MG12 Conference UNESCO - Paris- July 13-17th 2009 · pi gi wi i el gi wi ii i ... loss angles describing the structural losses of the suspension wires. 11iv; 1 ... •the naïve estimation

MG12 Conference – UNESCO - Paris- July 13-17th 2009

27/07/2009MG12 Conference – UNESCO - Paris- July 13-17th 2009

q3q2

q1

M3M2

M1

M1 marionetteM2 mirrorM3 recoil mass

[*] Bernardini A., Majorana E. ,Puppo P. ,Rapagnani P. , Ricci F., Testi G. "Suspension last stages for the mirrors of the Virgo interferometric gravitational wave antenna." Rev. Sci. Instr. 70, no. 8 (1999): 3463.

M1

M2

M3

27/07/2009MG12 Conference – UNESCO - Paris- July 13-17th 2009

2 2 2 2 2 2 2 2 2

1 1 2 2 3 3 1 1 1 2 2 1 2 3 3 1 3

1 1 1 1 1 1( ) ( )

2 2 2 2 2 2M x M x M x M x M x x M x x

2

2 ; 1, 2, 3/ 4

wi i

i

vi

i

r Y

Li

M

2 2 2

1 1 1

2 2 2

1/21/2

1/2

2 2

22

/ ; /

; 2, 3

2 ;42

p g w t t mario T

pi gi wi

i

el

gi wi

i i

i

i i ii i

el wi

i

M M

i

kg

L M

n Y Ik I r

L

w i

i

i

r

L

Y

i

in

27/07/2009MG12 Conference – UNESCO - Paris- July 13-17th 2009

27/07/2009MG12 Conference – UNESCO - Paris- July 13-17th 2009 5

2 2 231 2

1 1 2 1 2 3 1 3

1 2 3

1 1 1( ) ( )

2 2 2d

x xE M M MQ Q Q

x x x

In the model the losses of the system are defined by the mechanical quality factors of the oscillators. This is the most classical approach which comes from a very general treatise of the problem, and that can be found on classical texts like and in the papers like those of Callen and Greene [*] in which the quality factor is related to the decay times of the uncoupled oscillators. The mechanical quality factors include both the structural and the viscous dissipation mechanisms that could be present on the suspension system

1/21 11

1 1 1 1

1 1

/ ;

; 2, 3

vh t

i

i i i vhi

Q D Q

Q D Q i

Qvhi , Qvvi viscous quality factors Di pendulum dilution factors

Fi(w), Fiv(w) loss angles describing the structural losses of the suspension wires.

1 1 ; 1, 2, 3iv

iv iv vviQ Q i

[*] Callen H.B., Greene F. «On a Theorem of Irreversible Thermodynamics I-II.» Physical Review 86-88, (1952). 1, ..., 3

i i iQ

27/07/2009

6

3

2

321

ththIII

ththIII

ththththI

fF

fF

fffF

Generalized forces related to the uncoupled stochastic thermal forces

Equations of motion with thermal stochastic forces

MG12 Conference – UNESCO - Paris- July 13-17th 2009

2; 1, 2, 3

b i

thi

i

k T mf i

2 1

2

4 ˆ( ) R e{( ( ) ) }, 1, ..., 3b

thi ii

k TX Z i

1

2

3

ˆ

thI

thII

thIII

F

F

F

Z

X

X

X

Thermal noise of an element of the branched system via the FDT theorem

[1] Majorana E., Ogawa Y., PLA 233 (1997): 162-168.[2] Rapagnani, P. ,Il Nuovo Cimento, 1982.

m

FY

QY

m

FY

QY

m

FY

QY

th

th

th

2

0

02

00

0

0

0

2

2 2

03 0

2

03

11 12 13 2 2 2

02 31 03 02

21 22 23 2 2 2 2 2

02 21 02 02

31 32 332 2

03 03

2 2 2 2

03 03

1 1

1 2

0 2 0 2

3

1 1

ˆ

1

ˆ ˆ

Y X m M

Y X m M

Y X m

1

10

0 2

3

3

ˆ

th thI

th thII

thIIIth

F F

m M

F F

m MM

FF

Mm

2 2 2 2 2 2 2 2 2

0 0 0 0 0

1 1 1 1 1 1

2 2 2 2 2 2m y m y m y m y m y m y

0 0 0

2 2 20

0

0

1 1 1

2 2 2( , , , , , )

i in idy y yE m m Cross m Q y y y y

Qym

Q Qy

This kind of approach were already applied to the double oscillator system in the papers [1] and [2].The Lagrangian in can be expressed as a sum of independent quadratic terms through thesubstitution of the new coordinates of the normal modes , ,

oy y y

E. Majorana, P. Tricarico, “Decay Times of an N-Normal-Mode System”, Il Nuovo Cimento, Sept 1993

The coupled quality factors can be found by imposing the equivalence between the dissipation function expressed in term of the normal modes and the coupled system one . We notice that in general, there is a cross-term in the dissipative function which can be neglected for systems with high Q’s but is not null. For this reason are called quasi-normal modes .

