Aspects of the GEO 600 Style Triple and Quadruple Pendulum Suspension Systems. 4 October 2002

Transcription

1 Aspects of the GEO 600 Style Triple and Quadruple Pendulum Suspension Systems 4 October 2002 Mentor : Dr Calum I. Torrie California Institute of Technology T By John Veitch University of Glasgow Caltech SURF Programme 2002 Caltech UID: Contact: v@student.gla.ac.uk

2 Advanced LIGO Suspension Research John Veitch Introduction 2 LIGO Suspensions 2 Experiments 3 Properties of Suspension Wire Determination of Breaking Stress 4 Determination of Young s Modulus 10 Blade Height Adjustment 13 Single Pendulum Experiment 16 References 19 Acknowledgements 18 Appendix 20 The work described in this report was undertaken as part of a SURF project for the LIGO group at Caltech during the summer of The project was supervised by Dr C. Torrie and 2

3 1.1 Introduction Gravitational waves are a prediction of Einstein s General Theory of Relativity, which describes the influence of gravity as a curvature of space-time. Gravitational waves produce a curvature in such a way as to stretch, then shrink space in the directions perpendicular to their propagation. The Laser Interferometer Gravitational wave Observatory (LIGO) project is a facility whose goal is to detect these gravitational waves and study them to gain information about the cosmos. [1] In order to detect these strains of space, and therefore gravitational waves, the LIGO project has developed, and continues to develop, highly sensitive interferometers situated in Louisiana and Washington state, which act in unison to eliminate local noise sources. A typical gravitational wave is expected to produce a strain of the order or lower, producing a change in length of roughly meters in the 4km arms of the detectors, a distance roughly one thousandth the diameter of a proton, so the interferometers must be extraordinarily sensitive. Such a sensitive instrument will be extremely susceptible to noise from non-gravitational sources, including photon shot noise, seismic noise and thermal noise in the suspension wires and mirrors that make up the interferometer. [2] The first generation of LIGO is already installed and is expected to begin taking scientific data in the near future. The next generation of LIGO, known as Advanced LIGO, is expected to come online in LIGO Suspensions In order to isolate against seismic noise, LIGO employs several stages of isolation. The active seismic isolation system removes most of the external vibrations. This is complemented by the passive seismic isolation system, which removes most of the lowfrequency noise. To further reduce noise, especially at higher frequencies, the optics in Initial LIGO are suspended as single pendulums on metal wire. This design will be extended to include triple and quadruple pendulums to meet the more stringent isolation requirements in Advanced LIGO. When the optics are suspended in this manner, the pendulum acts as a filter, which has a transfer function that falls as 1/f2 above its resonant frequency f0, as shown in figure 1. Initial LIGO is designed to detect gravitational waves of frequencies from a few tens of Hz up to a few khz. Therefore a LIGO pendulum with f0=~1hz will provide an isolation factor of 104 at 100Hz. To further improve isolation, a system of multiple pendulums in series, as used in Advanced LIGO, has a Figure 1: Transfer function of single pendulum with f0=~1hz transfer function that falls off as 3

4 1/f 2n, where n is the number of pendulums. Therefore a quadruple pendulum system for a main optic in Advanced LIGO provides an isolation factor of in the horizontal direction at 100Hz. In the vertical direction, however, the pendulum has a much higher resonance frequency caused by the stiffness of the wire, and due to unavoidable crosscoupling between the horizontal and vertical modes, the limiting factor is the vertical isolation. [3] To reduce the frequency of the vertical oscillations, Advanced LIGO will employ cantilever blades, which act as springs with a low spring constant, lowering vertical resonant frequencies and thus enhancing isolation in the detection band of the interferometer. This project will investigate aspects of both the cantilever blades and the wire by which the pendulums are suspended, and an analysis of a single pendulum suspension will be performed. 2 Experiments Over the course of the work conducted for the SURF project, several experiments were performed, in which various aspects of the suspension system were analysed. These included the measurement of the physical properties of suspension wire and comparison between types of wire and different clamping methods. Over the course of the summer, Dr C. Torrie developed a library of clamps for adjusting the height of the cantilever blades in the suspension, and this aspect was tested on different blades. For a single pendulum system with both single and double loops of suspension wires, and symmetric and antisymmetric crossed blades, the frequencies of oscillation were measured. In addition to the experiments described above, Dan Mason of the Rensselaer Polytechnic Institute conducted a computer analysis of the cantilever blades and their deflection under load, wrote a mathematical simulation of the bending of the upper pendulum stage caused by the mass of the stages below, and performed an investigation into the phenomenon of cold welding. [4] 4

