Vibration of nuclear fuel bundles

Nuclear Engineering and Design 35 (1975) 399-422 © North-Holland Publishing Company

VIBRATION OF NUCLEAR FUEL BUNDLES Shodi-sheng CHEN Components Technology Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Received 27 September 1975 Several mathematical models have been proposed for calculating fuel rod responses in axial flows based on a single rod consideration. The spacing between fuel rods in liquid metal fast breeder reactors (LMFBRs) is small; hence fuel rods will interact with one another due to fluid coupling. The objective of this paper is to study the coupled vibration of fuel bundles. To account for the fluid coupling, a computer code (AMASS) is developed to calculate added mass coefficients for a group of circular cylinders based on the potential flow theory. The equations of motion for rod bundles are then derived including hydrodynamic forces, drag forces, fluid pressure, gravity effect, axial tension and damping. Based on the equations, a method of analysis is presented to study the free and forced vibrations of rod bundles. Finally, the method is applied to a typical LMFBR fuel bundle consisting of seven rods.

1. Introduction Flow-induced vibrations of fuel rods have been extensively studied both experimentally and analytically; a brief review is given in ref. [ 1]. The objectives of those studies are to understand the dynamic characteristics of fuel rods subjected to axial flows, to predict fuel rod displacements at a given flow velocity, and to minimize the detrimental effects of flow-induced vibrations. Several mathematical models have been proposed for calculating fuel rod responses; those models are based on the vibration of a single rod not coupling with the rods surrounding it. In a typical design of LMFBR fuel rods, the spacing between fuel rods is small. The vibration of an element will interact the surrounding ones because of fluid coupling. Therefore, any motion of an element in a fuel bundle will excite all other elements and fuel rods will respond as a group rather than as a single element. To the author's knowledge, such coupled modes* of fuel bundle vibration have not been analysed. To model flow-induced vibration o f fuel bundles, fluid coupling effects must be included. The objectives of this paper are to present a general method of analysis for coupled vibration of a group of parallel circular cylinders subjected to fluid flows and to develop a model for fuel bundle vibration. A detailed study is presented for a typical LMFBR seven-rod bundle. In addition to nuclear fuel bundles, other structural components consisting of a group o f circular cylinders, ranging from heat exchanger tubes, piles, parallel pipelines, and bundled transmission lines, frequently experience vortexexcited oscillations, fluidelastic instability, and other types of flow-induced vibrations. Many investigators have studied the dynamics of various types of structural components consisting of multiple cylinders. Those include two parallel cylinders [ 2 - 4 ] . two cylinders located concentrically and separated by a fluid [ 5 - 8 ] , a row of cylinders [ 2 , 9 - 1 3 ] , and a group of cylinders [ 14 20]. Those studies have revealed several complex fluid structural interaction characteristics. Despite the progress being made on the dynamics o f multiple cylinders in a liquid, a general method of analysis is not available. The method of analysis presented in this paper is not only applicable to fuel rod vibrations but is also useful in the study of other structural components.

* A coupled mode refers to a natural mode in which all cylinders in a group vibrate at the same frequency with definite phase relations among the cylinders. On the other hand, an uncoupled mode refers to a natural mode in which only a single cylinder is oscillating, while all others are stationary. 399

S.-S. Chen / Vibration of nuclear .fuel bundles

400 2. Added mass coefficients

2.1. Formulation and solution The added mass coefficients for a row of circular cylinders have been studied previously [13]. The study is based on the potential flow theory. The same method of analysis is applied for rod bundles. Consider the motion of a group of k circular cylinders vibrating in an ideal incompressible fluid, as shown in fig. 1 The axes of the cylinders are prependicular to the x - y plane. Let R / b e the radius of cylinder/and (x/, y/) be the local coordinates associated with cylinder f. The velocity potential associated with the motion of cylinder ], assuming all other cylinders are stationary, can be written

(1)

(9/: n=l~ r.n ) ~'ajn cosnO] + bin sinnOj), 1

where r/and O/are cylindrical coordinates referred to cylinder], and ajn and bfn are arbitrary constants to be determined. The total field at a point in the fluid consists of the partial fields generated by all cylinders, i.e. k

®: S o,..

(2)

/=1

All ~j can be written in terms of the local coordinates associated with cylinder i using the following relationships [13]"

cosnO]=(_l) n ~ r.n!

(n+m

l)!r[ n

m=0 m!(n - 1..)lRn+m q

cos[m0 i

(m + n) ~bi/] ,

O

©O O ©O ©O I

)X

k

Fig. l. A group ot k circular cylinders vibrating in a liquid.

S.-S. Chen / Vibration of nuclear fuel bundles

401

and oo

sinnOj_(_l)n+ 1 ~

(n+m-1)!r m sin[mOi-(m+n)~ij], m=0 m!(n - 1)!R.n. +m tl

r.n

]

(3)

where.R//is the distance between the centers of cylinders i and j, and ~i/is the angle between the x axis and the vector from the center of cylinder i to that of cylinder/. Let the superscript i denote the variable written in terms of the local coordinates associated with cylinder i. Therefore, k ~i = ¢i + ~ * ~ { , j=l /

(4)

where 12" denotes the summation f o r / f r o m 1 to k e x c e p t / = i. Using eqs. (1) and (3) gives ~,

cb~=~

o~ ( - 1 ) n ( n + m -

~

1)!R/n+lr m

{a/nCOS[mO i

m!(n - 1)!R.n. +m tl

n=l m=0

(m+n)~i/l

bjnsin[mOi-(m+n)~i/]}.

(5)

The velocity components of cylinder i in the x and y directions are aui/Bt and 8oi/3t, respectively. The fluid velocity component in the r direction is U. In terms of the local coordinates of cylinder i,

U i : a~i/ar i .

(6)

At the interface of the cylinders and fluid, the following conditions must be satisfied:

Uqri=Ri = (~ui/at) cos 0 i + (ao i/at) sin 0 i ,

i = 1,2, 3 ..... k .

(7)

Substituting eq. (6) into (7) and using eqs. (4) and (5), ain and bin are determined as follows:

ain=~[~i.t--~-+~inZ atl,

and

bin=NL6inl~-+[3inl-~],

(8)

where Otinl, [JIM, "/inl and 6in l are solutions of the following equations: k

(-n)e%l + ~ *

~

(n + m -

1)!Rn- 1Rjm+l

/=1

m=l (n - 1)!(m - 1)!Rg.+m

~,

oo (n + m - 1 ) ! R n - I R jm+l

+

/=1

m=l

k ,

~

(n

1)!(m-1)!R~l .+m ! n-1

(-1)m{ot/m I cos [(m + n)~ij ] + 6jml sin [(m + n) ~ij ] } = 6nl6il,

(-1)rn{~jm l sin [(m + n)~i/] - 6jm l cos [(m + n)~i/] } = O,

m+l

(_n),linl+j~ 1 ~ (n + m - 1 ) ! R i l R 7 'T (-1)m{TimlC°S[( m +n)t~iJ]+ "= =1 (n - l)!(m -1)!Rn. +m " ~3jmlsin[(m + n)t~iJ]}=O' t]

~k,

oo ( n + m _ l ) . R iI n - 1 R;m+l (--n)[Jjnl + j=l m= - - i 1) ~(ni m - - - - 1 , . _ . . ~ ~ (--1)m{7jmlsin[(m + n)t~iJ] -- 13jmlC°S[(m + n)~iJ]} = 6nl6il q i , l = 1 , 2 , 3 .... ,k,

n = 1 , 2 , 3 ..... ~.

