2011 super spring modelling

Superspring Digitally Controlled Feedback System in the Vibration isolation design used for the 72m resonant cavity.
1 | P a g e D e p a r t m e n t o f I n s t r u m e n t a t i o n , 2 0 0 6 - 2 0 1 0 , C U S A T
Dept. of Instrumentation,
Cochin University of Science and Technology
Superspring Digitally Controlled Feedback system in the Vibration Isolation
Design used for the 72 m Resonant Cavity
Submitted by: Supervisors:
Siddartha S Verma Prof. David G. Blair
Dr. Chunnong Zhao
Dr. Jean Charles Dumas
Summer Internship,
8th
January – 10th
March 2010.
School of Physics, AIGO, UWA, Perth

Acknowledgement
I would like to acknowledge and thanks Prof. David Blair for selecting me for the Summer Internship
work in AIGO, and giving me an opportunity to work and learn in Gravitational Wave Future
Detectors Instrumentation. I would like to thanks my Head of the department, Dr.
K.N.Madhusoodanan and my faculty members, for their valuable suggestions, and allowing me to
gain an experience in such an interesting field of gravitational wave detection. I’d also like to thanks
Dr. Chunnong Zhao for guiding me and providing very useful insights used for my research work in
AIGO. Also, I’d like to thanks Dr. Jean Charles Dumas who was always available for any help in
technical fields and literatures. He’d also make my stay at AIGO site at Gingin very useful for work.
Then, I’d like to say my thanks to Andrew who was instrumental for my stay and research work in
AIGO, and taking me to his accommodation on some of the weekends.
Last, but not the least, I’d like to thanks my parents for supporting me and helping me for doing the
Summer Internship and B.Tech Final year thesis project work.

Declaration
This is to declare that the project on “Superspring Digitally Controlled Feedback system in Vibration
Isolation Design used for 72m Resonant Cavity”, which was a part of Summer Internship in AIGO,
School of Physics, The University of Western Australia – Perth, Australia during the period of 8th
January to 10th
March 2010, which also fulfils my requirement of the Final Year Thesis work for my
Bachelors in Technology (B.Tech), Instrumentation 2006 -2010 in Dept. of Instrumentation, CUSAT,
Kochi – 682022, is a bona fide record of work done by me under the guidance of Prof. David G. Blair,
Dr. Chunnong Zhao and Dr. Jean Charles Dumas, AIGO (Gingin), School of Physics, M013, The
University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009.
Author:
Siddartha S Verma
Dept. of Instrumentation,
CUSAT, Kochi, Kerala – 682022
B.Tech, 8th
Semester, Instrumentation, 2006 - 2010

Certification
This is to certify that the project and thesis work done on “Superspring Digitally Controlled Feedback
system in Vibration Isolation Design used for 72m Resonant Cavity” by Siddartha S Verma, Dept. of
Instrumentation, CUSAT, Kochi – 682022, as a part of Summer Internship program conducted by
AIGO, School of Physics, The University of Western Australia, Perth during the period of 8th
January –
10th
March 2010, is a bona fide record of work done by him, under our guidance. He has finished his
project on time and submitted the report to us.
Prof. David G. Blair
Dr. Chunnong Zhao
Dr. Jean Charles Dumas
AIGO, School of Physics
The University of Western Australia,
35 Stirling Highway,
Perth, Crawley, Western Australia 6009

Preface ---------------------------------------------------------------------------
This section is being appended to the report in order to elucidate more about the Gravitational
Wave Astronomy and its onset as a main stream Astronomy in recent years. This section would also
describe the author’s background and interest in this field, and also the brief account of work and
contribution made by the author.
Gravitational Wave (GW) detection would lead to the direct confirmation of Einstein’s General
Theory of Relativity given by Einstein in 1916. They have been predicted long before but not have
been detected yet. GW are the disturbances in the curvature of space time caused by the motion of
matter. We can think of space time as a fabric that can bend or curve when we place an object on it.
But, the space time is 4 dimensional. The sources of GW could be binary orbit of black holes, a merge
of 2 galaxies, 2 neutron star orbiting each other, or any other object on a very exotic and unexplored
territory [3].
Gravitational Wave detection measures very small change in the fabric of space-time.
Interferometry is the technique which researcher uses to measure the stretches in the space time.
The Lasers require the test mass to be at a large distance from each other. It is the laser which
makes the continuous measurement. Since the masses are free to move, the distance between the
masses will fluctuate. A brief account of the
Coming from an engineering background, the author’s interest in Gravitational Waves comes from
the inclination towards fundamental science research.
This literature report is a summary of the work done during the Summer Internship period of 8th
January – 10th
March, in Australian International Gravitational Observatory (AIGO), which is operated
in Gingin, Western Australia by the School of Physics, The University of Western Australia, Perth. My
work was essentially related to the Superspring Digitally Controlled Feedback System in the
Vibration Isolation Design used in 72m resonant cavity. Much more about the work has been
explained later in Introduction section and later chapters of the report.
I’d now like to brief some of the issues I had to deal with during my internship period. They are, viz.:
 Getting a grasp of the current Vibration Isolation being used in Gravitational Wave research
and the specific one used in AIGO.
 Investigating the Superspring concept to be used for vibration isolation.
 Learning more about the LabView and MATLAB schemes currently being used for the
vibration isolators.
 Selecting an appropriate model which could be the best approximation of the vibration
isolation used in AIGO.
 Calculation of several transfer functions for the system and calculating the stability for each
of them.
 Optimizing the PID controllers for Superspring feedback mechanism and using other filters if
necessary.
 Checking the vacuum conditions necessary for getting good results.
 Implementation on the LabView control schemes currently being used for the Vibration
Isolators, and making necessary modification and check up for appropriate functioning.

Some revival in the hardware like actuators, sensors and channels, in the isolators could be
necessary after check up.
 Taking into consideration any coupling happening in the performance, and optimizing the
controller parameters for different degrees of freedom such as X, Y, Rotation, Tilt.
 Taking data from the spectrum analyser and analysing it for enhanced performance.
 Doing appropriate modifications if needed.
The author got help and advice regarding the suitability of the model and its implementation from
Dr. Jean-Charles Dumas and Dr. Chunnong Zhao. The main control schemes which were currently
operational in the main control computer in the control room near vibration isolator had been
developed by Jean-Charles. It is a sufficiently complex and difficult scheme to get a grasp on.
Due to the time constraints and focus of the group on other necessary obligations, like the AIGO
International Conference, getting the cavity locked before the conference and research proposals,
the project work implementation and check up had to be delayed for some amount of time. The
group would continue to work for the final performance of the Superspring after the authors work.

Abstract
The Superspring digitally controlled feedback mechanism for the vibration isolators used in the 72 m
resonant cavity has its origin in the reduction of residual motion of freely suspended test masses [1].
A super spring is a concept of electronic termination in which a 30cm spring can be made to behave
as a spring which is 1km or longer in length [2]. In our approach, 2 controllers have been designed
which are able to provide the feedback signals to the inverse pendulum stage and stabilize it by
reduction of differential motion. First of all, the position sensors between the inverse pendulum and
Roberts linkage pre-isolation stages is used to measure their respective displacement in 3 horizontal
degrees of freedom (X, Y and Yaw). The sensors being used are shadow sensors which consist of a
LED and 2 photodiodes mounted on the inverse pendulum, and an interlocked mask which is
attached to the Roberts Linkage, which partially blocks the light from the LED from the 2
photodiodes. The feedback of (x2-x1) and (x1-x0) signals from the sensors, along with appropriate
controllers C2 and C1, to the inverse pendulum actuators allow the net inverse pendulum frequency
to be lowered. This would also lead to the stability of control mass and then the test mass. The
model also takes care of the stability of the system, and is implemented on the control scheme
which is already being used for the isolators. The present limit of 3.3nm residual motion at 1Hz
frequency has been planned to be reduced to approximately 0.1 nm through implementing
Superspring digitally controlled feedback mechanism and other novel mechanism of Euler-Lacoste
vertical vibration stage [1]. Currently, a 80m optical cavity of finesse of about 750m is operational in
AIGO. The cavity goal is to work for higher finesse such as 15000 in vacuum, by which optical bars
and quantum measurement scheme experiments would be possible [1]. This requires a 10 fold
improvement in low frequency residual motion.

