A pendulum seismometer which consists of a massive boom free to rotate about a hinge, can be described in terms of the mass `M` of the boom, a distance `L` between the hinge end and the centre of gravity of the boom, a moment of inertia `K`, a damping coefficient `D`, and a spring which exerts a restoring moment `U phi` for an angular rotation `phi` about the hinge. The moment of inertia is taken to include the effect of any levers or other masses which may be constrained to move with the boom, the definition being such that `phi` is associated with a total kinetic energy of 1/2 `K`(d`phi`/d`t`)^{2} .

The equation of motion is

where `chi` is the component of earth displacement in the direction of free motion of the pendulum. The equation is valid only for small motions for which we can neglect the curvature of the path of the centre of gravity, and can assume linearity in the damping moment, and the restoring moment of the spring. Certain other design conditions must also be satisfied (Byerly, 1952). Equation [1] may be reduced to the form

The angular rotations of the boom are the same as those which would be executed by an equivalent pendulum; consisting of a point mass at the end of a weightless boom of length `l`. The constant `l` is called the 'reduced pendulum length' and its applicability is illustrated in Fig. 3.1

In some seismometers, the mass `M` is constrained to move along a fixed axis instead of swinging about a hinge. Let the displacement along this axis be `z`, and represent the effect of coupled auxiliary masses by introducing a total inertial mass `M`', such that the kinetic energy of the oscillating system is 1/2 `M`'(d`z`/d`t`)^{2}. This leads to a reduced equation of motion

We can now reintroduce the concept of 'reduced pendulum length' by writing

which puts the equation of motion in the same form as that of the swinging-boom instrument.

The response to a harmonic ground motion

Inspection of equations [2] and [3] shows that for high frequencies of earth motion `phi` tends to the limit `chi`/`l`, and `xi` tends to `pi`. Hence `phi` follows the ground displacement.

For very long periods of oscillation, equation [2] approximates to the form

which gives output proportional to earth acceleration.

Changes in `phi` are normally recorded as linear displacements of a light spot or pointer, for which we imagine a lever arm `l`' such that the indicator deflection is `l`' `phi` The ratio` l`'/`l` is called the 'static magnification' `V-overbar`.

We can express the dynamic response of the seismograph by writing

so that the actual magnification `V` for any given period of earth motion becomes

Response curves for various values of `beta` are given in Fig. 3.1a, Fig3.1b, and Fig3.1c, expressed as a function of the ratio of ground period to pendulum period. Note the symmetry of the curves in Fig. 3.1b which express the indicator deflection as a function of particle velocity in the ground.

In these instruments the output from the seismometer is taken through an electromagnetic transducer. Let the 'flux linkage' `G` of the transducer be such that it generates an e.m.f. of `G`(d`phi`/d`t`) when the boom is deflected. It can then be shown that the passage of a current `I` through the coil will react on the field to produce a force of moment -`GI`. The equation of motion of the pendulum is

The seismometer coil will be connected to a galvanometer through a network of resistances which can always be represented by the simple T configuration of Fig. 3.2. Here, the resistance `R` in the arm AB is taken to include the resistance of the seismometer coil, the resistance `r` of the arm BC includes the resistance of the galvanometer, and the single resistor `S` in OB allows for the effect of all the parallel shunt resistors which may exist in the actual configuration. The equation of motion of the galvanometer is

where `theta` is the galvanometer deflection, `k`, `d` and `u` are the moment of inertia, open-circuit damping and suspension stiffness respectively, `g` is the flux linkage and `i` is the current in the coil.

When the coil oscillates, an e.m.f.-`g`(d`theta`/d`t`) appears in the circuit. By the application of Kirchoff's Laws to the network it may be shown that

The moments -`GI` and -`gi` appear in the equations of motion of the seismometer and of the galvanometer, and it is convenient to regard the parts of these moments which involve dphi/dt and d`theta`/d`t` respectively as parts of the respective damping constants.

This is done by writing

for the galvanometer. The reduced equations of motion are

The periodic solution of equations [5] and [6] is obtained by eliminating `phi`, setting `chi` equal to the real part of the complex equation

and substituting for the derivatives of `Chi` and `Theta`. The solution is

On extracting `theta` as the real part of `Theta`, converting particle velocity to particle displacement by introducing the factor `omega`, and setting the trace displacement `a`_{0} = 2A`theta`_{0}, we obtain for the magnification

The quantity `sigma`_{s}`sigma`_{g} in equation [7] represents the 'galvanometer reaction' or coupling coefficient, which is often called `alpha`^{2}. This terminology has already been used in Section STA 1.