(9)

S.-S. Chen / Vibration of nuclearfuel bundles

402

The fluid forces acting on the cylinders can be calculated from fluid pressure p: p=

p (0~,/at),

(lO)

where p is the fluid density. The two components of fluid force acting on cylinder i in the x and y directions are

H i and Vi, respectively; Hi=_ f 0

2~rpl R i cos " OidOi, ]ri =Ri

2~r

Vi=_ f pi RisinOidOi" 0 ri =Ri

(11)

Using eqs. (4), (5), (8), (10) and (11) gives k (~12

Hi= -Plr l=l ~

(

02Ul 0201~, u i l -at- 2 + ail at 2 ]

~ ( R i + R I ~ 2 ( 02ui 02oi \ l/i: -prr l = l \ ~ - - - ! r i l -Ot - 2 + fiil ~t2 ) '

(12)

where

~iI-(Ri+Rl)z4R2 -aill - /=1 ~ n=a\Ri/!

(-1)n n(~jnl C°S [(n + 1)ffi/] + 6/nlsin[(n + 1)ffij])

[jil =-(Ri4R2i + RI)2

(-1)n

--[Jill-- /=1 n=l ~ (~7i/') {

~'~

4R2 ---Till °il=(R i +Rl)2 j=l n=l

ril-(Ri+Rl)2

-Sill-

~

n=l

(RR/].)n+l

n(Tjnlsin[(n + 1)~ki/] - [jjnlCOS[(n + 1)~ij]) ,

} (-- 1)n n(TjnlC°S[( n + l)ffi/] + [jjnlsin[(n + 1)ffi/']) ,

R/ n+l(_l)njt(a/.nlsin[(n+l)~i/]_6illCOS[(n+l)~2i/])

(13)

in which Otil, [jil, Oil and ril are called added mass coefficients and otii, [jii, °ii and Tii are self-added mass coefficients which are proportional to/~he hydrodynamic force acting on cylinder i due to its own accelaration, while the others are mutual-added mass coefficients which are proportional to the hydrodynamic force acting on a cylinder due to the acceleration of another cylinder.

2. 2. Reciprocal relations Consider the case where, in a group of k cylinders, cylinder i is moving with a velocity (aUi/Ot)ei, where ei is a unit vector, and all other cylinders are stationary. The mathematical solution for the fluid velocity potential is given by a, = (aUi/Ot)~i,

(14)

where ~i is the solution to the following problem: V2~bi = O,

Vq~i "n = 8(pi/On =ei.n = n i

onSi,

V~i'n = 0

on all otherS/-

(j--/=i),

and certain infinity conditions, where n is a unit vector normal to the cylinder surface, and S i is the surface of

(15)

S.-S. Chen / Vibration of nuclear fuel bundles

403

cylinder i. The hydrodynamic force acting on cylinder I in the direction of e l is given by Fli=(ffPOinldSl)

O2Ui/~t2).

(16)

Similarly, consider the case where all cylinders are stationary except cylinder l moving in the direction of e l with a velocity aUl/at. The fluid velocity potential is

=(but~at)~l.

(17)

where q~l is the solution of the following problem: 7201 = 0,

~]q~l"n = a~l/an = e I • n = n I

on St,

7¢/. n = 0

on all otherS/

(/4: I).

(18)

The hydrodynamic force acting on cylinder i in the direction o f e i is Fil = ( f f

P¢tnidXi)

(19)

(O2Ul/Ot2).

Si Using eqs. (15), (16), (18) and (19) gives Fil = - ~[il(a2 Ul /at2),

Fli = -.Yli(3Z ui /at2 ),

(20)

where 7it = - P

ff

dSi ,

vii= - o f f

Si

a¢l dS-l"

(21 )

Sl

Note that 4~i and ¢1 are two harmonic functions. According to Green's theorem, ff

act ¢i-~- dSo=

So

r r a¢i JJ~t ~ dSo

(22)

So

holds for any surface S O enclosing a region in which ~72q~i and V2¢l are zero. Consider the region between the surface S r enclosing ]~k=lSj and apply eq. (22) to the functions ¢i and el:

~faia~-~ntdS-l¢f-lg-YSl

Si

a¢i dS t"=ffSr(¢iaac--~tn-~t--~n) ~¢i~ dS r .

(23)

Let S r go to infinity. The integral over S r must vanish. Therefore, a¢i f f ~iaJ-~ndSt=f f ~t-g-ydSt. Sl

(24)

Si

It follows from eqs. (21) and (24) that "/i1 = ~1i"

(25)

Let Uie i = uie x

and

Utet = Oley ,

(26)

404

S.-S. Chert / Vibration o]' nuclear fuel bundles

then fron~ eqs. (12) and (20), eq. (25) is reduced to Oil = rli.

(2 7)

Similarly, let U i e i and U 1 e I be equal to other components of the cylinder displacements. It is shown that ail = ali

and

13il = 131i"

(28)

Eqs. (27) and (28) are the reciprocal relations. Physically, these mean that the hydrodynamic force acting on cylinder i in the e i direction due to a unit acceleration of cylinder I in the e l direction is equal to the hydrodynamic force acting on cylinder l in the e I direction due to a unit acceleration of cylinder i in the e i direction. The added mass coefficients ail , flil, °il and ril can be combined into a single added mass matrix 7pq, where

[],q

I tR/+R''2' RiR,,2 a;, ;

°'' (29)

2 I /Ri+R, 2

IR,+R, x2 r

Since 7pq is symmetric, for a group of k cylinders, there are k ( 2 k + 1) independent added mass components. It is possible to find a group of 2k principal axes such that ~[pq = 0

for p :/= q.

(30)

Let the eigenvalues and eigenvectors of [Tpq] be pp and {ap} (p = 1,2, 3, ..., 2k) respectively; one has the relation [Tpq] {aq} = Up{ap}.

(31)

As an example, consider the case of two cylinders with the same radius R. Assume that the cylinders are located on the x axis. In this case, it is found that [13] Oil = 7"]1 = 0,

a12 = a21 =-1312 = -1321-

a l l = a22 = 1311 = 1322,

(32)

Hence, the added mass matrix can be written all

[Tpq] = 0 rrR2

fi

a12

a12 ' a l l 0 0

kO

0

0 0 all --a12

~

.

(33)

--a12 a11A

It is found that the principal values of the added mass matrix are ,Ul = p r r R Z ( a l l -- a12),

P2 = p r r R Z ( a l l - a12),

P3 = PrrRZ(al i + a12),

P4 = prrR2(al 1 + at2). (34)

Those values may be considered as effective added masses for a group of cylinders. The physical meaning is as follows. In a group of cylinders consisting of identical cylinders, the natural frequencies of the group are equal to the frequency of the cylinder in vacuo multiplied by a factor { m / ( m + pp)}l/2, where m is cylinder mass. This will be shown in subsection 4.2. When all cylinders are identical, the principal values of added mass matrix are proportional to the principal values of added mass coefficient matrix. In this case, it is only necessary to consider the added mass coefficient ma-

$.-S.