5 Properties of Suspension Wire 2.1 Breaking Stress Introduction The suspension wire used in LIGO and advanced LIGO for the upper stages of the pendulum is an important part of the pendulum assembly. Suitable wires must be chosen for each pendulum. Due to the large 40kg masses to be suspended in Advanced LIGO, it is important that the wire be able to bear its load safely. To ensure this safety, the wires are only ever loaded with 1/3 of their breaking stress. The method that is used to clamp the wire onto the pendulum can also affect its strength. In this experiment comparisons were made of the breaking stress of wires across three manufacturers, different types of wire and different clamping methods Theory The breaking stress of a material can be measured directly by applying stress to the material until it yields. In this experiment the wire to be tested was used to suspend a platform, upon which mass could be loaded. The weight of the mass under gravity, then, provided the force with which to stress the wire. If we equate the equations for breaking stress:- F F B. S = = 2 A πr where F is the force on the wire and r is the radius, and weight:- F = mg Where m is the mass and g the acceleration due to gravity, the expression for Breaking Stress below is easily obtained:- mg B. S. = Equation 1 2 πr Fig 2: Simple Clamp Fig 3: Machined clamp Three types of clamp were tested in this experiment. The simple clamp was designed to be the simplest possible version of a clamp, with no special measures taken to reduce the damaging effects of the clamp, such as crushing of the wire. It consisted simply of two metal plates which bolted together with the wire in between. The machined clamp was designed to improve upon the basic clamp. This was achieved by reducing the tolerances of the hole seperation and diameter, having the faces fly-cut to ensure that they were perfectly parallel and placing a groove through the centre of the clamp to better 5

6 define the position of the wire. The machined clamp is similar to the clamps used in certain parts of GEO 600. The third clamp was designed to minimise the deformation of the wire as it is clamped, since the more the wire is crushed the smaller its cross-section and the easier it is to break. This was achieved by wrapping the wire round a cylinder before clamping it, where the cylinder takes up most of the load. 6

7 2.1.3 Procedure In the Breaking Stress experiment, music wire and stainless steel wire of diameters ranging from 0.2mm to 0.85mm were tested. The sources of wire were the Malin Wire Company, the California Fine Wire Company and the Knight Precision Wire Company. For this experiment, a length of wire was cut using wire cutters and clamped at both ends. Care was taken to ensure that the wire ran through the middle of the clamp, so that the angle at which the wire leaves the clamp is the same on each occasion. The upper clamp was then attached to a small 5-tonne capacity crane [see Picture 1], and a clear plastic cylinder attached around the assembly to ensure safety against the snapped wire recoiling and harming someone. For wires with diameters less than 0.3mm, mass in kilograms was added to the lower clamp in the form of cylindrical known weights on a hook (see Picture 1). For thicker wires of diameter >0.3mm a greater mass was required, and this was provided in the form of lead bricks Picture 1: The Breaking Stress apparatus of known mass, which were added to a metal platform suspended from the clamp. Both the hook and the platform were weighed beforehand, and their masses included in the calculation. Mass was added at first in large units, and as the theoretical breaking stress was approached, the units were made smaller, using the cylindrical masses in range 200g-2kg and finally the very small weights in range 5g-100g until the wire finally failed. This allowed the point at which the wire reached its ultimate tensile stress to be measured to a high degree of accuracy. The final mass suspended was recorded in a spreadsheet. The broken wire was then removed from the clamp, and the clamp inspected to ensure that the breaking of the wire had not damaged it. A new length of wire was cut from the spool and the procedure repeated until the spreadsheet had sufficient data to calculate an average result and standard error on the result. Wires that showed inconsistent breaking points were tested more often to reduce the overall error in the result. 7

