact and a pin joint can be counted the same way for one position and not for a range.

The shorthand becomes a mistake at exactly the point where the word “instantaneously” is dropped. The equivalent linkage is different at every position — its pivots move, because the centres of curvature move — so a chain of arguments that builds on it for a range of motion is building on a different mechanism at every step.

That is the same failure as claims two and three, arriving with better credentials. The correction is the same: the substitution is exact in velocities, right to second order in positions, and wrong at third order by an amount with an exponent attached to it.

The two that are worth remembering #

If only two survive the reading, they should be these.

The centre is where the velocities vanish, and nothing else vanishes there. That covers claims one, three and four at once, and it is the shortest correct statement of what the object is.

It belongs to a pair of bodies and it moves. That covers claims two and six, and it is the one that prevents a number quoted in a catalogue — a roll centre height, a lever position — from being read as a fixed property of a machine.

Claim five stands on its own because it is the one where the correct statement is genuinely useful and the incorrect one is genuinely close: a velocity ratio is not a force ratio, and the step between them is an assumption about an ideal mechanism that the site’s practice field spends its time undermining.

One last observation about why this particular object attracts so many over-readings, because the reason is structural rather than a matter of carelessness. The instantaneous centre is a point, and a point is the most concrete thing a geometry can hand anybody: it can be drawn, marked, pointed at and built at. Every one of the six mistakes is an attempt to give that point a persistence it does not have — to make it a pivot, a fixed feature, a centre of curvature, a place where a force acts. None of those is an unreasonable thing to want, and all of them are ways of asking a point to be a part. The honest description is the opposite: the pole is a property of an instant of a relative motion, it belongs to a pair of bodies rather than to either, and it moves as fast as the mechanism does. Anything true of it is true for no length of time at all. That is a difficult thing to hold on to while looking at a drawing where the point sits still on the page, which is precisely why the six survive.