Chen / Vibration of nuclear fuel bundles

405

trix. It can be seen from eqs. (34) that the principal values of the added mass coefficient matrix are/a~ ( = a l l - a12),

j[,/t

l

2 ( = a l l -- a12), #3 ( = a l l + °(12) and #4 ( = a l l + °t12)"

2. 3. Numerical results Based on the analysis, a computer program AMASS is developed for calculating added mass coefficients. This program can be used to calculate all elements of added mass matrix for a group of cylinders in which the cylinders may have different diameters and may be arranged in an arbitrary pattern. Added mass coefficients are in terms of series solutions. A finite number of undetermined coefficients are determined by inverting the matrix formed by truncating the infinite sets of eqs. (9). The added mass coefficients for four identical cylinders and a seven-rod bundle are shown in fig. 2 as functions of the number of terms taken in the calculations. In fig. 2, the cylinders are closely spaced (the g a p - r a d i u s ratio is equal to 0.1) and the convergent rates are relatively small. As the gap between the cylinders increases, fewer terms are needed. In the following calculations, n is taken to be ten. Added mass coefficients depend on Ri, Rij , ~ij and k. For a group of identical cylinders, arranged at equal distance, added mass coefficients depend on the gap only. Fig. 3 shows the added mass coefficients as functions o f g a p - r a d i u s ratio for a three-cylinder bundle. For large G/R, Otii and ~ii approach one, and all other approach zero. As G/R decreases, the magnitudes o f all added mass coefficients increase. Fuel bundles may be arranged in a hexagonal pattern or in an array. The self-added mass coefficient for the

4.0

I

I

I

I

I

I

3.0 2.0

I

I

I

I

I

I

I

I

4

.

0

~

~x

t.l_

j/

,.,

G 1.2

',=,

,,=, 2.5~-/

p~

~ L5

]/

'.-' 2.0 X

-

_

-

1.0

0.6 0-n2 = r2 ~

0.4

1.0

0.2

0

B a r

0.5 i I 3 4

I l 5 6 n

i 7

I 8

I 9 I0

0

i 2

I ] 3 4

t 5

] 6 n

I 7

I 8

] 9 I0

ol 0

q"~-0.2

I

,-'~1

I

I

0,4 0.6 0.8 1.0 GAP-RADIUS RATIO (G/R)

Fig. 2. Variations of added mass coefficients with the number of terms n used in calculations. Fig. 3. Added mass coefficients of three circular cylinders as functions of gap-radius ratio G/R.

I 1.2

1.4

4.0 ~ - ~

I

[

]

[

l

4'0

ii]

I

[ ~

[

]

T

~

.....

i,

t

\

,oi

o, _ ( ! ) q ) q )

'"""",...

8 2.0

2.0

=

=

B.

,0-

1.0-- o

°

THEORY EXPERIMENT(MORETTI 8 LOWERY)

~

• } o

o~ 0

0.2

0.4

I _ ~ l ~ LZ L4

0.6 0.8 LO GAP-RADIUS RATIO (G/R)

, O0

~.S

~

THEORY EXPERIMENT

I 0.~

~

I 0.4

( M O R E T T I 8 LOWERY)

I

I

I

I

I

0.6

0.8

1.0

1.2

1.4

1.6

GAP-RADIUS RATIO( G x / R )

Fig. 4. Theoretical and experimental values of added mass coefficients c~Â1 for a seven-rod bundle. Fig. 5. Theoretical and experimental values of added mass coefficients c~] 1 and ¢311 for a nine-rod array

i¸ - -

[

r~

i

•

i

1 I

I

I

I

@G

,=,8

I

i

2R

z uJ

G 11

i

5

"#1

~7

/

/~Y2'Y-3

L~ L~

j

7

6

~6

-YI

~z,Y3 Cb ,.=:::E

t

~3 '

~,~,,.,.,

3

,

,

\ ~,,' ' ,.

rll y

y

½

o.s o8 I.o 12 1.4 Is o GAP-RADIUS RATIO (G/R) Fig. 6. Eigenvalues o f added mass coefficient m a t r i x for a seven-rod bundle. o4

Fig. 7. Eigenvalues of added mass coefficient matrix for a nine-rod array.

02

L.__J

~ #li,#l 2

Yi5 Yj4

YI7' Y'I8

I 02

' ' I0

"-.

YI6

o

' •

,

p9,Y- 0 .

.,

'if8

~

J

i

L

1

J

0.4 0.6 0.8 1.0 12 GAP-RADIUS RATIO (G/R}

14

/6

S.-S. Chen / Vibration of nuclear fuel bundles

407

Table 1 Ratio of the largest principal value of added mass coefficient matrix to the largest self-added mass coefficient for two rod bundles Rod bundle

Gap-radius ratio (G/R)

Self-added mass coefficient (a11)

Principal value of added mhss coefficient matrix (~'1)

ratio (ta'l/all)

Seven-rod bundle

0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 1.4 1.6

3.9355 2.6110 2.0680 1.7704 1.5840 1.4577 1.3006 1.2098 1.1526 1.1145 1.0880

7.6758 5.0770 3.9751 3.3447 2.9307 2.6358 2.2418 1.9896 1.8145 1.6860 1.5880

1.9504 1.9445 1.9222 1.8892 1.8502 1.8082 1.7237 1.6446 1.5743 1.5128 1.4596

Nine-rod array

0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 1.4 1.6

3.1697 2.2330 1.8347 1.6105 1.4672 1.3687 1.2443 1.1714 1.1251 1.0941 1.0724

7.4923 4.9331 3.8542 3.2403 2.8391 2.5547 2.1768 1.9364 1.7701 1.6484 1.5558

2.3637 2.2092 2.1007 2.0120 1.9350 1.8665 1.7494 1.6531 1.5732 1.5066 1.4508

central element is o f interest. Figs. 4 and 5 show the values of the coefficient as functions of g a p - r a d i u s ratio as well as the experimental data obtained by Moretti and Lowery [21]. In both cases Oll = "/'11 = 0. However, i311 is always equal to a l l in a hexagonal bundle and/311 is equal to a l l only for G x -- Gy in an array. The analytical resuits and experimental data (the experimental data presented in fig. 5 are for G x = Gy) agree well over the range of parameters t e s t e d except that the experimental values are consistently higher. The principal values of added mass coefficient matrix for a seven-rod bundle and a nine-rod array are presented in figs. 6 and 7 as functions of g a p - r a d i u s ratio. Note that the principal values are not distinct; t'here are five repeated roots in each case. This is due to the symmetry of arrangement. If the rods are arbitrarily arranged and are of different sizes, the eigenvalues will be distinct. The largest eigenvalues of the added mass coefficient matrix ~u] is of particular interest since it is associated with the lowest natural frequency of coupled modes. The ratios of/1'1 to the largest value of the self-added mass coefficient Oll are presented in table 1. The ratio depends on the g a p - r a d i u s ratio G/R, rod number and rod arrangement. Moretti and Lowery [21] suggest that 2Oll + 1 can be used as an upper limit for/2]. From table 1 it is found that/a] < ( 2 O l l + 1) except for the nine-rod array at G/R = 0.1.

3. Equations of motion of a group of circular cylinders in axial flow Consider a rod in a group o f k circular cylindrical rods immersed in a fluid flowing at a velocity V parallel to the z axis (fig. 8). The rod has linear density (mass per unit length) m, flexural rigidity/~7 and total length l. A small

S.-S. Chert / Vibration of nuclear/uel bundles

408

/ T

x

(m 02UI+ V o + FN _

Hi_ Fp-f ) ~Z

Ot 2

2, /

0

Fig. 8. A group of circular cylindrical rods in axial flow.

z

Fig. 9. An element 8z of a cylinder.

element 6z of the rod is shown in fig. 9, where tile elastic forces, hydrodynamic forces, damping force and excitation forces associated with the motion in tile x--z plane are given. In fig. 9, T is the axial tension, Q is the shear force, M is tile bending moment, m(a2u/at2)6z is the inertia force of the rod (u is the rod displacement in the x direction), f 6 z is the external force acting on the rod surface, andl mg is the rod gravity. These forces are quite obvious; however, the others require some explanations.

3.1. Fluid pressure It is known that the effect of uniform fluid pressure on a cylindrical rod is equivalent to applying an axial tension pA, where p is the fluid pressure and A is the cross-sectional area of the rod. Newland [22] has determined the pressure effect for nonuniform pressure; the force acting on the rod is

Fp = (~/Oz ) [pA (Ou/az)] .

