Download 15.1 Seismometer Response The frequency response of a

15.1 Seismometer Response
The frequency response of a seismograph (meaning the entire seismometer and recording
system) is primarily governed by the seismometer (inertial pendulumn) simply because it is the
most non-linear part of the system. Amplifiers can be designed to give a flat or constant gain
response over a wide frequency range, but the mechanical properties of an seismometer preclude
equal detection (sensitivity) over a wide frequency range unless they employ some complicated
mechanical feedback networks.
Inertial Seismometer Response: Consider a simple suspended mass on a spring with
dashpot damping .
We will consider the inertial properties of the mass M and the relative displacement between the
ground and the mass.
For small displacements of the mass, (x'-x), in terms for small angles, (x'-x) ≈ l φ. Now summing
the mechanical moments:
1) Torque due to angular acceleration of the mass, M, is:
2
ΓM = M a l = M l d x'
dt2
15.1.1
3
but since x' - x = l φ, then x' = x + l φ, so the torque due to the mass can be written in terms
of x and φ giving:
2
d2φ
ΓM = M l d x + l
dt2
dt2
15.1.2
2) Torque due to spring: The restoring force F is proportional to the displacement (x'-x):
Force = -k(x'-x) , where k is the spring constant , giving:
Γs = - k l (x'-x) = -k l2φ (a torque due to the spring )
15.1.4
3) Torque due to the damping mechanism for a dashpot: For small velocities the force on a viscous body is proportional to the velocity, d(x'-x) , then:
dt
FD = -η velocity = -η
dφ
d(x'-x)
= -η l
dt
dt
15.1.5
where η is the damping coefficient. Then the total torque due to the return action of the
dashpot is:
ΓD = -η l2
dφ
dt
15.1.6
Now according to Newton's second law, the sum of the moments will equal the angular acceleration of the mass, giving:
ΓM = Γ s + ΓD
15.1.7
d2 φ
dφ
2
Ml d x + l
= -k l2 φ − η l2
dt
dt2
dt2
15.1.8
or after rearranging terms 15.1.8 becomes:
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M l2
d 2φ
dt2
+ l2 η
2
dφ
+ k l2 φ = −M l d x
dt
dt2
15.1.9
And defining common terms:
2
k,
Ωο = m
η=
η
c
, 2 β Ωο =
m
2 M Ωo
where Ωo is the natural frequency of the undamped system. If we substitute L for l then 1/L is
referred to as the "reduced" pendulum length. Substituting for the above terms and dividing out
common terms in 15.1.15 gives an important inhomogeneous differential equation for the seismometer response.
φ + 2 β Ωο φ + Ωο φ = −1 x
L
2
a frequency and time dependent equation.
15.1.10
This is the basic differential equation of a simple harmonic oscillator driven by an external force,
such as ground motion. The coupling to the driving force term, on the right side of 15.1.10 is:
x
L
In addition, if there is a moving coil transducer, i.e., a coil moving in a magnetic field which is
responsive to velocity because of Lenz's law, then an additional term is added to the right side of
15.1.10 giving:
2
φ + 2 β Ωο φ + Ω ο φ = x - G I
L
15.1.11
where I is the current from an external source, and G is a constant.
Possible external current sources are:
a) pure resistance load -- prospecting type seismometers.
b) galvanometer -- a recording device that has its own differential equation of motion.
We now want to solve 15.1.10 for φ as a function of frequency, the spectral response, and for t, the
time response due to a steady state input and a transient input.
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