Every seismometer responds to acceleration of the earth; meaning the mass moves
relative to the case because the case is accelerated.  In thinking about Newton's
2nd law, you can see that it is only an acceleration that "starts" anything of
dynamics type.  Although we develop kinematics by taking the derivative of position
to get velocity and the derivative of velocity to get acceleration; this approach
can easily confuse one into getting the 'cart before the horse'.  State variables
of nature (name used by dynamicists for position, velocity, and acceleration)
evolve only in one direction -- just opposite to our first exposure to calculus --
acceleration is the fundamental kinematic variable.  The position (deflection) of
the mass is directly proportional to the acceleration for a critically damped
instrument—as long as the rate of change of the acceleration (specified in terms of
frequency) is less than the natural frequency of the instrument.  Even if the
frequency of the drive is greater than the natural frequency, it is still a
straightforward matter to correct for the attenuation. Thus the mass movement of
every seismometer directly measures only one thing--acceleration. Of course the
movement of the seismic mass can be used to describe the earth's velocity by
integrating the output of a sensor that measures mass position (since the mass
deflection is directly proportional to acceleration, at least for drive frequencies
lower than the eigenfrequency).
 Before I say anything further about this, let me point out that it is a reasonable
assumption that whatever ‘decent’ linear sensor we use (one not worthless because
of friction), it has no significant influence on the evolution of the state
variables as long as we are not using force feedback that involves an actuator.  Of
course most of the commercial instruments use force-balance with an actuator, so
the arguments that follow must be modified to describe them.
 The most common older instrument is not measuring the mass position; rather by
means of a Faraday's law detector (time changing magnetic flux; i.e., magnet/coil)
it is measuring the time rate of change of position.  Since the position is
determined by the acceleration--simply stated, such a sensor measures the
derivative of the ground's acceleration, which is called the 'jerk'.  For harmonic
motions of the ground (sine or cosine), the velocity of the ground motion is given
by omega times the displacement of the ground (same as derivative of position to
get velocity), where omega is two-pi times the frequency.  Similarly the
acceleration is equal to omega times the velocity.  Thus there is a simple
connection between the amplitude of the mass displacement Am and the ground
amplitude A for drive frequency below natural frequency.  It is Am = omega squared
times A divided by omega0 squared, where both A and Am are in meters.  Omega0 is
two pi times the natural frequency of the instrument.  For a jerk detector the
relationship between Am and the ground velocity amplitude V in meters per second is
given by omega times V divided by omega0 squared.  One can get a better feel for
all of this by looking at instrument transfer functions; the best graphics that I
know about are provided on John Lahr’s website at
http://jclahr.com/science/psn/response/plots.jpg
Of these six cases, the upper right graph is the fundamental one.  It applies to
the output from our VolksMeter; labeled lctst in the data on Larry Cochrane’s
webpages.  The lower right graph is what gets output from the VolksMeter after
integration; labeled lctst1 on Larry’s pages.
 There is a subtlety in the use of these graphs to describe the old-style jerk
sensor (and for that matter the broadband units, since their output is similar,
except they use force-balance by means of an actuator to lower omega0).  Although
the expression ‘velocity sensing’ is used to describe such an output, bear in mind
what I said above.  It measures the velocity of the seismic mass motion, not the
velocity of the ground.  Since the mass motion is proportional to the acceleration
of the ground, what the instrument is measuring is the jerk of the ground (third
derivative of the displacement of the ground).  On the basis of the harmonic drive
assumption, this third derivative can be related to the first derivative, which is
the time rate of change (velocity) of the displacement of the ground.  In other
words, for a monochromatic harmonic drive (and a Hooke’s law mechanical oscillator
if one were to exist), the lower middle graph is a valid representation of how such
a sensor’s output voltage responds to ground velocity.  When the frequency of the
drive is changing, this is no longer as good a sensor to describe what the earth is
doing as compared to a position sensor.  Not only is there a loss of information
with the ‘velocity’ sensor (its inability to see offsets, which ‘get removed except
for transients’ because of the derivative); there are also even some ‘false data’
consequences.  Let me give you a brief explanation of the artifacts that can be
introduced.
 The potential energy function of a seismometer is not a parabola, the so-called
harmonic oscillator potential.  Hooke’s law does not apply to real systems because
of defect structures.  At low energies (the place now begging for better data) the
parabola is ‘modulated’ with fine structure; i.e., the potential is ‘ragged’ rather
than smooth.  When a seismometer with this fine structure is driven by an
acceleration of the earth, the simple connection between ground velocity and
Faraday-law sensor output voltage is no longer trivial.  The nonlinear nature of
the dynamics gives rise to a host of complications.  Deconvolution is no longer
possible because superposition (the standard approach for solving linear equations
of motion) is no longer valid.
 Having spent much of my career studying these nonlinearities, I am probably the
only person with connections to seismology, who is intimately familiar with the
challenges through direct experience.  I intend to generate some simulations of
representative cases (ones relevant to free earth oscillations) to illustrate some
of the ‘pathologies’ associated with ‘velocity’ sensing.  The artifacts generated
by a non-ideal (nonlinear) oscillator are worse for a ‘velocity’ sensor than for a
‘position’ sensor.  Not only that, the noise equivalent power spectral density is
also worse at low frequencies.  Thus the velocity sensor introduces more in the way
of artifacts having frequencies above the signal of interest, while at the same
time having a poorer signal to noise ratio when it comes to the actual frequency of
interest.