(35 )

3. 2. Hydrodynamic force The hydrodynamic forces acting on a cylinder are given in eqs. (12) for stationary fluid. The resultant force in the x direction is given by k

Hi =

pnR 2 ~ [~il(a2ul/at 2) + Oil(a2Ul/at2)]. l=l

(36)

When the fluid is flowing with a velocity V, the hydrodynamic force can be calculated following Lighthill [23] : k l=l

S.-S. Cben / Vibration of nuclear fuel bundles

409

3.3. Viscous damping force The force representing the viscous damping effect can be expressed as F D = C(Ou/at),

(38)

where C is an effective viscous damping coefficient. This coefficient can be determined experimentally by exciting the cylinder in a stationary fluid. 3.4. Dragforces The forces acting on a single rod set obliquely to a stream of fluid were discussed by Taylor [24]. For rough cylinders, he proposed the following expressions for normal and tangential drag forces: F N = o R V 2 ( 6 ' s i n O + C ' s i n 20)

and

F L=pRV2Ccos0,

(39)

where 0 is the angle between the axis of cylinder and flow direction, and C' and C' are drag coefficients associated wiht skin friction and pressure, respectively. Taylor's model is developed for a single cylinder in an infinite fluid. In a group of cylinders, mutual interaction will occur and the drag forces will be different from a single cylinder. To the author's knowledge, no general method of solution and experimental data are available for drag forces acting on a group of cylinders. Therefore, Taylor's model is used for multiple cylinders. The drag forces in eqs. (39) can be interpreted as interference drag. For small oscillations, the angle of incidence can be approximated by 0 = (1/V)[(Ou/Ot) + V(Ou/Oz)].

(40)

Using eqs. (39) and (40) and neglecting the high order terms yields FN=ORVC[(Ou/Ot)+ V(Ou/Oz)]

and

FL =O R V 2 ~ .

(41)

Having all the force components, we are in a position to derive the equations of motion. The equations for translational equilibrium in the z and x directions, and rotational equilibrium are (OT/~z) - m g + F L = O,

aQ aZ

F N + H i + F p + F L a U-+~

~-~[T~U~bz ~ az ] F D - m - a2u - =2

(42,43)

and Q = -(bM/~z).

(44)

The rod material is postulated to obey a stress-strain relationship of the Kelvin type. Classical beam theory gives M = El(a2u/3z 2) + lal(O3u/ataz2),

(45)

where/a is the internal datmping coefficient and I is the moment of inertia. Using eqs. (44) and (45) gives Q = - E l ( a 3 u / 3 z 3 ) - lal(a4u/ataz3) •

(46)

Integrating eq. (42) and using eq. (41) yields T(z) = T(l) + (oR V2C - mg)(l - z),

(47)

41 o

S.-S. Chen / Vibration of nuclear fuel bundles

where T(l) is the axial tension at the downstream end. Substituting eqs. (46) and (47) into eq. (43) yields k

ElO4U+ol ~Su +PzrR2 ~ az 4 atOz 4 l=1 X

-

mg+

il

+V

A x-+pRVC

Ul+Oil

+

+V-~

+

+m

Ot2

v - [T(1)-mg(l-z)+pRV2C(l

=f.

z)+pA]

(48)

Eq. (48) is the equation of motion of cylinder i in the x - z plane. Similarly the equation of motion in t h e y z plane can be derived. For convenience, a subscript p will be used to denote variables associated with cylinder p in the x - z plane, while p + k in the y - z plane. For example, Up(Z, t),Lplp, Cp andfp are the displacement, flexural rigidity, damping coefficient, and excitation in the x direction, and the corresponding quantities in the y direction are Uk+p, Ek+plk+p, Ck+p and fk+p. Using these notations, we obtain the equations of motion for a group of k cylinders as

~4Up O5Up ~ ~fpq -~+ V Eplp ~ z 4 + laplp ~t~z 4 + q=l ap

aUp

2Uq

_ [ OUp

- [Tp(l) -- ( m p g - pRpV2Cp)(l-- z) + pAp] Oz 2

aUp

OUp

O2Up p, q -- 1,2, 3 ..... 2k.

(49)

The appropriate boundary conditions associated with the equations of motion are: forz = 0,

kpup + EplpO3Up/az 3) =

0;

Cp(aUp/~Z) -Eplp(O2Up/OZ2) = O;

(50)

and for z = l,

kpup - Eplp(a3Up/OZ 3) = O, t

t

Cp(Ottp/OZ)+ Eplp(~2ttp/OZ 2) = O, t

"

t

where Cp and Cp are torsional spring constants, and kp and kp are displacement spring constants. Various forms of equations of motion for a single cylinder in axial flow have been derived [25]. Those equations have been successfully employed to study the responses of single rod to flow excitations. On the other hand, there has been only one attempt by Paidoussis to derive the equations of motion for a group of cylinders [26]. Since the hydrodynamic coupling effect is not included in the equation, it cannot be used to study coupled vibrations of a group of cylinders. Eqs. (49) include the fluid coupling; hence, it can be used for studies of both stability and response problems of fuel bundles.

4. Analysis

4.1. Flowing fluid In this analysis, it is assumed that all cylinders are of the same length and have the same type of boundary conditions. Let

Up(Z,t): ~ hpn(t)t~n(Z), n=l

(51)

S.-S. Chen/ Vibration of nuclear fuel bundles

411

where ~n(Z) is the nth orthonormal function of the cylinders in vacuo, i.e. l

} f ~ m ( Z ) qjn(z)dz=6mn,

(52)

0 where l is the length of the cylinders. Using eqs. (49), (51) and (52) gives • e

•

apnqmhqm+bpnqmhqm+Cpnqmhqm=fpn,

p , q = 1 , 2 , 3 ..... 2k,

m,n= 1,2, .. 0%

(53)

where

apnqm = (mp 6pq + 7pq) 6mn,

bpnqm = [Oap/Ep)mp 6O2n+ pRp VCp + Cp]~pq 6mn + 2 g~[pq Cmn,

Cpnqm = mp 6O2n6pq 6mn - {[Tp(D + PoAp - (mpg - pRp V2Cp)1] dmn + [mpg -- pRp V2Cp - Pl Ap] emn}6pq (54)

+ V27pqdmn + (pRp V2Cp - mpg + P l Ap)6pq Cmn, 1 /(~m

Cmn = [ J - ~ 0

~n dZ,

.i, l 1~O2~m emn=TJo~z2~nzdz'

1j i ~ 2 ~ m

dmn =- - ~n dZ, 1 0 ~z2

1 l

IPn= 7fOf,p

dz,

and ¢.Opnis the nth natural frequency of cylinder p in vacuo. The fluid pressure has been assumed to have the form P = PO - Pl z, where P0 corresponds to the fluid pressure at the upstream end and Pl is the pressure gradient. Eqs. (53) consists of an infinite number of differential equations. However, typically only a finite number of equations are selected from case to case according to the desired accuracy. It is assumed that the maximum value for m and n in eqs. (53) is taken to be N. Eq. (53) may be written in matrix form: [B]{S) + [D]{S) + [K]{S} = {L},

(55)

where B, D and K are square matrices and S and L column matrices. The dimension is (2k X N). When the fluid is flowing, K is not necessarily symmetric and positive definite and D is not necessarily proportional to B or K. Eq. (55) can be written [6]{~) + [3](~) = {7},

(56)

where [a] =

[o ;1 [0 :] ,

[3] =

K

,

(~} =

,

~

{7) =

°/

•

(57)

L

Then the damped free vibration mode shapes and mode values are obtained by solving [Xa + 31 (X} = 0.

(58)

The adjoint eigenvalue system is [ x ~ r + ~ T] ( y ) = 0.

