• No results found

Modal Analysis of a Nuclear Reactor with Fluid Structure Interaction: Added Mass and Added Stiffness Effects

N/A
N/A
Protected

Academic year: 2020

Share "Modal Analysis of a Nuclear Reactor with Fluid Structure Interaction: Added Mass and Added Stiffness Effects"

Copied!
12
0
0

Loading.... (view fulltext now)

Full text

(1)

! !""# $ % &

'( ) * )+

',

*- ).

.

'

/

0 ,)- (

-

- . * .

'*1 ((.(

*( ((.(

,,*.

.,,.

2

,

3

4

! "# $

(

'

!"

# !

$

%

#&

'

())()* +

+,

%&

'

%&

$( #

) *.

&

" # $ ! " # $ " % ! # # & ' " ( ! ) • ! % # ! *" ' • ! * ! ' • $ ! * " # * " ! #

(2)

5* '(- '* ! ! % - # , " % # " ) $ % ! . $ % /// ' $ % 0% 1 # & . $ 0% 2 3 4 ! $ ! + 5& " - # * 62 ! " # , # & *" + # !# * . 0% . 0% ! . 0# ( 7 " " # % ! #

(3)

!5 ((.( .,,. ! # 1 " * % # ! % ! * # " 9 1 .

(

% %ϕ

)

0# ) ! 1 % # . ! 0 " # & " # !55 6 7 1 " # " ' ! ) • % % % • * " # # " ) " # & ! " . -$ " % // 0% ! # .5 % / ' ":; % / 0# , %

(

% %ϕ

)

. "7 % // 0# δΩ δΩ δΩ δΩ δΩ δΩ δΩ δΩππππ Γ Γ Γ Γ ΩΩΩΩ .5%ν%ρ 0 Ω ΩΩ Ω ΓΓΓΓ . %ρ+0 %ϕ δΩδΩδΩδΩσσσσ δΩ δΩδΩ δΩ & " + # # " # 1 !# < Ω ! ∂Ω =∂Ωσ ∪∂Ω π ∪Γ ! ∂Ω ! % ∂Ω ! Γ " # ! ∂Ω % ! Γ # %

(4)

0 . σ # ρ # ! ! 1 ) 0 . = ∂ ∂ + σ ω ρ Ω .51# 0 0 . ⋅ = σ ∂Ω .51# 0 = ∂Ω .51# 80 < Ω ! ∂Ω =∂Ω ∪∂Ω π ∪Γ ! ∂Ω π ! . ! 0% ∂Ω . " 0 Γ " # ! ∂Ω % ! Γ # % ϕ # ρ % # ! ! 1 ) = + ∂ ∂∂ ϕ ρ Ω .51# =0 ϕ ω ρ = Ω .51# 0 = ⋅ ∂∂ϕ

.51# >0 = ∂Ω .51# ?0 ! ! 1 # ϕ ω ρ σ . 0⋅ = Γ .51# 0 ⋅ = ⋅ ∂∂ϕ Γ .51# /0 51 . 0 * " ) Γ % # 51 ./0 * ) Γ % # 1 . 0 ./0 ! .2 "< % // 0# , ) .ω0 =

{

∈ .Ω 08% =

}

Ω ∂ . 0% .ω0 =

{

∈ .Ω 0% =

}

. 0 0 .ω ϕ = ϕ

{

∈ .Ω 0

}

% δ %δ %δϕ × × % ! * .@ - % / ?0) 0A 0% . . B 0 0% . . B 0 0% . . B 0 0% . . B 0 0% . . C 0 0% . . D 0 0% . . C 0 0% . . E E ϕ ω δ ω δϕ ϕ ω δ δϕ ω δϕ ω ϕ δ ω ω δ ω δ ω + + + + − × = + .51# 0 , 51# . 0% % ! * )

(5)

• =

⋅⋅ Ω Ω 0 . 0 . 0 % . δ σ ε δ % • =

⋅ Ω Ω δ ρ δ 0 % . % • =

Ω Ω 0 % . C ρδ δ % • =

Ω Ω 0 % . δ δ =

Ω Ω 0 % . B δϕ δϕ % • =

⋅ Γ Γ δϕ ρ δϕ0 % . B " # E B BE B B % # .@ % / 'F $ ! (" % / / ' "7 % // 0 " ! $ 51# . 0 ! * )           ×           − =           ×           0 . 0 . 0 . C B B B B 0 . 0 . 0 . C ω ω ω ω ω ω ω 51# . 0 !5!5, 8 99 & *" # + 8 # ! ! ! & *" *" % $ !

