A slow sensor makes the bearing cycle

A rotor’s inertia slows the jump and cannot make it ring closed one door and pointed at another. A self-heating oil film under a soft motor drive has two stable states, a cool one and a hot one, and a rotor that has to spin up cannot make the film oscillate between them, because the film and the rotor only ever push each other the same way. Only a link of the opposite sign could do it: a drive that lowers its torque as the film warms. The essay tried one that acted instantly, on the film’s own temperature, and found that it made the cool state ring and then settle.

No real protection acts on the film. A film in a bearing is tens of microns thick, and nobody puts a sensor in it. The sensor sits in the housing — a thermistor in the bearing cap, a bimetal strip on the motor frame — and the housing warms from the film over its own thermal time, which for a lump of cast iron is minutes. The film’s own diffusion time, across a gap of that thickness, is of the order of a hundredth of a second. So the protection reads a temperature that lags the one it is protecting by four orders of magnitude.

The natural guess is that this makes things worse only in degree: a slow sensor corrects late, a late correction overshoots, and an overshooting correction on a stable system rings for longer before it settles. That is how a delayed negative feedback behaves in every linear textbook. It is the wrong picture here, and the reason is that the film is not linear in the one respect that matters. It has two states for the same torque.

An S and a slow variable #

The film is the one the film that heats itself set up: oil sheared between two walls held at a fixed temperature, its viscosity falling exponentially as it warms, so that the heat it generates rises as it heats. Held at a fixed stress it runs away past a fold. Driven through a motor whose torque falls with speed, as in a motor turns the runaway into a jump, the runaway becomes a jump between two branches of steady states, and a soft drive — one whose torque droops little with speed — has an S-shaped set of them: a cool branch, a hot branch, and an unstable middle branch joining the two folds.

Everything here uses that film and that drive, with a droop of 0.04 and a stall group of 2 in the film’s own units, and adds one state. The housing has a temperature excess H, measured in the film’s temperature units, which relaxes towards the film’s mean temperature with a time constant $T_h$ counted in film diffusion times:

$$\frac{dH}{dt} = \frac{\bar\theta - H}{T_h}.$$

The protection derates the motor on H. The stall group, which sets how hard the motor can drive the film, falls as the housing warms:

$$\delta_s(H) = \delta_{s0}\,(1 - gH)^2,$$

with g the derating strength, 0.22 per unit of housing excess unless stated. That is all. No delay is written into the equations; the lag is the housing’s own heat capacity, as it would be in a machine.

The figure is the whole argument in one picture. For each housing temperature the film has the steady states of its derated motor, drawn as mean film temperature against housing temperature. Between housing excesses of 1.237 and 1.593 both a cool and a hot state exist; outside that window only one does. The cool branch is lost when the housing cools below 1.237, because the torque has then risen past the cool state’s fold, and the hot branch is lost when the housing warms above 1.593, because the torque has fallen below the hot state’s.

The dashed line is where the housing would be in equilibrium with the film, H = θ̄. A steady state of the whole machine has to lie on one of the branches and on that line at once, and neither branch meets it inside the window. The hot branch sits far above it — the hot film’s mean is near 2.7 while the housing is below 1.6 — so on the hot branch the housing is always warming. The cool branch sits below it, the film’s mean near 0.55 against a housing above 1.2, so there the housing is always cooling. There is nowhere to stop.

The loop the housing drives #

So the machine goes round. Start on the hot branch with the housing at 1.24. The film is hot, the housing warms towards it, the derating lowers the torque, and the hot state follows along its branch, cooling a little, until the housing reaches 1.593 and the branch ends. The film has nowhere hot left to be and falls to the cool branch in a few diffusion times — instantly, on the housing’s clock. Now the housing is warmer than the film and cools. The torque rises, the cool state warms a little along its branch, and at a housing excess of 1.237 that branch ends too, and the film jumps hot. The loop closes.

With $T_h$ large the film is always on one branch or the other, and the time the housing spends on each leg is a single integral: across the window, divided by how fast the housing moves, which is the gap between the film’s mean and the housing’s own temperature. At a derating of 0.22 the housing spends 0.305 $T_h$ warming on the hot branch and 0.448 $T_h$ cooling on the cool one, a cycle of 0.7535 $T_h$ with the film hot for 40.5% of it.

That is the calculation with the fast variable eliminated. The figure is the calculation without eliminating anything: the heat equation for the film, its exponential source, the motor, and the housing’s relaxation, marched together from rest with $T_h = 300$. The film’s centre temperature is a square wave — a relaxation oscillation, in the language of the circuit theorists who named it — swinging between 0.64 and 4.87 in its own units, and the housing is a sawtooth between 1.22 and 1.61, just outside the two folds. The cycle repeats every 259 diffusion times and does not decay. It is not a transient ringing down; it is where the machine lives.