Authors Contribution
The author was responsible for making a theoretical model for Superspring implementation, taking
real considerations, in the vibration isolator. Most of the work done by the author has been
summarized in the chapter 3. Please refer it for more details and the comparison between no
Superspring control and with control. The author successfully showed theoretically that the
Superspring digitally controlled feedback mechanism actually works in the vibration isolators and is a
promising model for gaining a residual motion of 0.1nm at 1Hz, which is essential for future
experiments in AIGO. The author also studied rigorously the Labview control mechanism which is
being implemented already by Dr. Jean Charles Dumas and specifically the section which is being
used for Superspring implementation. The Labview control for Superspring is almost working well, as
the actual data suggests but could be modified for better performances. Because of time constraints,
the author was not able to do much investigation in this line.
The author was also successfully able to show that the Superspring design is actually effective in
reducing the response of the system in the region of 1 Hz by few DB’s. This was done by
implementing the Superspring design in the main Labview control, and changing the parameters in
control for Superspring design. The author acquired the data and then processed it to give the
results which have been shown in chapter 3. Please refer the contents and the chapter for more
details and actual results.
Apart from the specific subject of Superspring’s, the author was able to gather knowledge about the
current contemporary research going in the field of gravitational waves detection, by attending the
international conference in AIGO and the workshop.

Contents ----------------------------------------------------------------------------
Acknowledgement 2
Declaration 3
Certification 4
Preface 5
Abstract 7
Author’s contribution 8
1. Introduction 10
1.1 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Existing and Future Gravitational Wave detectors . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 The Gravitational Wave signal for detection . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Resonant mass detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.3 Long Baseline Laser Interferometry Detectors . . . . . . . . . . . . . . . . . . . . 15
1.3 Different noises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 Vibration Isolation requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2. Vibration Isolation design in AIGO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Inverse Pendulum Preisolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 LaCoste Linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Roberts Linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Euler Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Self Damped pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Control Mass and Test mass stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3. Superspring Digital Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1 Superspring feedback for pre-isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Model for Analysis and its realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Block diagram of the Superspring Control . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.2 Superspring results from actual implementation . . . . . . . . . . . . . . . . . . . . . 45

3.4 Needs and benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 LabView Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4. Conclusion and Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Chapter 1
Introduction
1.1 Gravitational Waves
Gravitational Waves are one of the direct verifications of the Einstein’s General Theory of Relativity
predicted in 1916. 300 years ago, Newton predicted that time is one dimensional and space is 3
dimensional, with space having an infinite stiffness. Almost 100 years from now, Einstein predicted
that the Nature around us is not 3 but 4 dimensional. He also predicted that space doesn’t have
infinite stiffness, but it has a finite value. Because of the finite value of the stiffness, it became
possible to have deformations in space.
The finite stiffness value which General Relativity introduces can be represented by the equation
(1.1) [3],
(1.1)
Here,  is the stress energy tensor,  is the Einstein’s curvature tensor, c is the speed of light,
and G is the Newton’s Gravitational Constant. Very excitingly, we can compare this equation to a
very simple equation, F = Kx, where k is the spring constant and inherits the stiffness and x is the
displacement. The K for space is /8G, which has an approximate value of 10 , which explains
the stiffness of the space. This also elucidates that why Newton’s Laws are such a good
approximation of Einstein’s General Theory of Relativity.
The field of Gravitational Waves was considered to be very impractical for 40 years, until 1960’s
when Joseph Weber started work in this field [3], and made the first instrument which was called
Weber bar. This was the first stepping stone in this field.
One of the most famous examples applicable to the field of gravitational wave research is that, when
Maxwell discovered Electromagnetic Waves in the 20th
Century, he couldn’t have contemplated how
the world would evolve post that discovery. It was beyond perception, and as the maturity level of
the humans increased, it was conceived that Gamma, X-rays, visible rays, radio waves, infrared
waves, microwave, etc, were all different frequency spectrum in the Electromagnetic regime. In
parallel, we are on the verge of same kind of discovery within a time period of 10 years from now,
when we will start detecting Gravitational Waves. The human maturity in the field of technology has
developed enough for the detection to be possible. Once we start the detection of Gravitational
Waves from gravitational wave sources, we could expect to detect objects which are very exotic in
nature and have disguised the scientific community hitherto. Such objects and other predicted
objects would be of significant interest to the scientific community all over the world. Objects that
are obscured in the electromagnetic regime could first time become accessible for direct
observations.

There’s no direct evidence that Gravitational Wave exists, but it has been proved that they exist in
the binary pulsar system PSR 1913+16 (also known as PSR J1915+1606 and PSR 1913+16). It was
proved by Hulse and Taylor, when they measured a loss in energy from that system at a rate almost
equal to that which was predicted by Einstein for the energy loss by Gravitational Waves [4]. For this
work, they obtained Nobel Prize in Physics in 1993 [5].
Some of the very interesting phenomenon’s which may be discovered are:
 The stochastic background of Gravitational Waves present all over the Universe. It could be
a conglomeration of many important, weak and strong signals.
 It would reveal the behaviour and lead to direct observation of the black holes, which are
still unexplained through electromagnetic observations.
 Since Gravitational Waves interact very weakly with the ordinary matter, it could travel to
the vast reaches and expanses of the Universe. It is very much possible that we could
observe Gravitational Waves from the Big-Bang itself.
 Other possibility is that, we could discover a Cosmic Gravitational Wave Background
(CGWB), which would be similar to the existing Cosmic Microwave Background (CMB).
 More information about Black Holes could lead to much clearer understanding of life and
the Universe itself. It may tell us something more about Dark Matter and Dark Energy which
constitutes 97% of the Universe.
 R&D in the Gravitational Wave detection experiments has proven itself to have applications
in other areas of science and technology such as Quantum Entanglement, Optical Bars and
Optical Springs, Classical measurement of Quantum noise, very precise vibration isolation,
high power laser resonant cavities, and several others.
Some of the predicted sources of Gravitational Waves are- burst source, periodic sources, and
stochastic background sources. Burst systems include coalescing compact binary system, which
could be black hole, neutron star, or a combination of both. It could also be the formation of
neutron stars or black holes during supernovae collapse. Periodic sources include binary orbit of 2
black holes, a merge of 2 galaxies, or 2 neutron stars orbiting each other. As the wave reaches the
vicinity of Earth, they are very weak. A pictorial representation of the wave is shown in Figure 1 [4].
(a) (b)
Figure 1.1- (a) A binary star system generating gravitational waves. (b) An artistic view
of space time ripples in case of a binary system [6].

1.2 Existing and Future Gravitational Wave Detectors
1.2.1 The Gravitational Wave Signal for detection
The Gravitational Waves have proven themselves to be one of the stiffest and reluctant
phenomenon’s human kind has ever encountered. One of the other such challenging quests could
be the detection and the experimental verification of String theory and Loop quantum gravity, in
near future.
Gravitational Waves are like ripples in the fabric of space-time. When you drop an object in silent
shores of a lake, the ripples could travel to another end of the shore. The ripples are stronger for
massive object dropped, and less for lighter objects. Similarly, Gravitational Waves are produced due
to asymmetric movement of very massive objects around each other. Massive the object, faster they
would travel around each other [8], and would give more gravitation radiation would be given off.
The Gravitational Waves decrease in strength as they travel away from the source. Even though they
are weak, they travel unobstructed within the fabric of space-time. This is the reason, they are able
to reach the grand detectors on Earth and give us the information which light waves cannot give [7].
They diminish in strength, but never stops or slows down.
Because Gravitational Waves are so weak, scientists all around the world have to use an instrument
which is sensitive to such slight variation in space-time. The distances between objects will appear to
an observer as rhythmically increasing or decreasing as the wave passes through an object. The
effect will decrease as the distance of the observer from the source becomes larger. The strains
reaching earth could be as small as 1 in 10 . The current best upper limit thus found by LIGO (under
CALTECH), is 2.3 x 10 . Essentially Gravitational Waves can exist at any frequency, but very low
frequency could be very hard to detect and no detectable objects have been found credible. Stephen
Hawking and Werner Israel have listed a range of Gravitational Wave frequencies which could be
detected plausibly. It ranges from, 10 to 10 Hz [8].
An easy way of representation of the effect of Gravitational Waves could be perceived as in Figure 2
[3].

