Fused Silica Suspension Research at Caltech, Lately
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Transcript of Fused Silica Suspension Research at Caltech, Lately
LIGO-G020152-00-D
Fused Silica Suspension Research at Caltech, Lately
Phil Willems
LIGO/Caltech
(lots of help from V. Sannibale, V. Mitrofanov, J. Weel)
LSC Meeting, LLO
March 20-23, 2002
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Suspensions Apparatus
Automated fiber pulling lathe
Q-measurement rig
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Mode frequencies enable precise determination of fiber radius
2
2 1
2
π4
21
ρ2 Lk
n
Lk
P
L
nf
een
μm157fiberr
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Whammy Sidebands
16.65Hz
This allows measurement of Young’s modulus (74.5GPa).
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Temperature Shift of Unloaded Mode Frequencies Yields (dE/dT)/E
K/1052.1
/2
/
4
f
dTdf
E
dTdE
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The Frequency Shift of the Violin Modes Yields the Dilution Factor
)2/(1
)(2
1/00 ββuα
Dβuα
f
dTdf
nn
n
nn
n
D
β
f
dTdf
2
/
For this suspension,
In general,
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And the Agreement is Good
datao theory
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Obtain the Structural and Thermoelastic Losses from Unloaded Fiber Q’s
Best fit:
8
7
106.7φ
K;/109.3α
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Free fiber frequencies are very consistent with inferred diameter
4 2 /ρωκ
;01)κcosh()κcos(
L EI
LL
But now E=73GPa compared to 74.5GPa for violin modes;
good agreement with stress-strain law E=E0(1+5.75
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Put it all together to predict Q
theory, with NTE
theory, w/o NTE
theory, w/o NTE
and x10-7/K
data
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Conclusions
Suspension fiber dynamics can be precisely quantified
Very high Q’s possible Q’s not up to NTE level (and in fact more consistent
with LTE theory- yikes!), but this likely due to low-frequency excess loss like recoil damping
Work is ongoing
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Dumbbell-Shaped Suspension Fibers
A new technique for low vertical bounce frequency and low thermal noise
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The Apparent Tradeoff
Low thermal noise Optimum fiber diameter Smaller diameter increases
noise
Low bounce frequency Smaller fiber diameter better
optimum thermal noise
low bounce frequency
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The Ribbon Solution
Make the cross-section as small as needed to reduce bounce frequency
Set the ribbon thickness to increase dilution factor and push thermoelastic peak to higher frequency
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But Notice the Different Dynamics of the Two Types of Motion
Pendulum/violin motion
Dissipative bending motion concentrated near ends of fiber
Loss not very sensitive to middle section of fiber
Vertical bounce motion
Dissipative stretching motion distributed along fiber in inverse proportion to cross section
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This Suggests an Obvious Solution
Make the fiber the optimum thickness for low damping at the top and bottom, and thinner in the middle for low vertical bounce frequency-
a DUMBBELL shape
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The Equations of Motion
fiber. theofsegment thelabels where
;2
ρω4
;2
ρω4
);sinh()cosh()sin()cos()(
22
22
nEI
EIPPk
EI
EIPPk
zkDzkCzkBzkAzX
en
tn
ennenntnntnnn
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The Boundary Conditions
boundaryat continuous force
)()()()(
boundaryat continuous torque)()(
boundaryat smooth slopefiber )()(
boundaryat smooth fiber )()(
at top clampedrigidly fiber 0)0()0(
1212211111
122111
1211
1211
11
zXPzXEIzXPzXEI
zXEIzXEI
zXzX
zXzX
XX
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The Boundary Conditions
endfiber on forcearbitrary )(
massat zero slopefiber 0)(
boundaryat continuous force
)()()()(
boundaryat continuous torque)()(
boundaryat smooth slopefiber )()(
boundaryat smooth fiber )()(
333
33
2323322222
233222
2322
2322
GzXEI
zX
zXPzXEIzXPzXEI
zXEIzXEI
zXzX
zXzX
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How the Boundary Conditions are Derived
F=EI X '''(z1)-TX '(z1)
F=EI X '''(z1)-TX '(z1)
111
222
Two forces must approach each other as joining segment goes to zero length, otherwise that section undergoes infinite acceleration.A similar argument shows that the torques also become equal.
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Thermal Noise Spectra: Baseline Ribbon vs. Dumbbell Fiber
ribbon
dumbbell fiber
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Influence of Welded Pins on Suspension Q
This program can easily model the welded pins on the ends of suspension fibers (extreme dumbbell)
Good approximation if pins are relatively long and thin
Used this analysis to confirm Mitrofanov and Tokmakov’s estimate of Q due to lossy pins
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Result of Simulation
Mitrofanov/Tokmakov estimate:
– This is based upon estimate of force exerted on pin by fiber
Our result: this is substantially correct, although for thicker fibers the torque also plays a significant role
2
11
ω
4
pp
pv
Lm
MgQQ
pineforcepin
couplepin
LkX
X
2
3
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The Big Result
A reasonably good, reasonably short pin will not unduly influence Q or thermal noise
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Fused Silica Materials Properties Database
Many different sets of material parameters for fused silica are in use by the various LSC groups to design suspensions and predict thermal noise.
This leads to much confusion when comparing results between groups.
I became aware of this when analyzing violin mode data above.
We need a common set of values shared by the community to facilitate information transfer.
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Proposed Material Parameters
IN CONSTRUCTION
(I have lots of Syracuse data to digest)