The effect of galvanometer reaction is concentrated into the single term in the denominator of equation [7]. It is significant if `sigma`^{2} is an appreciable fraction of unity, and if `alphabeta` is large or `omega`_{0} is not very different from `Omega`_{0}. In many seismographs we can obtain a good approximation to actual performance by setting `sigma`^{2} = 0, in which case equation [7] factorises in the form

where `V-overbar` is analogous to the 'static magnification' of the simple mechanical seismograph, and `F _{s}`,

The quantities `F _{s}` and

Detailed theoretical treatment (see, for example, Coulomb and Grenet (1935), Kirnos and Borisevich (1961), Teupser (1958), Tobyas (1963) or Willmore (1960)) shows that any response characteristic which includes galvanometer reaction can be produced by at least one combination of seismometer and galvanometer with different periods and damping constants, acting through a loose coupling network for which `sigma`^{2} can be taken as zero. Thus the most obvious way of changing magnification (by changing the coupling constant) will change the shape of the response curve if the effect of reaction is significant in the high-gain condition, and the adjustments required to meet this requirement have been tabulated for some special combinations. Section INST 5.3 of this manual contains one such example. It is also possible to produce exact or approximate extinction of the galvanometer reaction by introducing a resonant shunt into the coupling circuit (Teupser, 1969).

The variable-reluctance transducer consists of a coil linked with a magnetic circuit which includes a variable air gap, so that changes in the gap alter the magnetic flux in the circuit, and hence generate an e.m.f. in the coil.

Essential differences in performance between variable reluctance seismometers and those of the moving-coil type arise from the fact that the transducer in the former case has a relatively high inductance. At high frequencies the effect of the inductance is to reduce the current which the system can deliver to the galvanometer circuit. At intermediate frequencies (i.e. for two or three octaves above the natural frequency of oscillation of the seismometer) the phase difference between the transducer velocity and the current in the coil has a significant effect on the equation of motion. The resulting effect is portrayed on Fig. 3.3. Approximate calibration curves for variable reluctance seismometers may be derived by the graphical method of Section 3.2 using Fig. 3.3 in place of Fig. 3.1a (Chakrabarty and others, 1964).

If the electrical output signal from the seismometer is to be electronically amplified before it is recorded, it is possible to use forms of transducers other than the conventional magnet-coil systems. Of these systems, the most used is the variable capacity or capacity-micrometer system, in which the electrical capacity between a plate attached to the moving mass of a seismometer and a plate attached to the frame is varied by the relative motion of the frame and mass. As the capacity varies with the separation of the plates, such a transducer produces an output dependent on displacement. This gives greater long-period response than a magnet-coil transducer, at the cost of requiring greater long-term stability.

The relative displacement of the plate is measured, in most cases, by making the capacitor part of a resonant circuit and then measuring the frequency change as the capacity is varied. Capacity-micrometer systems are amply described in electronics literature, and the transducers used in seismology are special examples of standard varieties.

Commonly in seismological applications a 'push-pull' system is used in which there are three plates with one on the frame (or mass) and two on the mass (or frame) and the capacity increases between one pair as it decreases between the other. The extreme stability required in seismological use places severe limitations on the electronics of the systems and particularly on the stability of the frequency-sensitive circuits, and feedback circuits to eliminate long-period drift are usually employed.

When the seismometer output has been obtained in the form of a high-level electronic signal, important new classes of seismograph can be constructed by directing the output signal into an auxiliary transducer attached to the pendulum. In this way, the output signal generates a force acting on the mass, and modifies the equation of motion. The most effective applications are those in which the feedback is negative, i.e. in which the effect of the fed-back force is to reduce the relative motion between the elements of the primary transducer. This invokes the well-known general properties of negative feedback systems, which are to improve stability, linearity and dynamic range as compared with open-loop systems using components of similar quality. The major general classes are as follows:

When the feedback is arranged to maintain almost constant separation between the transducer elements, then the electric current which generates the force must follow the wave-form of the input acceleration. Devices of this class are commercially produced in considerable numbers, and are known as force-balance accelerometers.

Signals proportional to the transducer velocity are readily generated by amplifying the output of moving-coil seismometers, and the fed-back force has the effect of increasing the apparent damping of the pendulum. If this source of damping is used as a means of dispensing with the need for a shunt impedance or other passive damping device, it allows the entire output of the transducer to be presented to the primary amplifier, instead of a reduced portion of that output. A more subtle benefit is that the elimination of dissipative elements from the electromechanical system removes the primary source of Brownian motion of the seismometer mass, and therefore opens the door to the development of very small seismometers.

If the feedback force is generated by differentiating the output of a velocity transducer (or by two stages of differentiation from displacement output by means of a second-order high-pass filter), feedback will simulate the addition of inertial mass or moment of inertia to the seismometer pendulum, and thereby lengthen the period. Such acceleration feedback must be combined with velocity feedback, because the increased inertia requires more damping.

The above methods are used singly or in combination in many modern instruments. All effective systems must conform to the basic requirements for feedback amplifiers (notably for the avoidance of oscillation outside the desired pass-band) which are discussed at length in electronic textbooks. Descriptions of particular instruments may he found, for example in Block and Moore (1970), Sutton and Latham (1964), Plesinger (1973) or Unterreitmeier (1973).

Date created: 1/7/97 Last modified: 9/9/97 Copyright © 1997, Global Seismological Services Maintained by: Eric Bergman bergman@seismo.com