Violin Modes
S2 Line Noise Investigation
S.Klimenko, F.Raab, M.Diaz, N.Zotov
(thanks to Gaby and Andri for discussions)
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Outline
¾ Tracking of violin modes in S2
¾ Work in progress
¾ Conclusion
S.Klimenko, LSC meeting, March 2003
S2 VM Monitoring
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The LineMonitor uses an approximation of the line
interference signal I(t) in the form
I (t ) = ∑ an ⋅ cos( Ψn (t )) = ∑ an ⋅ cos( 2πnft + φn )
n
n
¾ we assume that the harmonic’s amplitudes do not change much
during the integration time T (line width << 1/T)
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The LineMonitor estimates an, f and φn
¾ Since violin frequencies are known, measure amplitude and phase.
¾ Integration time – 1 min
¾ Line parameters are constantly measured and stored in trend files
¾ Two harmonics for each mode are monitored. Some first harmonics
and most of the second harmonics are not visible for 1 min
integration time
S.Klimenko, LSC meeting, March 2003
Violin Amplitude
z
z
z
z
z
Α – pendulum vibration amplitude
a – mirror motion amplitude (<<A)
V – amplitude measured by LineMonitor in ADC counts,
converted to a using calibration
m
ν = α/Α:
ν=
≈ 8.3 ⋅ 10 −6
πM
2
2
4
kT
ω
ω
0
0 Φ(ω )
Thermal noise: S (ω ) =
⋅
⋅
mω 02 ω (ω 02 − ω 2 )2 + ω 04Φ 2 (ω )
¾ quality factor
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m
Q = Φ −1 (ω 0 ) = ω 0 / ∆ω
Wire/Mirror motion:
S.Klimenko, LSC meeting, March 2003
kT
2
(
)
1820
f
A =
≈
mω 02
2
2
2 kT
(
)
15
.
1
mf
a =ν
≈
mω 02
2
M
H2 violin modes for S1 run
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Thermal noise from measured Q’s of H2 violin modes
¾ Fred: e-log 9/8/02
¾ The raw data (red circles) is compared to the estimated thermal noise
(blue curve)
S.Klimenko, LSC meeting, March 2003
a
2
~ 14 − 20mf
343.754
39,000
343.814
143,000
ETMX
344.051
70,000
ETMX
344.110
143,000
349.201
116,000
349.245
90,000
349.282
90,000
349.659
175,000
Calibrated Amplitude
1 + γ (t )(C0 ( f ) R0 ( f ) − 1)
a ( t ) = V ( t ) ⋅ R( t ) = V ( t ) ⋅
α (t )C0 ( f )
z
z
z
z
R0 – response function
C0 – sensing function
H0 – open loop gain
(0.376 @ 344 Hz)
γ,α – t-calibration
S.Klimenko, LSC meeting, March 2003
Average square amplitude
z
z
P(a) – Raylegh distribution
P(a2) – exponential
¾ slope s gives <a2>
a
2
4−π
=s
4
−1
a 2 = 15.8mf
S.Klimenko, LSC meeting, March 2003
Amplitude distribution
S.Klimenko, LSC meeting, March 2003
exponential
S.Klimenko, LSC meeting, March 2003
external excitation
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non-exponential tail
¾ can be excluded from the fit
¾ LineMon outlier triggers
could be used as veto for the
analysis.
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shifted Rayleigh distribution
log( P ( a 2 )) = s( a 2 + as2 )
¾
slope is not affected by shift
S.Klimenko, LSC meeting, March 2003
L1 Violin thermal noise
very preliminary
frequency , Hz
343.06
343.48
343.65
344.42
346.65
346.94
displacement noise,
mf
14.5
15.8
12.9
19.9
16.6
18.1
stat. error ~ 1%
expected noise for simple mechanical model:
15.0-15.5 mf (depends on M and ω)
S.Klimenko, LSC meeting, March 2003
Conclusion & Plans
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Thermal excitation of the violin resonances is observed
¾ measured noise: 13-20 mf
¾ expected noise: 15.0-15.5 mf
¾ Perhaps, more accurate mechanical model should be used for
better agreement between measured and calculated noise.
¾ Servo influence on the slope is not ruled out.
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Plans
¾ do analysis of H1 and H2 modes (Sergey)
¾ calculate thermal noise from measured Q-factors and compare with the
LineMonitor results (Fred)
S.Klimenko, LSC meeting, March 2003