## PSN-L Email List Message

Subject: Seismometer Sensitivity (and a 'gedanken')
From: Randall Peters PETERS_RD@..........
Date: Sat, 20 Aug 2011 11:17:37 -0400

```The sensitivity of a gravitationally restored pendulum is proportional to t=
he square of its natural period, as is now readily proven.
Many are familiar with the famous formula for the period of a simple =
pendulum; i.e., two pi times the square root of the ratio of
length to the acceleration of gravity.  This means that the square of the n=
atural period is directly proportional to the length of the pendulum.
Consider now the sensor used to measure the horizontal linear displacement =
of the bob, away from the equilibrium that corresponds to alignment
with the direction of the earth's field. The sensitivity of this earliest u=
seful seismometer depends on the
displacement of the bob.  This assumes, of course, that we understand it is=
much smarter to measure the bob's displacement, rather
than the angular displacement of the pendulum, that is proportional to its =
acceleration.  It is a simple exercise in the use of
Newton's second law, to show that the angular displacement is indeed propor=
tional
to the horizontal component of the acceleration of the case; i.e., to the s=
ize in meters per second per second of
the seismic disturbance causing the pendulum to move (for accelerations sma=
ll compared to little 'g'.  Incidently, it is only
the acceleration of the earth that causesany seismometer to respond.  As is=
routinely done by seismologists, we can specify the earth motion by means =
of a
transformation to equivalent velocity (for a given frequency of assumed ste=
ady state oscillation).  This can be done mathematically after measuring ac=
celeration directly (as with a VolksMeter).  It can also be done electronic=
ally, as is accomplished by the feedback network of a force balance instrum=
ent.
It should be understood, however, no matter how you specify the motion; it =
is the acceleration of the earth
(and only the acceleration) that causes the instrument to respond to begin =
with.  If you doubt this statement then
ask yourself the following question--what is fundamental,velocity or accele=
ration.  Or stated in another way, which came first, the velocity or the ac=
celeration (for any problem inmechanics).  Tell me velocity and you are lik=
ely to fail my physics course.
The simplest sensor to demonstrate invariance to scaling influence =
is a capacitive sensor that works on the basis of voltage change
due to gap spacing variation, and for which the total charge on the capacit=
or is fixed.  This is physically meaningful for an ideal
parallel plate capacitor, having small gap spacing, that has been charged t=
o a particular voltage and the charging source then
disconnected. Very small changes in its gap spacing then yield proprtionall=
y small voltage variations across the plates. As
I mentioned in a previous mailing, the output voltage is for this case prop=
ortional to the electric field between the plates. It also
depends directly on the amount of the gap change, and thus will measure a m=
iniscule displacement of the pendulum.
So then, what happens to the sensitivity of this pendulum as we mak=
e it longer?  The answer is easy to understand.  Assuming scale invariance
of the sensor (or use of the same detector), and for a given small accelera=
tion, the size of the voltage output by the sensor (at the position
of the bob) is directly proportional to the length of the pendulum, since t=
he pendulum's angular deflection is itself proportional to
the acceleration.  Moreover, because the period squared is proportional to =
the pendulum length, we arrive at the (generally, independent of
seismometer type) result that the sensitivity is proportional to the square=
of the natural period of the pendulum.
We can apply similar reasoning to the hypothetical situation of a c=
oiled spring seismometer hanging vertically.  This was the
genesis of the problem assigned to Lucien LaCoste by Prof. Romberg in a mec=
hanics class at the University of Texas.  He in effect
told Lucien that to obtain a high sensitivity for this spring operating as =
a component of a vertical seismometer; i.e., by fixing the top of the sprin=
g and hanging a mass on its
bottom (and measuring the displacement of the mass), the spring would have =
to be prohibitively long to function as an instrument capable of detecting =
small
motions (like waves from a teleseismic earthquake).  The ingenious solution=
provided by LaCoste was to mount a much shorter spring at an
angle, to provide mechanical amplification'. But he then had to invent the =
zero-length spring in order to avoid the problems of mesoanelastic complexi=
ty. His
solution was unique to my academic experience.  I don't know any other prof=
essor whose benefits (probably both monetarily and in other
forms of satisfaction) were improved by forming a company (LaCoste Romberg)=
on the basis of his student's ingenuity.
Although I have not here proven the same thing for a mass/vertical =
spring instrument as was done for the simple pendulum; it is
nearly as easy to show that the sensitivity of this vertical spring instrum=
ent is also proportional to its size; i.e., to the length of
the spring.  On the basis of this reasoning I have come to the following co=
nclusion. It might just be true, in general, that the sensitivity
of a seismic instrument involves the 'size' of the instrument proportionall=
y.  This would explain, for example, why a MEMS
accelerometer is much too insensitive to pick up teleseismic waves that we =
routinely see with our macro-sized instruments.
