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Subject: folded variant with feedback
From: Randall Peters PETERS_RD@..........
Date: Mon, 18 Feb 2008 08:21:59 -0500

What I'm envisioning is not significantly different than the folded pendulum in terms of the physics involved.
Before describing it in more detail, let me respond to your comments, Chris.
  Yes, one approach that has been used is to place a spring at the bottom to 'soften' the restoring force of gravity
acting on the pendulum.  Although in principle o.k., in fact it has been shown to be unacceptable, due to the
dastardly properties of springs.  There is no need for such a spring to accomplish the same result.
      In the case of the so-called folded pendulum, there are really two pendulums--one that is 'usual', the other that
is inverted.  The usual one of the pair behaves in normal manner; i.e., if disturbed, gravity restores it to
equilibrium.  The inverted one behaves in just the opposite manner, and provides for a much greater
linear response than is possible by using positive feedback by means of a spring at the bottom of a single pendulum.
   Because one pendulum is trying to 'restore' to equilibrium whereas the other one ('destoring') is trying to take
the system away from equilibrium--the net effect of these competing forces is a system with a longer period.
It can be taken all the way to infinite period and beyond (critical point in which conversion from stable equilibrium to
unstable equilibrium occurs).  Just like any long period seismometer, the material properties limit how far one can go in the
direction of long-period until it becomes operationally unacceptable (enough to make a preacher cuss)..
    The difference between what I've bee discussing and the usual folded pendulum without feedback is the following.
Instead of two obvious pendulums as with the folded, there is a single (usual) pendulum hanging from the drive component of the
feedback system which is itself functioning as the inverted pendulum.  In other words, the axis at the top of this
drive component (holding the pivot for the usual pendulum) is of approximately the same length as the primary pendulum.
As the pendulum swings to the right, its axis on  the drive (inverted pendulum) swings to the left.  If the inverted pendulum
were of infiinite length (horizontal motion as was first discussed as a feedback means) all that the drive would accomplish is
to excite the primary pendulum via acceleration.  On the other hand, for the two pendulums swinging in precise phase opposition,
the net effect is one of a single pendulum with a longer period.
    The phase opposition of the two pendulums is guaranteed in the case of the folded pendulum because the two are rigidly
connected.  Which pendulum is more effective in controlling the period dependends on how close the mass is on the horizontal
connecting boom to the one pendulum or the other.  Get too close to the inverted pendulum and the system
goes unstable (goes beyond the critical point).
   Where my idea differs from the traditional folded pendulum has to do with the 'connection' between the two pendulums.
There is no 'flexibility' of that connection in the traditional system.  With the feedback arrangement I've described, there
is variable 'coupling' determined by the nature of the feedback circuit's pole/zero architecture.  Control of the phase between
the two units should be for engineers given to this business 'what floats their boat'.
    I see again in one of Chris' statements the extreme difficulty most everyone of us has when it comes to conceptual
understanding of a seismometer.  Yes, Newton's first law says that an object at rest wants to remain at rest'.  This inertial
property of matter is often misunderstood because not enough attention is given to the part of the statement that I left off;
i.e., ...remain at rest unless acted upon by a force.
    Einstein showed us that there doesn't have to be a force acting directly on the seismic (inertial) mass.  Indeed, it is the
acceleration of the case that is responsible for response.  The mass is trying by Newton's first law to remain in place as the
case is moved.  But it cannot remain fixed!!!!!  As the case moves, there is an unbalanced force on the mass that results.  With
the pendulum, the mass trying to stay at a fixed point and the case moved to a different point--means that there is a deflection
of the pendulum.  There is no difference to be realized from this and some force applied directly to the inertial mass with the
case unmoved.  Einstein's principle of relativity says that we cannot distinguish between the two.
   One can think about the response in the following way.  When the case moves, the inertial mass tries to remain fixed, but it
cannot remain that way ostensibly for longer than 1/4th the period of the mechanical oscillator of which it is a part.  After
all, if the system did not oscillate, we're engaging in complete foolishness to talk about sensitivity being proportional to the
square of the natural period.
    One can acceptably estimate the amount of relative motion between mass and case as follows (I'm trying to avoid detailed
math for those of you who are frightened by it)  Allow me just one foundational feature that you must accept on faith if you
can't follow the math.  For an object moving at constant acceleration, the distance traveled goes like the square of the time
during which it accelerates.  Since acceleration of the inertial mass cannot be avoided as the result of case movement, we see
immediately that the amount of motion (instrument sensitivity) is proportional to the square of the period of the instrument.
Why, because for only about 1/4th of the period of the system can the mass be assumed to be moving with a 'constant'
    For those who want to believe that the inertial mass does not accelerate (total misunderstanding of the physics of Newton's
laws applied to a seismometer)--think about the following.  The inertial mass is incapable of functioning without oscillatory
motion (even though we try with critical damping to suppress the transient parts).  Oscillation means 'back and forth', which in
turn means acceleration that is also back and forth oppositely directed to displacement.  There can be no displacement of the
inertial mass relative to the case without a corresponding acceleration of the ineretial mass. It is not at rest, and never can
be totally at rest!  To place one's emphasis on the displacement as opposed to the acceleration is to 'get the cart before the
horese'.  Acceleration is fundament; displacement is not!
    How many variants of this discussion are necessary before folks finally GET IT (the physics).  Hey, you amateurs are not the
only confused ones.  Many of the professional seismologists with whom I've interacted do not have a conceptual understanding of
how a seismometer works.  It they did, they wouldn't 'worship the god of velocity sensing'.

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