Finite element analysis of suspension elements for gravitational wave detectors

The existence of gravitational waves was predicted by Einstein in 1916 in his General Theory of Relativity. Gravitational waves are considered as ripples in space-time, however there has been no direct evidence to prove their existence. Scientists around the world are engaged in the development of instruments to detect gravitational wave signals. Once these waves are detected new avenues will be opened up in the field of Physics and Astronomy. Gravitational waves are quadrupole in nature and produce a tidal strain in space. They interact weakly with matter making them difficult to detect. The strains expected for gravitational waves which may be detected on earth are of the order of 10-22 to 10-23 (over a frequency range from few Hz to a few kHz). A network of instruments based on long baseline Michelson interferometers is being developed around the world, which should be sensitive enough to detect such strains. The Institute for Gravitational Research in the University of Glasgow in collaboration with the Albert Einstein Institute in Hannover and Golm and the University of Cardiff has been actively involved in the research for the development of instruments and data analysis techniques to detect gravitational waves. This includes construction of a long ground based interferometer in Germany called GEO 600 having an arm length 600 m and strong involvement in the larger detectors of the LIGO (Laser interferometer gravitational wave observatory) project in USA having arm lengths of 4 km. An upgrade to LIGO called Advanced LIGO has been proposed and is currently under development with significant input from Glasgow. The work in this thesis discusses the finite element analysis of suspension elements for the Advanced LIGO detectors. The Advanced LIGO suspensions have a quasi monolithic quadruple pendulum design with final stages comprised of four fused silica fibres or ribbons suspending a 40 kg test mass mirror. Minimising the mechanical loss in the pendulum and hence lowering the thermal noise in the suspension system is a fundamental requirement for increasing the sensitivity of the detector and this has been studied here by analysing the suspension elements of the final stage of pendulum system. Thermal noise is one of the most significant noise sources for gravitational wave detectors. It arises due to the random fluctuations of atoms and molecules in the materials of the test mass mirrors and suspension elements and is related to mechanical loss in these materials. This thesis is devoted to analysis of aspects of mechanical loss in the final stages of the suspension system. This requires modelling the energy distribution in the suspension elements and hence estimating what is known as the ‘dissipation dilution’ of the pendulum. The dissipation dilution factor of a pendulum plays an important role in reducing the mechanical loss in the pendulum and results from the fact that most of the energy associated with the pendulum is stored in the lossless gravitational field of the earth. Mechanical loss can result from internal frictional effects in materials or external frictional effects or thermoelastic effect. The thermoelastic effect tends to be most important of these for the elements of the suspension system. Minimising the thermoelastic loss in the suspension elements is an important issue for reducing the over-all thermal noise. Before analysing the suspension elements of the Advanced LIGO suspension system, a prototype design was modelled in ANSYS®. This was of a single fibre or ribbon pendulum system suspending a 10 kg mass (which approximately makes a quarter of the Advanced LIGO pendulum model). At first, pendulum suspensions comprised of fibres or ribbons of uniform cross-sectional area were analysed and later on fibres or ribbons with tapered ends were included and studied for various taper lengths. The study of tapered ends in a fibre or ribbon is very important as any real system requires such necks for joining purposes. Through this model, important aspects of the pendulum design were studied which included the modelling of the strain energy distribution in the fibres or ribbons, with the help of which the dilution factor of the pendulum was estimated. The most important finding from this chapter was the extent to which the tapered ends reduced the dilution factor of the pendulum. The single fibre pendulum design was then extended to a two-fibre pendulum system comprising of a 20 kg mass. This model was studied as a step between the single fibre pendulum system of the last chapter and the four fibres/ribbons system for the Advanced LIGO detector. There are some other important issues in the two-fibre pendulum model which were observed while studying it and these are related to coupling of the pendulum and tilt modes which leads to reduced dilution factor. The importance of attaching the fibre or ribbons close to a plane through the centre of mass becomes very clear. The Advanced LIGO suspension system was then modelled. The focus was on comparing the performance of silica fibres and ribbons and the associated dissipation dilution factor of the pendulum system. Similar to the single fibre and two fibre pendulum cases, here the fibres or ribbons were first studied as having uniform cross section and then for tapered ends. Further the performance achievable using real fibres and ribbons fabricated in the laboratory was also evaluated. The strain energy distribution in these suspension elements was found in order to compare their performance and again it was found that the dissipation dilution factor was seriously impaired compared with that of uniform fibres or ribbons. Finally given the poor dilution factor which can realistically be obtained, it became very important to investigate whether the major factor contributions to the material loss, i.e. the thermoelastic effect could be reduced. This can be achieved by tailoring the stress level at the point where the fibres or ribbons bends. Realistic designs of fibres or ribbons were analysed and shown to be acceptable for use in the Advanced LIGO detector