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Message ID: 886     Entry time: Tue Aug 26 12:00:45 2008
Author: Jenne 
Type: Summary 
Category: PEM 
Subject: Transfer function of Ranger seismometer 
This finishes up the calibration that Rana started in elog # 881.

The calibration of the Ranger seismometer should also include:
2 zeros at 0 Hz
2 poles at 1.02 Hz

This comes from finding the transfer function between the mass's motion and the motion of the ground.
    ..
m * x  = (x_G - x) * k  + d(x_G - x) * b
                          dt

where
  • m = mass
  • x = displacement of the mass
  • x_G = displacement of the ground
  • k = spring constant
  • b = damping constant

This gives
x               w0^2  +  i*w*w0/Q
----    =    -----------------------
x_G           w0^2 + i*w*w0/Q - w^2

where
  • w0 = sqrt(k/m) = natural frequency of spring + mass
  • w = frequency of ground motion
  • Q = q-factor of spring + mass system = 1/2 for critically damped system

The readout of the system is proportional to
d  (x - x_G)          (    w0^2  +  i*w*w0/Q          )    .                    w^2               .
dt                 =  (  -----------------------  - 1 ) * x_G   =      ----------------------- * x_G
                      (   w0^2 + i*w*w0/Q - w^2       )                w0^2 + i*w*w0/Q - w^2
Since we read out the signal that is proportional to velocity, this is precisely the transfer function we're looking for. With w0 = 1.02 Hz and Q = 1/2 for the critically damped system, we have 2 zeros at 0 and 2 poles at 1.02.
ELOG V3.1.3-