Figure 1.2 – (a) The deformation of masses as the Gravitational Waves passes through
them. The masses at the diagonal position don’t moves and have no effect due to
Gravitational Waves. (b) The circular and cross polarization of Gravitational Waves.
Resembling to Electromagnetic Waves, the Gravitational Waves also have particular amplitude (h),
frequency (f), wavelength () and speed (c). The relation between them can be written down as- c
=f. The amplitude of the gravitational waves shown in the figure 2 is around 0.5. It can be
calculated by the formula, h = L/L [3]. Gravitational waves reaching Earth are far more weaker than
this, almost billion times with h 10 . The frequency f, with which the wave oscillates is 1 divided
by the amount of time between two successive maximum stretches or squeezes. The wavelength 
is the distance along the wave between point of maximum stretch or squeeze. The speed of the
Gravitational Waves is the speed with which a point on the wave travels. For Gravitational Waves of
small amplitudes, it is equal to c. The Electromagnetic luminosity depends on the 2nd
time derivative
of the electric dipole moment, but the Gravitational wave luminosity depends on the 3rd
time
derivative of the mass quadrupole moment. The extra derivative arises because gravitational wave
generation is associated with the differential acceleration of masses [3]. The different polarization of
Gravitational Waves can be shown as in Figure 3 [3].
Figure 1.3- the figure elucidates more about the 2 different polarizations of the
Gravitational Waves (a) ‘+’ polarization (b) ‘x’ polarization.
Some of the actual Gravitational Waveforms are shown in Figure 4.
Figure 1.4- some of the actual Gravitational Wave signal waveform modelled, and which
can be used for different filtering techniques existing in Gravitational Wave research (a)
“+” – polarization of the dominant harmonic l = 2, m = 2 wavemode of a gravitational

wave signal coming from a complete Binary Black Holes (BBH) merger simulation [9] (b)
predicted gravitational waveform from the in spiral of 10  Black Hole Binaries [3].
Scientists all over the world have been using mainly 2 kinds of Gravitational Wave detectors in past
and present. They are, Resonant mass detectors [4] and Long baseline Laser Interferometric
Detectors [4], [10].
1.2.2 Resonant Bar Detectors
A resonant mass detector is a suspended test mass, typically a cylindrical bar or a sphere which has
very high Q-factor. Due to the very high Q-factor, once the detectors are ringed by a resonant mode,
they will continue to ring for very large duration of time. J. Weber was the first to build this kind of
detector in 1960’s. When a resonant bar detector is affected by a Gravitational Wave, it will
resonate if the wave contains spectral components which match that of a fundamental mode. The
displacement is measured using displacement transducers and for getting rid of thermal noise, the
resonant mass detector is usually kept in cryogenic conditions of 200mK to 5K. These detectors have
good strain sensitivities at their resonant frequency in the order of h = 4 x 10 . The region of
bandwidth in which they work is generally very small like 10 Hz.
1.2.3 Long Baseline Laser Interferometric Detectors
Laser interferometric detectors are the more advance version of gravitational wave detectors, as
compared to the resonant mass detectors. They make use of a principle of Michaelson
interferometer which has existed for more than 100 years now.
Figure 1.5: the figure illustrates the interferometry configuration used in the gravitational
wave detection. The mirror at the end of each interferometry arms is called ETM or end
test mass.
The interferometric experiments which are currently going on in the contemporary research can be
listed as-
 AIGO (Australian International Gravitational Observatory) under ACIGA, which has its
members as The University of Western Australia, Australian National University, Adelaide
University, Melbourne University).

 LIGO (enhance LIGO, Advanced LIGO), stands for Laser Interferometric Gravitational Wave
Observatory under California Institute of Technology and MIT.
 GEO600 in Germany under Max Planks Institute of Gravitational Physics, (Albert Einstein
Institute).
 TAMA in Japan, to be succeeded by LCGT experiment which is basically based on 3 km
cryogenic arm detection.
 Virgo (Advanced Virgo)
 INDIGO (Indian International Gravitational Observatory), which has started its development
very recently, few months back. INDIGO is going to engage themselves first with a research
prototype of 30m length.
 LISA (Laser Interferometric Space Antenna) under NASA, which would be a space
experiment.
 Einstein telescope (futuristic detector)
 Big Bang Observer (futuristic detector)
 DECIGO (a space futuristic experiment)
The main principle of the interferometric arms can be understood as a simple 2 arms interferometry.
In case of 2 arms, when the laser combines again at the beam splitter after getting reflected from
the 2 end test mirrors, then their phase difference depends on 2L1-2L2. If the length of both the
arms is perfectly same, then we won’t get any fringe shift in the photo detector which is also present
in the gravitational detectors to measure the core signal. But if, the length of the arms varies by even
a very small amount, then we would get a fringe shift in the output which contains the gravitational
wave signal. Each arm would stretch and shrink due to the effect of gravitational waves as shown in
the figure 1.2. The strain sensitivity of such detectors could be increased by any amount in such a
detector in principle. But due to the effect of several noises like thermal noise, shot noise, radiation
pressure noise, seismic noise, gravitational gradient noise, standard quantum limit (which is imposed
by Heisenberg’s uncertainty principle) and also the curvature of the earth, it not possible to realize
the capacity of the experiment. In order to increase the light storage time, the light is allowed to
have multiple reflections between the two end mirrors, ITM and ETM. The diagram to illustrate this
phenomenon has been illustrated in figure 1.6 below.
Figure 1.6: Light storage time in each arm is synthesized to a much larger value by having
a resonant cavity between 2 mirrors in each arm in the Michelson Interferometer
configuration (courtesy- [4] Jean Charles Dumas)

In the detectors, AIGO (figure 1.7) has an 80 m research facility for doing experiments related to
gravitational wave detectors. Current 80m facility is to be upgraded to 4km detector by 2014. The
location of the site would be Gingin, 90 km from Perth, Australia. The Japanese TAMA project was
the first interferometric detector, 300m long underground, which was built underground in Tokyo.
The Japanese most recent project is to build a 3 km LCGT cryogenic detector, which will be preceded
by a pathfinder project CLIO [4]. The CLIO is a 40m cryogenic project for LCGT. The German GEO 600
detector is near Hannover. The 3 km VIRGO detector is near Pisa, in Italy. It is currently the most
sensitive detector at low frequencies [4] ~ 10 Hz. The LIGO project has presently 2 detectors, one at
Livingston, Louisiana and Hanford, Washington [4]. Each site has 4km detectors and they are
separated by a distance of 4000 km. The Hanford site also possesses a 2km interferometer in the
same vacuum conditions. The detectors work very well in the region of 100 Hz to few KHz. In future,
the plan is to make possible a global detector with increasing sensitivity and very high sophistication.
The AIGO project would enable the triangulation of the detector all over the world and improving
the resolution efficiently. The LISA project is a space project headed by NASA. It consists of 3
instruments in triangular configuration, around the earth. In India, INDIGO is developing a 30m
research facility by 2013, which is a precursor for an advanced detector in future. Figure 1.7 shows
all the present detectors around the world. The futuristic detectors like Einstein telescope, Big Bang
observer and the DECIGO are 3rd
generation detectors which are planned to be operational in next
few decades.
Figure 1.7: An aerial view of AIGO and LIGO at Hanford, Washington. (Courtesy AIGO,
ACIGA LIGO, CALTECH)
Figure 1.8: An aerial view of LIGO at Livingston and artistic view of LISA. (Courtsey
LIGO, CALTECH and NASA/CALTECH/JPL.)

