Randall,

I'd like to use an adaptation of your analogy to try to make a very 
non-rigorous argument, though I believe an accurate one, as to how a 
specific type of force-balance instrument deals with potholes in your 
road.  I am here making the assumption that your analogy can be applied to 
the spring force in a vertical device.  I will also assume that the pothole 
forces are very small compared with the spring force and just represent 
small variations from linearity.

I realize that your example probably applies to much lower frequencies and 
much smaller motions than the regime I am describing and it's not 
surprising that you can discover some 'interesting' issues down there.  It 
is probably true that there is a practical limit as to how much low 
frequency sensitivity you can build into a vertical (i.e. a device which 
depends on a spring).  I gather that most verticals give up once you get 
beyond a hundred seconds or so.

For the specific case of a force balance vertical designed to be flat to 
velocity, the feedback in the flat portion of the instrument response is 
dominated by feedback via the derivative branch.  I will assume that here 
we are using ideal components, and that the spring is the only non linear 
element.  We are monitoring the position of the test mass, greatly 
amplifying that measurement and differentiating it to achieve a large 
signal proportional to the velocity of the mass.  That signal we are then 
sending to a forcing coil which pushes on the mass so as to oppose its motion.

To begin with we are moving smoothly down the road and everything is more 
or less in balance.  Now we hit one of the potholes and the mass motion 
suddenly slows very slightly.  As a result, the derivitave feedback branch 
quickly responds by reducing its resistance to the motion, allowing the 
spring-mass to more easily rise out of the pothole.  Then when the velocity 
suddenly begins to rise as we come back onto the flat road, the feedback 
quickly increases the resisting force to help keep the velocity constant.

The feedback effectively makes the pothole appear shallower than it realy 
is.  Quantitatively, the apparent 'depth' of the pothole is reduced by the 
strength of the feedback (loop gain).  In a good design that should be over 
100 and possibly much greater.  The pothole appears to be only 1% or less 
as 'deep' as it was without feedback.  I should mention that the above 
discussion will also apply to any other small damping forces, linear or non 
linear.

Brett

At 08:44 AM 2/9/2008 -0500, you wrote:

>  There's another conceptual analogy that I have used.  Imagine yourself 
> on a gravel road
>having fine structure (not smooth, but with washboard features that always 
>develop over
>time).  As long as you move at the right speed (not too fast, not too 
>slow) the motion at
>these large levels allows one to 'negotiate' the road.  If in a shallow 
>depression, one
>can travel back and forth (first forward, then in reverse) 'skating' over 
>the 'fine
>structure'.  If however, you get too slow near the bottom, you will get 
>hung up in one of
>the localized minima.  This is precisely what happens with a force-balance 
>seismometer
>when trying to observe low energy earth motions.



>    Randall


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