A warm bush stops the bearing hunting

A slow sensor makes the bearing cycle built a relaxation oscillator out of three ordinary parts. An oil film whose viscosity falls as it warms heats itself; under a motor whose torque droops a little with speed it has two steady states, a cool one and a hot one, for the same drive — an S-shaped set of them, the subject of a motor turns the runaway into a jump. A thermal protection reads the housing, which warms towards the film over minutes, and derates the motor as it warms. With a derating of the right strength neither state can last: the hot film warms the housing until the derated torque loses the hot state, the film drops cool, the housing cools until the torque recovers past the cool state’s fold, and the film jumps hot again. The bearing hunts, on the housing’s clock, with an amplitude that belongs to the film.

That calculation held the film’s walls at a fixed temperature while the housing drifted, and said so as the first thing a real machine would change. A bush is part of its housing. As the housing warms the film’s own walls warm with it, and a warmer wall makes a thinner, faster, hotter film — which warms the housing further. The previous essay’s closing question was whether that second link, positive where the derating is negative, widens the band of protections that hunt, narrows it, or closes it altogether by making the housing part of the runaway.

A warm wall is a different drive #

The question has an exact answer in one line of algebra, and the line is worth having because it turns a new problem into an old one. The film’s temperature excess θ obeys a heat equation with a source that grows exponentially with θ, and its walls are now at $\theta_w = \beta H$, where H is the housing’s excess and β is the share of it that reaches the bush. Write the film’s temperature as the wall’s plus a rise, $\theta = \theta_w + \varphi$. The rise obeys the same equation with walls at zero and its source multiplied by $e^{\theta_w}$ — the oil everywhere is thinner by that factor before it has heated itself at all.

The motor enters through the same factor twice. Its torque falls with speed, and the film’s speed at a given torque goes as the mean of $e^\theta$, which is $e^{\theta_w}$ times the mean of $e^\varphi$. Carry that through and the film on warm walls is exactly the film on fixed walls under a drive whose stall group and whose droop are both multiplied by $e^{\beta H}$:

$$\delta_s' = \delta_s\,e^{\beta H}, \qquad k' = k\,e^{\beta H}.$$

The first factor is the positive feedback the question expected: the warm bush pushes the film harder, against the derating’s attempt to push it less. The second is the one nobody asked about. A larger droop is a stiffer drive — one whose torque gives way more steeply as the film thins — and the S exists only for drives softer than a critical droop, the cusp, which for this film is 0.0738. The essay’s motor has a droop of 0.04. A bush warm enough that $e^{\beta H}$ reaches 1.85 takes the drive to its cusp, and at the cusp the film’s two states merge into one.

Every steady state, fold and cusp computed in the earlier essays carries over with those substitutions, with the film’s mean temperature raised by the wall’s. That was checked against a film marched directly with its walls held at an excess of 0.3: its steady mean, 0.77987, against the substituted drive’s 0.77988.

The S rises and shrinks #

The figure draws the film’s two branches of steady states against the housing’s temperature for three couplings, with the derating held at 0.22. With fixed walls both branches exist between housing excesses of 1.237 and 1.593, and the cycle runs round that window. With the bush taking 15 per cent of the housing’s excess the branches rise — the film is sitting on warmer walls — and the window moves to 1.517–1.700, narrower by half. At 30 per cent it has moved to 1.833 and shrunk to 0.013 wide.

Two things move the window, and the substitution separates them. The rising stall group means the housing has to be warmer before the derating has cut the drive to either fold, so the window slides to higher H. The rising droop means the two folds themselves — the drive at which the cool state is lost and the drive at which the hot state is lost — approach each other, because the S is flattening towards its cusp. The first alone would move the cycle without destroying it. The second is what closes it.

The line where the housing is in balance with the film cuts neither branch inside any of the windows: on the hot branch the housing is still always warming and on the cool one always cooling. So while the window exists, the bearing still hunts. The coupling does not give the housing anywhere to stop. It takes away the thing the film jumps between.

The folds meet at the cusp #

Following the two ends of the window as the coupling rises makes the mechanism plain. The cool branch ends, going down, at a housing excess that rises from 1.237; the hot branch ends, going up, at one that rises more slowly from 1.593. They meet at a coupling of 0.324. At that point the housing excess is about 1.85, $e^{\beta H}$ is 1.82, and the stiffened droop is 0.073 — the cusp.

