Randy,
With your instrument, for frequencies of drive that are above its char=
acteristic frequency (reciprocal of its period), the output is indeed propo=
rtional to the velocity of the ground on which the instrument sits; i.e., t=
o the first time derivative of ground displacement. Thus by taking the der=
ivative of the output voltage from your sensor in that regime you would obt=
ain the acceleration that is used in calculating the PSD. On the other han=
d, for ground oscillation that is at lower frequencies than the instrument'=
s natural frequency, the output is proportional to the third time derivativ=
e of the ground displacement. To get the acceleration in that regime requi=
res an integration of the output voltage. This happens because the transf=
er function of the instrument rises as 1/f toward the characteristic freque=
ncy and then falls as 1/f in going away from it toward high f. A 'connecti=
on' with the calculus is realized by noting that a derivative corresponds t=
o multiplying by f, whereas an integral to dividing by f.
Conventional seismometers (even those with feedback) generally operate t=
his way so as to mimic the old standard of voltage generated from relative =
motion of a coil and magnet in accord with Faraday's Law. As Chris Chapma=
n has frequently pointed out to this list-serve-for sensitive instruments, =
always let the coil be the part that moves with the inertial mass of the in=
strument, not the magnet system; which should be placed at rest on the fram=
e.
To calculate a proper PSD one must correct the Fourier transform of =
a signal for this frequency dependence of the instrument's transfer functio=
n. One can at the same time this correction is applied, also 'adjust' the =
math for the difference between your sensor and the type of sensor I prefer=
(as in the VolksMeter - output voltage proportional to ground acceleration=
below the characteristic frequency).
Much confusion exists because of the critical influence of the instru=
ment transfer function. In your instrument, the voltage generated by the r=
elative motion of coil and magnet is proportional to the time derivative of=
the displacement of your inertial mass relative to the case. On the other=
hand, with the VM, the voltage generated by its fully differential capacit=
ive sensor is instead proportional to the displacement itself, of the mass =
relative to the case. For either case, the displacement of the mass relati=
ve to the case is proportional to ground acceleration, when the frequency o=
f that acceleration is below the natural frequency of the instrument.
So in answer to your question-because of its complicating influence, =
the best way to get to acceleration (valid for all frequencies) with your i=
nstrument (for purpose of PSD calculations) is to correct your FFT with a p=
roperly formed transfer function. I can help you (and others if they are i=
nterested) to master the method of doing this using Excel. I have a veste=
d interested in doing so-since with a 'network' of instruments generating s=
uch records, we could hopefully have a better means for earthquake predicti=
on based on the paper that I previously mentioned.
Randall
Randy,
With your instrument, fo=
r frequencies of drive that are above its characteristic frequency (recipro=
cal of its period), the output is indeed proportional to the velocity of th=
e ground on which the instrument sits; i.e., to the first time derivative o=
f ground displacement. Thus by taking the derivative of the output vo=
ltage from your sensor in that regime you would obtain the acceleration tha=
t is used in calculating the PSD. On the other hand, for ground oscil=
lation that is at lower frequencies than the instrument’s natural fre=
quency, the output is proportional to the third time derivative of the grou=
nd displacement. To get the acceleration in that regime requires an i=
ntegration of the output voltage. This happens because the tran=
sfer function of the instrument rises as 1/f toward the characteristic freq=
uency and then falls as 1/f in going away from it toward high f. A &#=
8216;connection’ with the calculus is realized by noting that a deriv=
ative corresponds to multiplying by f, whereas an integral to dividing by f=
..
Conventional =
seismometers (even those with feedback) generally operate this way so as to=
mimic the old standard of voltage generated from relative motion of a coil=
and magnet in accord with Faraday’s Law. As Chris Chapma=
n has frequently pointed out to this list-serve—for sensitive instrum=
ents, always let the coil be the part that moves with the inertial mass of =
the instrument, not the magnet system; which should be placed at rest on th=
e frame.
=
To calculate a proper PSD one must correct th=
e Fourier transform of a signal for this frequency dependence of the instru=
ment’s transfer function. One can at the same time this correct=
ion is applied, also ‘adjust’ the math for the difference betwe=
en your sensor and the type of sensor I prefer (as in the VolksMeter –=
; output voltage proportional to ground acceleration below the characterist=
ic frequency).
=
Much confusion exists because of the critical influence o=
f the instrument transfer function. In your instrument, the voltage g=
enerated by the relative motion of coil and magnet is proportional to the t=
ime derivative of the displacement of your inertial mass relative to the ca=
se. On the other hand, with the VM, the voltage generated by its full=
y differential capacitive sensor is instead proportional to the displacemen=
t itself, of the mass relative to the case. For either case, the disp=
lacement of the mass relative to the case is proportional to ground acceler=
ation, when the frequency of that acceleration is below the natural frequen=
cy of the instrument.
=
; So in answer to your question—because of its=
complicating influence, the best way to get to acceleration (valid for all=
frequencies) with your instrument (for purpose of PSD calculations) is to =
correct your FFT with a properly formed transfer function. I can help=
you (and others if they are interested) to master the method of doing this=
using Excel. I have a vested interested in doing so—sinc=
e with a ‘network’ of instruments generating such records, we c=
ould hopefully have a better means for earthquake prediction based on the p=
aper that I previously mentioned.
=
; Randall
&=
nbsp;
=