(59)

The solution of eqs. (58) and (59) can be achieved by standard procedures. Assuming that the modal matrice obtained from eqs. (58) and (59) are [P] and [A 1, respectively, let

{~) = [P]{W).

(60)

412

S.-S. Chen/ Vibrationo] nuclearfuel bundles

Substituting eq. (60) into (56) and using the biorthogonality condition, one has

lo/]{w} + [y]{w} = [~TI{-~},

(61)

where ~' and/3' are diagonal and hence eq. (61) is uncoupled and easily solved. The eigenvalue obtained from eq. (58) is equal to i~2, where ~2 is the natural frequency of coupled modes. The dynamic behavior of the system is determined by ~2: (1) when f2 is real, the system performs undamped oscillations; (2) when ~ is complex having a positive imaginary part, the system performs damped oscillations; and (3) when ~2 is complex having a negative imaginary part, the system loses stability by buckling or flutter. If postinstability characteristics are of interest, a nonlinear theory including large deformation has to be developed. However, in the stable range, system responses can be calculated from eqs. (51), (60) and (61).

4.2. Stationary fluid Considerable simplification can be made when fluid is stationary and internal damping, gravity effect and fluid pressure are nelgected. In this case, eqs. (49) become

Eplp o4up OUp a2Up --2q~ 1 O2Uqat 2 - .[p, - - 4 + Cp -ffi- + mp ~t2 = 3'pq

p, q = 1, ~, 3,

.....

2k.

(62)

Following the procedure as in the case of flowing fluid, using eqs. (51), (52) and (62) yields 2k

mphpn+

,ypq n+2mp~pnWpn

n+mpco2nhpn=Jpn,

p,q= 1,2,3 ..... 2k,

=

where copn, a s defined previously, is the nth frequency of cylinder

a = 1,2,3 ..... ~ , (63)

p in vacuo and

l

~pn = Cp/2mp copn

Cpn

~n dz.

(64)

0 Note that eqs. (63) can be applied to all values of n. For each n, there are 2k equations which are coupled. However, there is no coupling among the equations for different n. This is true for a group of cylinders with the same type of boundary conditions and of the same length. For a given n, eqs. (63) may be written

[B]{iz') + [D]{/~) + [K]{h} = (L}.

(65)

For free vibration, neglect the damping and forcing terms and let {h } = {,q } exp (i~2 t).

(66)

Natural frequencies and mode shapes can be calculated from the undamped homogeneous equations [K]{h} = ~2 [B]{h}.

(67)

K is a diagonal matrix and M is symmetric. Let [P] be the weighted modal matrix formed from the columns of eigenvectors. It is easily shown that

[pTBp] = [I]

and

[pTKp] = [A],

(68)

where [1] is an identity matrix and [A] is a diagonal matrix formed from the eigenvalue ~22. When all cylinders are identical, and have the same properties in the x and y directions,

{Onp= COn, mp = m,

p = 1,2, ..., 2k.

(69)

S.-S. Chen / Vibration of nuclear fuel bundles

413

In this case, eqs. (67) may be written

["[pq](hq) = lAp(lip),

(70)

where Up = m (6o2 - ~2)/~22.

(71)

Eq. (70) is identical to eq. (31) i.e. the eigenvectors of the added mass matrix are the same as the mode shapes of the couplbd modes and the eigenvalues of the added mass matrix are related to the natural frequencies by eq. (71). Corresponding to each eigenvalue/.tp, the natural frequency of the coupled mode is ~2p = [m/(m + Up)] 1/2 (.On.

(72)

Eq. (72) shows that the natural frequency of the coupled mode is reduced in proportion to [m/(m +/ap)] 1/2. This is is similar to that of a single structure in a liquid; therefore,/ap may be interpreted as an effective added mass. The response to an excitation can be calculated from eqs. (65) and the associated initial conditions. For practical consideration, the damping matrix [ D ] may be assumed to be proportional to the stiffness matrix [ K ]. In this case eqs. (65) can be reduced to a set of 2k uncoupled modal equations by letting {h) = [P](W),

(73)

and premultiplying eqs. (65) by the transpose [pT], the result being

[pTBp] ((~) + [pTDp I {I~) + [pTKp I {W} = [PT 1 (L}.

(74)

In eqs. (74), the square matrices on the left-hand side are diagonal matrices. Thus, each equation reduces to that of a single oscillator and has the form 2k

(75) where ~pn is the frequency of coupled mode and r~jn is the corresponding damping ratio. Eq. (75) is easily solved and the displacement and other quantities of interest can be calculated from eqs. (51). I f D is not proportional to the stiffness matrix, a damped vibration mode superposition method, as shown in the previous section for flowing fluid, can be used.

4.3. Numerical examples The numerical results presented in this section are based on steel tubes whose outside radius is 1.270 cm (1.5 in.), wall thickness 0.1588 cm (0.0625 in.) and length 1.27 m (50 in.) and are simply supported at both ends. In each group of tubes under consideration it is assumed that all tubes are identical. Fig. 10 shows the normal modes of two groups of cylinders consisting of three and four cylinders for n = 1 where G = Gx = 0.127 cm (0.05 in.), and Gy = 0.1905 cm (0.075 in.). For each n, there are 2k normal modes for a group of k cylinders. Fig. 11 shows the frequencies of three tubes as functions of the gap G. As the spacing decreases, the frequencies of lower modes decrease while those of the higher modes increase. When the spacing increases, all frequencies approach that of a single tube in an infinite fluid. Note that there are two repeated frequencies in fig. 11. Corresponding to the second and third frequencies, there are two modes. The modes presented in the figure are orthogonal to each other. However, those are not the only sets of solution; a linear combination of the two modes of the set also satisfy the conditions of orthogonality. Figs. 12 and 13 present the response of a group of four tubes when tube 1 is subjected to an excitation

414

S.-S. Chen / Vibration of nuclear fuel bundles 40

©

I

I

)

/ ? _ 34.003 Hz

-?

_i

22.097Hz

29,505 Hz

30

Gx

G

i_li

33.721 Hz

.3 3-

28.405 Hz

~

~e 7:~ ~*i 2 9 . 9 1 3 Hz

35.876 Hz 36~814

30.809 Hz

\

I

t

/i

:, ....

HZ

,/

/ J

33,347 HI

36,173 HI

31.639 Hz

38.395 Hz

I

O.S

1.0

1.5

GAP (in.)

Fig. 10. Normal modes of three and four cylinders vibrating in a liquid. Fig. l 1. Natural frequencies of a group of three cylinders as functions of gap distance.

gl sin 03 t in the x direction, uniformly distributed along the tube. The tubes are of the same size and simply supported at both ends. The magnification factor is defined as the ratio of the displacement at midspan to that of the deflection of tube 1 in the x direction to a static load of the same magnitude. In fig. 12, G x = 0.127 cm (0.05 in.) and Gy = 0.1905 cm (0.075 in.), the tubes are close to one another and the coupling effect is significant. It is seen that although only tube 1 is subjected to external excitation, there is a significant response in the y direction and the response of tube 4 are comparable with those of tube 1. In fig. 13, G x = 2.54 cm (1 in.) and Gy --- 3.81 cm (1.5 in.), the coupling effect is much smaller. For those tubes not directly excited, the response are small. In this case, the response of tube 1 in the x direction is similar to that of a single tube. To illustrate tile general characteristics of tube bundles to axial flow, fig. 14 shows the complex" frequency (fR + i l l ) plotted as an Argand diagram of a two-tube system. The frequency is obtained from eq. (58), in which )~ = i(27r)(fR + ifI)'

(76) .