It is worth being exact about why this counts as a surprise. The warm bush is a positive feedback in the sense the question meant: a warmer bush makes a warmer film, which makes a warmer bush. Positive feedbacks in thermal systems usually destroy stability; this one destroys an oscillation instead, because the oscillation needed an S and the feedback’s effect on the drive’s shape outruns its effect on the drive’s strength. The bearing hunts because its film is bistable, and a warm wall makes the film less bistable. That is a statement about the film’s thermal boundary, not about the protection, and it would hold for any slow variable that warms the walls.

The cycle shortens to nothing #

While the cycle exists, it runs as before: the film on one branch, the housing drifting across the window at a rate set by the gap between the film’s mean temperature and its own, then a jump. The period is the housing’s time to cross the window twice, and it falls with the window: 0.7535 housing time constants with fixed walls, 0.484 with the bush at 15 per cent, 0.084 at 30. The share of each cycle spent hot barely moves, from 0.405 to 0.42, because both legs shorten together.

A short period is not yet a gentle cycle. For most of the range the jumps are nearly as large as with fixed walls — the film still leaves one branch for the other — and they shrink to nothing only close to the cusp, where the two branches merge. What shrinks first is how long the housing takes to cross a narrowing window, so near closure the bearing hunts fast, with the jumps’ own delays at each fold, which a lost steady state still holds the film measured, becoming a large part of each cycle.

The band of protections that hunt closes #

The practical question is not about one protection but about all of them. With fixed walls, deratings from 0.162 to 0.300 per unit of housing excess make the bearing hunt: gentler ones let it settle hot, stronger ones hold it cool. The figure computes that band at each coupling, by following the quasi-static machine from a cold start and bisecting on the derating at each edge.

The band narrows from both sides. At a coupling of 0.2 it runs from 0.182 to 0.279, at 0.3 from 0.205 to 0.254, and a little past 0.3 it is gone. The closure is the same cusp: whatever the derating, the window of housing temperatures the cycle would need sits where βH is large enough to stiffen the drive past its cusp, and no protection can move the window to cooler housings without also losing the hot state before the cycle starts. Past a bush coupling of about a third, no derating of any strength makes the bearing hunt.

That inverts the design lesson of the earlier essay. There, the band of hunting protections was a range to avoid, and nothing about the protection’s time constant could avoid it. Here a machine’s own construction — how well its bush is tied thermally to its housing — decides whether the band exists at all. A bush that follows its housing closely is a bearing that cannot hunt.

Where the bearing settles #

What the bearing does instead is settle, and the question’s last clause — whether the housing becomes part of the runaway — has a precise answer too. Past the cusp the film has one steady state at every housing temperature, and the housing drifts until it is in balance with it. The figure follows that balance as the coupling rises from 0.34 to 0.95.

The housing settles warmer the more closely the bush follows it: at an excess of 1.92 for a coupling of 0.35, 2.20 at 0.5, 3.30 at 0.9. The film’s rise above its own walls falls at the same time, from 1.30 to 0.33, because the protection has derated the motor to hold the balance: at a coupling of 0.5 the drive’s stall group is 27 per cent of its full value, at 0.9 it is 7.5 per cent. In that sense the housing has joined the runaway — the machine as a whole runs hotter the tighter the coupling. But it does not run away. The derating’s strength is set per unit of housing excess, and as the housing warms it takes the motor away faster than the warm bush can give the film back; the balance always exists below the housing excess at which the derating reaches zero. The bearing ends warm, steady, and slow.

What an operator would see, and what the rotor adds #

A hunting bearing announces itself: a housing temperature that saws up and down on a period of minutes and a motor current that steps between two levels with it, as the earlier essay described. The settled bearing past the cusp announces nothing. Its housing sits steady and warm, its current steady and low, and the only sign that anything is wrong is that the machine delivers a quarter of its rated torque. That is a harder fault to see than a hunting one, and it is the fault a well-coupled bush produces: the protection has won, by running the drive at a fraction of its rating for as long as the machine is switched on. Re-seating a bush so that it follows its housing more closely — which sounds like an improvement to heat removal — can turn a bearing that hunted into one that quietly underperforms, and the difference between them is a cusp crossed, not a setting changed.

The rotor, the machine’s third clock, still changes nothing about which side of the cusp the bearing is on. A rotor’s inertia slows the jump and cannot make it ring showed that the rotor and the film only ever push each other the same way, so the rotor can delay a jump but cannot create or remove one. Past the cusp there is no jump to delay. The rotor sets how fast the settled state is approached and nothing else.

