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Evaluation of thermal noise in monolithic suspensions with non-cylindrical fibres

Evaluation of thermal noise in monolithic suspensions with non-cylindrical fibres. Francesco Piergiovanni. Firenze/Urbino Virgo Group. Monolithic Suspension Team. ET-WP2 Thermal noise meeting - Roma 28/02/09. Monolithic Suspensions for Virgo+. Upper clamps. Cylindrical SiO 2 fibers.

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Evaluation of thermal noise in monolithic suspensions with non-cylindrical fibres

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  1. Evaluation of thermal noise in monolithic suspensions with non-cylindrical fibres Francesco Piergiovanni Firenze/Urbino Virgo Group Monolithic Suspension Team ET-WP2 Thermal noise meeting - Roma 28/02/09

  2. Monolithic Suspensions for Virgo+ Upper clamps Cylindrical SiO2 fibers Typical size: d = 300mm L =700 mm Lower clamps

  3. Fext Fluctuation – Dissipation Theorem m x k How to estimate thermal noise in cylindrical fibers x F Taking into account only horizontal displacements

  4. How to estimate thermal noise in cylindrical fibers The bulk equation for an elastic rod 4° order equation  Analytic Solution  4 boundary conditions

  5. How to estimate thermal noise in cylindrical fibers How to estimate thermal noise in cylindrical fibers Silica bulk loss angle Surface contribution Thermoelastic contribution Effective thermal expansion coefficient

  6. Cylindrical fiber loss angle Length: 700mm Diameter: 400mm 500mm A diameter increment produces: • The thermoelastic peak moves to lower f • The magnitude of the peak decreases • The surface loss angle decreases (behaves as 1/r) • The dilution factor decreases Good Bad • Two constrains for maximum diameter: • Violin modes frequencies • Bouncing mode frequencies • A constrain for minimum diameter: • Maximum safe working stress

  7. Energy distribution along the fiber at 0.5Hz Only the first and the last few mm’s of the fiber contain a not negligible energy The energy is stored in the bending region Bending length = 1mm

  8. The neck of the fiber Fibers produced with CO2 laser pulling machine at EGO present a neck The fiber neck The bending length of the produced fibres was measured to be about 4mm Diameter transition is placed in a region where a large amount of energy is stored Loss angle and stiffness are different from those of cylindrical fibres

  9. The neck of the fiber The elastic bulk equation There is no analytic solution We try to use a numerical solver for boundary value problems: MATLAB bvp6c • Collocation method starting from a guess solution • The analytic solution for a cylindrical fibre is taken as guess solution • Automatic choice of the mesh points driven by the residual values • Six order polynomial interpolation on each subinterval of the mesh

  10. The neck of the fiber Gaussian + exponential necks profiles Similar to the produced Virgo+ test fibers

  11. The neck of the fiber Bending of the fiber at 20Hz Preliminary results • Some evidences of a bad mesh choice for very sharp diameter variation • The statistical meaning of residual values should still be investigated in • order to understand the precision of the solution

  12. Optimized fibers for AdVirgo It is possible to produce fibers with two heads (some mm’s length) with suitable diameter to minimize the thermal noise at low frequency (Cagnoli, Willems, Phys. Rev. B, 65) Fiber heads thicker than the central part The diameter of the central part of the fiber determines the frequencies of the violin and bouncing modes Bending points

  13. Optimized fibers for AdVirgo Diameter of about 0.82mm minimize the thermoelastic loss

  14. Optimized fibers for AdVirgo 0.8mm diameter minimize the overall thermal noise at 20Hz

  15. Optimized fibers for AdVirgo Loss angle contributes for a L=50-600-50 mm d=0.8-0.4-0.8mm fiber

  16. Optimized fibers for AdVirgo For sharp diameter transition the fiber is made by 3 cylindrical segments Elastic equation for each segment 4 constants for the solution of each segment = 12 constants 4 boundary conditions + 8 matching conditions = 12 linear equations For sharp transition it is possible to solve the equation analytically

  17. Optimized fibers for AdVirgo Horizontal thermal noise of the mirror stage for a 40kg mirror

  18. Optimized fibers for AdVirgo Horizontal thermal noise of the mirror stage for a 40kg mirror

  19. Model of the last 3 stages of the Virgo suspension (Marionette - RF - Mirror) m1 = marionette mass m2 = reference mass m3 = mirror mass

  20. Model of the last 3 stage of the Virgo suspension The contribute of the marionette and the reference mass is relevant below 5 Hz

  21. Model of the last 3 stage of the Virgo suspension Optimized fibers vs GWINC with ribbons

  22. Comparison between cylindrical and optimized fibres

  23. Conclusions • A MATLAB function to calculate suspension thermal noise in AdVirgo configuration was written. Cylindrical and 3-cylindrical segments fibres can be handled. The function is ready to be implemented in a GWINC-like code. • A MATLAB function for thermal noise numerical calculation for general shaped fibres (even for not analytical profiles) is ready. BUT It is necessary to check the validity of the results with other BVP solvers and also with finite elements analysis Work in progress…

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