8 2.1.4 Results & Analysis Wire from the three manufacturers mentioned in section was tested in this experiment. The California Fine Wire Company provided 0.2mm and 0.34mm diameter Elgiloy wire [Graph 1]. Graph 1: Breaking Stress of Elgiloy Wire 2.50 Breaking Stress (GPa) BS (manufacturer) BS (simple) BS (round) BS (machined) mm Elgiloy 0.34mm Elgiloy Graph 2: Breaking Stress of Music Wire 3.50 Breaking Stress (GPa) BS (manufacturer) BS (simple) BS (round) BS (machined) mm California Music Wire 0.35mm California Music Wire 0.22mm Malin Music Wire 0.30mm Malin Music Wire 0.35mm Malin Music Wire 0.40mm Malin Music Wire The Malin Wire Co. provided a range of music wire, which was tested along with music wire from the California Fine Wire company [Graph 2]. It was found that, although the manufacturers specifications were similar, in tests the Malin wire outperformed the California Wire at 0.3mm diameter. At 0.35mm diameter, however, both wires gave similar results for the basic clamp, but the California Wire had a higher breaking stress when used with the round clamp. 8

9 The third company to provide wire was Knight Precision wire manufacturers. They provided music wire and Stainless Steel wire [Graph 3]. The results show that the Knight wire always performed below the manufacturer s specifications. Wire from Knight Precision was already used in suspension systems at the GEO 600 gravitational wave detector in Germany and at the MIT Quad, which is in development. However, there is no risk to these existing installations due to the safety precaution of only loading wires to one third of their breaking stress, which is still well below the breaking stress determined in these experiments. Graph 3: Breaking Stress of Knight Precision Wire 2.90 Breaking Stress (GPa) BS (manufacture BS (simple) BS (round) BS (machined) mm Knight 0.51mm Knight 0.54mm Knight 0.60mm Knight Stainless 0.80mm Knight 0.85mm Knight The Effect of Clamping on Breaking Stress In all three sets of data, it can clearly be seen that the type of clamp used has a strong effect on the results of the breaking stress experiment. In every case, the use of the round clamp improves the breaking stress value over the simple clamp. The machined clamp shows an improvement from the simple clamp, as would be expected with the addition of the precautions described in section The lower performance of the simple clamp is attributed to compression of the wire in the clamp, decreasing its cross-sectional area, and therefore reducing its ability to bear weight. This effect is reduced in the round clamp, as described in section

10 Effect of Kinks on Breaking Stress When working with fine wire, i.e. diameter 0.22mm, caution must be taken to avoid bending the wire and causing a kink. Experiments were performed on two types of kink which are common in the laboratory when working with wires. The first was a kink caused by the wire being bent, for example over the sharp edge of a table, and pulled straight again. It was found that such a kink dramatically reduces the breaking stress of the wire. [Graph 4] The second type of kink occurs frequently when working with spools of wire, whereby the wire is caught in a loop and the loop pulled closed, resulting in the pinching of a small section of wire into a circle. When this type of kink was tested, it was found that the wire was so weakened that the addition of any significant mass caused the wire to break at the kink. These results indicate that kinked wires should not be used when assembling pendulums, as they cannot provide the safety margin that is required Conclusions Breaking Stress (Pa) 2.50E E E E E E mm Elgiloy Graph 4: Effect of kinks on breaking stress 0.2mm Elgiloy (kink) 0.34mm Elgiloy 0.34 mm Elgiloy (kink) 0.2mm Elgiloy 0.2mm Elgiloy (kink) 0.34mm Elgiloy 0.34 mm Elgiloy (kink) This experiment has shown that the choice of clamp used when suspending the pendulums can have an effect on the breaking stress of the wires used. Based on the results of this experiment, a new clamp was designed by Dan Mason to combine the benefits of the machined and round clamps. This involved combining the small size of the machined clamp with the rounded surface of the round clamp [Fig 5]. This clamp will be produced and tested by Dr Calum Torrie in the winter of Figure 5: Combined Clamp This experiment has also demonstrated the need for testing of wires prior to their use in the suspension systems, since the values of breaking stress supplied by the manufacturer are not always accurate. This inaccuracy is compounded by the choice of clamp in suspending the wire. It was found that the Malin Music wire performed well, having both a comparatively high breaking stress, and results with the round clamp showing an agreement with the manufacturer s value to within 20% at maximum, and a much smaller margin in the cases of 0.22 and 0.3 diameter wires. Further work may be conducted to compare this to the performance of California Fine Wire Co. music wire. 10

