From a number of comments and questions that have appeared on the list serve over the last month or so,
it appears that some of you might benefit (if I were to write it) from something booklet-like titled,
"The Physics of a Seismometer" ??
     The appropriate level of mathematics (language of physics) is something I haven't yet decided.
There are important issues that could be addressed, independent of the math (unless proof is required for
unbelievers), such as the following:

    Seismometer misconceptions

     There is a great deal of misunderstanding when it comes to the physics of seismometers.  The biggest
single contributor to confusion involves 'period' of the instrument.  Only for a simple (ideal) pendulum
(horizontal seismometer) or a simple (idealized) vertical spring holding a mass (vertical
seismometer)--does the inherent (mechanical) sensitivity of the device depend quadratically on the
natural period of oscillation; i.e., proportional to T squared.  This is the natural period T of harmonic
motion if the unit were not dampened to prevent oscillation, as is done with virtually all instruments
other than some I find (evidently as a 'heretic'), quite useful.
     The sensitivity to the ground's acceleration (ONLY thiing that any seismometer responds to) is only
'half' the story.  The instrument's sensitivity to its own structural changes is also proportional to
T-squared.  Because the instrument is under considerable stress by attempting to statically support at
equilibrium the inertial mass required for it to function--the structural changes can not be ignored for
any truly useful instrument.  In particular, creep never ceases, and even miniscule varaiations in
temperature can have a large effect.  Trying to eliminate the structural influence as compared to the
acceleration influence is the GREAT challenge of any instrument design.
     The tradeoff that is part of the design must weigh mechanical benefit versus electronics benefit.
Keep in mind that linear electronics by iteself (WITHOUT force feedback) can never influence the
instrument in a quadratic manner (as implied by the very word LINEAR).  In other words reducing the
corner freqeuency of the passive electronics (devoid of an actuator to provide feedback) can NEVER be as
influential as lowering the natural frequency (lengthening the period) of the mechanical oscillator
itself.   Because electronics establishes a lower threshold of detectability (due to 1/f noise from the
amplifiers and also white (frequency independent) noise due to ADC bit resolution) there is a vastly
greater benefit from mechanical improvement than there is from electronics improvement.  That shouldn't
come as a great surprise.  After all, you could have perfect electronics, but if the mass doesn't move
under the influence of ground acceleration, then the seismometer will not respond.  At low levels that is
exactly what can prevent earthquake detection.  It has nothing to do with the motion being below the
threshold established by the noise of the electronics; it has everything to do with the system being
'latched' in a metastability that derives from internal friction that operates at the mesoscale.  When
seismologists talk about nonlinearity of the mechanical system (which is religiously avoided), what they
are discussing is elastic anharmonicity (nonlinearity)--undesirable distortions at large levels of
motion.  Also very important but unknown until recently, are the influences of damping anharmonicity that
derives from internal friction (changes in the defect structure of the spring) and which operates at the
other extreme; i.e., at low levels.  For information about damping anharmonicity, consult the article
titled "anharmonic oscillator" that I wrote for the 10th Ed. of the McGraw Hill Encyclopedia of Science
and Technology
     Practical seismometers are rarely configured with an appearance even remotely similar to the simple
harmonic oscillators of idealized type mentioned above (pendulum or mass/spring).  Their performance is
governed by properties due to their "compound" nature.  Many of the commercial instruments employ
"force-balance", in which the inertial mass is constrained (by means of an actuator that is part of a
force-feedback network) to execute very little motion in response to earth acceleration.  Instead of
monitoring the motion of the inertial mass relative to the case, what is monitored is the error signal of
the electronics required to keep the mass from moving.  In the case of force-balance this is indeed a
large force, since it is used to not only keep the mean position of the mass from changing; but it is
even great enough to prevent oscillation.  In other words, the feedback is tailored to provide the
desired near critical damping.  With the forcing function of the actuator being large enough to provide
this damping, it is also true that the feedback can be designed (in a manner analogous to PID controller
design) to yield an instrument that behaves (by reason of the feedback) as though it were a simple
oscillator with a much longer natural period.  Keep in mind that the period lengthening results only from
the feedback force supplied by the actuator.  Electronics without feedback can never accomplish the same
thing!  Note also that it is the strong feedback force that results in a quadratic improvement in the
idealized sensitivity--because the "effective mechanical natural period" has been increased.
     One might then think that force-feedback is the answer to every problem; but it is not!  The
additional complexity and cost are only part of the matter.  There must be a small amount of inertial
mass motion for the electronics to be able to generate an error signal.  This is not always possible.
The damping anharmonicity that I mentioned 'wars against' the force-balance concept.  An alternative
approach which has merit and which I alone appear to have used, is the following--instead of (i) a 'hard'
feedback force (commercial standard) that is able to greatly alter the properties of the equivalent
mechanical oscillator, use (ii) a 'soft' feedback force that keeps the system from migrating out the
range of acceptable motion, but which allows the instrument to 'seek its own best equilibrium' while
'skating over the metastabilities of its real as opposed to idealized harmonic potential'.

Randall