Axial Flow Induced Vibration of a Rod Supported by Two Translational

7/28/2019 Axial Flow Induced Vibration of a Rod Supported by Two Translational

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A xia l - f low - indu ced Vibra t ion of a Ro d Suppor ted by Tw o T rans la t iona lSprings

H. S. Kang, K. N. Song, Y. H. Jung

Fuel Manufac tur ing T echnology Developm ent , Korea Atom ic Energy Research Ins ti tu [email protected] i . re .kr

A B S T R A C T

An axial-f low-induced vibration model was proposed for a rod supported by two translat ional springs at both ends.

For the der iva t ion of the v ibrat ion model , the normal mode method was used to so lve the random vibra tion problem of th

cylinder subjected to the randomly fluctuating pressure acting on i ts surface by axial f low. The first natural frequency and

mode shape functions for the f low-induced vibration, so-called FIV, model were derived by using Lagrange's method based

on the single mode approximation. The vibration displacement at reactor condit ions were calculated by the proposed mode

for the spring-supported rod and by the previous model for the simple-supported(SS) rod. The result ing vibration

displacem ent for the spring-suppo rted rod wa s larger than that of the SS rod, and the discrepancy b etween both

displacements was much larger at low flow velocity than at high flow velocity. The vibration displacement for the spring-

supported rod appeared to d ecrease with the increase of the spring constant .

I N T R O D U C T I O N

PW R fuel rods are exposed to reactor coolant of high flow velocity. I t is known that the vibration of the fuel rod(FR) is

generated by the coolant f low, and the fret t ing-wear by the vibration is frequently found on the surface of i t. This problem o

the FR is not on fluidelast ic instabil i ty that causes excessive vibration and fai lure in short t ime but on turbulence-induced

excitat ion that generates small amplitude and may cause long-term frett ing-wear damage. The fret t ing wear by this sub-

cri t ical vibration is generally accepted as a root cause of a fuel rod fai lure which is not known well yet . Since the FIV

obviously g enerates the relat ive mo tion betwe en the F R and spacer grid, that can lead to fret t ing dam age o f the FR, i t is very

important to understand wha t is the actual vibration behavior o f the fuel rod supported by spacer grids. The spacer grid ha

several springs in a cell in order to support the fuel rod flexibly. Therefore, the supporting m ethod for the fuel rod is actually

not a simp le support but a spring support . I t was repo rted that the spring con stant of the spacer grid gav e a significant effec

to the moda l param eters o f the FR[1 ] .

How ever, m ost works on FIV o f the FR w ere associated with the SS cylinder. In this study, therefore, the spring support

effect on the FIV of the cylinder was surveyed as com pared w ith the SS. In order to derive the FIV m odel for the spring-

supported cylinder, the author 's previous wo rk[2] w as com pletely accepted, in which a procedure was described to solve the

random vibration prob lem o f the cylinder subjected to the randomly fluctuating pressure acting on the cylinder surface by

axial f low. The cylinder vibration mo de w as obtained by using L agran ge's me thod based on the single mode(1 st mod e)

SMiRT 16, Washington DC, August 2001 Paper # 1526 Transactions,


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approximat ion . F or the numer ica l ca lcula tions , the spr ing cons tants were se lec ted wi th in the range of those o f t

comm ercia l spacer gr id spr ings , and the mater ia l proper t ies of the cyl inder and f low condi t ions for the PW R reac tor w

used.

M A T H M E T I C A L M O D E L A N D N U M E R I C A L C A L C U L A T I O N

1 . M a t h e m a t i c a l M o d e l

The fo l lowing Equat ion O f M ot ion(EO M) is bas ica l ly used wi th the except ion o f the axia l force, which w as proposed

the previous s tudy[2] .

EI c34y (m fV 2) c32Yc3x4 + c3x 2

+ C c3y c32y~ + M ~ - q ( x , t )c3t c3t2

(1)

Where ,mys the added m ass of f lu id ,

M is the to ta l ma ss(sum of themyand rod mass)

Ot i s the v iscous d amp ing force which w as equated to the sum of the forces F~

def ined as previous s tudy[2].

