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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)