A major development in this direction has been the AIGO international conference which took place
in AIGO from 22nd
to 24th
February, 2010.
Figure 1.7: the figure illustrates all the detectors all over the world [4].
1.3 Different noises
The noise which occurs in gravitational wave detectors can be summarized as seismic noise,
gravitational gradient noise, thermal noise, laser noise and quantum noise [4]. Figure 1.8 elucidates
the effect of noise at different frequency spectrum, due to different noises.
Figure 1.8: the figure elucidates the effect of different noises and the limitations caused by
different noises on gravitational wave sensitivity.
1.4 Vibration Isolation requirements

Because of the seismic noises existing when a gravitational wave detector is built, the detector has
to be isolated from them. This necessitates the use of vibration isolation. The mirror which are
suspended from the vibration isolators, have to be essentially stationary with respect to each other.
The tilt can also occur due to the curvature of the earth if the distance between the mirrors is very
large. If the mirrors are not kept stationary with respect to each other, the locking of the optical
cavity could not be achieved. This becomes essentially difficult, because the order of magnitude one
has to care about is > 10 . In vibration isolation, we can talk about 2 prototypes. One is active
isolation and the other is passive isolation.
Active isolation means a feedback mechanism using a servo loop in the test mass stage. The motion
of the test mass with respect to the suspension platform is sensed, and a feedback is given by an
actuator to the platform.
Passive isolation is another method used for vibration isolation. It incorporates the use of
pendulums and mass spring system. The overall system acts as a low pass filter which filters out the
frequencies higher than resonant frequency f0 of the system as (f0/f) .
1.5 Control Systems
For implementing the control systems in the vibration isolators, a Sheldon instrument DSP board [4]
is used. It is a SI-C33DSP board on a OCI bus, based on a 150 MHz Texas Instruments TMS320VC33
using a mezzanine board SI-MOD6800 to provide 32 input channels (16 bit ADC), 16 output channels
(16 bit DAC), and digital inputs and outputs. The usage of the channels has been explained in the
Table 1. The input channels are basically used for shadow sensors, vertical and horizontal readouts,
and 2 for injecting any analog signal. The output mainly consists of channels used for actuators and
current power supplies.
In between the DSP and the control components, an intermediate analog system is used for
amplification and filtering the input and output channels [4]. The signal from the photodiode is
distributed as 2 inputs on the DSP board. The control signal is then distributed from the DSP board to
the corresponding channels on the control circuit. The control circuit contains antialiasing filters and
a high speed current amplifiers before distribution to the actuator coils.
The algorithm used in the control scheme and the graphical user interface is realized using LABVIEW.
The built in libraries provided (for the Labview) by the Sheldon Instruments has to be integrated
using proper guidelines before being able to use it with normal Labview software.
Table 1: The table illustrates the input output channel distribution in the DSP [4].

The control scheme in the vibration isolators at AIGO has to deal with very low frequencies effects
[4]. The Superspring feedback mechanism improves the response of the system at low frequencies.
The control also has to maintain the test mass alignment which it does with the help of control mass
sensors. Most of the system is passively controlled with a little bit of active damping.
The various degrees of freedom of each stage has been illustrated in the Table 2 below.
Table 2: The table illustrates the various degrees of freedom associated with each stage
[4].

Chapter 2
Vibration isolation design in AIGO
The vibration isolation design in AIGO uses 9 stage cascaded system [12]. Figure 3.1 illustrates more
about the design of the 9 stages in the isolator. Out of the total 9 stages, there are 3 stages for pre-
isolation. It includes the inverse pendulum stage, Robert’s linkage stage and the la-Coste vertical
stage. From this pre-isolation, the horizontal triple self damped pendulum stage and the vertical
Euler spring stage is held. Some of the topics in this chapter are not associated to author’s primary
work, so it has been explained very briefly. This could rather help to grasp a holistic approach
towards the isolators and the author’s work in supersprings.
2.1 Inverse pendulum preisolator
More about this stage has been elucidated well in the chapter 3.
2.2 Lacoste linkage
The La-coste linkage has its resonant frequency at around 100mHz [11]. It has the function of
providing vertical preisolation. It also has a large dynamic range. The combination of coil and magnet
provides the actuation for the positioning and damping for the inverse pendulum and Lacoste stage.
The heating of the coil springs in the Lacoste stage helps in compensating for vertical slow
temperature drift. The figure for the Lacoste linkage has been given in chapter 3.
2.3 Robert’s Linkage
The Robert’s linkage is suspended from the 3D structure as shown in figure 3.3. It is nested along
with the 2 pre-isolators [11]. Basically, it is a cube frame suspended by 4 wires hung from the
Lacoste stage. The geometry of the Robert’s linkage helps in keeping the load at almost flat
horizontal level. This makes the gravitational potential energy almost independent of the
displacement and minimizing the restoring force which results in a low resonance frequency.
2.4 Euler springs
The Euler springs are used for low frequency vertical isolation. It attenuates the vertical components
of the seismic noise [12]. It has been shown in figure 2.1

Figure 2.1: The figure illustrates the structure of the Euler spring.
2.5 Self Damped pendulums
It is one of the isolation stages in the 9 stage vibration isolators. Essentially, 3 self damped pendulum
stages are used, each of mass as 40kg’s. It consists of viscously coupling different degrees of
freedom of the pendulum mass as illustrated in figure 2.2.
Figure 2.2: the figure illustrates the structure of the self damped pendulum [12].
The intermediate mass is pivoted at its centre of mass. The comb like magnets is mounted to the
frame. These magnets are coupled with copper plates which create a viscous damping through eddy
current coupling, which eventually reduces the Q factor of the pendulum at specific normal modes.
2.6 Control mass and test mass stage
Figure 2.3 illustrates the structure of control mass and accompanying test mass. The control mass is
an 30 kg mass stage. It is used to hang the test mass from itself using niobium ribbons with
dimensions as 25,000 mm thick, 3 mm wide, and 300 mm long.

Figure 2.3: the figure illustrates the structure of control mass along with the test mass
stage [12].
The control mass is suspended from the vibration isolation system using a single wire. It also
contains actuators and sensors which are used to access the 5 degrees of freedom. The 5 degree of
freedom includes translation in all 3 dimensions, yaw and pitch.

Chapter 3
Superspring Digital Feedback Control (SDFC)
3.1 Superspring feedback for preisolation
SDFC is a novel feedback method [1, 4, 11, 12] which has been implemented in the AIGO vibration
isolators, and is expected to decrease the residual motion at 1Hz from ~ 5nm now, to ~ 0.1 nm at 1
Hz which will make several future experiments in AIGO possible. The experiments like strong optical
bars, and using it in quantum measurement schemes that supersedes the quantum limit posed by
Heisenberg uncertainty principle, is only possible after the implementation and full realization of the
goals by SDFC. Originally [2], Superspring concept can be understood as the electronic termination of
an example 30 cm long spring such that the mass suspended from it behaves as if the spring were
1000m or even longer in length. This concept helps in pre-isolation in case .
At present, the 2 vibration isolators which are operational in AIGO have a residual motion of 5nm at
1Hz. This residual motionNB1
is good for locking a cavity of finesseNB2
 750. The cavity is
approximately 72m in length and the laser used is Nd:YAG laser. During the next stage of operation,
an optical cavity of finesse of around 15,000 is required to demonstrate experiments like optical bars
and quantum measurement schemes. For working with this high finesse, an optical cavity and cavity
is hard to attain. This requires a 10 fold improvement in the low frequency residual motion, at
around 1 Hz. Keeping the cavity locked for a long time, and to keep it stable is also a challenge.
Figure (3) [11] explains more about the residual motion characteristics currently and with SDFC
realization.

