[Server-sky] paper on vibrational modes of a disk (2)

[Keith Lofstrom]

----- Forwarded message from Keith Lofstrom <keithl at kl-ic.com> -----

> On Fri, Mar 27, 2009 at 09:31:19PM -0700, Howard Davidson wrote:
> > There are some nice compact equations towards the beginning.
> > 
> > In general a stiff low density material, such a silicon or beryllium, 
> > with have rather high frequency modes of small amplitude.
 
On Fri, Mar 27, 2009 at 11:26:31PM -0700, Keith Lofstrom wrote:
> Thanks Howard!  Of course, these are for a centrally secured disk,
> but the analytical methods will need only slight modification for
> a moving center.

That gave me the "shape" of things.  I don't (yet) know the alpha
for the unconstrained disk floating free, but I am guessing it will
be around 1.0 for the lowest frequency mode.  The frequency is 
proportional to thickness divided by diameter squared, and the M
factor is a velocity (approximately the speed of sound over 3 ).
So for this very thin, very wide disk, I can expect vibrational
modes above 1.5 Hz.   If I "turn on" acellerations much more 
slowly than that, I can hope to avoid exciting them.

The next trick will be figuring out damping.  If something is vibrating
in a vacuum with no mechanical coupling to anything else, I have a
sneaky hunch that the only losses will be due to (very small) 
nonlinearities in the stress-strain curves turning vibration into
heat - the stress will be tiny, so the damping will be tiny.  Doppler
variation of light pressure will also damp the disk, but I have a hunch
that it will be a fantastically small effect.  Hence, I will probably
need to actively measure the strain with distance measurement (to another
server-sat) and remove it with timed impulses on the thrusters.  

The good thing is that the /reason/ to remove the vibrations - keeping
the antennas aligned - is being measured with the same elements whose
variation I care about, while the model is slow and computable and
characterizable in tables.

A lot of the measurements would be easier optically; counting fringes
and such.  Having some kind of optical measurement (and communication)
capability would be very useful.  The vibration modes are reminiscent
of Zernike numbers for lenses and mirrors.  I wonder what the
capabilities of flat optical systems are, assuming I can etch surface
gratings on top of optical emitters and sensors?

Keith

PS - I assume the Allerton Press copyright notice means I can't put the
Makaeva et. al.  PDF on the server-sky website.  I expect I will just
include the pointer to the Springer verson ($$$!) or should I say (€€€!)
and to the worldcat reference (a few dozen university libraries have it)
and hope people can find it from that.

http://www.springerlink.com/index/6N2L3636G2H64W56.pdf
http://www.worldcat.org/oclc/259507786&referer=brief_results

Oh, to have a Stanford nearby ...

-- 
Keith Lofstrom          keithl at keithl.com         Voice (503)-520-1993
KLIC --- Keith Lofstrom Integrated Circuits --- "Your Ideas in Silicon"
Design Contracting in Bipolar and CMOS - Analog, Digital, and Scan ICs