The parameters used in computation are as follows: gap distance G = 0.127 cm (0.05 in.); axial tension Tp(/) = 0.0; fluid pressure P0 = 0.0; pressure gradient P l = 0.0; drag coefficient Cp = 0.01 ; internal damping coefficient ~t = 0.0; and external damping coefficient Cp = 6.8948 N sec 1 m - 2 (0.001 l b sec - l in.-2). The first four modes are shown in fig. 14 where the number in the figure indicates the flow velocity in 30.48 m/sec (100 ft/sec). Note that all natural frequencies are located in the upper half of the complex plane when the flow velocity is small, and the system performs damped oscillation in all modes. The effects of the flowing fluid are to reduce the natural frequencies and to contribute to damping; the damping effect is due to drag force. As the flow velocity increases, the real p a r t f R of the first mode becomes zero, and the imaginary p a r t f I becomes negative; the system loses stability by buckling. With a further increase in the flow velocity, the instability becomes

I

I

I

I

I

0~,- x ~

[\ /1 ]~

OLO .LO .LO .LO .LO .L~.~ (),

A

~,--~' a, // / :,

~

I--

L

I

I01

I

__

L

TUBE

i

l

i

I

I

]

I

L

I

I

/V V ~'-~.J x DIRECTION

\

7~

o

~

I

I

-J

L

5 . 4 g,s,nw~

L

i

TUBE

2 I

t

0

l

~y ~-~ Gx

I

j I

i

A/~

i

J

i

]

I

G

~,~

4

3

TUBEI

~L~

3

I y DIRECTION

3

z

i

9i A

4

5

=

-~G x I

0

I

TUBE 4

2

o!

TUBE I y DIRECTION I

L

3

I

2b

/--,.

TUBE 4

TUBE 4 DIRECTION [

I

TUBE 4 ~ R E C T I O N I

0o

I0

0

50 55 60

15 20 25 30 35 40 45 FREQUENCY (Hz)

,;

5

,5

20 25 30 35 40 45 FREQUENCY (Hz)

Fig. 12. Frequency response of four tubes for Gx = 0.127 cm (0.05 in) and Gy = 0.1905 cm (0.075 in). Fig. 13. Frequency response of four tubes for Gx = 2.54 cm (1 in) and Gy = 3.81 cm (1.5 in).

748 13.57.

6.8

I 58

6.7 7.5 15.6

6.7

I 7.641~3. ~

3.9

5.76

,6\ ,X5 6 - - MODE 5 ~

6.8 7.5

I65

L

•.,-~ o

'3.9 13.62.

7.0 7.48

I

7

MOOE4

~

/

~?~',o

-

",o ~o

,J751 ~2~'~-7.55 ~V

,3.8 -4 r

I

.o~ "-%° MODE2 ~ 7

46.65

' 5.76 -2 13.61

r

8

-6

2

-8

.7.65

-I0 20

I

I

I

J

I

40

60

80 fR

I00

120 140 160

I

Fig. 14. Complex frequencies of two tubes, simply supported at both ends in axial flow.

50 55 60

416

S.-S. Chen / Vibration of nuclear fuel bundles

flutter-type. The second mode exhibits similar characteristics as the first mode, In the flow velocity range considered, the third and fourth modes are stable. However, as the flow velocity is further increased, they might become unstable. The effects of various parameters on the dynamic characteristics have also been studied. Internal and external damping forces obviously contribute to damping and axial tension tends to increase the systems natural frequencies. The effects of fluid pressure and drag force are as follows: (1) Increasing fluid pressure tends to increase natural frequencies; its effect is the same as applying an axial tension. Therefore, natural frequencies increase with P0 and decrease with p 1. (2) In the practical ranges of flow velocity, drag forces contribute to damping. A system with higher drag coefficient Cp will have a larger damping factor. 5. Vibration of a typical LMFBR fuel bundle The LMFBR fuel is stored in stainless steel tubes with helical wire wraps around the outside surface. The parameters describing a typical design of LMFBR fuel rod are given in table 2. A seven-rod bundle in sodium, as shown in fig. 15, is considered in this study; vibrations in stationary sodium and flowing sodium are presented. In calculations, the stiffness of fuel and wire wraps are neglected; however, the masses are included. As an approximation, the rods are assumed to be simply supported at both ends and the length is equal to the spacer pitch. The system parameters calculated from table 2 and other parameters used in computation are given in table 3. 5.1. A d d e d mass coefficients

The added mass coefficients are computed by the program AMASS. The results are presented in table 4. Several general characteristics are noted: (1) The self-added mass coefficient for rod 1 is the largest. (2) al 1 and ~11 are the same. Calculations also have been made for the self-added mass coefficient of rod 1 in other directions. It is found that the self-added mass coefficient of rod 1 is independent of the direction of motion. (3) all and ~il are symmetric, and Oil = rip This has been proved in subsection 2.1. (4) aii and ~ii are always positive and larger than one; while oii and 7"ii may be negative. For example, o 11 = 0, i.e. the motion of rod 1 in the x direction does not induce any force acting on itself in the y direction. The principal values of the added mass coefficient matrix are given in table 5. Note that the ratio of the largest principal value ~] and a l l is 1.8554.

©OO Fig. 15. A seven-rod bundle.

S.-S. Chen / Vibration of nuclear fuel bundles

417

Table 2 Fuel rod parameters Outside diameter Wall thickness Rod-to-rod pitch Spacer wire diameter Spacer pitch Fuel density Stainless steel density Young's modulus for stainless steel at 1000°F

0.5842 cm (0.23 in.) 0.0381 cm (0.015 in.) 0.7264 cm (0.286 in.) 0.1422 cm (0.056 in.) 30.48 cm (12 in.) 9.4708 g/cm 3 (0.342 Ib/in. 3) 7.9754 g/cm 3 (0.288 lb/in. 3) 15.8579 × 101Opa (23 × 106 psi)

5. 2. Free vibration in a stationary liquid

The effects of fluid pressure, damping and drag force are temporarily nelgected to understand more clearly the effect of fluid coupling. The fundainental frequency of the individual rod is 65.949 Hz in vacuo and 63.257 Hz in sodium. First, consider the case that only a rod in the group vibrates, while all other rods are stationary. This type of motion is known as uncoupled vibration. The frequency of this type of vibration can be calculated using the self-added mass coefficients. For example, the frequency for rod j vibrating in the x direction is given by the following relationship: frequency of uncoupled mode for rod / = (frequency in vacuo for rod j) X

mass of rod ] ]1/2 ~ mass of rod / + t~)). X displaced mass of fluid by r o d / ]

(77)

The frequencies of uncoupled modes for the seven-rod bundle are given in table 6. The frequencies of coupled modes can be calculated from eq. (67). However, when all rods are identical, as shown in subsection 4.2, the following proceudre can be used to calculate the frequencies of coupled modes: frequency of coupled mode = (frequency in vacuo) X (

mass, of rod ~ 1/2 mass of rod +/alp X displaced mass of fluid ] "

The results are given in fig. 16.