1 (32y LF - - 2 P D V Z C : c3x2 ( 2 - x ) + Z m : V

+ - p D C D .2 a t

~2Y

~3xOt

c3y 1 c3y+ p D V z C : ~ x + 2 p D V C : Ot

(2 )

T he Cfand Cn in Eq. (2) are the profi le and skin drag respectively for a rod. Since i t is well known that the rod in axi

f low is weak damping, the damp ing te rm in equat ion (1) can be neglec ted for s impl ic ity in obta in ing na tura l f requencies

mode shapes .

Consider ing the uni form mass of a cy l inder and length l suppor ted on equal spr ings of to ta l s t iffness k , as shown

Fig 1 .

M ,I

2

Fig . 1 A Cyl inder suppor ted by equal spr ings a t both ends


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The d e f l e c ti o n equ a t i o n o f t he cy l i n de r was a s sume d t o be

y(x , t ) - 1 ( x ) q l ( t ) + ~ b2 ( x ) q 2 ( t ) (3 )

The spa t ia l equat ions~ 1 - - sin(erx / l)an d ~b2 - 1.0 could be chosen for the equat ion (3) .Us ing Lag range ' s eq ua t i on , t he f o l l owi ng we re o b t a i ned .

4 2ql - l - - -q2-'[-( O l l q l - 0

2. . . . 2

- -q l +q2 + (022q2 - 0

( 4 )

(5)

wh ere 0)11 - - m q- ; n a tu r al f r equency o f the b eam on r i g id suppo r t s

k0 ) 2 2 = - - ;n a t u r a l f r e qu e n c y o f r ig id beam on sp r i n gs

m

Solving these equat ions , the na tura l f requency for the cy l inder suppor ted on the two spr ings can be obta ined f rom

equat ions as fo l lows:

2 2 2T2(--On -- 0) 22 - - 2

(T+ 1 )+ ~(T _ 1)2 32- + 7 ~ - T(6)

w h e re T - ( 0 ) l l ]0 ) 2 2

The de f l e c t io n e q ua t i on ca n be t a ke n a s f o l lo wings ;

y ( x , t ) - q k ( x ) q ( t ) - ( b +sin

(7 )

whe re ,

7g'X) q ( t ) (8)

1

b - ~ r r - 1 ) g T- 1) 2 + -~ 5 -T8 (9 )

2 . N a t u r a l F r e q u e n c y a n d M o d e S h a p e C o m p a r i s o n w it h S S R o d a n d S p r i n g - S u p p o r t e d R o d

For num er ica l ca lcu la t ion , mater ia l p roper t ies and ge om etry of the fue l rod , and spr ing cons tan t range o f

comm erc i a l space r g r id s p r i ng s w e re u sed a s s umma r i z ed i n Tab l e 1 . Th e f ue l r od was a s su m ed t o be i n PWR rea cto r. T

the tempera ture and ve loc i ty range o f the coolant w as assum ed to be 310 C, and 5 to 8 m/s . S ince th is ca lcu la t ion

ba sed on t he s i ng l e mod e ap p r ox ima t i on , on ly t he f i r s t mo de s hape was ca l cu l a t ed an d c ompa red a s shown i n F ig . 2 .

ca lcu la t ion resu l t s we re sum ma r ized in Table 2 . Natura l f req uency in case of the spr ing suppor t w as appeared to incr

gradu al ly up to tha t of SS wi th the increa se o f the spr ing cons tan t . Th eore t ica l ly, the SS can b e cons idered as the sp


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support by the infinite spring constant. Therefore, the calculation results for the spring-supported rod were judged to be

reasonable.