11 2.2 Young s Modulus Introduction The suspension wire used in LIGO contributes to the way the pendulum behaves. This occurs because the pendulum s resonant frequencies in the vertical direction are dependent on the Young s Modulus (Modulus of Elasticity) of the wire, as described below. [1] Therefore, when trying to model the behaviour of a pendulum, it is vital to know accurately the Young s Modulus of the wire used. In this experiment the Young s Modulus was measured using two different methods, and compared with the theoretical values supplied by the manufacturer. Fig 6: Vertical Measurement of Young s Modulus Theory As already mentioned, the Young s Modulus was measured in two ways. The first method used the wire as a spring and measured the vertical bounce frequency. The second involved measuring the rotational oscillation frequency by suspended a known moment from the wire. Vertical When a mass is suspended from a length of wire, it can be made to oscillate with an angular frequency, ω :- k ω = 2 πf = Equation 2 m where m is the mass suspended, f is the frequency, and k, the spring constant is given by:- 2 Eπr k = Equation 3 l where E is the Young s Modulus, r is the radius of the wire and l is the length of wire used. Combining equations 2 and 3 and rearranging to find E, the Young s Modulus, gives:- 2 4πmlf E = Equation 4 2 r The length, radius and frequency can then all be measured, as described in section

12 Rotational When a moment is attached to the end of a piece of wire, and a small torque applied, it will rotate back and forth with a frequency f such that: - k R 2 πf = Equation 5 I Where I is the moment of inertia attached to the wire, and k R is a constant given by:- 4 Eπr k R = Equation 6 4(1 + σ ) l Fig 7: Rotational measurement of Young s Modulus Where E is Young s Modulus, l the length of wire and σ is Poisson s Ratio, taken as σ=0.19 for steel. The values of l, r and I are determined by experimental setup, and Poisson s Ratio for steel was taken as σ= Procedure Vertical A length of wire between 6m and 8m long was clamped using the basic clamp [see figure 6], and the clamp attached to the high-bay crane. This length was required to allow an oscillation large enough to measure easily and to provide a low enough frequency to count accurately. The crane was then raised up and the free end of the wire clamped with the same basic clamp. A mass was then suspended from the wire with a hook, and the total mass m recorded. A gentle force was applied in the vertical direction, which caused the mass to oscillate vertically. The frequency of these oscillations was measured by timing 10 oscillations and using the equation f=10/t where T is the time measured. This measurement was repeated several times for each wire and an average value taken. The crane was then lowered and the length of the wire measured to find l. Rotational A length of wire approximately 15 cm long was clamped at the top and suspended off the edge of a table. This length was chosen to produce oscillations which had a short enough period to measure. The other end was clamped to an object of known moment of inertia I. The lower object was rotated through a small angle along the vertical axis and released. It was seen to slowly turn and stop, then turn back to its initial position. [See fig 7] The period of this oscillation was measured by timing 10 such movements and using the formula f=10/t. 12

