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Subject: Dampng anharmonicity and Seismometry
From: Randall Peters PETERS_RD@..........
Date: Sat, 09 Feb 2008 08:44:28 -0500

Too all:
    For reason of its length and other properties, I previously resisted supplying to the
listserve the comments which follow.  It appears now appropriate to mention them.

Damping Anharmonicity and Seismometry

The fine structure of the otherwise smooth (Hooke’s law) potential invalidates the linear
(viscous) damping model used to describe simple harmonic oscillation.  There are few
mechanical systems that even come close to being in agreement with it.  This is the case
even at substantial amplitudes of oscillation, if the frequency is low—which means that
nonlinear processes dominate the damping of seismic instruments in the regime where many
now with frustrations want to go to study the earth.
 An important feature involves ‘self-similarity’, the hallmark of fractal (complex,
chaotic) systems.  Thus there are properties of even the (huge) earth itself that are
similar to what is found in a (small) seismometer.  In some respects it is conceptually
easier to envision what goes on inside the earth.
 Our planet is like a multiply-cracked hard-boiled egg.  The influence of the tidal
forces of sun and moon are vitally significant to its dynamics.  One way to think about
this is as follows.  Roll the hard boiled egg between your hands; as the shell fragments
undergo rapid snap, crackle, pops (ala subduction of plates in the earth), the.egg must
itself oscillate after each event.  A ‘local ping’ of the egg will cause oscillations
that persist longer than the oscillations caused by the ‘tidal’ rolling.  In similar
manner, a large earthquake is followed by long lived eigenmode oscillations.  For
example, the earth rang like a bell for weeks after the great Andeman-Sumatra
earthquake.  To believe that it does not also ‘ring’ due to rapid relaxation after a
snap, crackle, or pop is to ignore the physics.
 Those of us trained in solid state physics know that the earth must oscillate all the
time (due to its temperature) over the full range of admissible states; i.e., the
so-called ‘density of states’.  It was Einstein’s analysis of the heat capacity of solids
along these lines that constituted one part of the modern physics revolution of the last
 So what is the primary difference between the two oscillation cases just mentioned.
First of all, as was noted, the large amplitude motions in which the system ‘skates’ over
the fine structure ‘bumps’ is more ‘monochromatic’ (longer lived).  Just because the
coherence time of the lower level ones is much shorter doesn’t mean they are
non-existent.  It means they are harder to observe.  With my cumulative spectral power
(CSP) analysis they are much easier to study.  As compared to the power spectral density
(PSD) approach, it is much better suited to the manner in which the eye/brain is able to
assess information.  The CSP allows fine structure of frequency domain type to be readily
seen without having to resort to the more complicated ‘waterfall’ methods of conventional
spectral type.
 There’s another conceptual analogy that I have used.  Imagine yourself on a gravel road
having fine structure (not smooth, but with washboard features that always develop over
time).  As long as you move at the right speed (not too fast, not too slow) the motion at
these large levels allows one to ‘negotiate’ the road.  If in a shallow depression, one
can travel back and forth (first forward, then in reverse) ‘skating’ over the ‘fine
structure’.  If however, you get too slow near the bottom, you will get hung up in one of
the localized minima.  This is precisely what happens with a force-balance seismometer
when trying to observe low energy earth motions.
 The force-balance instrument is without equal for looking at earthquakes all over the
world, but in my opinion it will never yield insights into the physics I’ve been recently
researching.  For example, the VolksMeter allows me to look at diurnal and seasonal
changes of importance.  Most recently I’ve discovered that there is energy exchange
between the eigenmodes and the microseisms.  Nobody to my knowledge has previously
postulated this.  I have also with this pendulum seen in Larry Cochrane’s data the
terdiurnal tide having a period of 8 hours.  The 12-h and 24-h components are easily seen
with a variety of instruments like strain gauges; but the terdiurnal component was
previously seen almost solely in meteorological (upper atomosphere) measurements using
 In the case of mechanical oscillators, the potential well in general has features having
some similarity to the various analogies that I’ve mentioned.  The details of the fine
structure have not been worked out; since internal friction is not understood from first
principles.  I have postulated that individual ‘grain boundary’ regions (ala Chris
Chapman’s comment concerning the analogy with magnetic domains) become altered when
strain energies exceed certain thresholds.  If that 'quantum' postulate could be proven,
it would probably result in a Nobel prize; however, the challenge of reproducibility in
experiments is Herculean (and nobody other than myself is to my knowledge yet trying to
prove the matter).  The problem requires extreme patience, because the frequencies
required for study are so low.  Thus the lifetime of the investigator comes into play.
 Concerning the friction ‘force’—
Linear damping in the equation of motion causes the friction force to be sinusoidal and
shifted in phase by 90 degrees from the sinusoidal displacement (proportional to
velocity).  It also requires that the quality factor Q as a function of natural frequency
f  be of the form Q proportional to f.  Seismometers DO NOT conform to this!  Gunar
Strekeisen was apparently the first person to observe this nonconformity, while he was a
grad-student working with a LaCoste spring.  What he observed was Q proportional to
f-squared.  This quadratic dependence is the hallmark of hysteretic damping (that
engineers have known about for a long time, but for which physicists are almost
universally ignorant).
 What I have shown through experimentation is that the friction force is not sinusoidal.
It is more nearly an attenuating square wave!  Its amplitude falls off as the amplitude
of the displacement decreases, thus giving rise to an exponential decay.  Just because
the decay is exponential, does not mean the damping is linear!!  Only the fundamental of
the Fourier component of this square wave friction is important in establishing the Q.
Thus the nonlinear damping masquerades as linear even though it is far from being so!
Among other of its features, there is no damping redshift; i.e., the natural frequency
does not decrease as the amount of damping increases.

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