## PSN-L Email List Message

Subject: Pendulum Q
From: Roger Sparks rsparks@..........
Date: Sun, 21 Jan 2007 12:14:28 -0800

```Good Morning All,

Of course removing the damper will improve the sensitivity of a
seismometer.   Yes, it takes power to move a pendulum from the rest
position, and earthquakes provide that power.  Once moving, the pendulum
is considered to have stored energy in the form of velocity (which is
measured in relation to stationary surroundings).

A pendulum will continue to receive energy from ongoing waves as time
passes, and the pendulum (if friction free and undamped) will integrate
the positive and negative motions into an instantaneous velocity.   The
instantaneous velocity would be the sum of all the impulses over the
time period since the instrument was constructed (a very long time).
This is only true for an ideal pendulum.

An ideal pendulum can be digitally approximated easily from a damped
pendulum.   This is done by summing all the readings from the moment of
first "switch on".   This is not too practical.

More practical,  we can recognize that even an undamped pendulum will
have friction from hinges, air and sensor interactions.   It will have a
decay curve as the energy from an initial impulse is lost to friction.
The pendulum will also have a natural frequency which is constant for
any dimensionally stable system.  Both the decay curve and pendulum
frequency can be modeled in software.  Once modeled, the effect of each
discrete impulse can be subtracted from (or added to) the pendulum's
ongoing output signals.   Assuming we are using velocity detector, the
subtracted result would be instantaneous acceleration, the added result
would be the displacement.

To use this method, the pendulum's period and Q would be critical
measurements.   Any difference between the model and actual instrument
response would result in a digitally introduced error.   In practice,
the errors should be small over the time periods of interest.

Larry has made available in Winquake a feature called "Integration".
The help file has the following for this function:

*/"Integrate Data Alt+I/*

Used to integrate the data set. Integration converts acceleration to
velocity and velocity to displacement."

This brings up a very important point--what is the pendulum sensor
reading?   Is it acceleration, velocity, or displacement?

I need help here.   I think we could make an acceleration detector with
a strain gage, which changed resistance under strain.   Magnetic sensors
are all velocity detectors because they move through a changing magnetic
field.   A capacitive detector is a displacement detector because the
output signal is the result of relative positions.

Back to Larry's integration feature.   He seems to have anticipated at
least part of this discussion by providing the integration feature.   By
integrating the signal over what ever time period desired, we accurately
model the output of an undamped pendulum.   The only difference is that
about 1/2 the power of the earthquake signal is lost in the damping
system of the damped pendulum.  This reduces the signal to noise ratio,
but only the noise from seismometer and electronics, not noise from the
actual seismic signal.

It is interesting to notice that the process of display can change the
character of the signal.   For example, a displacement display would
represent the relative position on a chart.   The larger the difference
in position, the larger the display difference.   But the sensor must
have had an acceleration and velocity as it moved, so the display is
actually an integrated sum of the acceleration and velocity
components.   The only thing that has really changed from velocity or
acceleration detection is that the phase of the displayed result has
changed in time.  If  acceleration or velocity is measured, we can
predict where we are going.   If displacement is measured, we know where
we are at the instant of measurement.

Roger

*//*

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