13 2.2.4 Results & Analysis As in the breaking stress experiment above, wires supplied by the California Fine Wire Company, Malin Wire Co and Knight Precision were tested. The results are shown in graphs 5, 6 and 7. As can be seen in all three graphs, the two methods of measurement give results which are in good agreement, mostly within 10% and all within 19%. The values supplied by the manufacturers are generally also within 20% of the measured values. The unusually high Young s Modulus of the 0.45mm diameter Knight precision wire could be the result of a different type of wire being confused with the music wire. If this were the case, then the values measures would differ from the manufacturer s information, which was given for music wire, would not be accurate. Young's Modulus (GPa) Young's Modulus (GPa) Graph 5: Young's Modulus of Elgiloy Wire 0.2 mm Elgiloy 0.34 mm Elgiloy Graph 6: Young's Modulus of Malin Music Wire 0.22 mm Malin 0.30 mm Malin 0.35 mm Malin 0.40 mm Malin Vertical Rotational Manufacturer Vertical Rotational Manufacturer Conclusions When modelling the behaviour of a pendulum suspension, it is important to know the Young s modulus of the wire being used. This experiment shows that the manufacturer s values must be verified for any wire used in LIGO. For example, the 0.45mm Knight wire could have been used and the Young s Modulus taken as 207 GPa in the model, causing the model to give erroneous results. Young's Modulus (GPa) Graph 7: Young Modulus of Knight Precision Wire mm Knight 0.45 mm Knight 0.51 mm Knight 0.54 mm Knight 0.55 mm Knight 0.6 mm Stainless Knight 0.8 mm Knight 0.85mm Knight Vertical Rotational Manufacturer 13

14 3.1 Introduction to Blade Deflection The method used by LIGO to isolate the mirrors from the seismic noise of the earth is to use a seismic stack to remove most of the vibrations from the ground, and then suspend them as a single pendulum from the stack, where the pendulum acts as a filter to remove noise at frequencies away from its resonance. This provides good isolation in the horizontal directions, where the resonant frequency is below the operational range of LIGO, but in the vertical direction the resonant frequency is much higher. In order to lower the frequencies, and hence improve vertical isolation, the pendulums incorporate cantilever blades, which bend flat under the weight of the components in the stages below. The pendulum s resonant frequency can then be damped actively using negative feedback electronics at the uppermost stage of the triple pendulum. In order to achieve the required amount of damping, several constraints must be placed on the system. One of these is that the break-off point of the various stages must not be more than 1mm away from the horizontal line through the center of mass. This ensures a stable and wellcoupled pendulum. [2] Each stage of the pendulum system apart from the final one has its own particular set of blades designed and manufactured for it. The final stage does not have cantilever blades because a high-q material is required to give a sharp, well-defined resonance peak at the final stage. To ensure that the constraints on the pendulum system described above are met, several techniques are used. Firstly, a set of well matched blades are chosen by manufacturing the required amount and spares. A well matched set are then chosen from a tabletop experiment. Secondly, methods of adjusting the tip of the blade have been introduced, including a library of clamps designed by Dr Torrie, and Russell Jones at the University of Glasgow, each cut so that the blade sits at a different angle, from 0 to 2.5. There is also the ability to add and remove mass from the pendulum stage below. This allows a set of blades to be designed to deflect flat under a certain load. The addition or removal of a small amount of mass will then cause the blade to bend up or down by a small amount. In this experiment the effect of such a mass change will be analysed to provide a guide to how much is required to move the blade by a desired amount. 3.2 Experiment For each type of blade, there is a theoretically calculated mass for which the blade should be approximately flat, meaning that the tip is at the same level as the clamped part (See figure 8). 14

15 Figure 8: The blade bent so that the tip is a distance y above the fixed end A cantilever blade from the British-German interferometer, GEO 600, on loan from Russell Jones of Glasgow University, was used as a reference blade. This blade was used to gather data on how the blade behaves under a varying load, since its properties are well known. This set of reference data could then in turn be used to calibrate blades whose properties are not well known by adding or removing small amounts of mass until the blade was flat. This experiment was performed by fixing the reference blade to the optical bench, using a clamp of 0 angle, akin to those used in GEO, and adding the 8kg suggested by theory as the mass required to flatten it. Further mass was then added in 50g increments, and a height gauge used to measure the distance y between the two ends of the blade. Once the blade was flat, the effect of loading it beyond flatness was investigated by adding more mass. The results of this experiment are shown in Graph 1 below. It should be noted that the effect of changing mass is nearly linear around the flat point. 15