Figure 3: (a) The measured integral residual motion of the cavity [xrms = ∫ ( )

, where x is the
displacement spectrum [m/√ ]. It is at the nanometre level above 1 Hz. Note that the
measurement is limited by laser noise above 2Hz [12], due to free running laser. (b) A model with
the same feedback scheme as used for the measurement. (c) The modelled performance with an
optimized preisolation feedback scheme, using the Superspring concept. (d) A modelled
performance if no feedback was implemented. (e) Assumed input spectrum.
3.2 Experimental realization
The SDFC can be experimentally realized in the vibration isolators by monitoring the signal from the
position sensor of the Robert’s linkage which is used to control the inverse pendulum stage. The low
level of active feedback forces allows us to suppress the incoming noise. For knowing more about
the design and geometry of the Inverse Pendulum and the Roberts linkage, please see Figure 3.1
[12].
Figure 3.1 – Full Vibration Isolator system and the schematic that shows the different
stages of pre-isolation and the multipendulum stage with a test mass at the bottom of the
chain.
The signals from the inverse pendulum and the Roberts linkage are processed using the transducers
in those stages. The position sensors which have been mounted between the inverse pendulum and
Robert’s linkage pre-isolation stage are used to measure their respective displacements in the 3
horizontal degrees of freedom (X, Y, and Yaw). Figure 3.2 and 3.3 [12] shows the physical design of
the inverse pendulum stage, which is used for horizontal pre-isolation and the Robert’s linkage
which is used for horizontal and vertical pre-isolation respectively. The optical position sensors are

practically used for this purpose. Figure 3.5 explains more about the sensors which are being
currently used in AIGO.
Figure 3.2: First stage horizontal and vertical pre-isolation. The pre-isolator combines 2
ultralow frequency stages- (a) the horizontal inverse pendulum (b) the vertical LaCoste
linkage.
Figure 3.3: (a) This figure shows the principle of Robert’s linkage in which the load is
suspended from a point P, which stays in the same plane for variation in the position of
the point C and D. (b) this figure shows the actual 3D cube shaped design used in AIGO
suspension system.
These sensors from the inverse pendulum stage and the Robert’s linkage stage are used to provide a
signal (x2-x1) and (x1-x0) as explained in the figure X4. Also the configuration of actual sensors and
actuators are shown in figure X5 and X6 respectively. The shadow sensors are also used to monitor
the position of several stages with the help of local control system.
These sensors are simple device which consists of a LED which shines an infrared beam onto 2
photodiodes 40mm away. The width of each diode is 10mm. A long obstruction of width of around
10mm is attached to the stage to be monitored. The obstruction if placed perpendicularly to the 2
sensors, such that the shadows of the obstruction forms fall equally on both photodiodes. The

resulting current generated from each of the photodiode is almost proportional to the area
illuminated. As the obstruction moves across the sensors, the differential current from the
photodiode signals gives the actual signals. The response is nearly linear. After the extraction of the
differential photocurrent from the sensors, these signals are amplified and converted to a signal
which is nearly in the range of 10 V. This is for the ADC module of the DSP board from which digital
control systems takes the output readout. The dynamic range of these optical shadow sensors are in
the range of 10 /√ .
The actuators which are used in AIGO vibration isolators are of magnetic coil type and are
represented in figure X6. Each section of the actuator contains coils. There are 2 versions of
actuators which are currently operation in the isolator. One is a larger than the other. The larger
ones are used for position control; drift correction, and damping the ultra low frequency normal
modes [12]. It consists of a design in which coils have ~ 1600 turns of 0.25 wire to form a coil
diameter of ~ 65-80 mm, and a resistance of ~ 115 . The magnet which is used is a  20 x 10mm
dimension, neodymium boride which results in a force of ~ 160mN with a current of 100mA driving
the actuator (50mA each coil, connected in parallel). The smaller version of the actuator which is
used at the control mass has a coil diameter of 25 – 30mm, made of ~600 turns of 0.25 mm wire.
These coils have a resistance of ~ 37  and are paired with a  10 x 10 mm magnet. The magnetic
field within the actuators is almost uniform within 1% in the central 10mm of its range. This allows a
large dynamic range in the control system.
(a) (b)
Figure 3.4: A simple 2 pendulum stage explaining the pre-isolation feedback using shadow
sensors. A shadow optical transducer measures the displacement (x2-x1) which is then fed
back to the 1st
stage. Also, it is added with an active damping signal proportional to (x1-
x0) which is also measure using shadow optical transducer at inverse pendulum and the
ground. The force F1 has been modelled as, F1 = C1(x1 – x0) + C2(x2-x0). (a) This
figure shows the simple schematic on which the superstring configuration can be applied.
The load is shown as such without any actual displacement. (b) This figure shows that the
load has a mass = m3 and has a displacement = x3.
m3 x3

(X5) (X6)
Figure 3.5: The shadow optical transducer is a simple device, where a LED shines a beam
onto 2 photodiodes, and an intermediate obstruction is attached to the part to be
measured.
Figure 3.6: The magnet coil actuator used in the vibration isolator design used in AIGO.
A magnet mounted on an isolation stage is placed in the center of 2 coils that are
mounted on the support frame.
The stage which is represented in figure 3.4 with a displacement x0 is in reality the flexure or the top
of the pre-isolator stand as represented in the figure 3.1. The stage with mass m1 represented in the
figure 3.4 is in reality the inverse pendulum stage and m1 = mass of the inverse pendulum. The
displacement x1 is the displacement of this stage m1. The stage with a mass of m2 represented in
the figure 3.4 is in reality the Robert’s linkage stage and has a mass m2. The displacement of this
stage is shown as x2 in the figure 3.4. The 3rd
stage which is shown in the figure is essentially not a
single stage at all. It has been represented as a single stage but is essentially a sum of several stages.
Basically, it is a sum of several stages as follows: Self Damped Pendulum1, Self Damped Pendulum2,
Self Damped Pendulum 3, Control Mass, and test mass stage. The table showing several stages in the
vibration isolator is shown in Table X1.
Stage Frequency (f)
mHz
Q value (Q) Mass (m)
Kg
Length (l)
m
Inverse Pendulum 80 30 370 NA
Robert’s Linkage 260 100 40 NA
Self Damped Pendulum1 g/L1 20 60 0.6
Self Damped Pendulum2 g/L2 20 45 0.6
Self Damped Pendulum 3 g/L3 20 45 0.6
Control Mass g/L4 100 36 0.3
Test mass g/L5 10 4 0.2
Table 3.1: this table shows the values of frequencies, Quality factor, Mass and length as applicable in
each stage.
A simple equivalence could be used between the spring constant and the length is Kx = mg. Also, K =
m . The value of the spring constant K can be better represented as K = K0 (1+ 1/Q). The 1/Q term

is also called as the loss factor or . Therefore, Q = 1/. For more notes on damping and its
modelling, please go to NB3 (after the end of chapter 3 and reference [4]).
The sensors which are used for generating the differential signals (x2-x1) and (x1-x0) are mounted on
the Robert’s linkage and the Inverse pendulum respectively. The inverse pendulum has 8 photodiode
sensors- A1, A2, B1, B2, C1, C2, D1 and D2. We generate a differential signal from each pair of
photodiodes to obtain signals A, B, C, and D. More about the configuration of the sensors on Inverse
pendulum has been explained in the figure X7. After extracting signals such as A, B, C and D from the
4 sensors at each side of the inverse pendulum, we try to obtain X, Y and Rotational (Rot) signal from
them. X is simply (B-D)/2, Y is (A-C)/2 and Rot is obtained by multiplying all the signals and then
averaging them out. This information would be necessary for understanding, once we get to the
actual Lab View control scheme of the Superspring. In case of Robert’s Linkage, we have 4
photodiodes, or 2 pairs of diodes. They can be said to be – X1, X2, Y1 and Y2. The x and Y degree of
freedom information can be extracted from these 4 signals as, X = X1-X2 and Y = Y1-Y2. To get a
better understanding of the Robert’s Linkage sensor positions, please refer to figure 3.8.
Figure 37: This figure elucidates more about the sensors which we use on the inverse
pendulum stage to carry out the Superspring operation. They help in extracting the signal
(x1-x0). X, Y, and Rotational Degree of freedom variations are calculated as discussed
above in the paragraph [11]. Also note the actuators which are operate along with the
sensors, which are used for final actuation for Superspring operation. F1 = C1(x1- x0) +
C2(x2-x1) is the external force which is made to act on this stage with the help of these
actuators..

