[Server-sky] paper on vibrational modes of a disk (3)

[Keith Lofstrom]

On Sat, Mar 28, 2009 at 05:46:14PM -0700, Howard Davidson wrote:
> I would expect that the amplitude of any drumhead modes would be much to 
> small to see at RF.  I am much more concerned by secular drifts and slow 
> angular excursions of the whole object.

Indeed, whole object movements need to be well controlled.  Remember that
I have very fine control of the rotation, and that the rates of movement
tend to be very slow compared to measurement and computation, and we are
in the "pure physics zone" - wind gusts and environmental perturbations
are not a problem, tidal forces vary slowly, and even the impulses of the
radiation flux tends to average out over a largish object like this.

I don't think I can do much accurate distance or angle measurement within
one server-sat.   However, I can measure between server-sats, and my RF
buddies assure me that I can control oscillators within a fraction of a
degree.  If I do frequent calibrations, and average over many cycles, I'm
pretty sure I can reduce time/phase measurement errors down to very low
levels.  On one of my past projects I was delivering digitally controlled
arbitrarily timed pulses within 10 femtoseconds over a 1.2 nsec window
using ramp timers (synthesizing tester sampling windows).  That was per
pulse - averaging over billions of narrow band sinusoidal oscillations
has got to be better.

I hope that these have two positioning servo systems - one built into
the hardware, capable of gross position control, and one in uploadable
software (stored in flash) that is frequently improved as better software
comes along.  The software is in control most of the time, unless the
hardware control senses things are getting way out of hand.  So I expect
the positioning to get better and better as the ground engineers find
better algorithms and characterize the system better.

It occurs to me that there is something to learn from the LIGO mirror
positioning system.  This is actually very much like that, only at 
millimeter wavelengths rather than micron wavelengths.  If I can get
anything like the same pico-fringe behavior, I'm set.  Heck, a
milli-fringe between server-sats would work for me.

> You can make remarkably good measurements of the flexure of a silicon IC 
> by integrating in some cleverly laid out strain gauges. In general you 
> can resolve much less than one fringe of light.

I will have to look into that.  The feeling I get is that paperthin 
silicon will not generate much strain when curled by milliradians, the
difference between inner and outer radii is too small.  But it is worth
a look, and I do know how to make very good measurements with CMOS.

> If you want to do in plane optical sensing of defections you probably 
> want to used Bragg Gratings and guided waves. There are some very 
> sensitive fiber optic sensors that depend on being able to see how much 
> the fiber has been stretched compared to a reference section. this can 
> be done as planar integrated waveguides.
> 
> I would not try for any of that kind of technology for this system right 
> now. the power required and the heterointegration would add a lot of 
> complexity, and perhaps not a lot of function.

You are right - on the first ones, I will do all that I can with RF and
microwaves on the first systems.  Later on, we will need optical systems
of some sort to do backhaul between arrays.  That will be version 3
(version 2 never works).

> If there are any polymers in the stack you will have damping. Shear in 
> the polymer goes right to heat.

The good thing about polymers is that the modulus is typically very low,
so they won't flex the stack thermally.  I do wonder about their vacuum
and wide temperature range survivability, but hopefully there are some
suitable polymers that I can coat very thinly (micron thick) on there.
They might also help prevent van der Waals welding between serversats
during deploy - I'm told Teflon is good for low van der Waals force.

> I do not have a clear idea of whether or not you can post the paper 
> under "fair use".

I assume not, I will just post the links and hope people can get access
to a decent library.  

> Here is a website with animations of drumhead modes:
> http://www.kettering.edu/~drussell/Demos/MembraneCircle/Circle.html

Good stuff.  Actually, I am most interested in learning how they make
the animated GIFs - if there is an easy-to-use open source tool for
that, and I can do a lot of drawing with it.  For now, I plan to
generate big heaps of coordinates with Perl and gnuplot, or just with
subroutines in gnuplot.

Of course, the rim is moving up and down, too, so I expect to see
something like the (0,1) mode with a node partway in from the rim,
going at a somewhat higher frequency.  The lowest frequency mode will
likely be a curling mode, like:
                               _______
 \         /   ___________    /       \    ___________   \         /
  \_______/                  /         \                  \_______/

(forgive the ugly ASCII art, and round them in your mind)

> Somewhere I do not remember there is probably a theorem on how much of 
> the energy from a small impulse couples into free body motion, and how 
> much excites internal modes.

I remember something like that too.  There are electrical analogies for
many mechanical systems, and when I think about it a bit I will probably
find one that my limited mathematical imagination works on.  I think it
can be approximated with Fourier decomposition of the pulses and shaped
edges used - the "DC" component does the desired work, the sinusoids do
the excitation.

Keith

-- 
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