The force that is really an acceleration

This site’s founding claim is that everything adds to nothing. A body at rest has forces on it that sum to zero, and the whole of statics follows.

A body going round a curve is not at rest, and its forces do not sum to zero. They sum to $ma$, pointing at the centre of the curve, and there is nothing to be done about that except to notice which side of the equation it is on. D’Alembert’s move, from 1743, is to put it on the other side:

$$\sum F - ma = 0$$

call $-ma$ a force, and hand the whole apparatus of statics a problem it can now solve. The force is the acceleration, written on the other side, and its only claim to existence is that the sums cancel with it there.

Which free body produced the number #

The vehicle, cut free of the track, with its weight, the rail or tyre forces, and the inertia term drawn as though it were a load.

It is worth being pedantic about the bookkeeping, because the error this invites is not subtle and it is common. Once $-ma$ is on the free body, the free body is in equilibrium and the right-hand side is zero. Writing $\sum F - ma = ma$ counts it twice; writing $\sum F = 0$ without the term at all describes a body in a curve that nothing pushes. The move is all-or-nothing.

With the term in place, the arithmetic is ordinary statics. Resolve along the track’s surface: the inertia term contributes $mv^2/R\cos\theta$ and the weight contributes $-mg\sin\theta$, so the force the rail must supply per unit weight is

$$\frac{v^2}{gR} - \tan\theta$$

— the cant deficiency, a pure number with no mass in it. The speed at which it vanishes is $\sqrt{gR\tan\theta}$, which is what the curve was set out for: 73 km/h on the curve drawn, at six degrees of cant.

Below that speed the deficiency has the other sign and the rail is pushed the other way, which is why a slow train on a fast curve wears the inner rail. The free body is a choice, and choosing the vehicle rather than the track is what makes this one line.

The inequality with no mass in it #

Overturning is the case worth dwelling on, because the result is the opposite of everybody’s intuition and it is one line.

The vehicle tips when the resultant of weight and inertia passes outside the wheel on the outside of the curve. Take moments about that wheel: the inertia term $mv^2/R$ acts at height $h$ and the weight $mg$ acts at half the track $b/2$. The mass is a factor of both, so it cancels, and the condition is

$$\frac{v^2}{gR} > \frac{b}{2h}$$

A loaded lorry and an empty one overturn at the same speed. So do a full tanker and an empty one, provided the load’s centre of gravity is where the vehicle’s is. What decides is $b/2h$ — the track width over twice the height of the centre of gravity — and nothing else about the vehicle at all.

Which is why the interventions that work are geometric. Lower the load; widen the track; and, for a tanker, put baffles in it — because a partly full tank has a centre of gravity that moves outward as the vehicle corners, which raises the effective $h$ and is the reason a half-full tanker is the dangerous one. The liquid has a period of its own is the dynamic version of the same shifting mass.

Sliding, which does have a material in it #

The other way off the curve is sideways, and the comparison between the two is instructive.

The vehicle slides when the lateral force exceeds friction: $v^2/gR > \mu + \tan\theta$, roughly. That inequality does not contain the mass either — friction is proportional to normal force, which is proportional to weight — but it does contain $\mu$, which is a property of two surfaces and of the weather.

So the two failure modes are separated by a comparison of two pure numbers: $b/2h$ against $\mu$. A vehicle with $b/2h$ below $\mu$ slides before it tips, which is the safer of the two and is why racing cars are wide and low. A vehicle with $b/2h$ above $\mu$ tips first, which is a high-sided lorry on a dry road.

And it means the same vehicle changes failure mode with the weather. On ice it slides; on dry tarmac it tips. The force that is whatever it needs to be is what friction is doing in that inequality, and it is the only term in this essay that is not geometry.

The flywheel, where the size cancels out of the stress #

The cleanest instance of an inertia load in a structure rather than a vehicle is a spinning ring, and its result is one of the most surprising in the collection.

Take a thin ring of radius $R$ spinning at $\omega$. Each element of it has mass $\rho A\,R\,d\theta$ and acceleration $\omega^2R$ inward, so the inertia load is a uniform outward radial line load $\rho A\omega^2R$ per unit length. A ring under a uniform radial load carries hoop tension of load times radius — the force that is only a radius — so

$$N = \rho A\omega^2R^2, \qquad \sigma = \rho\,\omega^2R^2 = \rho v^2$$

where $v$ is the rim speed. The radius has cancelled.

A flywheel’s bursting stress depends on its rim speed and on nothing else. A 250 mm wheel and a 4 m wheel of the same steel burst at the same 200 m/s, one at 7,600 rpm and the other at 480. And the energy stored per unit mass is $\tfrac{1}{2}v^2$, which is the same quantity again — so the whole design of a flywheel is a competition between one material property, $\sigma/\rho$, and nothing else at all.

That is why flywheels are made of composites rather than steel: not because they are stronger, but because $\sigma/\rho$ is several times better, and $\sigma/\rho$ is the entire specification. The ranking belongs to the load case is the general form of that argument.

The cant a track is set out at, and the two speeds it is not #

A railway curve’s cant is a single number and the trains on it are not, which makes the setting-out a compromise with a name.

Cant is chosen for an equilibrium speed — the speed at which the deficiency vanishes and the passengers feel nothing sideways. A fast train exceeds it and runs at a cant deficiency; a slow freight train runs below it, at a cant excess, with the resultant leaning inward. Both are limited, and for different reasons: deficiency by passenger comfort and by the lateral force on the rail, excess by the wear on the inner rail and by the risk of a stopped train overturning inward in a high wind.

