Re: [ATM] Polishing / Figuring Simulator

[Mike Lockwood <melockwo@uiuc.edu>]

Hi,

Mark Holm wrote:
> Another principle that might work into the calculation is that the 
> upper disc is not accelerating in rotation.  This means that the 
> torques have to balance.  The torque of the overhanging part isn't 
> too hard to get. The torque of the pressure distribution is the 
> integral of the first moment of the pressure distribution.  I think
> this torque condition gives you the slope function I mentioned 
> above, but am not 100% certain.

A few months back I worked on a related topic.  I thought a good way
to test out my assumptions about rotary polishing (that is, using a 
75% diameter lap on a slow arm with a fast spindle like the 
Mirror-O-Matic, or like Zambuto's machine's do) would be to compare my 
predicted rate for the rotation of the lap with what I observed.  This 
rotation is highly sensitive to lap condition, though.

I wrote some code to find the speed of the lap that minimized the
torque that causes the lap to rotate.  I assumed equal pressure over
all the lap, which does not hold when the lap overhangs, but I did it
anyway because I wanted to get the rest of the simulation working.
Torques were calculated and summed computationally (I did not develop
an equation - I did it numerically) for small parts of the lap that
were in contact with the mirror.

As I computed it, the incremental torque was proportional to the
difference in speed of the lap and the mirror for one small spot on
the lap in the direction normal to the radius.  That is, only the
component of the velocity difference that is normal to a line from the
spot on the lap being considered to the center of the lap (which is
the radius or moment arm) contributes to the torque.  Since the 
friction coefficient is the same over all the lap, and since I assumed 
equally distributed pressure, those did not need to be known to 
determine lap rotation speed relative to the mirror.

So, basically what I did for each offset (the off-centeredness of the
lap) was to find the total torque for the lap running at a variety of
speeds.  Then I found the minimum of those torques, and that was the
speed at which the lap would rotate for that offset.  By repeating 
this process, I found the predicted rotation speed for a wide range of 
offsets.

I ended up with a plot of lap rotation speed as a function of lap 
offset.  From what I recall of the results, they predict the reversal 
of the lap rotation direction at some point, but the real lap spun 
faster, likely due to my bad assumption of uniform pressure over the 
whole lap.

If someone has a closed-form expression for the pressure on an
overhanging pitch lap as a function of overhang, or even just a good
approximation, I would like to see it.

I find that it doesn't take long for me as a human to adapt to the lap
I am currently working with, but some adaptation is usually necessary.
I can feel what it is doing and I can estimate the effect by that
feeling.  I bet an automated system can make a fairly good mirror, but
I don't think it would be as smooth as a hand-figured surface, or one
made by a less sophisticated machine in capable, practiced hands.

	Mike Lockwood

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