(

% %ϕ

)

* + % * ! 1 )

(6)

0 . 0 % % . 0 % . 0 % . 0 % . 0 % . 0 . 0 % % . 0 % . 0 % . 0 % . 0 % . 0 % % . 0 % . 0 % . 0 % . 0 % . 0 % % . 0 % % . 0 % % . 0 % % . 0 % % . θ θ ϕ θ θ ϕ θ ϕ θ ϕ θ θ θ θ θ θ θ θ " " " " " " " " " " " " " " " " " " " " " " " " ⋅                 − + ⋅                 +                 =                

≥ ≥ .51# 0 51 . 0 * *" . 0% . ≥ 0 " . ≥ 0# . 0 . = 0# .G * #% / /0% ) • 2 % • 2 # $ ! Φ 51# . 0 % ! )       ×       + =       ×       − − − − 0 . 0 . B C B B C B B C B B C B 0 . 0 . C ω ω ω ω ω .51# 80 H % ! . " 0 ! . #% 80% ! )

(

ω

D

+

A

+

)

×

.

ω

0

=

.51# =0 , 51# . =0% $ ! % ) − =ρ .51# 0 * 51 # . 0% . 80 . =0# & % % ! # 51 . 0 % ! 51 # . 80 . =0 % ! ! ( # !5:5 6 1 66 6 8 51 . =0 * % # ! ! # & % ! # + = ! ! # 1 % # .== I(0 % . 8 I(0 # ! >>J # J $ ! 1 #

(7)

K ) == I( K8 ) 8 I( # $ ! µµµµ =8# 8/# 8 8# # = # /# 8 # >#8 = 8 # # " # + * # 1 # ! " .&* % 0) • ) ! # 1 > I(% ! ! .== I(0# * 8 > . + # 80% ! 1 ! % # • ) % # 1 ?/ I(% ! ! 1 .== I( 8 I(0# # " > J . 0' 1 % ! # . 1 % 0 " # + !% ! " #

(8)

K ) > I( K ) ?/ I( K8) I( K=) I( K ) / I( K>) 8 I( # $ ! % % µµµµ ?/#= > #=8 > 8 # 8# = # 8#? #? 8# ? 8= # #/

(9)

:5 ((.( ,,*. .,,. , ! % # & % ( 1 $ ! ! $ # " # ! # * * " % # " ! " # ( % ! # :55 8 8 6 ! ! $ & % ! " # ! ! . % // 0# $ ! *

Ω Ω ∂ ∂ = Ε ρ % * ! * % $ ! .: % / 0) Γ ⋅ − Ω ⋅ + Ω ⋅ = Ε

Γ Ω Ω & &0 . 0 . 0 . 0 . ε σ ε σ .51# >0 , 51# . >0%ε . 0 % ! )         ∂ ∂ + ∂ ∂ = 0 . % ε .51# ?0 0 . ε % ! )

⋅∂ ∂ ∂ = % 0 . ε .51# 0 0 .& σ ! & ' # 51# . >0 ! ) •

Ω 0 . 0 . ε σ ! * % •

⋅ Γ Γ & & % •

⋅ Ω Ω &0 . 0 . ε σ " % $ " # * δ

(

' + #

)

= ' * ! * )

0

.

0A

.

D

0

.

+

+

&

σ

π

=

&

&

.51# /0 π *% σ " *# & * 51# . /0%

(10)

$ &# $ 0 B . B 0A . B D +& σ − π & = # . 0 B & $ &B $ # $ ! % 9 ! ! )

(

ω

.&0

D

+

A

+

D

+

&

.

