Transfer functions of mechanical and electromagnetic seismometers

  

Figure 6: Amplitude response (this means, responsivity to steady-state 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 second-order corner at the normalized frequency 1.

According to Eq. (7)and (8), Eq. (25) can be rewritten as

(s² M + s R + S) Z = F -s² M X (26)

or

Z = (F/M - s² X) / (s² + s R/M + S/M) (27)

We arrive at the same result, expressed by the Fourier-transformed quantities and $j \omega$ in place of s, by simply assuming time-harmonic motions $x(t) = \tilde X e^{j \omega t} / 2 \pi$ and $z(t) = \tilde Z e^{j \omega t} / 2 \pi$ as well as a time-harmonic external force $f(t) = \tilde F e^{j \omega t} / 2 \pi$. Eq. (25) then reduces to

$$(- \omega^2 M + j \omega R + S) \tilde Z = \tilde F + \omega^2 M \tilde X$$ (28)

or

$$\tilde Z = (\tilde F/M + \omega^2 \tilde X) / (-\omega^2 + j \omega R/M + S/M)$$ (29)

By checking the behaviour of $\tilde Z(\omega)$ in the limit of low and high frequencies, we find that the mass-and-spring system is a second-order high-pass filter for displacements and a second-order low-pass filter for accelerations and external forces (Fig. 6). Its corner frequency is $f_0=\omega_0/2\pi$ with $\omega_0=\sqrt{S/M}$. 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 $\omega_0$, the ground motion $\tilde X$ is amplified by a factor $\omega_0 M/R$ and phase-shifted by $\pi/2$. The imaginary term in the denominator is usually written as $2 \omega \omega_0 h$ where $h = R / (2 \omega_0 M)$ 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 $\tilde U$. 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; $\tilde U = -E j \omega \tilde Z$; the negative polarity is deliberate) we get

$$\tilde U = -j \omega E (\tilde F/M + \omega^2 \tilde X) / (-\omega^2 + 2 j \omega \omega_0 h + \omega_0^2)$$ (30)

from which, in the absence of an external force (i.e. f(t)=0, $\tilde F=0$), we obtain the frequency-dependent complex response functions

$$\tilde H_d(\omega) := \tilde U / \tilde X = -j \omega^3 E / (-\omega^2 + 2 j \omega \omega_0 h + \omega_0^2)$$ (31)

for the displacement,

$$\tilde H_v(\omega) := \tilde U / (j \omega \tilde X) = -\omega^2 E / (-\omega^2 + 2 j \omega \omega_0 h + \omega_0^2)$$ (32)

for the velocity, and

$$\tilde H_a(\omega) := \tilde U / (-\omega^2 \tilde X) = j \omega E / (-\omega^2 + 2 j \omega \omega_0 h + \omega_0^2)$$ (33)

for the acceleration. With respect to its frequency-dependent response, the electromagnetic seismometer is a second-order high-pass filter for the velocity, and a band-pass 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.