Brett,
The paper mentions the Homestake Mine as the site of preliminary test=
s of their instrument. Presumably, the measurements done there were at low=
-enough depths to eliminate many of the tilt noises associated with horizon=
tal instruments located on the surface. Then later it is mentioned that th=
ey were doing some measurements at the INFN Gran Sasso National Laboratory.=
I don't see any way the latter site could compare favorably to the former=
one, unless their Italian Lab is located a 'fair' depth underground.
About deconvolving 'signals' with their instrument. They are not looki=
ng at earth motions the same as a seismologist. Your 'interest' and theirs=
are probably 'diametrically opposite'. You would like to 'see all vibrati=
ons of the earth' (assuming you could distinguish among the multitude of ty=
pes), whereas they would like to build a vibration isolation stage (really,=
really fancy spring isolation device) that disallows them from seeing any =
vibration (other than the specific 'one' associated with earth motion that =
results from a gravitational wave interacting with our planet. Thus they =
restrict their attention to the 'achilles heel' of their challenge-the back=
ground noises free from earthquakes or man-made disturbances (like the fell=
ing of trees by lumberjacks near the Livingston, LA LIGO site).
I don't know the details of their interest in the folded pendulum. C=
an only conjecture that they might be trying to have a 'right kind of instr=
ument' associated with their Michelson Interferometer that is designed to m=
easure dimensional changes in an arm length (at a level of one part in 10^2=
0+). Maybe they can then hopefully distinguish between motion due to a gra=
vitational wave as opposed to motion of 'natural noise' type. Or maybe the=
y just are trying to insure that the earth noise that they wish did not exi=
st could actually be lower (at least at some times) than what has been meas=
ured by seismologists. You will notice from their Fig. 6 that the interfer=
ometer measurements gave a curve lying at a lower level than anything else,=
including previous 'standards' of seismologists.
But I am concerned about whether there may be something of an 'apples =
to oranges' comparison. Their 'power' unit is a weird convention, involvin=
g meters per root Hz, rather than the seismic convention of m^2/s^4/Hz. In=
cidentally, all through the paper they appear to have been sloppy (typo's, =
I assume, in the printed parts rather than the ordinate of Fig. 6 which are=
stated as m times root Hz rather than per root Hz).
So I am not sure whether what you have done to get to acceleration noi=
se from their data in dB levels is actually going to correspond properly to=
what you would estimate using your instrument. It should also be noted =
that when calculating spectral 'powers' (no matter the convention) using th=
e FFT, there is an important subtlety that has been mentioned here not too =
long ago. In other words, the spectral bin widths associated with all the =
algorithms I know about are not 1 Hz (rather the delta frequency between th=
e adjacent discrete points of the calculation). These are usually much sma=
ller than 1 Hz, and depend on the Nyquist frequency and the number of point=
s used in the calculation.
What really captured my attention from their paper has to do with what=
they observed when the instrument was evacuated, and shown in their Fig. 4=
.. At atmospheric pressure, it looks like their instrument is pretty well d=
escribed by the simple harmonic oscillator with viscous (linear) damping. =
But at lower pressures, you see that the quality factor becomes quadratic, =
rather than near linear, in the dependence on eigenfrequency. This is the =
hallmark of nonlinear damping that derives from internal friction. Apparen=
tly, because of the monolithic nature of their instrument, the influence of=
defects is less than with most instruments.