σ

π

0A

)

×

.

ω

.&0

0

=

.51# 0

0

.ω.&0 ω. &0 ) 1 & " # +

$ % B0 .& = ω $ &B% # $ ( 1 # + !% ! ! % . # π σ 0# !5:5 6 1 66 6 + !% ! & ! ! # * + = % ! % + # ># 0 .& σ # ! π σ + = # + = % 51# . 0# 8 1 ! ! # ! !

(11)

( ! & !" #$ % $ % K #? / I( #? I( K ?/#= = I( ?/#=> I( K> 8 # 8 I( 8 #/? I( + ? 1 ! # # # # # # # 8 = > # # ( ! $ ! + " % 0 .ω.&0 σ σ =− × π =− π × .ω.&00 # ) )σ & ! % $ % * # ! ! $ #

(12)

&5 '* )- '* & ! " # < " !# & % ! ! # ! ! " $ . #% 0 + ! $ ! % ! # & ( ! $ ! " # + !% $ " ! # ! " ! - * # $*').(4. .* ! $ # :L,<<74M% :<&G,F, NL O5 * ! 5& 2 3 9 + - 5 # .,. .* . &* % +#% . 0% P Q P 1 # , 9 % I Q#

@ % O#N#% . / 0% + 5 4 5 & % 4 "I % 5 ! #

@ - % N#% . / ?0% + + , 4 % 3 & % # % # 8/"> # % #% . // 0% 3 < + 5 & % H 6 # 2 % #% < % N#<#% . // 0% & 3 % # 5 % :# #% . / 0% & 4 + + " , % N G % # ?/% # % # ?"/ # + (% #N#% . /? 0% 5 < 1 2 , % N 5 , % # >?"?8# : % #N#% . / >0% G # , # C * P % 2 5 C5 P + % # >/% 5 # $ % N#% . ///0% + " , 4 % + 5 & @ 5

& # & @ # + 5 & 2 % # 8 % # 8 "= #

$ % N#% . ///0% 5 1 $ & ) +5 @5 & % @ . // " // 0% + 5 & 2 % # 8 % # 8" =# $ % N#% . 0% + 5 G 2 & 5 % @ . //="// 0% $ G % # ?% # 8/"># % I# N"4#% 7 % #% . // 0% + , % H 6 # -$ % #% % #:#% . // 0% < ( & & L : ( 5 4 % , N 3 5 % # 8 % # /" >#

% :#% :; % 4#&#% . / 0% & + 5 + & + "

, & # N G % # 8% # 8% # ?" #

% N#+#% < P% #% < % 2#% 4 *% @#% . 80% < < +

2 + " & % 4 G 4 % # = =% # ?"/8#

% N#+#% @ % 2#% < P% #% . 0% & 3 ! + "

, % 4 G 4 % 2 % N ?" % #

G *% 4#% #% . / /0% & & & %

# & % & =" % / /#

F $ ! (% 7# #% % #<#% . / /0% + 5 # @ + <

References

Related documents

Here, an N-S based finite volume solver is coupled with a finite element based linear and nonlinear structural solvers to study the nonlinear aeroelastic characteristics of the

This gives six more unknowns therefore closure (turbulence) models have to be applied. In CFD codes there are many turbulence models available but unfortunately

There is a significant change in response at the soil column frequency which clearly shows that the effect of the SSSI on the lighter surface founded building can be

All the time histories are statistically independent and are applied simultaneously and analysis are performed for the ½ SSE (OBE). The generated FRS is smoothened

This paper summarizes the efforts in applying the installed coupling software to demonstrate/investigate fluid-structure interaction (FSI) between pressure wave and

In this problem, dierent electric voltages were applied to the piezoelectric material and the vibration of this plate at dierent times was considered... was observed that by

In this study dynamic fluid structure interaction analysis was carried out by combining Computational Fluid Dynamics (CFD) and Computational Structural Dynamics (CSD) solver

This project was largely aimed at gaining a basic understanding and better overview of the fundamental structural behavior of the AGARD 445.6 wing under