The numbers are worth having because they show how small the margins are. Actual cant is limited to about 150 mm on 1,435 mm track — six degrees — because a train stopped on the curve must not be uncomfortable or unstable. Cant deficiency is limited to about 110 mm, a further four degrees. Together they allow $\tan\theta \approx 0.18$, so the maximum lateral acceleration is about 0.18 g and the fastest speed on a curve of radius $R$ is $\sqrt{1.8R}$ metres per second — 190 km/h on a 1,500 m curve, and no faster whatever the train.

That single inequality is why high-speed lines are straight. The speed goes as the square root of the radius, so doubling the speed needs four times the radius, and the radius is bought in land. It is a geometric constraint on a national scale that comes from one term on one free body, and tilting trains exist entirely to move it — by tilting the body rather than the track, they raise the deficiency the passenger feels without raising the force on the rail.

The load that swings out, and keeps swinging out #

Slew a crane, or start a fairground swing, and the load hangs out at an angle. The naive calculation takes the radius as drawn and finds $\tan\theta = \omega^2r_0/g$. It is too small, and the reason is a feedback.

Swinging out increases the radius, which increases the inertia force, which swings it out further. The equilibrium angle is the root of

$$\tan\theta = \frac{\omega^2(r_0 + \ell\sin\theta)}{g}$$

and it is always larger than the uncoupled answer. For a load slung 12 m below a jib at 18 m, slewing at 0.35 rad/s, the naive angle is 12.7° and the true one is 14.7° — a 17% growth in radius over the un-swung value.

That matters because a crane’s duty is quoted at a radius, and the radius the operator reads off the jib is not the radius the load is at. The overturning moment is the load times the true radius, so a chart read at the drawn radius understates the moment by the same 17%. Balanced, and four times as heavy is the counterweight arithmetic this feeds into, and the feedback is the reason slewing rates are limited rather than because of anything the machine cannot take.

A structure that never moves, carrying an inertia load #

It would be easy to file all of this under vehicles. It should not be, because the largest inertia loads most structural engineers ever design for are applied to buildings that go nowhere.

An earthquake is an inertia load and nothing else. The ground moves; the building’s mass resists being moved; and the force on every floor is that floor’s mass times its own acceleration, put on the free body by exactly the move this essay opened with. A response spectrum is a table of accelerations, and the “equivalent static force” method is d’Alembert’s principle with a coefficient — the spectrum is not a load is what that coefficient is a summary of.

The same is true of a crane’s hoist, a lift’s emergency stop, a machine’s out-of-balance, a vehicle impact and a blast. In every case the free body is a mass and its own acceleration, and in every case the design question is what acceleration to use rather than what force.

Which gives a way of sorting the loads on a structure that is more useful than dead-and-imposed. Some loads are applied and some are inertia, and the two behave differently under nearly everything: an applied load does not change when the structure is stiffened, and an inertia load does, because stiffening a structure changes its period and its period decides its acceleration. A stiffer building attracts more earthquake force, and there is no analogue of that for its own weight.

Where the model stops #

The motion was steady. A body going round a curve at constant speed has a purely radial acceleration; one that is also accelerating along the curve has a tangential term as well, and the resultant inertia force is no longer horizontal or radial. Braking on a curve is the case, and it is the one that produces most of the incidents.

The body was rigid. A vehicle on suspension rolls outward as it corners, which raises the effective height of its centre of gravity and lowers the overturning speed — often by a fifth. A rigid-body calculation of a sprung vehicle is unconservative in exactly the term it is most sensitive to.

The rotation was slow enough to be treated statically. A slung load reaching its equilibrium angle does so by swinging, and it overshoots: the dynamic amplification on a suddenly applied slew is up to a factor of two on the change in angle. Twice the deflection, for the same load is that factor, and it applies here to a load nobody thinks of as suddenly applied.

The pavement was assumed to hold the wheels. A vehicle on a curve delivers its lateral force to whatever it is standing on, and a crane or a stacker on a slab delivers it to the slab — which then has to carry a horizontal force applied at the top of a wheel and resisted at its own supports. That is an ordinary structural problem which is very often forgotten, because the load case is filed under “vehicle” rather than under “lateral”.

And the frame was assumed to rotate uniformly. A body in a genuinely accelerating rotating frame has a Coriolis term as well as a centrifugal one — a force proportional to velocity relative to the frame, at right angles to it. For a crane’s trolley moving in or out while the jib slews it is real and is what makes the load’s path a spiral rather than an arc.

The generalisation #

The habit worth taking away is to notice which side of the equation a term is on, and to move it deliberately rather than by reflex.

D’Alembert’s move is one instance. There are others in this collection that have the same shape: a prestress is a load or a resistance depending on which side it is written, and the load put on backwards is entirely about choosing; a settlement is a displacement or a moment field, and the support that moved is the same choice; a temperature change is a strain or a force. In every case the physics does not care and the arithmetic does, and putting a term on the wrong side is the commonest way to count it twice or not at all.

The second thing to carry is the test this essay applied three times: look for the symbols that cancel. The mass cancelled out of the overturning inequality; the radius cancelled out of the flywheel’s stress; the material cancelled out of the cant. Each cancellation is a statement that a whole family of structures behaves identically, and each is worth more than the number it came from — because a number is about one design and a cancellation is about all of them.