Two similar seismometers connected in series with a load exhibit two independent or “normal” modes of motion. In the first mode the two coils move so that the voltages generated are almost equal and in phase, producing a large current in the circuit and considerable damping of the motion. This is the mode normally considered in designing the external damping circuit for series seismometers. In the second mode the coils move nearly 180° out of phase, produce a relatively small current in the circuit, and, consequently, experience very little damping in addition to their open circuit damping. Strong initial excitation of this mode can produce a sustained oscillation damaging to later parts of the seismic record.
The usual mathematical description of this system, i.e., two harmonic oscillators coupled through their damping terms, readily yields approximate expressions for the size and damping of the load current in the case of nearly identical seismometers with little internal damping. For example, two such seismometers connected to a load producing large damping for the first mode will exhibit a damping of only (ω1 − ω2)2/4ωf in the second mode. Here ω1 and ω2 are the angular frequencies of the two separate seismometers, ω is the average angular frequency, and f is the fraction of critical damping for the first mode.
Two seismometers connected in parallel can have considerable electromagnetic damping in both modes of oscillation.
For n seismometers in series, there are n − 1 modes which may be poorly damped. The frequencies of these modes are distributed so that one lies between each adjacent pair of the original uncoupled frequencies. The damping in each mode is of the order of n(ωk − ωk+1)2/8ωf.
The steady-state characteristic can be readily examined using the seismometer equivalent circuits. The result is strongly dependent on the mode of excitation. For example, if two similar seismometers are connected in series but only one is excited, the frequency characteristic shows both a maximum near the rms of the two seismometer frequencies and a minimum at the frequency of the seismometer not excited.