Table 3 System parameters used in calculations Rod moment of inertia (Ip) Rod mass per unit length (mp) Rod length (l) Flexural rigidity (Ep Ip) Axial tension (Tp(l)) Internal damping coefficient (~p) External damping coefficient (Cp) Fluid pressure (Po) Fluid pressure gradient (Pl) Drag coefficient (Cp) Sodium density

2.4474 X 10 -3 cm4 (5.88 X 10 -5 in.4) 0.255,2 kg/m (0.0143 lb/in.) 30.48 cm (12 in.) 9.3245 Pa (1352.4 psi) 0.0 0.0 2.0684 N sec/m 2 (0.0003 lb sec/in. 2) 3.4474 X 10 s Pa (50 psi) 1.6287 Pa/cm (0.6 lb/in. 3) 0.01 0.8205 g/cm 3 (51.2 lb/ft 3)

(78)

418

S.-S. Chen / Vibration o f nuclear fuel bundles

Table 4 Added mass coefficients for a seven-rod bundle

[eql ] =

1.6042 0.1405 0.1405 -0.5140 0.1405 0.1405 -0.5140

0.1405 1.2753 -0.4376 -0.1383 0.0152 0.0843 0.1846

0.1405 -0.4376 1.2753 0.1846 0.0843 0.0152 -0.1383

-0.5140 0.1383 0.1846 1.3709 0.1846 -0.1383 0.1497

0.1405 0.0152 0.0843 0.1846 1.2753 0.4376 0.1383

0.1405 0.0843 0.0152 0.1383 -0.4376 1.2753 0.1846

0.5140 0.1846 -0.1383 - 0.1497 -0.1383 0.1846 1.3709

[aill =

0.0 -0.3779 0.3779 0.0 -0.3779 0.3779 0.0

-0.3779 0.0552 -0.1282 -0.1667 -0.0952 -0.0382 0.4874

0.3779 0.1282 -0.0552 --0.4874 -0.0382 0.0952 0.1667

0.0 0.0904 0.2310 0.0 0.2310 0.0904 0.0

0.3779 0.0952 0.0382 0.4874 0.0552 -0.1282 0.1667

0.3779 -0.0382 0.0952 0.1667 0.1282 0.0552 -0.4874

0.0 0.2310 0.0904 0.0 0.0904 -0.2310 0.0

[r/l ] =

0.0 -0.3779 0.3779 0.0 -0.3779 0.3779 0.0

-0.3779 0.0552 0.1282 -0.0904 -0.0952 -0.0382 0.2310

0.3779 0.1282 -0.0552 -0.2310 0.0382 0.0952 0.0904

0.0 --0.1667 -0.4874 0.0 0.4874: 0.1667 0.0

-0.3779 0.0952 -0.0382 - 0.2310 0.0552 0.1282 -0.0904

0.3779 0.0382 0.0952 0.0904 0.1282 --0.0552 -0.2310

0.0 0.4874 0.1667 0.0 -0.1667 0.4874 0.0

[~il] =

1.6042 0.2958 -0.2959 0.3587 -0.2958 -0.2958 0.3587

-0.2958 1.3390 0.3920 0.0101 -0.0947 -0.2125 -0.2302

-0.2959 0.3920 1.3390 -0.2302 ¢ -0.2125 -0.0947 -0.0101

0.3587 0.0101 -0.2302 1.2434 -0.2302 0.0101 0.0701

-0.2958 0.0947 - 0.2125 -0.2302 1.3390 0.3920 0.0101

0.2959 -0.2125 -0.0947 0.0101 0.3920 1.3390 --0.2302

0.3587 -0.2302 0.0101 0.0701 0.0101 -0.2302 1.2434

Tile m o d e shape o f an u n c o u p l e d m o d e is the same as t h a t o f an isolated rod. H o w e v e r , the m o d e shape for a coupled m o d e is m u c h m o r e c o m p l i c a t e d . The n o r m a l m o d e s for the seven-rod b u n d l e are given in fig. 16. N o t e that there are 14 natural f r e q u e n c i e s w h i c h c o r r e s p o n d to a single f r e q u e n c y in the case o f an isolated rod. A m o n g 14 frequencies, there are five pairs o f r e p e a t e d f r e q u e n c y . These are associated w i t h the s y m m e t r y o f a r r a n g e m e n t . The lowest f r e q u e n c y is associated with the m o d e in w h i c h the central rod is s t a t i o n a r y and all

Table 5 Principal values of the added mass coefficient matrix for a seven-rod bundle. ul = 2.9765 U2 = 2.6800 u3 = 2.6800 /~4 = 1.7771 /as = 1.5803 u6 = 1.5803 u7 = 1.0420

u8 = 1.0420 ,u9 = 0.7840 .Ulo = 0.7840 #11 = 0.6194 /.t12 = 0.4860 tal3 = 0.4860 /214 = 0.3764

S.-S. Chen / Vibration of nuclear fuel bundles

419

Table 6 Natural frequencies of a seven-rod bundle (uncoupled vibration). Rod number

1 2 3 4 5 6 7

Natural frequencies assuming all other rods are stationary Vibration in the x direction

Vibration in the y direction

61.782 62.573 62.573 62.340 62.573 62.573 62.340

61.782 62.417 62.417 62.651 62.417 62.417 62.651

peripheral rods vibrate in the direction along the radial line connecting the centers of the rod and the central rod. In this mode, since the fluid has to be displaced considerably, the effective added mass is large. On the other hand, associated with the fourteenth mode, all peripheral rods vibrate perpendicularly to the radial lines and are in phase; the added mass effect is much smaller

5. 3. The effects o f sodium f l o w Vibration frequencies for rod bundles subjected to fluid flows can be calculated from eq. (58). Table 7 gives the vibration frequencies for the seven-rod bundle for several flow velozities. The real and imaginary parts of frequencies, fR and fI (see eq. 76), are given by the upper and lower numbers in each group; fR and f l are related

2

5 8 . 7 8 3 HZ

59.394

61.839

63.151

64.242

3

4

5

591394

61.379

611839

63.811

63.611

63.151

64.599

64.599

64.896

Fig. 16. Normal modes of a seven-rod bundle vibrating in a liquid.

X-X Chen / Vibration of nuclear fuel bundles

420

Table 7 Natural frequencies of a seven-rod bundle at several flow velocities. Flow velocity

0

25 ft/sec

50 ft/sec

75 ft/sec

100 ft/sec

1st mode

59.361 0.51261

59.094 0.55803

58.287 0.60298

56.920 0.64733

54.957 0.69097

2nd mode

59.979 0.52333

59.738 0.56974

59.011 0.61574

57.783 0.66125

56.023 0.70615

3rd mode

59.979 0.52333

59.738 0.56974

59.011 0.61574

57.783 0.66125

56.023 0.70615

4th mode

61.983 0.55890

61.825 0.60856

61.350 0.65802

60.551 0.70723

59.416 0.75614

5th mode

62.448 0.56731

62.309 0.61773

61.890 0.66799

61.186 0.71805

60.188 0.76787

6th mode

62.448 0.56731

62.309 0.61773

61.890 0.66799

61.186 0.71805

60.188 0.76787

7th mode

63.773 0.59164

63.685 0.64427

63.422 0.69682

62.982 0.74928

62.361 0.80162

8th mode

63.733 0.59164

63.685 0.64427

63.422 0.69682

62.982 0.74928

62:361 0.80162

9th mode

64.439 0.60406

64.376 0.65781

64.190 0.71151

63.878 0.76516

63.438 0.81874

10th mode

64.439 0.60406

64.376 0.65781

64.190 0.71151

63.878 0.76516

63.438 0.81874

1 lth mode

64.875 0.61226

64.828 0.66674

64.691 0.72120

64.461 0.77562

64.138 0.83000

12th mode

65.235 0.61907

65.202 0.67417

65.104 0.72924

64.941 0.78430

64.712 0.83933

13th mode

65.235 0.61907

65.202 0.67417

65.104 0.72924

64.941 0.78430

64.712 0.83933

14th mode

65.535 0.62478

65.513 0.68039

65.448 0.73599

65.341 0.79157

65.190 0.84714

to the m o d a l damping ratio ~', which is given by

= ( f l / f R ) / [ 1 + ( f i / f R ) 2 ] 1/2.

(79)

Table 7 shows that f R decreases with increasing flow velocity, while fI increases with flow velocity, i.e. as the flow velocity increases, natural frequencies o f all modes decrease and damping ratios increase. The decrease in natural frequencies is due to fluid centrifugal force, while the increase in damping is attributed to fluid drag force.

