Evaluation of thermal noise in
monolithic suspensions
with non-cylindrical fibres
Francesco Piergiovanni
Firenze/Urbino Virgo Group
Monolithic Suspension Team
ET-WP2 Thermal noise meeting - Roma 28/02/09
Monolithic Suspensions for Virgo+
Upper clamps
Cylindrical SiO2 fibers
Typical size:
d = 300mm
L =700 mm
Lower clamps
How to estimate thermal noise in
cylindrical fibers
Fluctuation – Dissipation Theorem
X ( ) 
x

2
Y    i
m
k
4kBTemp
Fext
 Y 
x  
Fext  
Fext    ( K    m 2 ) x  
F ( L,  )
K   
X ( L,  )
x
F
Taking into account only horizontal displacements
How to estimate thermal noise in
cylindrical fibers
The bulk equation for an elastic rod
-ω2ρS X= - EI XIV + T X''
4° order equation  Analytic Solution  4 boundary conditions
K 
EIp  p 2   2    cos  pL   p sin  pL  
2 p cos  pL    p 2   2  sin  pl 
T  T 2  4  S 2 EI

2 EI
T  T 2  4  S 2 EI
p
2 EI
How How
to estimate
to estimate
thermal
thermal
noisenoise
in cylindrical
in
cylindrical
fibersfibers
E  E0 1  i 
bulk  4.1 1010
Silica bulk loss angle
S  bulk m ds
1
r

th  
1  ( ) 2
Surface contribution
Thermoelastic contribution
 T  Temp

  E0   

ES

 C
2
Effective thermal expansion coefficient
1 dE

 1.52 104 K 1
E dTemp
Cylindrical fiber loss angle
Length: 700mm
Diameter: 400mm 500mm
A diameter increment produces:
The magnitude of the peak
decreases
Good
The thermoelastic peak moves
to lower f
The dilution factor decreases
A constrain for minimum diameter:
Two constrains for maximum diameter:
Maximum safe working stress
 Violin modes frequencies
Bouncing mode frequencies
Bad
The surface loss angle
decreases (behaves as 1/r)
Energy distribution along the fiber at 0.5Hz
Only the first and the last few mm’s of the fiber
contain a not negligible energy
The energy is stored in the bending region
Bending length = 1mm
The neck of the fiber
Fibers produced with CO2 laser pulling
machine at EGO present a neck
The fiber neck
The bending length of the
produced fibres was
measured to be about 4mm
Diameter transition is placed
in a region where a large
amount of energy is stored
Loss angle and stiffness are
different from those of cylindrical
fibres
The neck of the fiber
The elastic bulk equation
  S ( z ) X ( z )   E  I ( z ) X ( z )   TX II ( z )
2
II
II
There is no analytic solution
We try to use a numerical solver for boundary value problems: MATLAB bvp6c
•
•
•
•
Collocation method starting from a guess solution
The analytic solution for a cylindrical fibre is taken as guess solution
Automatic choice of the mesh points driven by the residual values
Six order polynomial interpolation on each subinterval of the mesh
The neck of the fiber
Gaussian + exponential necks profiles
Similar to the produced
Virgo+ test fibers
The neck of the fiber
Bending of the fiber at 20Hz
• Some evidences of a bad mesh choice for very sharp diameter variation
• The statistical meaning of residual values should still be investigated in
order to understand the precision of the solution
Optimized fibers for AdVirgo
It is possible to produce fibers with
two heads (some mm’s length) with
suitable diameter to minimize the
thermal noise at low frequency
Fiber heads
thicker than
the central
part
Bending
points
(Cagnoli, Willems, Phys. Rev. B, 65)
The diameter of the central part of
the fiber determines the frequencies
of the violin and bouncing modes
Optimized fibers for AdVirgo
Diameter of about 0.82mm minimize the thermoelastic loss
Optimized fibers for AdVirgo
0.8mm diameter minimize the overall thermal noise at 20Hz
Optimized fibers for AdVirgo
Loss angle contributes for a L=50-600-50 mm d=0.8-0.4-0.8mm fiber
Optimized fibers for AdVirgo
For sharp diameter transition the fiber is made by 3 cylindrical segments
Elastic equation for each segment
 2  Si X  z   EIi X IV ( z)  TX ''( z)
i  1, 2,3
4 constants for the solution of each segment = 12 constants
4 boundary conditions + 8 matching conditions = 12 linear equations
For sharp transition it is possible to solve the equation analytically
Optimized fibers for AdVirgo
Horizontal thermal noise of the mirror stage for a 40kg mirror
Optimized fibers for AdVirgo
Horizontal thermal noise of the mirror stage for a 40kg mirror
Model of the last 3 stages of the Virgo
suspension (Marionette - RF - Mirror)
m1 = marionette mass
m2 = reference mass
m3 = mirror mass
2
 0   k1  k2  k3   m1
  
 k2
0
F 
 k3
  
x3
Y  i
F
k2
k2   2 m2
0
  x1 
 
0
  x2 
2
k3   m3   x3 
 k3
Model of the last 3 stage of the Virgo
suspension
The contribute of the marionette and the reference mass is relevant below
5 Hz
Model of the last 3 stage of the Virgo
suspension
Optimized fibers vs GWINC with ribbons
Comparison between cylindrical and
optimized fibres
Conclusions
 A MATLAB function to calculate suspension thermal noise in AdVirgo
configuration was written.
Cylindrical and 3-cylindrical segments fibres can be handled.
The function is ready to be implemented in a GWINC-like code.
A MATLAB function for thermal noise numerical calculation for general
shaped fibres (even for not analytical profiles) is ready.
BUT
It is necessary to check the validity of the results with other BVP solvers
and also with finite elements analysis
Work in progress…