122

2 2

3 31 1 2 2

2 2 2 2

1 2 2 3 3

122

2

3 31 1 2 2

2 2 2 2

1 2 2 3 3

2 2 2 42 4

1 3 00 3 3 01 1 2 2

0 0 0 2 4

1 3 3 3 2

1

mm mQ m

Q Q Q

mm mQ m

Q Q Q

mm mQ m

Q Q Q

1

2 2 2

0 1 31 3

2 2

21 2 3

1

the expressions are completely equivalent to those ones found in the reference [*].

, ,o

y y y

27/07/2009MG12 Conference – UNESCO - Paris- July 13-17th 2009

9

thththFTFTFTX

YYYX

)()()()(

)(

230022212

23022212

Normal modes mass matrix

The calculation is quite complex…

we have to remember that the stochastic thermal forcesfthi of the free oscillators are uncorrelated

2

3

2

3

2

2

2

2

2

1

2

1

2

2)()()()(

nthnthnththTfTfTfX

)()()()()()()()()()()()()(

)()()()()()()()()()()()()(

)()()()()()()()()()(

23313302221232111133

23313202221222111122

23310222121111

TNNTNNTNNT

TNNTNNTNNT

TNTNTNT

n

n

n

This equation differs considerably by the “naïve treatment” of the thermal noise in which the calculation is performed by a simple quadratic sum of the thermal terms of the normal modes.

Using the transformations above we find:22

2 21 22 0 23

2 2 22 2 2

21 22 0 23

* *

21 22 0 0

* *

21 23

* *

22 23 0 0

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

thX Y Y Y

Y Y Y

Y Y Y Y

Y Y Y Y

Y Y Y Y

then we obtain the following expression:

The main difference is the presence of the cross-term due to the correlation between the normal forces that in the naïve treatment are erroneously neglected. Only in the condition in which the modes are orthogonal (the cross-term in the dissipation function is null) the cross-term is null and the two expressions coincide. The stochastic cross-coupling is in this case identically zero.

We can express the mirror oscillator cohordinate in terms on the normal eigenvectors by finding how it is related to the normal forces:

1 1

0ˆ ˆ ˆ ˆ ˆ;

ˆ

ˆ

th thI

th thII n

th thIII

n

F F

F N F N M M

F F

M

M

1 2 3

2

3

2; 1, 2, 3

thI th th th

thIII th

thIII th

b i i

th i

i

F f f f

F f

F f

k T Mf i

Uncoupled oscillators mass matrix

27/07/2009MG12 Conference – UNESCO - Paris- July 13-17th 2009

Hzm /2

• the naïve estimation in the off-resonance zone at high frequencies is quite different, and in some cases can give a wrong estimation of the thermal noise.

• the Normal Mode Treatment is in full agreement with the FDT treatment

Hz

27/07/2009MG12 Conference – UNESCO - Paris- July 13-17th 2009

June 26th, 2007INFN 11

• In presence of low dissipative mirror suspensions, the contributions of the other last stage suspension elements to thermal noise of the mirror cannot be neglected.• A new thermal noise estimation must be done by including the viscous and internal dissipations of the marionette and recoil mass pendulum.• It turns out that the marionetta’s mechanical pendulum losses give a non negligible effect via its recoil, in the off-resonance high-freq. range. • A more detailed description for Virgo+ and AdV cases can be found in [*].

[*] VIR-015C-09, F. Piergiovanni, M. Punturo and P. Puppo, The thermal noise of the Virgo+ and Virgo Advanced Last Stage Suspension (The PPP effect).

0

0

• We have calculated the thermal noise of a branched system of oscillators through the normal mode representation.

• This model describes well the mechanical behavior of the last stage suspension system like in the Virgo interferometer and is consistent with the predictions coming by the fluctuation-dissipation theorem.

• In the evaluation of a multimodal system thermal noise it is important to take into account the cross correlation term among the various modes which is neglected in a naïve treatment;

• In presence of low dissipative mirror suspensions, the contributions of the other last stage suspension elements to thermal noise of the mirror cannot be neglected and a new thermal noise estimation must be done by including the viscous and internal dissipations of the marionette and recoil mass pendulum.

Further Steps

The developed formalism can be useful to estimate the thermal noise of a system in a thermally steady state and with the stages at different temperatures.

This study can be important for an evaluation of the thermal noise of the suspensions in a 3rd generation g.w. interferometers.