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Subject: transfer functions
From: Randall Peters PETERS_RD@..........
Date: Mon, 04 Feb 2008 17:37:12 -0500

    I see nothing wrong with the Excel sheets that you've generated--at least in
terms of the relative shapes of the transfer functions expressed in terms of
either (i) ground acceleration, or (ii) velocity, or (iii) displacement (the
system state variables).  It is critically important to understand, however, that
these three transfer curves are not all equally important.  Acceleration is what
results in velocity, which in turn results in displacement.  The only way to treat
the dynamics is by means of Newton's 2nd law, which is in terms of acceleration.
Moreover, the only state variable responsible for moving the inertial mass
relative to the case of a seismometer is ACCELERATION.  Even in the case of tilt,
the response is due to acceleration.  In this special case of tilt, it is the case
moving relative to the essentially stationary pendulum that gives rise to a
response.  And the amount of response is determined by the component of the
earth's field (little g = 9.8 m/s^2) that is perpendicular to the pendulum.  Where
the tilt is different from the usual acceleration response is that it has no
frequency response associated with it.

Now about your analysis approach (which is conventional):  it is a steady state
analysis, assuming harmonic excitation.  The only excitation mechanism is (as I've
tried to emphatically point out) -- ground acceleration.  The velocity is obtained
from this acceleration by means of integration, and in turn the displacement is
obtained by integrating the velocity.  ONLY if the harmonic excitation persists
for a long enough interval of time to be reasonably monochromatic--is it possible
to obtain an approximation for the aforementioned integrals by simply dividing a
given expression by the angular frequency.

I can give you a good example of where this does not work.  With the VolksMeter I
routinely see near-discontinuous displacements (due to tilt).  Sometimes they are
a step and sometimes they are a pulse (bistability).  They generally happen at
levels close to that of the noise (whether ground or electronic).  If one takes
the derivative of such a signal (what one would see with the conventional
detectors), they are almost never observable--even though they are clearly visible
in the raw data.  This is for two reasons: (i) the derivative only yields two
spikes that look much like noise, and (ii) the derivative is fundamentally a
'noise-producing' operation.

What has confused so many people derives from the nature of the detectors
employed.  For example, the Faraday law detector (magnet/coil) that has been used
for many years, does not really measure the velocity of the ground.  The output
that it generates is proportional to the velocity of the inertial mass relative to
the case of the seismometer.  Since the mass movement is proportional to the
acceleration of the earth, the common detector is thus really responding to the
derivative of ground acceleration (the jerk).  If one wants to really see an
output that is a proper representation of ground velocity, then take a look at the
VolksMeter's integrated signal.

The world is hung up on steady state, linear system analysis--not recognizing that
it has limitations.  One can't simply operate with a convenient but aritificial
transfer function (such as 'velocity' or 'displacment') and get the right answer
all the time by means of the simple transformations involving the angular
frequency.  The proper treatment iinvolves actually doing the integrals!   Even
the pole/zero description that is routinely mentioned is one that at its
foundation embraces the transient consequences of changes that are too short-lived
for the steady state assumptions to be valid.  Just because the instrument is
near-critically damped does not mean that there are no transient features!


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