ATM Pressure distribution between discs

[mdholm@telerama.com]

Hi ATM's

I am thinking about a program to calculate the wear patterns of different 
polishing strokes to help make hand figuring a bit more predictable.  I know 
that in principle, polishing wear is a very complex phenominon, and that there 
are a lot of polishing variations that would be hard to model.  I intend to 
start out simple:

    Only straight line motion, or motion that can be modeled as a series of 
straight lines.  No spinning.  I will assume that spinning motion is 
negligable.  Not strictly true, but, for most hand work, probably close.

    The tool doesn't deform.

    No added weight or pressure.

    No velocity effects.  Wear is proportional to distance traveled in contact 
x pressure.

    Both the mirror and lap are circular.  For now, I will not deal with star 
laps, etc., though in the future I would want to add that since is is a very 
common technique.

    The top disc has uniform mass distribution.

    The mirror and lap can have different diameters.

I don't intend, at the outset, to get too mathematically sophisticated about 
this.  If it takes a lot of stupid arithmetic, that is what computers are for.  
If I can get a coordinate system, boundary conditions and a function for each 
area element, the computer can do the grunt work.

The tricky part of this is dealing with the case where the top disc overhangs.  
I think I can deal with the projected area part, but I am a bit stumped on the 
pressure distribution.  Those of you who took Statics may even have done this 
problem as an excercise.  Since I never took Statics, I am a bit at sea.  I 
think there are two conditions I have to meet.  1. The sum of the pressure x 
area elements has to equal the weight of the upper disc.  2. The sum of the 
pressure moments has to equal the negative of the sum of the weight moments.  I 
need to get the pressure at each area element where the discs are in contact.  
I will assume, to start, that the top disc doesn't deform.  I expect to stay 
well away from the teetering case where pressure would go infinite given the 
ridgidity assumption.

Anybody feel up to helping me with this one?  I am moderately fluent in 
calculus, vectors and physics.

(I realize also that wear may be a nonlinear function of pressure, but I have 
to start somewhere, and a linear approximation may be good enough.)

Mark Holm
mdholm@telerama.com