Table 1 Material and Geometry data

Name Property Value(310 C)

Rod

EI 14.03 N/ m 2

Mass per length 0.728 kg/m

Span length 0.522 m

O.D / I. D 9.5 / 8.22 mm

Spr ing Constant 50-~400 kN/m

Table 2 1 st Natural Frequency of SS and Spring Support

Case 1 st Natural Frequency (Hz)

SS-SS 25.35

K = 50 kN/ m 23.52

SpringSupport K = 200 kN/ m 24.86

K = 400 kN/ m 25.10

3 . F l o w - i n d u c e d - v i b r a t i o n A n a l y s i sThe response of the cylinder to the turbulence axial flow was derived at previous study[2]. The result is written as

follows;

1.0

"~ 0.5

.. i

0.0

" ~ K=400 kN/m

0. 0

R o d l e n g t h

Fig. 2 First Mode Shapes of the SS Rod and the Spring-supported Rod


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o o o o

d ~ y y ( ~ , ~ ' , c o ) -4 d 2~ ~ ~ ( ~ ( ~ " ) H m co)H ~ (co)z 2 g/(co)J n em = l n = l

where, Oyy ; Displacem ent PSD

H n - -

E 1 - m f g 2 - M ( o 2 + i 2 ~ M c o c o .

H~ ; Conjugate of H ,

; Frequency Response Function

J n ; Joint acceptance[2 and3]

Z 2 - 1 f 2 ' ~ f 2 ~ 4o [ c d [ o O'[ lcos COS. Io Ex p -0 .2 7 5 - O ' O dO ' dOU

~((.O) ; Power spectral density o f wall pressure[2]

(10)

Vibration d isplacement of the rod can b e obtained by integrating the equation (10) from zero to infinite on the fre

domain. The dam ping mo del and pow er spectral density of wall pressure discussed in the previous study[2] was use

The frequency response function multiplied by its conjugate was calculated according to spring stiffness, and its

are illustrated in Fig. 3.

1 E-5

1 E-6

1 E-7

1 E-8

1 E-9

1E-10

1E-11

1E-12

1E-13

1E-14

I : . : . .. .. .. .. .. . . . . : " ' " ' ' ' ' 1

50 kN/m, v=5 m/s. . . . . 5 0 kN / m , v = 8 m / s

400 kNkn , v=5 m/s. . . . . 4 0 0 k N / m , v = 8 m / s

, I ~ I ~ I , I ,

2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0

Frequency(rad/s)Fig. 3 Frequency Response Function for Spring Support

4 . N u m e r i c a l C a l c u l a t io n R e s u l t s a n d D i s c u s s i o n .

U s i n g t h e d e r i v e d e q u a t i o n s i n s e c t i o n 2 .1 a n d t h e d a t a i n Ta b l e s 1 a n d 2 , t h e d i s p l a c e m e n t P S

f o r b o t h t h e S S r o d a n d t h e s p r i n g - s u p p o r t e d r o d w e r e c a l c u l a t e d f o r f l o w v e l o c i t y o f 5 m / s a n d 8


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a n d t h e i r r e s u l t s w e r e d e p i c t e d i n F i g . 4.

11E'8E - 9 . . . . i

1E-10

15-11!

E 1E-12Ev 1E-13ao913_ 1E-14

t-

1E-15Em 1 E - 1 60

Q. 1E-17

D 1 E - 1 8

1 5 - 1 9

. . . . . . . . I . . . . . . . . I

- - O - - S S -S S , v =5 m / s I % ~- a - s s - s s , v = 8 rn / I

k=50 N/m, v=8 m/s I I

1 E - 2 0 . . . . i10

' I i I I I I I I I I I I i i I I I

1 O0 1000

Frequency(rad/s)

Fig . 4 D i s p l a cem e n t P SD o f B o th SS R od a nd Sp r i n g -sup p o r t R od

The leve l of the d isp lacem ent PSD goes up as the f low ve loc i ty increases and the spr ing cons tan t decreases whi le t

resonant f requencies seem to have no change . The PS D leve l of the spr ing-suppor ted rod i s s ligh t ly h igher than tha t of t

SS rod a t the same f low ve loc i ty.