   Randall

psn-l-digest-request@.............. wrote:

> .------ ------ ------ ------ ------ ------ ------ ------ ------ ------.
> | Message 1                                                           |
> '------ ------ ------ ------ ------ ------ ------ ------ ------ ------'
> Subject: Re: Pendulum Q
> From:    ChrisAtUpw@.......
> Date:    Wed, 24 Jan 2007 20:33:55 EST
>
> -------------------------------1169688835
> Content-Type: text/plain; charset="US-ASCII"
> Content-Transfer-Encoding: 7bit
>
>
> In a message dated 24/01/2007, Brett3mr@............. writes:
>
> Let me  borrow Einstein's thought-experiment box, the one with
> no windows that is  big enough to let me get inside. Sitting in the box, I
> coat the  bottom with thought-experiment ice, the kind that has no
> friction. Then I  set a 1Kg brass weight on the ice in the center of the
> floor and wait for  the thought-experiment earthquake to occur, which
> happens right on  time. Being that the quake is conveniently close, I
> observe that the  mass appears to be moving, which I record with my video
> camera. The  question is, what am I observing when I plot the motion of the
> weight? And then, can I tell anything about the nature of the motion  of
> the box (i.e. ground velocity or acceleration) from analyzing the  weight's
> apparent motion inside the box? Also, I observe that, since  there can be
> no (horizontal) force acting on the weight because of the  ice, it will be
> seeing no (horizontal) acceleration at  all.
>
> Hi Brett,
>
>     You are recording both the position and the time of  relative movements
> of the mass. From this you can infer the lateral acceleration  and velocity of
> the frame of reference, if you know that the mass is not  being accelerated.
>
>     Regards,
>
>     Chris Chapman
>
> -------------------------------1169688835
> Content-Type: text/html; charset="US-ASCII"
> Content-Transfer-Encoding: quoted-printable
>
> 
> 
> 
> 
>  Arial"=20
> bottomMargin=3D7 leftMargin=3D7 topMargin=3D7 rightMargin=3D7> e_document=20
> face=3DArial color=3D#000000 size=3D2>
> 

> 
In a message dated 24/01/2007, Brett3mr@............. writes:
>  style=3D"PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: blue 2px solid"><=
> FONT=20
>   style=3D"BACKGROUND-COLOR: transparent" face=3DArial color=3D#000000 size=
> =3D2>Let me=20
>   borrow Einstein's thought-experiment box, the one with 
no windows that=
>  is=20
>   big enough to let me get inside. Sitting in the box, I 
coat the=20
>   bottom with thought-experiment ice, the kind that has no 
friction. The=
> n I=20
>   set a 1Kg brass weight on the ice in the center of the 
floor and wait=20=
> for=20
>   the thought-experiment earthquake to occur, which 
happens right on=20
>   time. Being that the quake is conveniently close, I 
observe that=20=
> the=20
>   mass appears to be moving, which I record with my video 
camera. The=20
>   question is, what am I observing when I plot the motion of the=20
>   
weight? And then, can I tell anything about the nature of the mot=
> ion=20
>   of 
the box (i.e. ground velocity or acceleration) from analyzing the=20
>   weight's 
apparent motion inside the box? Also, I observe that, si=
> nce=20
>   there can be 
no (horizontal) force acting on the weight because of the=
> =20
>   ice, it will be 
seeing no (horizontal) acceleration at=20
> all.

> 

> 
Hi Brett,
> 
 
> 
    You are recording both the position and the tim=
> e of=20
> relative movements of the mass. From this you can infer the lateral accelera=
> tion=20
> and velocity of the frame of reference, if you know that the mass is no=
> t=20
> being accelerated.
> 
 
> 
    Regards,
> 
 
> 
    Chris Chapman
>
> -------------------------------1169688835--
>
> .------ ------ ------ ------ ------ ------ ------ ------ ------ ------.
> | Message 2                                                           |
> '------ ------ ------ ------ ------ ------ ------ ------ ------ ------'
> Subject: Re: Pendulum Q
> From:    Brett Nordgren 
> Date:    Wed, 24 Jan 2007 21:22:44 -0500
>
> --=====================_49664656==_.ALT
> Content-Type: text/plain; charset="us-ascii"; format=flowed
>
> >Hi Brett,
> >
> >     You are recording both the position and the time of relative
> > movements of the mass. From this you can infer the lateral acceleration
> > and velocity of the frame of reference, if you know that the mass is not
> > being accelerated.
> >
> >     Regards,
> >
> >     Chris Chapman
>
> Chris,
>
> Thanks for your comment, that's exactly what I was trying to demonstrate.
>
> On the one hand, there is the derivative relationship between ground
> displacement, velocity and acceleration and on the other, the practical
> problem of constructing a device to measure them.  My 'experiment' was an
> attempt to untangle the two issues.
>
> Regards,
> Brett
> --=====================_49664656==_.ALT
> Content-Type: text/html; charset="us-ascii"
>
> 
> 
Hi Brett,

>  

>     You are recording both the position and the time of
> relative movements of the mass. From this you can infer the lateral
> acceleration and velocity of the frame of reference, if you know that the
> mass is not being accelerated.

>  

>     Regards,

>  

>     Chris Chapman


> Chris,


> Thanks for your comment, that's exactly what I was trying to
> demonstrate.  


> On the one hand, there is the derivative relationship between ground
> displacement, velocity and acceleration and on the other, the practical
> problem of constructing a device to measure them.  My 'experiment'
> was an attempt to untangle the two issues.


> Regards,

> Brett
>
> --=====================_49664656==_.ALT--
>
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