16 Graph 1: Deflection of blade (mm) vs. mass suspended (kg) Deflection (mm) Mass (kg) Another way of altering the height of the blade is to use specially angled clamps to hold it in place. These clamps were produced with increments of 0.5 from 2.5 to 0 (flat). They were then tested with the reference blade to see how much of an effect they had. The reference blade was loaded such that it was flat with the flat clamp attached to the optical bench, which was horizontal. The clamp was then substituted with the first angle clamp of 0.5 pointing up, and the same mass loaded on the blade. The vertical displacement y of the tip was recorded, and the procedure repeated with the clamp in the reverse position, so it pointed down. This was performed for all five clamps, and the results are shown in Table 1 below. Clamp Angle (degrees) Vertical displacement of tip (mm) Table 1: Clamp angle against displacement of tip 16

17 3.3 Procedure for Adjusting Blade Height These two methods of altering the blade height were combined into a technique that could be used in the LIGO suspension system to fine-tune the adjustment of the blades. To perform a correction of the blade height, the offset from zero is measured, then the closest value in Table 1 is selected. Once the clamps have been changed, the offset is measured again, and a small amount of mass is added or removed to bring the blade into horizontal position. This procedure was performed in the laboratory on a set of four blades that were rejected for use in a suspension system because they were not well matched. Through the addition and removal of mass and the use of the angled clamps, all four blades were flattened. Two of the blades flattened well with only minor adjustments in suspended mass (5g and 15g) after the angle clamps were used. However, the other two blades were not so well matched. The first of these two required the use of a 2.5 clamp angled down and the addition of 450g to bring it into alignment. The second was out of alignment in the opposite direction, and required a 2.5 up clamp and the removal of 270g. Normally, the blades would be better matched to begin with using the selection method outlined above, and would not require such large adjustments. These blades were deliberately mismatched. This demonstrates how the technique can be used to flatten the blades without adding significant mass to the lower stages of the pendulum. 3.4 Conclusion This experiment has demonstrated that it is possible to fine-tune the position of the LIGO suspension blades, as required for proper control of the system, by a series of simple operations that can be performed easily on site. It has also provided data on the library of angle clamps available, which may be used as a lookup table for use when adjusting blades comparable to those used in this experiment. However, the addition and removal of masses such as 450g or 270g cannot be accomplished practically in the pendulum system. A solution to this problem may be to extend the range of the angle clamps to 3.0 or above. 17

18 4.1 Introduction As described in section 1.2, Advanced LIGO will use a system of spring blades to reduce the vertical resonance frequency of the pendulum, and due to the greater test masses used, the blades must be long enough to support the weight. [5] To accommodate these long blades, a design was produced by Dr Calum Torrie which involves installing the blades in a crossed configuration, reducing the footprint of the blades to meet constraints. However, the existing model of the pendulum setup does not take into account the effects that this antisymmetric arrangement of the blades will produce. Therefore, this experiment was designed to examine the effects of the new blade configuration in comparison to the previous, symmetric setup with no blades. The experiment was performed for situations where the mass was suspended by both two wires and four wires. The work carried out here is an extension of that done in the summer of 2000 by a summer student from the University of Hanover, who worked with Dr Torrie at the University of Glasgow. 4.2 Theory A single pendulum, as shown in Figure 9, has 6 degrees of freedom, and therefore six mode frequencies. These degrees of freedom are: Longitudinal motion parallel to the x- axis; sideways motion parallel to the y-axis; vertical motion parallel to the z-axis; Roll rotation about the x-axis; Tilt rotation about the y-axis; Figure 9: Diagram of a Single Pendulum Rotation rotation about the z-axis. [2] The origin of the axes is taken as the centre of mass. The modes are dependant on the stretching of the wires, which are assumed to act as linear springs, and also on the flexing of the cantilever blades to which the wires are attached. A model of a single pendulum [Appendix], which was written by Dr Torrie using the Matlab programming language, takes the physical parameters of the system, including the Young s Modulus of the wire as established in section 2, and produces predictions of the 6 mode frequencies of a symmetric pendulum. It can produce results for both 2-wire and 4-wire configurations. These were compared with the experimental results for both the symmetric and crossed-blade setups. 18

