Title:

Double pendulums for terrestrial interferometric gravitational wave detectors

This thesis is written for the threefold purpose of: 1. Advocating the use of double pendulums, in preference to single stage pendulums. to suspend the test masses used in terrestrial interferometric gravitational wave detectors. 2. .Justifying the necessity of developing a comprehensive dynamic model of a double pendulum suspension. This requires consideration of all six degrees of freedom (per mass) of the coupled system. 3. Demonstrating that there is sufficient confidence in the modelling techniques to proceed to the design and construction of a double pendulum suspension for use in a full scale gravitational wave detector. The author believes that the use of double pendulums to suspend the test masses of a terrestrial interferometric gravitational wave detectors offers an accessible means of obtaining the high sensitivity required to record gravitational waves using such a detector. The following is a summary of how this assertion is justified in the course of this thesis. Chapter 1 is an introduction to gravitation and the postulation of the existence of gravitational waves. Potential sources and the expected signal strengths are discussed. This forms a guide to the sensitivity that is required by a gravitational wave detector. An overview of the subject of gravitational wave detectors is found in Chapter 2. Various types of detector are described: these include: resonant mass detectors, laser interferometers and also the proposed ESA space mission LISA. Noise sources that limit the sensitivity to gravitational waves of interferometric detectors are introduced in this chapter and many are discussed later in this thesis. Chapter 3 is a brief presentation of the noise floor of the Glasgow prototype interferometric gravitational wave detector (measured on 6th March 96). The origin of the detector noise in different frequency regions is discussed. It is shown that the sensitivity of the detector could be improved at frequencies below 200 Hz by a more careful design of pendulum. The formulation of a dynamic model of the pendulum suspension is presented in Chapter 4. This describes how the normal mode frequencies and Q factors of the pendulum suspension can be obtained from the equations of motion. It also describes how a statespace model of the suspension can be formulated. Statespace modelling techniques are used through out this thesis for assessing the transfer functions and servo control of the double pendulums considered. It is noted that a single stage pendulum is a special case of a double pendulum. Thus, the performance of a single stage pendulum can be analysed for comparison. This chapter also contains the formulation of models for the resonant modes of the suspension wires and the vibrational modes of the test mass itself. In Chapter 5 the state space model of the double pendulum is used to analyse damping of the normal modes by an electronic servo. (Such control suppresses the resonant enhancement of noise sources.) This must be accomplished without introducing excess noise to the system. It is shown that the frequency of the normal modes of the pendulum determine whether this is possible. The need to achieve good damping puts constraints on the design of the pendulum. This necessitates a comprehensive model of a double pendulum. The results presented in this chapter are a validation of the design techniques employed. The servo control required to obtain the output signal from the interferometer is discussed in Chapter 6. The requirements placed on feedback elements are analysed in conjunction with the level of seismic noise that they require to accommodate. A split feedback topology is considered to assess the feasibility of using electrostatic actuation on the test mass instead of magnetic. The issues detailed in this chapter also place constraints on the pendulum design. A critical limit to the sensitivity of an interferometric gravitational wave detectors is the Brownian motion of the test mass and suspension. Chapter 7 is a comprehensive treatment of this noise source as it arises in a double pendulum. Concluding remarks are made at the end of Sections 7.3, 7.4 and 7.5 instead of a general conclusion at the end of the chapter. In Chapter 8 double pendulums are considered for the GEO 600 project. Chapter 9 forms a conclusion to the research presented in this thesis. The numerical code used to generate the dynamic model of a double pendulum is listed in Appendix A. This has been written for the 'MATLAB' environment. To do the full statespace modelling of the pendulum and servo electronics requires the 'Control Systems Toolbox'. Appendix B is a detailed derivation of Equation 4.8.
