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, 푇 is the stress energy tensor, 퐺 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 푐 /8G, 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 10푀 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 = ∫ 푥 (푓)푑푓, where x is the

displacement spectrum [m/√퐻푧 ]. 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 푚/√퐻푧 .

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 = m11

2, K20 = m222

and K30 = m332. Here, 3 = 푔/퐿 . 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-

푚1푥1̈ = 퐹1 −퐾1(푥1 − 푥0) − 푏1(푥1̇ − 푥0)̇ - 3.1

This gives, 푠 푚1푥1(푠) = 퐹1(푠) − 퐾1 푥1(푠) − 푥0(푠) − 푠푏1(푥1 − 푥0)

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) = 21 or = b1/ (2√퐾1푚1 ). The denominator equation becomes (푠 + 212+12). Here, = 푑푎푚푝푖푛푔 푟푎푡푖표.

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

Process Sensor

r e p

u

c b

(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;

푚1푥1̈ = 퐹1 −퐾1(푥1 − 푥0) − 푏1 푥1̇− 푥0̇ + 퐾2(푥2 − 푥1) + 푏2(푥2̇ − 푥1̇) - 1st stage - 3.4

푚2푥2̈ = 퐹2− 퐾2(푥2 − 푥1) − 푏2 푥2̇ − 푥1̇ + 퐾3(푥3 − 푥2) + 푏3(푥3̇ − 푥2̇)- 2nd stage - 3.5

푚3푥3̈ = 퐹3− 퐾3(푥3 − 푥2) − 푏3(푥3̇ − 푥2̇) 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.

퐹1 = 퐶1(푥1 − 푥0) + 퐶2(푥2 − 푥1); 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.

푥3푥2

=퐾3/푚3

푠 + 퐾3/푚3 3.8

푥2푥1

=퐾2(푚3푠 + 퐾3)

[(푚2푠 + 퐾2 + 퐾3)(푚3푠 + 퐾3) − 퐾3 ] 3.9

푥1퐹1

=1

푚1푠 + 퐾1− 퐾2 푥2푥1− 1

3.10

푂푝푒푛 퐿표표푝 퐺푎푖푛(푂퐿퐺) =퐶1 1− 푥0

푥1 + 퐶2 푥2푥1− 1

푚1푠 + 퐾1− 퐾2 푥2푥1− 1

3.11

푥2푥0

=[퐾2(−퐶1 + 퐾1)(푚3푠 + 퐾3)]

[{(푚2푠 + 퐾2 + 퐾3)(푚3푠 + 퐾3) −퐾3 }{푚1푠 − 퐶1 + 퐶2 + 퐾1 + 퐾2} − 퐾2(퐶2 + 퐾2)(푚3푠 + 퐾3)]

3.12



푥1푥0

=(−퐶1 + 퐾1)[(푚2푠 + 퐾2 + 퐾3)(푚3푠 + 퐾3) −퐾3 ]

[{(푚2푠 + 퐾2 + 퐾3)(푚3푠 + 퐾3) −퐾3 }{푚1푠 − 퐶1 + 퐶2 + 퐾1 + 퐾2} − 퐾2(퐶2 + 퐾2)(푚3푠 + 퐾3)]

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.

푥3푥0

=푥3푥2

푥2푥0

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;

퐹1 = 퐶1(푥1 − 푥0) + 퐶2(푥2 − 푥1);

And, we can write the values of controller as given in equation 3.15 and 3.16 below.

퐶1 = 퐾푔 퐾푝1 +퐾푖1푠

+ 푠퐾푑1 3.15

퐶2 = 퐾푔 퐾푝2 +퐾푖2푠

+ 푠퐾푑2 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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%