The v ibra t ion d isp lacement o f the rod in ax ia l f low can be obta ined by in tegra t ing the PSD shown in F ig . 4 f ro

0(zero) to inf in i te theore tica l ly. Ho wever, as repor ted not only in au thor ' s previous s tudy[2] bu t in many, i t was be l ieved th

a one-mode( lS t mode) approxim at ion wo uld be eno ugh to predic t the rod v ibra t ion in ac tua l engineer ing poin t of v ie

Therefore , in th i s s tudy, it was dec ided tha t the in tegra t ion range f rom 0(zero) to 800 rad /s (127 Hz) w as adopted to ge t t

v ibra t ion d isp lacement . The d isp lacements obta ined by doing tha t were i l lus t ra ted in F ig . 5 . The resu l t s came up to th

expecta t ion tha t the maxim um disp lacem ent was obta ined in the case of the cy l inder exposed a t the f low ve loc i ty of 8 m

and suppor ted on the spr ing cons tan t of 50 kN/m, which was the h ighes t f low ve loc i ty and sof tes t spr ing of a l l cases . A

resul t s were no rmal ized by the tha t of the SS rod a t 5 m/s ve loc i ty. The sof te r the suppor t spr ing i s , the la rger the v ibra ti

d isp lacement i s a t the same f low ve loc i ty. This spr ing effec t i s s t ronger a t the low f low ve loc i ty than a t the h igh f lo

ve loc i ty. As i t were , w hen the f low ve loc i ty i s 5m/s , the d isp lacem ent d iscrepancy be tw een the SS rod and the rod suppor t

by the 50 kN /m spr ing goes up to a lmost 20 %. O n the o ther hand, the d iffe rence reduces up to ha l f of i t when the f lo

ve loc i ty co mes up to 8 m/s . I t i s be l ieved tha t in the range of the low f low ve loc i ty, the na tura l f requency and mo de sha

are inf luenced mo re by the spr ing sof tness than by the increase of f low ve loc i ty.

C O N C L U S I O N

An ax ia l - f low- induced v ibra t ion(FIV ) mo del was proposed for a rod suppor ted by two t rans la t iona l spr ings a t bo th


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2. 0

-,-, 1.8t-(1.)Eo

1 .6

._~- o 1 . 4(1.)N

Eo 1.2

Z

1 . 0

- - - m - - S S - S S Ik = 4 0 0 k N / m Ik = 5 0 k N / m I

I i I i I , I

5 6 7 8

FI0w Vel0city(m/s)

Fig . 5 Vibra t ion D isplacement of the Cyl inder as a Funct ion of Mean Flow Veloci ty

ends. The first natural frequency and mode shape functions for the FIV model were derived by using Lagrange's method

based on the single mode approximation. The vibration displacement was calculated by the proposed model for the spring-

supported rod, and the result was com pared with that of the SS rod. I t was co ncluded that the vibration displaceme nt for the

spring-supported rod was larger than that of the SS rod, and the disp lacemen t decreased with increase of the spring constant

Although the vibration displacement due to axial f low appeared to be clearly dominated by the f low velocity and considered

to be small in quanti ty, in case of the PW R rod supp orted by the soft spring, 20 % more d isplaceme nt than the SS rod may

be possibly app reciated in the range of the f low velocity o f 5 to 8m/s, which i s known to be the ve loc i ty range o f the PW R

reactor coolant.

A C K N O W L E D G E M E N T

This pro jec t has been car r ied out under the nuclear R&D program by MO ST.

R E F E R E N C E S

[ 1 ]

[2 1

[3 1

H.S.Ka ng, et al ., "A Study on the Vibrational Beh avior of the Fuel Rods Con tinuously Sup ported by a Rotary and

Bent Spr ing Sys tem, "Korea Sound and Vibra tion Socie ty, May 1998, pp . 454-460.

H.S. Kan g, et al . , "A Study on the Axial-f low-ind uced Vibration o f Fuel Rod, ' Tansactions o f the 15 h International

Conference on SMiRT, Vol. 2, pp. 341- 348, Seoul, Korea, August , 15-20,1999.

S.S. Chen an d W. Wa mbsgan ss, "Parallel-f low-induced vibration o f fuel rods," Nuclear Engine ering and Design 18,

1972, pp .253-278.

Documents

Axial Flow Induced Vibration of a Rod Supported by Two Translational