Resonances, quality factors in LIGO pendulums: a compendium of formulas Gabriela González December 11, 2001 1 Introduction and notes There are simple formulas for the different pendulum and violin modes, obtained from simple mechanical models, that I have collected in the summary following this introduction. There are also some formulas for quality factors that follow from from some hand-waving arguments, and also from some more sophisticated models, that are also presented in the summary. I used the model in [1] and [2] to calculate the modes frequencies and quality factors. This model uses a mass (with a finite size and moment of inertia) suspended by an anelastic wire. This means the suspension wire has a finite cross section, and a complex Young modulus, of the form E(1 + iφ w ), where φ w = 1/Q w would be the loss measured for the free, unclamped wire. The model used in [1] was useful for a mass suspended by a single wire; it was extended in [2] for the case of a mass suspended by two wires (or a loop), as in LIGO masses. The model used allows the calculation of thermal noise of the suspended mass in its six different degrees of freedom, using the Fluctuation-Dissipation Theorem. But it also allows, by zooming on resonances, the numerical calculation of frequencies and quality factors of normal modes. In general, and in the absence of asymmetries, we find coupled equations for the pendulum and pitch modes and for the side and roll modes; vertical and yaw modes are uncoupled from all others. That means, for example, that the spectral density of brownian motion of pendulum motion will show a peak at the pitch mode, but that the spectral density of the vertical mode will not show a peak at any of the other five normal modes. However, all modes will show peaks at the violin modes. I have used Matlab programs to calculate the results shown here. These programs, and the formulas in the summary, use as parameters the ones in LIGO Technical Document T970158-06, also detailed later in this report. Some more particular points worth mentioning: The model for the wire loss coupling into the pendulum modes uses as a parameter the distance = EI/T, which represents the distance over 1
the wire bends near the clamps. For the LOS LIGO parameters used, this distance is 1.4 mm. The pendulum and pitch modes are close enough in frequency that the predicted quality factors are not even close to the approximate formulas in the table. This coupling depends, among other things, on the pitch distance h and the elastic distance. There is an eternal question ([2],[3], for example) about a factor of 2 in the predicted Q for the pendulum mode: is the dilution factor L/ or 2L/?? There are many physical and mathematical ways of guessing it (does the wire bend at both top and bottom, or just at the bottom? is it the ratio of spring constants or potential energies?), but the correct factor for the pendulum Q is 2L/ : this would correspond to the measured ω 0 τ/2, where ω 0 is the frequency of the mode in radians, and τ is the amplitude decay time. This dilution factor and its relationship with the dilution factor that matters (the one we use to calculate thermal noise in the gw band) depends on whether the mass is hanging on one, two or more wires. See below for more confusion. If if the losses are only due to the wire anelasticity parameterized by a loss factor φ w, then the thermal noise in between the pendulum frequency and the first violin mode is approximately x 2 (f) 4k BT 0 ω 2 0φ Mω 5 where φ = ( /L)φ w. Did we lose a factor of two again? No: for a mass suspended by a loop, the effective Q to calculate the thermal noise in the gravitational wave band is half the Q of the pendulum mode. (That said, that large pendulum mode is not achieved anyway unless the pendulum frequency is far enough away from the pitch mode.) The violin modes show some anharmonicity due to the elasticity of the wire. This is most visible in the loss factors than in the frequencies themselves. The approximate formulas for the modes,taking into account (some) anharmonicity, are ( T nπ ω n = 1 + 2 ( ) 2 ( ) ) 2 nπ ρ L L + 4 + L L Q n = L (1 + 4 L + 2L (nπ)2 ) These formulas are only valid for mode numbers up to about 15-20, but the formulas break down after that. (Thanks to Phil Willems to point out an extra term in this formula). 2
Warnings: I am not 100% sure I have been consistent with the nomenclature in drawings, formulas and Matlab programs, you may find l where it should L or the way around. For LIGO parameters, it doesn t matter much (2% difference). The measured frequencies do not correspond too closely with the formulas or the model predictions, we could use these to fit in reverse the wire and mass parameters to the measured values. The model does not take into account wedges. 2 Summary Mode approx ω 2 0 calc freq(hz) approx Q calc. Q pendulum g L h = (2π0.75Hz)2 0.77 Hz L Q w = 326 Q w 28.6 Q w pitch 2T h J y = (2π0.65Hz) 2 0.66 Hz 2(h+ ) Q w = 14 Q w 24.6 Q w yaw side 2T ab J = zl (2π0.48Hz)2 al 0.49 Hz b Q w = 44 Q w g L h = (2π0.75Hz)2 0.72 Hz L Q w = 326 Q w 45 Q w 305 Q w roll 2EAb 2 J xl = (2π18.5Hz)2 18.5 Hz Q w Q w vertical 2EA ML = (2π13.1Hz)2 12.7 Hz Q w Q w violin1 violin2 violin3 T ρ ( π l )2 = (2π317Hz) 2 L 319 Hz 2 Q w = 163 Q w T ρ ( 2π l )2 = (2π634 Hz) 2 L 638 Hz 2 Q w = 163 Q w T ρ ( 3π l )2 = (2π952Hz) 2 L 958 Hz 2 Q w = 163 Q w 160Q w 153Q w 143Q w 3 Definitions Distances L, h, l, a, b as shown in figure 1. l 2 = (L h) 2 + (b a) 2. Mass M Wire tilt angle α = tan 1 (b a)/(l h) = sin 1 (b a)/l = cos 1 (L h)/l Tension T = Mg cos α/2 3
Elastic distance: = EI/T Wire loss factor φ w 4 Parameters used Matlab File ETMparameters.m %ETM parameters (T970158-06) M=10.3; %mass, kg R=12.5e-2; %optics radius, m H=.1; %optics thickness, m J=M*(3*R^2+H^2)/12; %moment of inertia for pitch and yaw, kg*m^2 Jx=M*R^2/2; %moment of inertia for rotation g=9.80; %m/s^2 rhow=7.8e3; %steel density, kg/m^3 r=.31e-3/2; %LIGO wire radius, m A=pi*r^2; %wire area cross section rho=rhow*a; %mass per unit length l=.45; %vertical distance from top of wire to com h=8.2e-3; %vertical distance from wire end to com l=.45; %vertical distance from top of wire to com b=sqrt(r^2-h.^2);%half distance between bottom attachment points E=2.1e11; %Young s modulus (steel) I=.25*pi*r^4; %area moment of inertia phiw=1e-3; %loss angle for wire EI=E*I*(1+i*phiw);%complex parameter used in formulas a=33.3e-3/2; %half distance between top attachment points L=sqrt((b-a).^2+(l-h).^2); %wire length alpha=atan((b-a)./(l-h)); %wire angle with vertical T=M*g*(l-h)/(2*L); %tension in each wire With these parameters, =1.4mm. References [1] Brownian motion of a mass suspended by an anelastic wire G.I. González and P.R. Saulson, J. Acoust. Soc. Am. 96,207-212 (1994). 4
Figure 1: LIGO pendulum suspension, with a single wire loop, attached slightly above the center of mass and angled towards the center. 5
[2] Suspensions thermal noise in the LIGO gravitational wave detector, Gabriela González, Classical and Quantum Gravity 17(21),4409 (7 November 2000) (gr-qc/0006053). [3] Damping dilution factor for a pendulum in an interferometric gravitational waves detector G. Cagnoli, J. Hough, D. DeBra, M.M. Fejer, E. Gustafson, S. Rowan, V. Mitrofanov, Phys. Lett. A 272, p 39-45, 2000 6