What the warm bush has done to the S is what turning a single control does to any cusp. The drop falls at a fold because its hanging state is lost at one; a pattern of convection cells remembers how the heat was turned up because two of its states coexist over a range of the control. Both would lose their hysteresis if a second parameter could be moved to their cusp, and in the bearing the bush’s coupling is exactly that second parameter — one the designer sets, once, by how the bush is mounted.

Marched in full #

Everything above uses the quasi-static reduction: the film always at a steady state for the current housing temperature. The figure marches the whole machine instead — the heat equation for the film’s rise, with a source term for its walls warming, the derated and stiffened drive, and the housing’s own relaxation — from rest, with a housing three hundred film diffusion times slow.

At a coupling of 0.2 the film still jumps between its branches, every 153 diffusion times, against a quasi-static period of 110: the ghost of each fold holds the film longer when the window is narrow, because the housing crosses less of it before the jump is due. At 0.4 the film rises, overshoots once and settles, its housing at 2.011 against the quasi-static balance of 2.010. The reduction’s prediction — cycle on one side of the cusp, settle on the other — is what the full equations do.

What was checked #

With the coupling set to zero the machine must be the earlier essay’s bearing, and it is: a period of 0.75351 housing time constants against 0.75353 computed there. The substitution was checked against a film marched with its walls held warm, to one part in a hundred thousand. At a coupling of 0.1 and a housing a thousand diffusion times slow the marched period is 645 against a quasi-static 588, the ten per cent excess the folds’ delays contribute at that time constant. The quasi-static machine also had to be taught one thing the grid it walks on can hide: a balance point lying between the last grid point on a branch and the fold where the branch ends. At a derating of 0.31 with fixed walls the cool branch’s balance sits at 0.878, just inside its fold, and the machine now looks there before letting the film jump. The tests refuse a bush warmer than its housing, a housing with no time constant, and a tolerance of zero.

What the picture cannot show #

One share for the whole bush. The wall’s temperature is a fixed fraction of the housing’s excess, instantly. A real bush has its own thermal mass between the film and the housing, so its temperature lags the housing’s, and a lag between two slow variables can reintroduce an oscillation the instantaneous link removes. The calculation here is the limit of a bush much faster than its housing.

Both walls alike. A journal bearing’s shaft and bush warm differently: the shaft is cooled along its length and the bush by its housing. A film with walls at different temperatures has an asymmetric profile, and the substitution that makes the problem exact needs a single wall temperature.

A lumped housing and a smooth derating, as in the earlier essay, and an exponential viscosity law, which the film that heats itself adopted for its tractability and which real oils follow only over a limited range of temperature.

Why the positive link loses #

Feedback arguments are usually made about signs: a negative link stabilises, a positive link destabilises, a negative link with delay oscillates. The bearing shows how little the signs decide on their own. The oscillation here needs three things — a slow variable, a push back from it, and a fast variable with two states — the same three that make a compressor surge when the plenum decides — and the warm bush attacks the third, which no sign argument sees, because it acts on the shape of the film’s response rather than on its level. The same substitution applies wherever a boundary’s temperature enters an exponential source — a chemical reactor whose wall follows its jacket, a thermistor whose substrate warms with it — and in each the boundary moves the drive’s shape as well as its strength. The warm boundary makes the runaway easier to start and harder to be bistable about.

Who worked it out #

The exponential film and its fold are Nahme’s, from 1940, and Frank-Kamenetskii’s in thermal explosion, where the effect of a boundary temperature on the critical condition was part of the theory from the start: a warmer wall lowers the critical parameter by the factor $e^{\theta_w}$ used here. The cusp, where two folds meet and bistability ends, is the simplest catastrophe in Thom’s classification of 1972, and the interplay of a fast bistable variable with a slow one pushing back is the relaxation oscillator of van der Pol, FitzHugh and Nagumo. What is new here is the observation that a boundary temperature changes both of the drive’s parameters at once, and that the second change wins.

Still open: a bush with a mind of its own #

The instantaneous bush is the calculation’s weakest assumption, and removing it adds a third slow variable. A bush whose temperature relaxes towards the housing’s over its own time — faster than the housing, slower than the film — gives the machine two slow clocks, and a lag between the bush and the housing means the stiffening arrives after the heating it should have prevented. The next calculation gives the bush its own time constant and asks whether the band of hunting protections, closed here above a coupling of a third, reopens when the bush is slow enough; and whether a machine with a slow bush can hunt on two periods at once, the film jumping on the housing’s clock while the window itself breathes on the bush’s.