S.-S. Chen / Vibration of nuclear fuel bundles

421

As can be seen from table 7, the effect of flow on natural frequencies is small for practical flow velocity range, say 7.62 m/sec (25 ft/sec) to 15.24 m/sec (50 ft/sec). If the flow velocity is increased to a certain limit, in this case about 270 ft/sec, the rod bundle becomes unstable by buckling. However, such a high flow velocity will not be encountered in practical situations. Therefore, instabilities associated with parallel flows, in general, are of little concern in the design of practical system components even in the case of rod bundles, except when the unsupported length of the rods is long. 6. Conclusions This paper presents a general method of analysis for coupled vibration of a group of circular cylinders vibrating in liquids; particular emphasis is placed on fuel bundle vibration. This has been the first method ever developed for analysing rod bundle vibration in liquids which can be applied to a group of cylinders arranged in any pattern. With this method of analysis the response of a system consisting of a group of cylinders subjected to fluid flows can readily be analysed. In particular, the mathematical models for fuel bundles subjected to parallel flows and heat-exchanger tube banks subjected to cross flows can be refined by incorporating the interaction effect which all existing models fail to take into account. Efforts are being made in developing better mathematical models for both parallel- and cross-flow induced vibrations of tube bundles. The computer code AMASS can be used to calculate the added mass coefficients for a group of cylinders with different diameters. In the code it is necessary to solve a system of linear equations. The number of equations depends on the number of cylinders and number of terms used in the series solution. The current version of AMASS can be used for many practical system components. The equations of motion derived for fuel bundles include fluid coupling, damping, drag force, centrifugal force, Coriolis force, fluid pressure, gravity effect and axial tension. These equations can be used for analysing the response of fuel bundles to flow noises and establishing stability limits. Several forms of the equations of motion were presented previously; only in one study was the equation developed for fuel bundles [26]. However, the fluid coupling effect is not included; it can be used for uncoupled vibration only. For coupled vibration, eq. (49) will give correct results. Based on the method of analysis for added mass coefficients and the equations of motion for rod bundles, free and forced vibrations of a group of rods are analysed. Numerical results are presented for rod bundles consisting of two, three, four and seven rods. It is noted that in all cases the lowest frequency of coupled modes is always lower than the lowest frequency of uncoupled modes. The implication is that using uncoupled modes for mathematical models of rod bundles may not be conservative. An experiment designed to verify the theory has been planned. A series of tests involving several groups of tubes in stationary fluid and flowing fluid will be performed. One set of tests for two tubes vibrating in stationary water has been completed. Preliminary results show that the theoretical results and experimental data are in good agreement.

Acknowledgement This work was performed under the sponsorship of the Division of Reactor Research and Development, US Energy Research and Development Administration. The author wishes to express his gratitude to Drs. M.W. Wambsganss and T.T. Yeh for their commonts on the report.

References [1] S.S. Chen, Parallel-flow-inducedvibrations and instabilities of cylindrical structures, Shock Vib. Dig. 6 (10) (1974) 2-12. [2] J.L. Livesey and C.F. Dye, Vortex excited vibration of a heat exchanger tube row, J. Mech. Eng. Sci. 4 (4) (1962) 349-352.

422

S.-S. Chen/ Vibration of nuclear/uel bundles

[3] J.F. Wilson and H.M. Caldwell, Force and stability measurements on models of submerged pipelines, Trans. ASM1C .I.l..nf:. Ind. 93 (1971) 1290-1298. [4[ S.S. Chen, Dynamic response of two parallel circular cylinders in a liquid, J. Pressure Vessels Technol. 97 (2) (1975) 77 83. [5] S.S. Chen, Free vibration of a coupled fluid/structural system, J. Sound Vib. 21 (4) (1972) 387- 398. [6] F. Cesari and S. Curioni, Velocity influence on the free vibrations in a coupled system fluid structure, 2nd SMiRT Conf., Sept. 1973, Paper E5/3. [7] S.S. Chen, Dynamics of a rod-shell system conveying fluid, Nucl. Eng. Des. 30 (1974) 223 233. [8] S.S. Chen and G.S. Rosenberg, Dynamics of a coupled shell/fluid system, Nucl. Eng. Des. 32 (1975) 302-310. [9] B.W. Roberts, Low frequency, aeroelastic vibrations in a cascade of circular cylinders, Mechanical Engineering Science Monograph, No. 4, Sept. (1966). [10] J.H. Connors, Jr, Fluidelastic vibration of tube arrays excited by cross flow, Flow induced vibration in heat exchangers, ASME 4 2 - 5 6 (1970). [11 ] R.C.F. Dye, Vortex-excited vibration of a heat exchanger tube row in cross-flow, Proc. Int. Sym. Vibration Problems in hrdustry, Apr. (1973) paper 417. [ 12] R.D. Blevins, Fluid elastic whirling of a tube row, Presented at the Pressure Vessels and Piping Conference, Miami Beach, Florida, 24-28 June, 1974; ASME Paper No. 74-PVP-29. [13] S.S. Chen, Vibration of a row of circular cylinders in a liquid, ANL-CT-75-34 (1975); also to be presented at Fifth National Conference on Mechanical Vibrations, ASME, 1975. [14] A.D.K. Laird and R.P. Warren, Groups of vertical cylinders oscillating in water, J. Eng. Mech. Div., Proc. ASCE 89 (EM 1) (1963) 25- 35. [15] A.D.K. Laird, Flexibility in cylinder groups oscillated in water, J. Waterways Itarbors Div., ASCE 92 (WW3), Aug. (1966) 69-85. [ 16] Y.N. Chen, Flow-induced vibration and noise in tube-bank heat exchangers due to yon Karman streets, J. Eng. Ind. 90 (Series B), Feb. (1968) 134-146. [ 17 ] T. Shimogo and T. lnui, Coupled vibration of many elastic circular bars in water, Proc. 21 st Japan National Congress for Applied Mechanics (1971) 495 505. [ 18] M.M. Zdravkovich, Flow induced vibrations in irregularly staggered tube bundles, and their suppression, Proc. Int. Sym. Vibration Problems in Industry, Apr. (1973) Paper 413. [19] S. Mirza and D.J. Gorman, Experimental and analytical correlation of local driving forces and tire tube response in liquid flow induced vibration of heat exchangers, 2nd SMiRT Conf., Sept. 1973, Paper F6/5. [20] Y.N. Chen, Fluctuating lift forces of the Karman vortex streets on single circular cylinders and in tube bundles, Part 2 criterion for the fluidetastic orbital vibration of tube arrays, ASME Paper No. 73-Det-146 (1973). [21] P.M. Moretti and R.L. Lowery, Hydrodynamic inertia coefficients for a tube surrounded by rigid tubes, Presented atASME PV and P Nat. Congress, San Francisco, Calif., June1975, Paper No. 75-PVP-47. [22] D.E. Newland, Research note: deflection equation for the buckling of an elastic column subjected to surface pressure, J. Mech. Eng. Sci. 15 (1) (1973) 73-75. [23] M.J. Lighthill, Note on swimming of slender fish, J. Fluid Mech. 9 (2) (1960) 305-317. [24] G. Taylor, Analysis of the swimming of long and narrow animals, Proc. Roy. Soc. London A214 (1952) 158-183. [25] S.S. Chen and M.W. Wambsganss, Parallel-flow-induced vibration of fuel rods, Nucl. Eng. Des. 18 (1972) 253--278. [26] M.P. Paidoussis, Dynamics of cylindrical structures subjected to axial flow, J. Sound Vib. 29 (3) (1973) 365-385.