Download Resonances, quality factors in LIGO pendulums: a

Resonances, quality factors in LIGO pendulums:
a compendium of formulas
Gabriela González
December 11, 2001
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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 + iφw ), where
φw = 1/Qw 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
p
parameter the distance ∆ = EI/T , which represents the distance over
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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
à /∆ 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
ω0 τ /2, where ω0 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 φw , then the thermal noise in between the pendulum frequency
and the first violin mode is approximately
x2 (f ) ∼
4kB T0 ω02 φ
M ω5
where φ = (∆/L)φw . 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
s
Ã
µ ¶2 µ
¶2 !
T nπ
2∆
∆
nπ∆
ωn =
1+
+4
+
ρ L
L
L
L
∆
Qn =
L
¶
µ
∆
∆
2
1+4 +
(nπ)
L
2L
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).
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• 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
approx ω02
Mode
calc freq(Hz)
approx Q
calc. Q
pendulum
g
L−h
= (2π0.75Hz)2
0.77 Hz
pitch
2T h
Jy
= (2π0.65Hz)2
0.66 Hz
2(h+∆)
Qw
∆
yaw
2T ab
Jz L
= (2π0.48Hz)2
0.49 Hz
aL
b∆ Qw
side
g
L−h
= (2π0.75Hz)2
0.72 Hz
L
∆ Qw
roll
2EAb2
Jx L
vertical
2EA
ML
violin1
T π 2
ρ(l)
= (2π317Hz)2
319 Hz
L
2∆ Qw
= 163 Qw
160Qw
violin2
T 2π 2
ρ( l )
= (2π634 Hz)2
638 Hz
L
2∆ Qw
= 163 Qw
153Qw
violin3
T 3π 2
ρ( l )
= (2π952Hz)2
958 Hz
L
2∆ Qw
= 163 Qw
143Qw
3
= (2π18.5Hz)2
= (2π13.1Hz)2
L
∆ Qw
= 326 Qw
= 14 Qw
28.6 Qw
24.6 Qw
= 44 Qw
45 Qw
= 326 Qw
305 Qw
18.5 Hz
Qw
Qw
12.7 Hz
Qw
Qw
Definitions
• Distances L, h, l, a, b as shown in figure 1. l2 = (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 = M g cos α/2
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• Elastic distance: ∆ =
p
EI/T
• Wire loss factor φw
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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).
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Figure 1: LIGO pendulum suspension, with a single wire loop, attached slightly
above the center of mass and angled towards the center.
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[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
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