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I've been reading into air bearing recently and keep coming across the term "pre-load". References online and wikipedia both state that a pre-load helps stiffen the bearing such that their is less variation.

I'm guessing the pre-load ensures the air-gap is maintained/little variation.

However, I'm having a difficult understanding how adding some force (pre-load) would actually help with this.

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The reason is to prevent instability. What is instability? Let's have look at a simple shaft-journal bearing system.

Here is an ideal system wherein a shaft is rotating clockwise inside a journal bearing. As you can see the eccentricity is zero. Notice the gap between the shaft and the journal is filled with oil, or air in case of air bearings.enter image description here

Suppose the whole system rotates unloaded in the beginning. Now duo to some irregularities (bumps, impacts ...) the eccentricity between the shaft and the geometrical center of the journal bearing increases. Here you can see indeed the shaft moves upward to the right, the direction of the eccentricity is $\theta = 38.78°$:enter image description here

Now the pressure distribution tends to rotate the shaft in counterclockwise direction, the pressure in narrow split is the highest. The resultant of pressure in narrow split is $F$. If the bearing is not preloaded then this force $F$ makes the system unstable by disrupting the natural rotational motion of the shaft. By preloading the bearing in all directions of course (because this force $F$ happens to have random directions because the irregularities happen randomly) compensate the force $F$ and the system stays stable.

If the angular speed of the shaft is low enough and the viscosity is high enough then the disruption goes away automatically and no preload is needed, but that's not the case for high angular velocities in combination with low viscosity. The viscosity of air is low enough and the air bearing usually are used in high rpm, so the preloading is indispensable.

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I suspect that by 'pre-load' they mean putting a flow restriction between the fluid pressure supply and the bearing. Doing this is essentially adding a (large) fluidic resistor between the pressure supply (analog of voltage source.) This may seem wasteful, dropping all that pressure through that restriction, but the result is a flow rate that's basically LargePressure/LargeFluidResistance; this gives you almost constant flow, -- because there's almost nothing you can do at the bearing that would be nearly as much flow resistance as that (seemingly wasteful) upstream restriction, the flow stays quite constant (with respect to motions at the bearing), making bearing act mechanically stiff.

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