Figure 3.8: The control of Robert’s Linkage through the Shadow sensors photodiodes and
also the heating of the four suspension wires [11]. These sensors are important because
they are used in the Superspring operation to extract the signal- (x2-x1). We need to take
care in the LabView control scheme that these signals have proper signs and are operated
with appropriate controllers and filters.
3.3 Model for analysis and its realization
Other control schemes for La-Coste and Control Mass are not significant for Superspring operation
and have been explained in section 2.6. Also, for knowing more about the control implementation
and degree of freedom in case of Inverse Pendulum and Robert’s Linkage, please refer section 2.6.
The Super-spring operation has been explained in figure 3.4. But the masses m1, m2 and m3 are not
to scale and could lead to confusion. For getting an accurate idea of the Superspring model the
author has tried to use a more conspicuous model, please refer figure 3.9.
Figure 3.9: The figure representing actual 3 stage masses with corresponding masses of
understandable sizes. The biggest mass m1 is the Inverse pendulum stage 1 which is
hanging from the flexure point which is denoted as the ground stage. The displacement of
the 1st
stage can be denoted as x1. The smallest stage 2 with the smallest mass m2 is the
Robert’s linkage stage. The displacement of 2nd
stage is denoted by x2. The intermediate
mass stage 3 is the sum of all the stages existing in real condition in the isolator. The
mass m3 can be said to the approximate sum of self damped pendulum 1 (m=60kg), self
m1
m2
m3
x1
x2
x3
Ground level
Stage 1
Stage 2
Stage 3

damped pendulum 2 (m= 45 kg), self damped pendulum 3 (m = 45 kg), control mass (m
= 36kg) and test mass (m = 4kg). For more details about 3.1, please refer to table 3.1.
Table 3.1 can also be referred for more details about the resonant frequencies, Q-factors,
and length information.
The author has tried to represent the pendulum model shown in figure X9 as an equivalent spring
model. In case of spring models, we need to convert the lengths to appropriate spring constants. The
spring constant model is much easier to handle as compared to the figure X9 pendulum model. The
author has tried to explain the spring model in figure 3.10.

Figure 3.10: the figure represents the simplified spring model which can be used for
Superspring design. The mass os the 1st,
2nd
and 3rd
stage of the model is m1, m2 and m3
respectively. The biggest mass m1 is the Inverse pendulum stage 1 which is hanging from
the flexure point which is denoted as the ground stage. The displacement of the 1st
stage
can be denoted as x1. The smallest stage 2 with the smallest mass m2 is the Robert’s
linkage stage. The displacement of 2nd
stage is denoted by x2. The intermediate mass stage
3 is the sum of all the stages existing in real condition in the isolator. The mass m3 can be
said to the approximate sum of self damped pendulum 1 (m=60kg), self damped
pendulum 2 (m= 45 kg), self damped pendulum 3 (m = 45 kg), control mass (m = 36kg)
and test mass (m = 4kg). For more details about X1, please refer to table X1. Table X1
can also be referred for more details about the resonant frequencies, Q-factors, and length
information. B1, b2 and b3 are three damping coefficients
In reference to the figure X10, we can say that the spring constants of the springs between 3 stages
could be said to be K1, K2 and K3. The spring constants could be written in the form of K = K0(1+i).
Here the K0 would be the principal spring constant of the stages, given as K10 = m11
2
, K20 = m22
2
and K30 = m33
2
. Here, 3 = / . L is the length of the length of the 3rd
stage spring.
The process of deciding the length of the 3rd
stage was a bit tricky, because in reality it is not a single
stage but has several stages like Self Damped Pendulum 1, Euler Spring 1, Self Damped Pendulum 2,
Euler Spring 2, Self Damped pendulum 3, Euler Spring3, Control Mass and Test Mass. After taking
into consideration, like centre of gravity of the entire equivalent set up, the author came to a
b1
M1
M2
M3
X1
X2
X3
F1, b1 F2 F3
X0
K1 K2 K3
, b2 , b3

conclusion that a length L = 1.2m could be usedNB4
. (NB4: However, there’s a scope of further improving this
parameters for future studies for much precise results.)
In order to elucidate a system shown in figure X10, one can try to grasp and understand it by not
including 3 masses, but by working with just a single stage. The author would explain it by a very
simple procedure as shown in next few paragraphs. Referring to figure 3.11, the mass of the m1 (the
first stage element) has been kept as 370Kg. Lets the resonant frequency  = 80mHz (as one can see
from Table 3.1 that it’s the resonant frequency of the 1st
stage of Inverse Pendulum). So,
Figure 3.11: A 1 stage spring mass system. The parameters have been kept as same as the
Inverse Pendulum stage in Superspring configuration.
The equation of motion in this system can be written as-
1 1̈ = 1 − 1( 1 − 0) − 1( 1̇ − 0)̇ - 3.1
This gives, 1 1( ) = 1( ) − 1 1( ) − 0( ) − 1( 1 − 0)
Taking Laplace transform, and solving, this gives us 2 transfer functions, x1/x0 and x1/F1. The x1/x0
transfer function is calculated keeping F1 = 0 and x1/F1 transfer function is calculated using x0 = 0.
He b1 is also known as damping coefficient of the particular stage.
Calculating the transfer functions, they are given as below-
=
( / )
( )
3.2
And, =
( / )
( )
3.3
Plotting the above 2 functions, one can get the Bode Plots. Bode plots are the plots which shows the
Amplitude and Phase response of a particular system, and can also be used to judge the stability of a
particular system. The criteria, one has to use while judging the stability while using Bode Plot is the
Phase Margin and Gain Margin conditions. One can also use other criteria like the Root Locus
method and Nyquist Method. Essentially, one has to keep a track of the performance of the system
M1
K1
X1
F1, b1
X0

too while checking the stability. A stable system with a very good performance is always better that
a stable system with a very bad performance. One has to allocate the performance according to the
desired working goals of the process. In the equation 3.2 and 3.3 written above, (K1/m1) = 1 and
(b1/m1) = 21 or  = b1/ (2√ 1 1 ). The denominator equation becomes ( + 212
+12
).
Here,  = .
Some of the responses for the above system could be investigated in the figure 3.12. They are
plotted for different values of b1, in case of x1/F1 and x1/x0 respectively.

Figure 3.12: the above 2 figures show the response of the x1/F1 and x1/x0 in case of
different values of damping ratio .
Some of the other important parameters to consider while working with transfer function are the
use of different controllers, and filters. Figure 3.13 illustrates the simple technique of selecting
appropriate filters for your experiment. Author feels that this concept is necessary when you work
with the actual response of the Superspring system.
Figure 3.12: (a) Illustrates the use of s and 1/s in a transfer function. (b) Illustrates the
difference between a PID control and a PID control with Ki = 0. (c) Illustrates the
difference between a PID control and a PID control with Kd = 0. (d) Illustrates the effect
of increasing the pole parameter of a low pass filter. Note that green line if x1/F1, blue
line is with a LPF 1/(s+1) and red line is with a LPF 1/(s+10).
Figure 3.13 (a) illustrates the effect of using a LPF, HPF on the response of a particular transfer
function, and figure X13 (b) illustrates the effect of using HOF with different pole parameters. Figure
3.13 (c) illustrates the use of LPF, HPF and PID to effectively bring down the response of the entire
system.
(a) (c)
(b) (d)

Figure 3.13: Effect of a LPF, HPF and combination of both has been illustrated in the
figure (a). In figure (b), we have different values of pole parameters for a HPF used with
x1/F1. Figure (c) illustrates the suppression of the entire response of the system using
LPF, HPF and PIDNB5
. (NB5: Note that one can try using any filtering scheme for getting the best
results.)
3.3.1 Block Diagram for Superspring Design, in the Vibration Isolators:
In order to draw a clear understanding of the control systems used for the Superspring process,
author tries to explain it using a general control loop for control systems. Figure 3.14 explains the
loop.
(a) (b)
(c)

Figure 3.14: A general control loop which can be implemented in a process control system
[13]. R is the disturbance, e is the error signal, c is controlled variable, p is the process
variable, and u is the actuating signal.
Referring back to figure 3.9 and 3.10, which is the actual model of the Superspring design, the
author has given a Block Diagram which can be used for realizing the goal of SDCF systems. The
signal from the ground has been denoted by x0. Signal from the Inverse pendulum (X) has been
denoted by x1. The signal from the Robert’s Linkage has been denoted by x2.
Controller Control
element
ProcessSensor
r e p
u
cb
(x0/x1), T1 (x2/x1), T2
C1 C2
(x1/F1)
+-
+-+-
x0 x1 x1’ x2
F1

