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Transfer functions of mechanical and electromagnetic seismometers
Figure 6:
Amplitude response (this means, responsivity to steadystate harmonic
ground motion) of a mechanical seismometer (spring pendulum, left) and
an electromagnetic seismometer (geophone, right). The normalized
frequency is the signal frequency divided by the eigenfrequency (corner
frequency) of the seismometer. All of these response curves have a
secondorder corner at the normalized frequency 1.

According to Eq. (7)and (8), Eq. (25) can be rewritten as
(s^{2} M + s R + S) Z = F s^{2} M X

(26) 
or
Z = (F/M  s^{2} X) / (s^{2} + s R/M + S/M)

(27) 
We arrive at the same result, expressed by the Fouriertransformed quantities
and
in place of s, by simply assuming timeharmonic motions
and
as well as a timeharmonic external force
.
Eq. (25) then reduces to

(28) 
or

(29) 
By checking the behaviour of
in the limit of low and high frequencies, we find that the
massandspring system is a secondorder highpass filter for
displacements and a secondorder lowpass filter for accelerations and
external forces (Fig. 6). Its corner frequency is
with
.
This is at the same time the "eigenfrequency" or "natural frequency"
with which the mass oscillates when the damping is negligible. At the
angular frequency ,
the ground motion
is amplified by a factor
and phaseshifted by .
The imaginary term in the denominator is usually written as
where
is the numerical damping, i.e. the ratio of the actual to the critical
damping. Viscous friction will no longer appear explicitly in our
formulae; the symbol R will later be used for electrical resistance.
In order to convert the motion of the mass into an electric signal, the
mechanical pendulum is in the simplest case equipped with an
electromagnetic velocity transducer (see subsection 3.7) whose output voltage we denote with .
We then have an electrodynamic seismometer, also called a geophone when
designed for seismic exploration. When the responsivity of the
transducer is E (volts per meter per second;
;
the negative polarity is deliberate) we get

(30) 
from which, in the absence of an external force (i.e. f(t)=0,
), we obtain the frequencydependent complex response functions

(31) 
for the displacement,

(32) 
for the velocity, and

(33) 
for the acceleration.
With respect to its frequencydependent response, the electromagnetic
seismometer is a secondorder highpass filter for the velocity, and a
bandpass filter for the acceleration. Its response to displacement has
no flat part and no concise name. The corresponding amplitude responses
are illustrated in Fig. 6.
Next: Design of seismic sensors
Up: Basic Theory
Previous: The mechanical pendulum
Erhard Wielandt
20021108