In the last week I put together a high-performance  MEMS-based seis=
mocardiograph.  The user-friendly, inexpensive (\$35) circuit
board sold by Dimension Engineering out of Akron, OH  (product number DE-AC=
CM3D) uses the Analog Devices ADXL330 3-axis accelerometer
chip, along with a buffer to make it user friendly.  The chip is micro-mach=
ined and works on the basis of differential capacitive sensing of the motio=
n of the MEMS cantilever.  It is easily calibrated by means of 90 degree ro=
tations (around the principal axes) and looking at how the analog voltage o=
utput
changes according to the direction of the earth's field relative to a given=
axis.  One is by this means operating on the
basis of deflections of the cantilever due to its weight. A similar chip is=
used in my Droid-x (Kionix KXTF9) to change the
display, when you flip the phone from one viewing orientation to another (t=
o keep things right-side up). I tried unsuccessfully to use the Kionix
accelerometer for SCG  purposes; but the 12-bit ADC employed is not fine en=
ough.
For purpose of our SCG studies, I calibrated the DE-ACCM3D system=
by the means just stated, with it connected to a 24 bit ADC (USB 4-channel=
unit) sold by Symmetric Research out of Kirkland, WA.  I measured the nois=
e limited sensitivity of the device in all three channels at about one part=
in 10,000
of the earth's field.  This is roughly 10,000 times worse sensitivity than =
the VolksMeter, which also uses an Analog Devices chip
(capacitance to digital converter, AD7745).  I proceeded then to wonder if =
the difference between them is simply due to a difference
in their size.  The pendulum length of the VM is about 0.3 m.  Although I c=
ould not find a spec. for the length of the cantilever
in the ADXL330, I would on the basis of my hypothesize, predict it to be of=
the order of not less than 30 microns.
A cantilever does not bend as far under the influence of acceleration as do=
es a pendulum swing for the same acceleration.
The mechanics of cantilever bending is quite complicated mathematically, so=
I didn't 'go there' for mmore precise calcuations.  But I would expect
that a factor of ten times 30 might be representative.  And in fact, I foun=
d a research article showing an early generation
MEMS device using a cantilever, whose length was about 500 microns.
I hope that some of you readers will find this long post interesting,
Randall
<!--
/* Font Definitions */
@font-face
{font-family:Calibri;
panose-1:2 15 5 2 2 2 4 3 2 4;}
/* Style Definitions */
p.MsoNormal, li.MsoNormal, div.MsoNormal
{margin:0in;
margin-bottom:.0001pt;
font-size:11.0pt;
font-family:"Calibri","sans-serif";}
{mso-style-priority:99;
color:blue;
text-decoration:underline;}
{mso-style-priority:99;
color:purple;
text-decoration:underline;}
span.EmailStyle17
{mso-style-type:personal-compose;
font-family:"Calibri","sans-serif";
color:windowtext;}
..MsoChpDefault
{mso-style-type:export-only;}
@page WordSection1
{size:8.5in 11.0in;
margin:1.0in 1.0in 1.0in 1.0in;}
div.WordSection1
{page:WordSection1;}
-->The sensitivity =
of a gravitationally restored pendulum is proportional to the square of its=
natural period, as is now readily proven.      Many are familiar with the famous formula=
for the period of a simple pendulum; i.e., two pi times the square root of=
the ratio of length to the acceleratio=
n of gravity.  This means that the square of the natural period is dir=
ectly proportional to the length of the pendulum. Consider now the sensor used to measure the horizontal linear di=
splacement of the bob, away from the equilibrium that corresponds to alignm=
ent with the direction of the earth's f=
ield. The sensitivity of this earliest useful seismometer depends on the displacement of the bob.  This assum=
es, of course, that we understand it is much smarter to measure the bob's d=
isplacement, rather than the angular di=
splacement of the pendulum, that is proportional to its acceleration. =
It is a simple exercise in the use of =
Newton's second law, to show that the angular displacement is indeed propor=
tionalto the horizontal component of th=
e acceleration of the case; i.e., to the size in meters per second per seco=
nd of the seismic disturbance causing t=
he pendulum to move (for accelerations small compared to little 'g'.  =
Incidently, it is only the acceleration=
of the earth that causesany seismometer to respond.  As is routinely =
done by seismologists, we can specify the earth motion by means of a <=
/o:p>transformation to equivalent velocity (for a =
given frequency of assumed steady state oscillation).  This can be don=
e mathematically after measuring acceleration directly (as with a VolksMete=
r).  It can also be done electronically, as is accomplished by the fee=
dback network of a force balance instrument.It should be understood, however, no matter how you specify the motion=
; it is the acceleration of the earth (=
and only the acceleration) that causes the instrument to respond to begin w=
ith.  If you doubt this statement then ask yourself the following question--what is fundamental,velocity or a=
cceleration.  Or stated in another way, which came first, the velocity=
or the acceleration (for any problem inmechanics).  Tell me velocity =
and you are likely to fail my physics course.          The simplest s=
ensor to demonstrate invariance to scaling influence is a capacitive sensor=
that works on the basis of voltage change due to gap spacing variation, and for which the total charge on the cap=
