PSN-L Email List Message
Subject: Re: Vertical BB derived from STM-8
From: Brett Nordgren brett3nt@.............
Date: Sat, 13 Mar 2010 21:16:35 -0500
Matt:
A complete answer to your excellent questions would need a couple of
dozen pages, so I will start with the easy one.
Flexures make the best hinges. In a feedback vertical, the boom
essentially remains stationary so they actually flex very
little. Their stiffness does raise the natural frequency before
feedback very slightly, but since that can be adjusted over a very
wide range, say a half second period out to infinity, by slight
adjustments to the length of the astatic support spring, the effect
of any slight stiffness of the flexures is totally negligible.
The Inyo Force Balance Vertical
design http://bnordgren.org/seismo/gif_images.htm was somewhat
influenced by Sean-Thomas' STM-8, and by the STS-1. It uses a
clamped-end leaf spring which has some advantages over the flexible
connections used in the STM-8.
You can think of the mechanics of such a design as having proof mass
which tends to remain fixed in inertial space with the ground and
instrument frame moving under it. The mass is supported by an
astatic spring, designed so that its supporting force changes very
little with ground motion, designed to largely offset the gravity
force acting on the mass. Then a force transducer, driven by the
feedback circuit acts on the mass in such a way that it is forced to
move exactly with the earth and the instrument's frame. In effect
you are observing the acceleration required to make the mass move
with the earth and then in effect integrating that to give a velocity
signal, which provides a very accurate measure of the instantaneous
velocity of the earth vibrations.
In drawing the feedback block diagram, the input is usually
considered to be ground acceleration which is summed with the
(negative) feedback acceleration force from the force
transducer. Any small difference between them results in a
microscopic motion of the mass which is observed by a sensitive
displacement sensor. Its output is amplified, frequency shaped and
applied to the P, I and D feedback branches, which in turn drive the
force transducer.
There are many good references to these designs. Some of the best
have been written by Erhard Wielandt who was instrumental in
perfecting the force-balance seismometer concept.
http://www.iris.edu/stations/seisWorkshop04/PDF/Wielandt-Design3.doc
The STS-2 manual has a good block diagram that encourages some study
http://bnordgren.org/seismo/sts-2_manual_page_18.jpg
In your derivation I didn't quite follow the action of the feedback
loop, though I may have missed it.
The equations I (and Sean-Thomas) used to compute the instrument
response are outlined in http://bnordgren.org/seismo/loop4.pdf and
there is some useful information in
http://bnordgren.org/seismo/feedback.pdf and
http://bnordgren.org/seismo/feedback_in_seismic_sensors3.pdf which
is based on the STM-8 parameters. In general, at low frequencies
(where the loop gain is high) the instrument response is very nearly
1/B where B is the combined transfer function of the feedback
branches. The instrument's high frequency rolloff occurs at the
point where the loop gain has fallen to 1, about 37 Hz in the STM-8,
and from there falls, matching the falling velocity response of the
spring mass before feedback.
We have so far been able to ignore the back EMF, though a thorough
analysis might be in order at this point. In general the coil
motions are slow and extremely small, though that clearly doesn't
prove anything, but only suggests that it might not be a big issue.
There is much much more in the way of information on this design, but
these should be a small start.
When you are ready for more please let me know.
Regards,
Brett
At 01:36 PM 3/13/2010, you wrote:
>I've been doing some conceptual design work on a vertical broadband
>sensor, and I have a few questions. By reading past archives of the
>PSN-L I noticed that Sean-Thomas Morrissey was a contributor to this
>list, and there was some discussion about his STM-8 vertical
>leaf-spring design.
>
>1.) My first question is about deriving the transfer function. I tried
>to derive it myself, but from first principles, but what I got isn't
>what is shown in Sean-Thomas' work. I think the problem is my
>understanding of what the input and output of the transfer function
>are. Here's my logic for the transfer function:
>
>You can't measure the position of the Earth's surface from a
>stationary reference frame, and you can't measure the position of the
>mass from a stationary reference frame. Only the relative position
>between the two can be measured. My first postulate is that the force
>acting on the mass only depends on the relative position of the mass
>and Earth's surface. Combining this postulate with Newton's F= ma, I
>get:
>
> s^2 * M * X(s) = F(s) * (Y(s) - X(s))
>
>Where M is the mass, X(s) is the position of the mass in a stationary
>frame, Y(s) is the position of the ground in a stationary frame. The
>only thing we can directly measure is Y(s) - X(s), so I believe the
>transfer function is from Y(s) to Y(s) - X(s)/
>
> Now X(s) = Y(s) - (Y(s) - X(s)), substituting in the left
>hand side of my above expression:
>
> s^2 * M * (Y(s) - (Y(s) - X(s))) = F(s) * (Y(s) - X(s))
>
> s^s * M * Y(s) = (F(s) + s^2 * M)(Y(s) - X(s))
>
> s^2 * M
> (Y(s) - X(s)) = ----------------------- Y(s)
> F(s) + s^2 * M
>
>Then F(s) would be k + s*eta + K_p + s * K_d + K_i /s where K_p, K_i,
>K_d are the coefficients of a PID controller, k is the mechanical
>spring constant, and eta is the mechanical damping, if any.
>
>When I plug this all in I get a somewhat simpler expression.that what
>Sean-Thomas had. So I went back on including that fact that the
>integrator pole is not zero, and the back EMF generated in the
>feedback coil, and wound up with something way more complicated that I
>am not going to type out here. Does anybody have any hints?
>
>2.) My second question is about the hinge. Is foil suitable or should
>I look at something like a knife edge or ball bearing hinge? I'm
>leaning strongly towards foil. How much of an impact does the
>flexibility of the foil really have on the period? Given that the
>spring constant of the main spring is probably much much larger.
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