The claim this answers, that a slow thermal protection can only delay a stable film’s settling, fails at its first step. There is no stable state of the whole machine for the film to settle into. A linear delayed feedback overshoots a target that exists. The housing here chases one that the film keeps moving to the other side of it.

Why the amplitude belongs to the film #

In a linear feedback loop with a delay, the size of an oscillation is a matter of gain: turn the gain down and the ringing shrinks, and below some gain it dies. Nothing like that is true of this cycle. Its amplitude is the height of the S — the difference between the hot and cool branches at the folds — and that belongs to the film and its motor, not to the protection. A centre temperature swinging by more than four of the film’s units is not a small correction left over from a feedback that nearly works. It is the full jump that a motor turns the runaway into a jump described, taken twice every cycle.

The protection’s settings decide something else. They decide where the window sits in housing temperature, how fast the housing crosses it, and — the question that matters to anybody who fits one — whether the cycle happens at all.

The figure runs the derating strength g across its range and computes the quasi-static cycle at each. Below about 0.16 there is no cycle, because the protection is too gentle: the housing can warm all the way to the hot film’s mean temperature before the torque has fallen enough to lose the hot state, so the hot branch crosses the equilibrium line and the machine settles there. The bearing runs hot for good, and the protection has done nothing but shift the hot state a little. Above about 0.31 the reverse happens: the derating is so strong that the cool branch meets the equilibrium line before the torque can rise past its fold, and the machine settles cool, with the motor running derated.

Between the two, the film cycles. The fraction of each cycle it spends hot falls steadily as the derating strengthens, from three quarters at the gentle edge to an eighth at the strong one, because a stronger derating pushes the window to cooler housing temperatures, where the hot film heats the housing faster and the cool film cools it more slowly. The period has a minimum near three quarters of $T_h$ in the middle of the range and grows at both edges, where one leg of the cycle takes the housing close to the equilibrium line and it creeps there, moving slowly because the film and the housing are nearly at the same temperature.

That shape — no oscillation at either extreme and a band of it between — is the practical content. A protection strong enough to matter and too weak to hold the film on its cool branch is exactly the protection that makes a bearing hunt, and there is no setting of its time constant that avoids it. A faster sensor only shortens the period.

The ghost at each fold #

The quasi-static cycle assumes the film leaves its branch the moment the branch ends. It does not, and a lost steady state still holds the film showed why. Just past a fold there is no steady state, but there is a place where the film’s rate of change is small, the ghost of the two states that collided there, and a film arriving at it lingers before it moves on. That essay measured the delay for a film pushed past its fold by a fixed overshoot. Here the overshoot is not fixed. The housing keeps drifting while the film lingers, dragging the fold further past it, so the film leaves sooner the faster the housing moves.

The figure marches the whole machine at housing time constants from 20 to 1,000 diffusion times and compares the period with the quasi-static prediction. At $T_h = 20$ the marched cycle is 27.1 diffusion times against 15.1 predicted, eighty per cent longer, and at $T_h = 1000$ it is 805 against 754, seven per cent longer. The ratio tends to one, as it must, but slowly, and the right-hand panel says how slowly: the excess of the marched period over the prediction grows as $T_h$ to the power 0.364.

A third is the exponent the theory of slow passage predicts. A fold approached by a parameter moving at rate ε delays the jump by a time of order $\varepsilon^{-1/3}$ in the fast variable’s own units — slower drift, longer lingering, but only as the cube root. Here the housing drifts at a rate proportional to $1/T_h$, so each fold adds a delay proportional to $T_h^{1/3}$, and a cycle made of two folds and two slow legs has a period of about 0.7535 $T_h$ plus a term of that shape. The measured 0.364 sits a little above a third because the smallest housing times are not yet asymptotic. The housing’s sawtooth runs from 1.22 to 1.61 rather than from 1.237 to 1.593 for the same reason: the housing keeps moving while the film hesitates at each fold.

For a real bearing the cube root is what makes the quasi-static cycle a good description. A housing time constant of a minute or two is some ten thousand film diffusion times, where the cycle is minutes long and the delay at each fold, extrapolated along the same power, is about a second. The film spends almost the whole cycle on one branch or the other, jumping between them faster than any sensor on the housing could register.

Three clocks, and which link does the work #

The rotor has not appeared in any of this, and it need not. Its time is between the film’s and the housing’s, and the essay before this one showed that its only effect is to add a delay to each jump, of the same kind as the ghost’s. It pushes the film the same way the film pushes it, so it cannot make or break the cycle. Every ingredient of the oscillation is elsewhere.

There are two. The first is the S: a fast variable with two stable states for one value of a slow one, which here is the film under a soft drive. The drive’s softness is not optional. A stiff drive, with a steep torque characteristic, has no fold at all, a single steady state at every housing temperature, and on that single branch the housing and the film find their equilibrium and stay there. A motor turns the runaway into a jump found the droop at which the folds appear; below that droop, no derating of any strength makes the machine cycle.

