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1 GG 5330/6330 Lab #1 1 1/26/04
Lab #1: Calibration Of An Inertial Seismometer
GG5530/6330 Spring Semester, 2004
Earthquake Seismology and Hazard Assessment
Due January 28, 2004
This lab will demonstrate the procedure for calibrating an inertial seismometer.  The objective
is to determine the free period, damping factors, and relative frequency sensitivities (or response)
of a modern moving coil seismometer (a Geotech model S-13) for displacement, velocity, and
acceleration.  For this lab, you are required to determine the free period, open circuit damping,
and relative frequency response for the seismometer using the attached data (determined earlier
from a calibration exercise).
1. Free period of the Seismometer: T0
To determine the free period of the seismometer, a DC pulse is applied to the calibration coil
and the resulting seismometer oscillation (undamped) is recorded.  From the data provided,
determine the free period of the S-13.
2. Damping
The damping of the seismometer is related to the rate at which motion of the spring-mass
system decays to zero.  The decay rate can be controlled by applying varying amounts of
resistance across the seismometer data coil, as demonstrated in lab.  The relationship between
damping factor, damping ratio, and overshoot is shown in Figure 1.  Using the data provided,
calculate damping factors for the open circuit (free period determination ) and applied resistances
ranging from 1000 W to 12000 W.
Overshoot--Most seismometers of this type are set to have a damping factor 0.7.  What is the
overshoot for our laboratory case?
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3. Frequency Responses
To determine the steady-state response of the S-13 seismometer, a sine-wave signal
generator (sinusoidally varying emf) is connected to the calibration coil that acts as a forced
harmonic oscillator.  The signal is varied over a range of frequencies say 0.05, 0.1, 0.5, 0.8, 1.0, 2,
4, 8, 10, 20 and 40 Hz that we will provide.
a. Measure and tabulate the peak-to-peak signal amplitude (in millimeters), recorded on the
lower half of the strip-chart record, for all values of the driving frequency.  Do this for the
overshoot value provided in the data that you acquire (strip chart recordings).
b. Plot the values of amplitude (normalized to the maximum amplitude) versus frequency on
log-log paper.  This is the response to a constant acceleration input because of the manner in
which we were driving the seismometer with a constant peak-to-peak voltage or force which
made the motion proportional to acceleration.
c. To get the acceleration response multiply each amplitude in step # b (before normalizing)
by 2pf and again plot the normalized response as above.
d. To get the displacement response, divide the amplitudes in step # b by 2pf (again before
normalizing) and plot the response.
4. Discussion of Frequency Responses
The governing differential equation for a mass-spring seismometer is of the form:
f + 2bWof + Wo
2
f = f(t)
where f(t) is the forcing function.
And:
Wo = 2pT0  and b = damping factor
T0 is the free period of the seismometer and f(t) is the displacement of the mass.
For our case, an appropriate forcing function has the form:
f(t) = eiwt       where w = 2pf
Note that I have used the symbol, w, for W0 compared to our lecture notes.  With this
definition, we can solve the differential equation for a solution of the form:
f(t) =H(w)eiwt
where H(w) is complex and H(w) is defined as the instrument response.
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For the 1:24 overshoot ratio case, plot the theoretical amplitude response |H(w)| versus
frequency (i.e., just plot it like your data).
Does the form of this plot match the form of any of the real data plots?  It probably won't.
Understanding the reason for this discrepancy is directly related to understanding just what a
seismometer of this type measures.
Now plot |H(w)/w| versus frequency.  This plot should match the form of your plot from
part 3-b above.  Note that H(w) was derived from an equation containing f(t), the displacement
function.  The velocity function is v(t) = f(t).  Transformed to the frequency domain this
becomes v(w) = wf(w).  Since we have seen that dividing our f(w) by w produces functions
similar to our raw data function we now know the seismometer was responding to velocity and
not displacement.  Therefore, your plot in Part 3-b is the response of a velocity sensitive device
to a constant acceleration.  Our original differential equation still holds, but we must realize what
the particular seismometer we are testing is sensitive to.
To make this a little more understandable, the function found by measuring the
seismometer output at different frequencies relates current in the calibration coil to voltage
appearing at the data coil terminals.  Since the peak amplitude of the current into the calibration
coil is held constant, then the peak amplitude of the force applied to the moving mass is constant
and the peak amplitude of the acceleration of the mass is constant.  The output voltage is
proportional to the velocity of the mass relative to the seismometer frame.
The transfer function is called the Hcc(w) in the seismometer transfer function derivation and has
the form:
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5.  Phase Responses
So far, we have only concerned ourselves with the amplitude responses of the
seismometer.  Seismometers also have phase responses.  The meaning of this phase response is
often not emphasized.
The instrument response H(w) can be written: H(w) - |H(w)|eiq(w) where q(w) is the
phase response.
Next find the phase response for x(t).  For the 1:20 overshoot case, plot q(w) versus
frequency.  Note that this is the theoretical phase response.
Consider what this phase response means.  For an input function of the form, eiwt, we consider a
function of the form:
|H(w)|eiq(w)ei(w)t
or
ei(w)t Æ SEISMO Æ |H(w)|eiq(w)eiwt
Manipulating the output response:
|H(w)|eiq(w)eiwt = |H(w)|e
iw( )t + q(w)w
So an arrival at the seismometer at the time t is not recorded until a time delay of:
( )t + q(w)w ;
i.e.,  if q(w) is proportional to w and the time shift is constant at all frequencies, then there is no
delay problem.  However, typically q(w) is not linearly related w.  In this case, different
frequency components of an arrival are recorded at different times.  In other words, the
seismometer can be a source of uneven arrivals in the recorded seismogram.
Using your theoretical phase response, determine if this is a significant problem in the S-13
seismometer.  Use only two frequencies:  0.1 and 10 Hz.
6. Seismometer calibration using a Java script
You can find this script at: http://www.ifg.tu-clausthal.de/java/sein/seisdoc-e.html