acitor is fixed.  This is physically meaningful for an ideal parallel plate capacitor, having small gap spaci=
ng, that has been charged to a particular voltage and the charging source t=
hen disconnected. Very small changes in=
its gap spacing then yield proprtionally small voltage variations across t=
he plates. AsI mentioned in a previous =
mailing, the output voltage is for this case proportional to the electric f=
ield between the plates. It alsodepends=
directly on the amount of the gap change, and thus will measure a miniscul=
e displacement of the pendulum. &n=
bsp;      So then, what happens to the sensitivity=
of this pendulum as we make it longer?  The answer is easy to underst=
and.  Assuming scale invarianceof =
the sensor (or use of the same detector), and for a given small acceleratio=
n, the size of the voltage output by the sensor (at the position of the bob) is directly proportional to the lengt=
h of the pendulum, since the pendulum’s angular deflection is itself =
proportional to the acceleration.  =
;Moreover, because the period squared is proportional to the pendulum lengt=
h, we arrive at the (generally, independent ofseismometer type) result that the sensitivity is proportional to the=
square of the natural period of the pendulum.          We can apply =
similar reasoning to the hypothetical situation of a coiled spring seismome=
ter hanging vertically.  This was the genesis of the problem assigned to Lucien LaCoste by Prof. Romberg in a=
mechanics class at the University of Texas.  He in effect =
told Lucien that to obtain a high sensitivity for =
this spring operating as a component of a vertical seismometer; i.e., by fi=
xing the top of the spring and hanging a mass on itsbottom (and measuring the displacement of the mass), the sprin=
g would have to be prohibitively long to function as an instrument capable =
of detecting smallmotions (like waves f=
rom a teleseismic earthquake).  The ingenious solution provided by LaC=
oste was to mount a much shorter spring at an angle, to provide mechanical amplification'. But he then had to inve=
nt the zero-length spring in order to avoid the problems of mesoanelastic c=
omplexity. Hissolution was unique to my=
academic experience.  I don't know any other professor whose benefits=
(probably both monetarily and in other=
forms of satisfaction) were improved by forming a company (LaCoste Romberg)=
on the basis of his student's ingenuity.          Although I have no=
t here proven the same thing for a mass/vertical spring instrument as was d=
one for the simple pendulum; it isnearl=
y as easy to show that the sensitivity of this vertical spring instrument i=
s also proportional to its size; i.e., to the length ofthe spring.  On the basis of this reasoning I have com=
e to the following conclusion. It might just be true, in general, that the =
sensitivity of a seismic instrument inv=
olves the 'size' of the instrument proportionally.  This would explain=
, for example, why a MEMS accelerometer=
is much too insensitive to pick up teleseismic waves that we routinely see=
with our macro-sized instruments. &nbs=
p;       In the last week I put together=
a high-performance  MEMS-based seismocardiograph.  The user-frie=
ndly, inexpensive (\$35) circuitboard so=
ld by Dimension Engineering out of Akron, OH  (product number DE-ACCM3=
D) uses the Analog Devices ADXL330 3-axis accelerometerchip, along with a buffer to make it user friendly.  T=
he chip is micro-machined and works on the basis of differential capacitive=
sensing of the motion of the MEMS cantilever.  It is easily calibrate=
d by means of 90 degree rotations (around the principal axes) and looking a=
t how the analog voltage outputchanges =
according to the direction of the earth's field relative to a given axis.&n=
bsp; One is by this means operating on thebasis of deflections of the cantilever due to its weight. A similar chip=
is used in my Droid-x (Kionix KXTF9) to change thedisplay, when you flip the phone from one viewing orientation =
to another (to keep things right-side up). I tried unsuccessfully to use th=
e Kionix accelerometer for SCG  pu=
rposes; but the 12-bit ADC employed is not fine enough.          &nbs=
p;For purpose of our SCG studies, I calibrated the DE-ACCM3D system by the =
means just stated, with it connected to a 24 bit ADC (USB 4-channel unit) s=
old by Symmetric Research out of Kirkland, WA.  I measured the noise l=
imited sensitivity of the device in all three channels at about one part in=
10,000of the earth's field.  This=
is roughly 10,000 times worse sensitivity than the VolksMeter, which also =
uses an Analog Devices chip(capacitance=
to digital converter, AD7745).  I proceeded then to wonder if the dif=
ference between them is simply due to a differencein their size.  The pendulum length of the VM is about 0.=
3 m.  Although I could not find a spec. for the length of the cantilev=
erin the ADXL330, I would on the basis =
of my hypothesize, predict it to be of the order of not less than 30 micron=
s.A cantilever does not bend as far und=
er the influence of acceleration as does a pendulum swing for the same acce=
leration.  The mechanics of cantil=
ever bending is quite complicated mathematically, so I didn't 'go there' fo=
r mmore precise calcuations.  But I would expectthat a factor of ten times 30 might be representative.  =
And in fact, I found a research article showing an early generationMEMS device using a cantilever, whose length wa=