Figure 3.15: Actual control system block diagram designed by the author to show the
actual working of the Superspring design.
From figure 3.9 and 3.10, we can write some of the general equations of the model. These are the
most general equations or a way of deriving them, which are in operation when we deal with the
transfer function of a process like Vibration Isolators used in AIGO or in any other Vibration Isolator
in any other gravitational wave detectors. From figure 3.9 and 3.10, we have the following
equations;
1 1̈ = 1 − 1( 1 − 0) − 1 1̇ − 0̇ + 2( 2 − 1) + 2( 2̇ − 1̇ ) - 1st
stage - 3.4
2 2̈ = 2 − 2( 2 − 1) − 2 2̇ − 1̇ + 3( 3 − 2) + 3( 3̇ − 2̇ )- 2nd
stage - 3.5
3 3̈ = 3 − 3( 3 − 2) − 3( 3̇ − 2̇ ) 3rd
stage- 3.6
In all these equations, author has assumed that F2 = 0 and F3 = 0. This is because, the force we get in
the control diagram drawn from figure 3.15, 3.9 and 3.10, we don’t have an effective external force
acting on the 2nd
and 3rd
stage. The external force is only present in the first stage of the model of
the Superspring design. The value of force F1 can be given as in equation 3.7. The b1, b2, b3 have
been assumed to be 0 in the Superspring modelling. This is because; we are not taking into
consideration viscous damping into the model.
1 = 1( 1 − 0) + 2( 2 − 1); 3.7
From the above 3 general equations, the author has derived all the transfer functions which are
necessary for getting appropriate results and approximate performances.
3
2
=
3/ 3
+ 3/ 3
3.8
2
1
=
2( 3 + 3)
[( 2 + 2 + 3)( 3 + 3) − 3 ]
3.9
1
1
=
1
1 + 1 − 2
2
1 − 1
3.10
( ) =
1 1 −
0
1 + 2
2
1 − 1
1 + 1 − 2
2
1 − 1
3.11
2
0
=
[ 2(− 1 + 1)( 3 + 3)]
[{( 2 + 2 + 3)( 3 + 3) − 3 }{ 1 − 1 + 2 + 1 + 2} − 2( 2 + 2)( 3 + 3)]
3.12

1
0
=
(− 1 + 1)[( 2 + 2 + 3)( 3 + 3) − 3 ]
[{( 2 + 2 + 3)( 3 + 3) − 3 }{ 1 − 1 + 2 + 1 + 2} − 2( 2 + 2)( 3 + 3)]
3.13
Also, we can derive x3/x0 by multiplying the equation for (x3/x2) and (x2/x0). It has been illustrated
in the equation 3.14 below.
3
0
=
3
2
2
0
3.14
When the author says that F1 = C1(x1-x0) + C2 (x2-x1), it means that C1 controller has been used for
the Local Control System and C2 controller has been used for the Superspring design. When C1 is on,
and C2 is off, it means that only Local Control Systems is enacting on the vibration isolator. When C1
is on and C2 is also on, it means C1 and C2 controller are working and enacting together to make the
Superspring design and implementation successful.
We can also remember that, we also have certain values for C1 and C2 as they are PID controllers.
For PID controllers, we have their transfer functions in the form of Kp, Ki and Kd. Kp is the
proportional gain, Kd is the derivative gain and Ki is the integral gain. To represent the actual
functioning of the external Force F1, we can repeat equation 3.7 as below;
1 = 1( 1 − 0) + 2( 2 − 1);
And, we can write the values of controller as given in equation 3.15 and 3.16 below.
1 = 1 +
1
+ 1 3.15
2 = 2 +
2
+ 2 3.16
For actual modelling the Superspring design, the author has plotted different responses occurring in
the Superspring framework at different stages and then comparing it with the responses with only
Local Control Systems or essentially only C1 on and C2 as off. Since the response x3/x2 and x2/x1
doesn’t have any control element in their modelled response transfer function, they can be drawn
absolutely, and do not need referential representation.

Figure 3.16: Response for (a) x3/x2 and (b) x2/x1 for Superspring design.
(a)
(b)

Figure 3.17: Response without and with control in case of x2/x0.
(a)
(b)

Figure 3.18: (a) illustrates the figure without and (b) illustrates with control in x1/x0.
(a)
(b)

Figure 3.19: (a) and (b) illustrates x1/F1 transfer function with no control and control in
Superspring design.
(a)
(b)

Figure 3.20: (a) and (b) illustrates control and no control in the Open Loop Gain in the
Superspring Design.
(a)
(b)

Figure 3.21: (a) and (b) illustrates the plot for control and without control in case of
x3/x0 response in Superspring design.
(a)
(b)

3.3.2 Few Superspring results from actual implementation
Once the Superspring model was modelled, the author tried to implement the design on the actual
Vibration Isolators. In order to work with the actual experimental set up, the vibration isolator lid
had to be closed. The Superspring control was tried to be implemented in the Vibration isolator
without the tank lid was open. But, when the lid is open, the experiment has many sorts of error like
damping error from airs, and other effects.
Figure 3.22: illustrates the actual procedure for closing the tank and creating a vacuum of
the order of 1mbar to 1 bar pressure, which is necessary for taking the implemented
Superspring data. In the figure it can be seen that the experimentalists have to take very
careful precautions before entering the Ultra Clean Room in which the tank and isolator is
located. It also shows the crane which is used for lifting up the lid and closing it very
precisely, without making any major external contact with the isolator.
After closing the lid, it takes 1 hr. for the isolator to reach a pressure of 1mbar. Taking care of proper
cooling conditions, we can switch the TMP pump on to go to very low vacuum conditions like bar.
However, our experiment testing for the Superspring implementation has been carried out at

around 1 mbar. Figure 3.23 tries to illustrate the improvement in performance when the author tried
to monitor the result from the Labview Control system for the Vibration Isolator.
Figure 3.23: The figure shows the result for the Y degree of freedom for the Control mass.
The readings were taken for the Superspring design on, and for different values of Kd2
values. The graph which shows the lower response has been taken for

3.4 Needs and benefits
In improving the low frequency response of the vibration isolator and enabling to work on a
resonant cavity with a higher finesse (~15000) than the present one (~750).

3.5 LabView Control
In this section, author tries to show some of the diagram of the actual control system which is being
currently used in the Control implementation of the vibration isolator. They are being shown in
figure 3.23.
Figure 3.24- Inverse pendulum control in Labview Control scheme.
In the figure 3.24 and 3.25, the output screen, the output panel and the monitor channel is being
shown.

(a).
(b)
Figure 3.25: All input overview and the global control overview. (Courtsey - AIGO)

4. Conclusion and Future work
In near future, the Superspring design will be tested for improving its functionality on the vibration
isolator, in order to improve the performance at low frequencies. After the fully functional
Superspring design, the isolator would be able to see the enhanced residual motion performance at
1 Hz, which is 0.1nm as compared to ~ 3.3nm at 1 Hz presently. Since it has been shown by the
author that the theoretical model and real data of the Superspring is actually improving the
performance of the isolator, some of the model parameters can be tested further to see the results.