19 4.3 Procedure The wire was prepared on a jig, and clamped at one end. The other end was attached to a 9kg mass to stretch it under the load it would experience in the experiment. In this way the length of the wire was kept constant in each case. Once the wire was stretched, it was clamped at the desired length and the excess wire removed. The wire was attached to the cantilever blades, which were fixed to the support structure for the pendulum. The wires were loaded with weights until the blades became flat, then the safety bar bolted on over the blades to prevent them recoiling and causing injury if the wire were to break. The weights were removed from the wires and the 18.78kg mass jacked up into position and clamped onto the wires, then the jack was lowered so the mass was hanging freely. To disassemble the apparatus the procedure was performed in reverse. Picture 2: Crossed blade pendulum setup Once the apparatus was set up, an accelerometer was attached to the mass and the output fed into an amplifier. The amplified signal was then passed on to a spectrum analyser which performed a fast Fourier transform on the signal and displayed the output. Each mode was excited by applying a small impulse by hand in the desired direction and allowing the pendulum to move for 128 seconds, over which period of time the spectrum analyser sampled the data. The output was then displayed on the analyser, showing a large peak at the resonant frequency of the mode in question. The same procedure was performed for all six modes with both the 2-wire and 4-wire, crossed and symmetric blade arrangements. 4.4 Conclusion The effects of using 2 or 4 wires, and using crossed or uncrossed blades was shown to be on the order of 3% between the four different setups. Further work is required in order to properly analyse all of the effects that contribute to the variation in frequency. 19

20 5 Conclusions The design and implementation of Advanced LIGO will require a lot of work and will pose challenges that must be met over the course of the project. The work described in this report has shown the importance of properly determining the qualities of the suspension system if it is to operate within the required tolerances of the LIGO project. It has been shown that the suspension wire must have its physical properties experimentally determined, as the values given by the supplier may be inaccurate to a significant degree (section 2). It was also shown that the library of angle clamps and addition and removal of small masses can prove very effective at making fine adjustments of the suspension system once it is assembled. Further work will have to be done in order to describe accurately the effects of the antisymmetric pendulum arrangement and create a computer model of the system. 6 References [1] [2] C. I. Torrie, PhD Thesis, University of Glasgow, 1999 [3] M. V. Plissi, C. I. Torrie et al, GEO600 triple pendulum suspension system: Seismic isolation and control, Review of Scientific Instruments vol 71 no. 6, June 2000 [4] D. Mason, Advanced LIGO Suspension Research SURF Report, 2002 [5] N A Robertson, Advanced LIGO Suspension System Conceptual Design, Pre-print paper LIGO DCC number T D 7 Acknowledgements I would like to thank Dan Mason for his hard work as my lab partner, Dr Calum Torrie for making it all happen, and Dr Mike Plissi and Prof Norna Robertson for their valued advice. 20

21 Appendix The Matlab computer model of single pendulum sing.m %Single pendulum %A.1 Model of a single pendulum %The file mcsing.m assembles the single pendulum model, the ABCD matrices for all degrees of freedom and returns the normal mode frequencies, for all six degrees of freedom, as outlined in chapter 4. % Maike July 2000 Summer Project on "2 wires to one blade" % mcsing.m clear all %****************************************************************** % all units are in S.I. g = 9.81; %****************************************************************** % FOR SUMMARY OF DEFINITIONS SEE MY THESIS % INPUT tx = 0.123; % thickness of mass tr = 0.08; % radius of mass den = 7800; % density of mass l3 = 0.264; % wire length %l3 = 0.2; %????????????????????????????????????????????? % (NB: - length defined as l in figure (4.1)) d = ; % height of break-off of wire above the c. of m.??????????????????????????????????? s = ; % 1/2 separation of wires X direction %s = 0.000; % for one wire off one blade s=0.000 t1 = 0.160; % 1/2 separation of wires Y direction t2 = ; R = 200e-6; % radius of wire (diameter 400 microns) Y = 2.09e11; % Young's Modulus of the wire N = 4; % Number of wires (2 or 4) %ufc = 0.987; % uncoupled mode frequency of cantilever stage(=0 for no cantilevers) ufc = 1.07; % % measured by John Veitch %1.95 with ms=3kg % NB:- uncoupled mode frequency- the frequency observed for a cantilever in a particular stage supporting only the mass of that stage 21