Since you mention Brownian motion, I will mention the following that i=
s not to my knowledge addressed anywhere other than maybe some of my papers=
.. The 'standard' calculation of performance threshold, as determined by at=
mospheric molecular motion-is an example of the use of a very famous concep=
t in thermal physics, called the equi-partition theorem. For a linear syst=
em (simple harmonic oscillator with viscous damping) this theorem says that=
every 'square term' in the Hamiltonian of the system provides one half kT =
of thermal energy to its (Brownian) motion. The problem with trying to est=
imate a meaningful threshold by this means, for systems that are nonlinear,=
is that the Hamiltonian of the nonlinear oscillator is not consistent with=
the premise on which the theorem is based; i.e., linearity. The very ter=
m harmonic means linear, and there are two types of anharmonicity (meaning =
nonlinearity)-elastic type and damping type. If you will type 'anharmonic =
oscillator' into Google, you will see that 'ChemWiki' speaks to these two t=
ypes that exist in nature. Everybody knows about the elastic type but almo=
st nobody the damping type-insofar as its significant influence. I am glad=
to see that at least the Chemists have started to pay attention.
It is tempting, because of our training, to think that mechanical syst=
ems operating at small levels must always be linear. My friend and colleag=
ue Tom Erber at Illinois Tech, showed before his retirement (involving many=
years of outstanding research in places where he chose as an outstanding t=
heoretical physics to 'dirty his hands' with experimental types) that the o=
nly place Hooke's law is ever obeyed is where displacements are so small th=
at they are measurable only with scanning microscopes (such as atomic force=
type). Just today I learned of a seminar given last week at the U. Minn. =
Physics dept. titled "Not so simple harmonic oscillator". The presenter (D=
r Proksch) was pointing out how AFM studies have shown that the dynamics of=
the 'tip' of such a microscope is not governed by a force 'constant'-meani=
ng that future interpretations of data will in some cases need to be 'adjus=
ted' for the nonlinear nature of that tip. I was circumstantially directed=
to the abstract of his seminar because of the article that I wrote (about =
15 years ago) titled, "The not-so-simple harmonic oscillator", publ. in Ame=
r. J. Phys. It speaks to the matter of nonlinear damping, for which there =
is no 'damping redshift' of the type that is seen with the linear SHO.
Randall
Brett,
The paper mentions=
the Homestake Mine as the site of preliminary tests of their instrument.&n=
bsp; Presumably, the measurements done there were at low-enough depths to e=
liminate many of the tilt noises associated with horizontal instruments loc=
ated on the surface. Then later it is mentioned that they were doing =
some measurements at the INFN Gran Sasso National Laboratory. I don&#=
8217;t see any way the latter site could compare favorably to the former on=
e, unless their Italian Lab is located a ‘fair’ depth undergrou=
nd.
About=
deconvolving ‘signals’ with their instrument. They are n=
ot looking at earth motions the same as a seismologist. Your ‘i=
nterest’ and theirs are probably ‘diametrically opposite’=
.. You would like to ‘see all vibrations of the earth’ (as=
suming you could distinguish among the multitude of types), whereas they wo=
uld like to build a vibration isolation stage (really, really fancy spring =
isolation device) that disallows them from seeing any vibration (other than=
the specific ‘one’ associated with earth motion that results f=
rom a gravitational wave interacting with our planet. Thus they=
restrict their attention to the ‘achilles heel’ of their chall=
enge—the background noises free from earthquakes or man-made disturba=
nces (like the felling of trees by lumberjacks near the Livingston, LA LIGO=
site).
&=
nbsp; I don’t know the details of their interest in the folded p=
endulum. Can only conjecture that they might be trying to have a R=
16;right kind of instrument’ associated with their Michelson Interfer=
ometer that is designed to measure dimensional changes in an arm length (at=
a level of one part in 10^20+). Maybe they can then hopefully distin=
guish between motion due to a gravitational wave as opposed to motion of &#=
8216;natural noise’ type. Or maybe they just are trying to insu=
re that the earth noise that they wish did not exist could actually be lowe=
r (at least at some times) than what has been measured by seismologists.&nb=
sp; You will notice from their Fig. 6 that the interferometer measurements =
gave a curve lying at a lower level than anything else, including previous =
‘standards’ of seismologists.