5. Bibliography (references)
[1]. Research proposal to Australian Research Council by Dr. Jean Charles Dumas, Project id-
DP110102576, Admin Organization – The University of Western Australia.
[2]. R.J.Rinker – Super Spring – A new Type of Low-Frequency Vibration Isolator, Phd Thesis,
University of Colorado at Boulder, 1983.
[3]. L Ju, D G Blair, and C Zhao. Detection of gravitational waves. Reports
on Progress in Physics, 63(9):1317{1427, 2000.
[4.] Jean-Charles Dumas – High Performance Vibration Isolation Design for a Suspended 72m Fabry-
Perot Cavity, Phd Thesis, School of Physics, the University of Western Australia at Perth, 2009.
[5] http://en.wikipedia.org/wiki/Hulse-Taylor_binary
[6] http://web.mit.edu/sahughes/www/Graphics/nsns.jpg
[7] http://imagine.gsfc.nasa.gov/docs/features/topics/gwaves/gwaves.html
[8] http://en.wikipedia.org/wiki/Gravitational_wave
[9] http://numrel.aei.mpg.de/People/personal_webpages/reisswig.html
[10] http://www.ligo.caltech.edu/
[11] Compact Vibration Isolation and Suspension for Australian International Gravitational
Observatory: Local Control System, J.C.Dumas, P.Barriga, C.Zhao, L.Ju and D.G.Blair, Rev. Scientific
Instruments, 2009
[12] ] Compact Vibration Isolation and Suspension for Australian International Gravitational
Observatory: Performance in a 72m Fabry Perot Cavity, P.Barriga, J.C.Dumas, A.Woolley, C.Zhao, and
D.G.Blair, Rev. Scientific Instruments, 2009
[13] Process Control Instrumentation Technology, 8th
edition, Prentice Hall India, by Curtis D.
Johnson, University of Houston. ISBN – 978-81-203-3029-0
[14] An Introduction to the Pound-Drever-Hall laser frequency stabilization, by Eric D. Black, LIGO-
P990042- A- D 04/04/00, California Institute of Technology and Massachusetts Institute of
Technology

6. Appendices
[1] Compact Vibration Isolation and Suspension for Australian International Gravitational
Observatory: Local Control System, J.C.Dumas, P.Barriga, C.Zhao, L.Ju and D.G.Blair, Rev. Scientific
Instruments, 2009
[2] ] Compact Vibration Isolation and Suspension for Australian International Gravitational
Observatory: Performance in a 72m Fabry Perot Cavity, P.Barriga, J.C.Dumas, A.Woolley, C.Zhao, and
D.G.Blair, Rev. Scientific Instruments, 2009
[3] The Matlab code used for converting the data from Spectrum Analyzer (.78D) to the format
which can be used in the MATLAB.
%----------------------%
% Convert the data %
%----------------------%
convert = 0; % Set as 1 to convert all the .78D files
if convert == 1
nsamples = 49; % Converts everything from 1 to 49
for i = 1:nsamples,
if i < 10
n = ['00' num2str(i)];
elseif i < 100
n = ['0' num2str(i)];
else
n = num2str(i);
end
%system(['SRT780.EXE ' SRS' n '.78D ' n '.txt /D /Oasc']);
system(['SRT785.EXE /Oasc /D /Cx,mag /Upk,lin,deg SRS' n '.78D ' n '.txt']);
end
clear('nsamples','i','n','convert');
end
%--------------------%
% Load the data %

%--------------------%
i = 8 ;
if i < 10
n = ['00' num2str(i)];
elseif i < 100
n = ['0' num2str(i)];
else
n = num2str(i);
end
filename = [n '.txt'];
temp = load(filename);
freq = temp(:,1);
Y = temp(:,2);
%-----------------------%
% Present the data %
%-----------------------%
% figure(1);
% clf;
loglog(freq,Y,'b-'); %hold on;
title('Spectrum');
xlabel('Hz');
ylabel('Y');
grid on;
hold on;
%

[4] MATLAB code used for modelling;
s = tf('s');
Kg = 150; % global gain
g = 9.8; % Value of acceleration due to gravity
L = 1.2; % Length of the 3rd stage
f1 = 0.08; % Resonant Frequency of 1st stage in Hz
f2 = 0.26; % Resonant Frequency of 2nd stage in Hz
w1 = 2*pi*f1; % Resonant Frequency of 1st stage
w2 = 2*pi*f2; % Resonant Frequency of 2nd stage
w3 = g/L; % Resonant Frequency of 3rd stage
m1 = 370; % Mass of 1st stage
m2 = 40; % Mass of 2nd stage
m3 = 190; % Mass of 3rd stage
Q1 = 30; % Q factor of 1st stage
Q2 = 100; % Q factor of 2nd stage
Q3 = 20; % Q factor of 3rd stage
K10 = m1*(w1^2); % Spring constant for 1st stage
K20 = m2*(w2^2); % Spring Constant for 2nd stage
K30 = m3*(w3^2); % Spring constant for 3rd stage
Kp1 = -0.33; % Proportional Gain of C1 controller
Ki1 = 0.5; % Integral Gain of C1 controller
Kd1 = 0; % Derivative gain of C1 controller
Kp2 = -0.67; % Proportional Gain of C2 controller
Ki2 = 0.1 ; % Integral Gain of C2 controller
Kd2 = 1000; % Derivative gain of C2 controller
K1 = K10*(1+i/Q1); % m1*w1^2*( 1+i/Q1), w1 = 80mHz, Q1 = 30
K2 = K20*(1+i/Q2); % m2*w2^2*(1+i/Q2), w2 = 260 mHz, Q2 = 100
K3 = K30*(1+i/Q3); % m3*w3^2*(1+i/Q3), w3 = g/1.2, Q3 = 20
% Kp1 = -0.33; % Proportional Gain of C1 controller
% Ki1 = 0.5; % Integral Gain of C1 controller

% Kd1 = 0; % Derivative gain of C1 controller
% Kp2 = -0.67; % Proportional Gain of C2 controller
% Ki2 = 0.1 ; % Integral Gain of C2 controller
% Kd2 = 1000; % Derivative gain of C2 controller
C1 = Kp1 + Ki1/s + Kd1*s;
C2 = 0; % Kp2 + Ki2/s + Kd2*s;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%Code for calculating the x3/x2 transfer function.%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(1)
a1 = K3;
a2 = (s^2*m3 + K3);
sys1 = (a1)/(a2); % calculates the x3/x2 transfer function.
w = logspace(-2,3,1000);
bode(sys1,w)
grid;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Code for calculating x2 / x1 transfer function%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(2)
a3 = K2;
a4 = (m2*s^2 + K2 + K3);
a5 = K3*K3/(m3*s^2 + K3);
sys2 = a3/ (a4 - a5) % calculates the x2/x1 transfer function
w = logspace(-2,3,1000);
bode(sys2,w)

grid;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Code for calculating x2/x0 transfer function%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(3)
a6 = K2*(-C1 + K1)*(m3*s^2 + K3);
a7 = [(m2*s^2 + K2 + K3)*(m3*s^2 + K3) - K3*K3];
a8 = (m1*s^2 - C1 + C2 + K1 + K2);
a9 = K2*(C2 + K2)*(m3*s^2 + K3);
sys3 = a6/(a7*a8 - a9); %calculates the x2/x0 transfer function
w = logspace(-2,3,1000);
bode(sys3, w)
grid
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Code for calculating x1/x0 tranfer function %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(4)
a10 = (-C1 + K1)*[(m2*s^2 + K2 + K3)*(m3*s^2 + K3) - K3*K3]
a11 = a8*a7;
a12 = a9;
sys4 = a10/(a12 - a11); % calculates the x1/x0 transfre function
w = logspace(-2,3,1000);
bode(sys4,w)
grid
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Code for calculating the x1/F1 transfer function %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(5)
a13 = 1/sys4; % This gives us the tranfer function x0/x1 which is the
% inverse of x1/x0 transfer function
a14 = sys2;
a15 = m1*s^2 + K1 - K2*(a14 -1);
%*(1 - a13)
sys5 = 1/a15;
w = logspace(-2,3,1000);
bode(sys5,w)
grid
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Code for calculating the Open Loop Gain TF %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(6)
a16 = C1*(1 - a13) + C2*(a14 - 1);
a17 = sys5;
sys6 = a16*a17;
w = logspace(-2,3,1000);
bode(sys6,w)
grid
%%%%%%%%%%%%%%%%%%%%%%%%%%%7%%%%%%%%%%%%%%%%%%%%
%%% Code for calculating the x3/x0 transfer function %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(7)
sys7 = sys1*sys3;

w = logspace(-2,3,1000);
bode(sys7,w)
grid
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

2011 super spring modelling

Recommended

Recommended

More Related Content

Similar to 2011 super spring modelling

Similar to 2011 super spring modelling (20)

More from Siddartha Verma

More from Siddartha Verma (13)

Recently uploaded

Recently uploaded (20)

2011 super spring modelling