22 %****************************************************************** % CALCULATIONS % TO GET FROM EXCEL SHEET!!!! m = 18.78; Ix = ; Iy = ; Iz = ; %m = den*pi*tr^2*tx; % mass %Ix = m*(tr^2/2); % moment of inertia (sideways roll) %Iy = m*(tr^2/4 + tx^2/12); % moment of inertia (longitudinal tilt) %Iz = m*(tr^2/4 + tx^2/12); % moment of inertia (rotation) kw = (N/2)*Y*pi*R^2/l3; % spring constant of the wire kc = 1/2*(2*pi*ufc)^2*m; % spring constant of the blade if (kc==0) k = kw; else k = kc*kw/(kc+kw); end si = (t2 - t1)/l3; % sin(w) co = (l3^2 - (t2 - t1)^2)^0.5/l3; % cos(w) %****************************************************************** % LONGITUDINAL AND TILT FREQUENCIES (L, RT) % 2 wires to one blade k=kw; % for one wire off one blade %k = kc*kw/(kc+kw) k11 = - m*g*d/iy - 2*k*s^2*(co)^2/Iy - m*g*d^2/iy/l3/(co) - m*g*s^2*(si)^2/iy/l3/(co); k12 = + m*g*d/iy/l3/(co); k21 = + g*d/l3/(co); k22 = - g/l3/(co); % matrix A = [ k11 k12 k21 k22 ]; % calculation of the frequency longtilt = (sqrt(abs(eig(a))))/2/pi %****************************************************************** % SIDEWAYS AND ROLL FREQUENCIES (T, RL) k = kc*kw/(kc+kw); s11 = - m*g*d/ix + m*g*t2*(si)/ix/(co) + m*g*d/ix*(t2*(si)/l3 - d*(co)/l3)- m*g*t2*(si)/ix/(co)*(t2*si/l3 - d*(co)/l3) + 22

23 2*k*l3*(si)*t2/Ix*(t2*(si)/l3 - d*(co)/l3) + 2*k*l3*d*(si)^2/Ix/(co)*(t2*(si)/l3 - d*(co)/l3) - 2*k*t2^2/Ix- 2*k*d*t2*(si)/Ix/(co); s12 = + m*g*d*(co)/ix/l3 - m*g*t2*(si)/ix/l3 + 2*k*(si)*t2*(co)/Ix+ 2*k*d*(si)^2/Ix; s21 = - g*(t2*(si)/l3 - d*(co)/l3) - 2*k*(si)^2/m/co*(t2*(si) - d*(co)) + 2*k*t2*(si)/m/(co); s22 = - g*(co)/l3-2*k*(si)^2/m; % matrix A = [ s11 s12 s21 s22 ]; % calculation of the frequency sideroll =(sqrt(abs(eig(a))))/2/pi %****************************************************************** % ROTATIONAL FREQUENCY (RZ) k = kc*kw/(kc+kw); ro = - m*g/l3/(co)*(s^2*(co)^2 + t1*t2) - 2*k*s^2*(t2 - t1)^2/l3^2; rot = + ro/iz; % calculation of the frequency rot = (sqrt(abs(rot)))/2/pi %****************************************************************** % VERTICAL MODES (Z) k = kc*kw/(kc+kw); ver = + 2*k*(co)^2 + m*g*(si)^2/l3/(co); vert = - ver / m; % calculation of the frequency vert = (sqrt(abs(vert)))/2/pi 23