The second is a slow variable that pushes back: the housing, warmed by the hot state and cooled by the cool one, moving the drive in the direction that destroys whichever state the film is in. Either alone is harmless. An S without the push-back is the bistable bearing of the earlier essays, sitting on whichever branch it happens to be on. A push-back without the S is the instant derating of the rotor essay, which rings and settles. Together they make the same machine as a compressor in surge, where the plenum decides whether the fast flow through the blades jumps between its branches while the slow pressure in the plenum drags it back, and as the neuron models of FitzHugh and Nagumo, where a membrane voltage with two stable states is pushed back and forth by a slow recovery variable.

The analogy with the drop at its fold, which the motor essay offered, can now be made exact. A pendant drop falls at a fold because nothing pushes it back once its hanging state is lost; a dripping tap is a relaxation oscillator only because the tube refills it. The housing is the refill. Without it the jump happens once.

Hysteresis without memory of the operator #

Bistable fluid systems usually show their two states as hysteresis under a control that somebody turns: a stall that comes on at one angle and goes off at another, as in the lift that arrives late, or a convection pattern that remembers how the heat was turned up. The loop in the first figure is the same loop. What is different is that nobody turns the control. The housing’s temperature is the control, and the film sets it — so the hysteresis loop is traversed automatically, at a rate set by a thermal mass, for as long as the motor is switched on.

That also says how to recognise the behaviour in a machine. A bearing temperature that is steady is either on one branch or the other, and nothing in its reading tells an operator which. A bearing temperature that alternates between two plateaus on a period of minutes, with the housing’s reading a slow sawtooth, is this cycle, and the two plateaus are the two branches.

What was checked #

The film’s mean temperature enters both the housing equation and the quasi-static integral, so it was computed from the closed-form steady profile and checked against a direct average of the profile at three states, agreeing to three parts in a billion. The quasi-static period was computed by integrating along each branch between the folds, and the marched period was measured between successive upward jumps after the first had settled. At $T_h = 1000$ the two agree to seven per cent, with the excess growing as the power law in the previous section. The calculation is also fed cases it must refuse: a housing with a negative time constant, a tolerance no average could meet, and a drive stiff enough to have no folds, from which no window of housing temperatures can be computed because there is no S to put one in.

What the model leaves out #

A lumped housing. One temperature for the whole housing is the simplest thermal mass. A real housing has gradients, and a sensor at its surface reads a filtered version of the temperature near the bush. More filtering lengthens the slow legs and does not touch the folds, so it changes the period and not the existence of the cycle.

Walls at a fixed temperature. The bush on either side of the film is held at the reference temperature throughout. In a machine the bush warms with the housing, and a warmer bush both heats the film and makes its viscosity lower from the start. That shifts both branches with H in a way the derating does not, and it could move the window or close it; it is the first thing a calculation for a particular machine would have to add.

A smooth derating. Many protections are switches rather than slopes: a bimetal cut-out that removes power at one temperature and restores it at another. That is a relaxation oscillator by construction, with its hysteresis in the switch rather than in the film, and it is a different and much older story. The point here is that a protection with no hysteresis of its own inherits the film’s.

A planar film. As in the essays before. A journal bearing’s gap varies round the shaft, and the shape of the gap sets its load; a hot film lets the shaft move, which is another state with links of its own.

Who worked it out #

The fold at fixed stress is Nahme’s, from 1940, and Frank-Kamenetskii’s for thermal explosion. Relaxation oscillations were named by van der Pol in 1926, for a triode circuit whose fast voltage had an S-shaped characteristic and whose slow variable was a capacitor’s charge. The reduction to two slow legs joined by fast jumps is the method of singular perturbation that grew out of that problem, and the delay at a fold approached at a finite rate, with its cube-root law, was worked out by Pontryagin’s school in the 1950s and by Haberman in 1979 for slow passage through a saddle-node. FitzHugh in 1961 and Nagumo in 1962 wrote the neuron model with the same structure. What is new here is only the assembly: that the thermal protection of a bearing, derating a soft drive on a slow housing temperature, is one of these circuits.

Still open: a bush that warms with the housing #

Every figure holds the walls of the film at a fixed temperature while the housing drifts. In a machine the bush is part of the housing’s thermal mass, so as the housing warms the film’s own boundary warms with it, and the whole S slides along the temperature axis as well as being pushed by the derating. The link is of the positive kind: a warmer bush makes a warmer film, which warms the bush. The next calculation lets the walls follow H, and asks whether that second slow link widens the band of derating strengths that cycle, narrows it, or — past some coupling — closes it altogether, leaving a machine that settles hot however hard the protection derates, because the housing has become part of the runaway.