But I am concerned about whether the=
re may be something of an ‘apples to oranges’ comparison. =
Their ‘power’ unit is a weird convention, involving meters per=
root Hz, rather than the seismic convention of m^2/s^4/Hz. Incidenta=
lly, all through the paper they appear to have been sloppy (typo’s, I=
assume, in the printed parts rather than the ordinate of Fig. 6 which are =
stated as m times root Hz rather than per root Hz).
So I am not sure whether what yo=
u have done to get to acceleration noise from their data in dB levels is ac=
tually going to correspond properly to what you would estimate using your i=
nstrument. It should also be noted that when calculating =
spectral ‘powers’ (no matter the convention) using the FFT, the=
re is an important subtlety that has been mentioned here not too long ago.&=
nbsp; In other words, the spectral bin widths associated with all the algor=
ithms I know about are not 1 Hz (rather the delta frequency between the adj=
acent discrete points of the calculation). These are usually much sma=
ller than 1 Hz, and depend on the Nyquist frequency and the number of point=
s used in the calculation.
=
What really captured my attention from their paper has to=
do with what they observed when the instrument was evacuated, and shown in=
their Fig. 4. At atmospheric pressure, it looks like their instrumen=
t is pretty well described by the simple harmonic oscillator with viscous (=
linear) damping. But at lower pressures, you see that the quality fac=
tor becomes quadratic, rather than near linear, in the dependence on eigenf=
requency. This is the hallmark of nonlinear damping that derives from=
internal friction. Apparently, because of the monolithic nature of t=
heir instrument, the influence of defects is less than with most instrument=
s.
=
Since you mention Brownian motion, I will mention the following that is not=
to my knowledge addressed anywhere other than maybe some of my papers.&nbs=
p; The ‘standard’ calculation of performance threshold, as dete=
rmined by atmospheric molecular motion—is an example of the use of a =
very famous concept in thermal physics, called the equi-partition theorem.&=
nbsp; For a linear system (simple harmonic oscillator with viscous damping)=
this theorem says that every ‘square term’ in the Hamiltonian =
of the system provides one half kT of thermal energy to its (Brownian) moti=
on. The problem with trying to estimate a meaningful threshold by thi=
s means, for systems that are nonlinear, is that the Hamiltonian of the non=
linear oscillator is not consistent with the premise on which the theorem i=
s based; i.e., linearity. The very term harmonic means linear, =
and there are two types of anharmonicity (meaning nonlinearity)—elast=
ic type and damping type. If you will type ‘anharmonic oscillat=
or’ into Google, you will see that ‘ChemWiki’ speaks to t=
hese two types that exist in nature. Everybody knows about the elasti=
c type but almost nobody the damping type—insofar as its significant =
influence. I am glad to see that at least the Chemists have started t=
o pay attention.
&nbs=
p; It is tempting, because of our training, to think that mechanical s=
ystems operating at small levels must always be linear. My friend and=
colleague Tom Erber at Illinois Tech, showed before his retirement (involv=
ing many years of outstanding research in places where he chose as an outst=
anding theoretical physics to ‘dirty his hands’ with experiment=
al types) that the only place Hooke’s law is ever obeyed is where dis=
placements are so small that they are measurable only with scanning microsc=
opes (such as atomic force type). Just today I learned of a seminar g=
iven last week at the U. Minn. Physics dept. titled “Not so simple ha=
rmonic oscillator”. The presenter (Dr Proksch) was pointing out=
how AFM studies have shown that the dynamics of the ‘tip’ of s=
uch a microscope is not governed by a force ‘constant’—me=
aning that future interpretations of data will in some cases need to be =
216;adjusted’ for the nonlinear nature of that tip. I was circu=
mstantially directed to the abstract of his seminar because of the article =
that I wrote (about 15 years ago) titled, “The not-so-simple harmonic=
oscillator”, publ. in Amer. J. Phys. It speaks to the matter o=
f nonlinear damping, for which there is no ‘damping redshift’ o=
f the type that is seen with the linear SHO.
Randall
=
=