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
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=
tional
to 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. As
I mentioned in a previous =
mailing, the output voltage is for this case proportional to the electric f=
ield between the plates. It also
depends=
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 invariance
of =
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 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 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 its
bottom (and measuring the displacement of the mass), the sprin=
g would have to be prohibitively long to function as an instrument capable =
of detecting small
motions (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. His
solution 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 is
nearl=
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 of
the 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) circuit
board so=
ld by Dimension Engineering out of Akron, OH (product number DE-ACCM3=
D) uses the Analog Devices ADXL330 3-axis accelerometer
chip, 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 output
changes =
according to the direction of the earth's field relative to a given axis.&n=
bsp; 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 (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=
10,000
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 dif=
ference between them is simply due to a difference
in 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=
er
in 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 expect
that a factor of ten times 30 might be representative. =
And in fact, I found a research article showing an early generation
MEMS device using a cantilever, whose length wa=
s about 500 microns.
&=
nbsp; I hope that some of you readers will find this long post intere=
sting,
Randall
=