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. |