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\title{Comparison between Air and Ball Bearings for GGG.}
\author{Raffaello Pegna}
\date{January, 19 2012}

\begin{document}
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%% \begin{abstract} 
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%% %\tableofcontents 
\section{Ball Bearings}
\label{sec:BallBearings}
In ball bearings the balls are in constant contact with and form a solid bridge between the bearing races.
Each race is a ring with a groove where the balls rest. The groove is
usually shaped so the ball is a slightly loose fit in the
groove. Thus, in principle, the ball contacts each race at a single
point. However, a load on an infinitely small point would cause
infinitely high contact pressure. In practice, the ball deforms
(flattens) slightly where it contacts each race. The race also dents slightly where
each ball presses on it. Thus, the contact between ball and race is of
finite size and has finite pressure. Note also that the deformed ball
and race do not roll entirely smoothly because different parts of the
ball are moving at different speeds as it rolls. Thus, there are
opposing forces and sliding motions at each ball--race
contact. Overall, these cause bearing drag and noise.

Super precision grade ball bearings are manufactured with typical $\sim100$~nm geometric tolerances: balls are manufactured with $\sim100$~nm
sphericity and $\sim500$~nm diameter tolerance. Very roughly, a 5~cm diameter ball bearing can then cause a shaft fitted to the inner race 
to tilt by some $\mu\rm{rad}$. Due to the diameter 
tolerance, different balls in a bearing rotates at different rates, giving rise to a sistem with complex periodicities, also at low frequencies. 
The complexity of the ball bearing can then result in high and low frequency shaft tilt noise in the $\mu\rm{rad}$ region.

\section{Air Bearings}
\label{sec:AirBearings}
Air bearings use a thin film of pressurized air to support
a load, the same way the puck on an air hockey table "floats" on
air. This type of bearing is called a "fluid film" bearing. Fluid film
bearings have no solid-to-solid contact under typical running
conditions; instead, a film of pressurized air forms a layer between the solid 
machine elements and
serves to transfer forces from one to the other.   The fluid is
able to transfer forces because as the fluid is pushed through the
bearing gap it generates a pressure profile across the bearing
area.
 Fluid film bearings
offer a number of advantages over mechanical bearings. First, because
there is no contact, air bearings do not suffer from wear or heat
generation due to friction. They also exhibit no starting or running
friction, even under their highest design loading. In addition, the
fluid film acts to center and average out small scale errors in the
components resulting in motion which can be more accurate than the
individual bearing components. Air bearings also offer much higher
stiffness than rolling element bearings because the air film fully
supports the components, as opposed to balls or rollers which have
point or line contact and are therefore limited due to Hertzian
contact stiffness.

Air bearings require very tight bearing gaps for proper operation 
($10 \, \mu \rm{m}$) 
which translates into extremely high accuracy requirements on the components.
Gas bearings have essentially zero friction at all speeds and because the tight bearing 
clearances demand high accuracy components this results in extremely high accuracy motion. 
Air bearings offer good noise performances because they have excellent
accuracy, precision and repeatability. They offer high accuracy
because the components are manufactured with extremely tight
tolerances and because of the air film's averaging effects. Typical
rotary runout can be as low as $10\, nm /\rm{cm}$ and tilt characteristics 
as low as $0.5 \, \mu \rm{rad}/\rm{cm}$. Because the air bearing 
has two surfaces and only two surfaces, the tilt noise is 
essentially concentrated at the rotation frequency,
while the noise performance is order of magnitude better at low frequency.
 
\end{document}

