structural system

Journal of Sound and Vibration (1972) 21(4) 387-398

FREE VIBRATION

OF A COUPLED

FLUID/STRUCTURAL

SYSTEM

S.-S. CHEN

Argonne National Laboratory, 9700 South Cuss Avenue, Argonne, Illinois 60439, U.S.A. (Received 4 January 1972)

The beam-like vibration of a fluid/structural system consisting of a cylindrical rod submerged in an ideal fluid enclosed by,a cylindrical shell is studied analytically. A method is proposed to tInd the natural frequency. Numerical results are presented to show the effects of various parameters, and an approximate expression is given for the fundamental frequency of the coupled system. 1. INTRODUCTION The problem of dynamic interaction between liquid motion and elastic deformation is of fundamental interest in a variety of applications: e.g., bending vibration of piping systems [l], flutter of plates and shells [2], response of fuel tanks [3], vibration of heat-exchanger tubes [4], etc. Accounts of some of the considerable work in this field have been published [3, 51. In this paper, a study of a simple coupled fluid/structural system is described. A method is proposed to find the natural frequency. First, the problem is formulated in terms of the uncoupled, beam modal functions of the rod and shell, and a fluid velocity potential. Next, the structural and fluid aspects of the problem are solved separately in terms of interaction forces. The structure-fluid boundary conditions are then used to obtain the frequency equation. The frequency equation is expressed in terms of the physical properties of the fluid and those of the structures. An approximate explicit expression is given for the fundamental frequency. This study was motivated by the problems in connection with the design of LMFBR (liquid metal-cooled fast breeder reactor) system components. For example, pump tank and internals, sodium isolation valves, control rods, and fuel assemblies are the same types of coupled fluid/structural systems as described in this paper. Determining the natural frequency of these systems is necessary in design to avoid the detrimental vibrations induced by flowing fluid, and rotating and reciprocating machinery. In some cases, the fluid is flowing, but the effect of the flow velocity on natural frequencies is small in practical reactor design [6]. Consequently, the flow velocity may be neglected in the study of free vibration.

2. PROBLEM STATEMENT Consider a fluid/structural system which consists of a cylindrical rod and a cylindrical shell located concentrically. The annular region is filled with incompressible frictionless fluid (see Figure 1). The rod radius is R and the inside radius of the shell is R. The rod and shell have the same length, 1, which is much larger than R. The beam-like vibration is studied: i.e., the rod and the outer shell are considered as Euler-Bernoulli beams. The governing equations of motion are &&na2y=p

ax4

387

at2

(1)

388

S.-S. CHEN

and

where EI is the flexural rigidity, y is the transverse displacement, x is the axial coordinate, t is the time, m is the mass per unit length, and P is the resultant force per unit length of fluid

Figure 1. A

rod submerged in an ideal fluid enclosed by a cylindrical shell.

acting on the structures.? The notations without bars denote rod variables while those with bars denote shell variables. As an example, simply-supported conditions are considered; thus, y=$=O

at

x = 0,

y=$=O

at

x = I,

j=g=O

at

x = 0,

2Y jj=&i=O

at

x = 1.

(4)

The forces P and P are attributed to fluid motion; therefore, the motion of fluid in the annular region has to be studied. Assuming an irrotational flow, we can define a velocity potential Y such that the fluid particle velocity vector can be expressed as V=V?P(V=z~,e,+v,e,+u~e~).

(5)

From the continuity equation for an incompressible fluid, Laplace’s equation is obtained: VZY=O.

(6) The fluid boundaries are the rod surface and shell inner surface; thus, the normal velocities of the fluid are equal to those of the structures : L;

r

=~cosB at at

v ==%ose r at

at

r = R,

r=R.

(7)

At the two ends, either fluid pressure or fluid velocity has to be specified. As an example, the ends are taken to be stationary; thus, V, = 0

at

x = 0,

U, = 0

at

x = 1.

t A list of nomenclature is given in the Appendix.

(8)

COUPLED FLUID/STRUCTURAL

SYSTEM

389

Finally, the forces P and P are given by

P(x,r)=-R~p(x,r,e,t),,_,cosede, 0

&x, t) = I?

7

p(x,

I,

0, t)l,_?r cos 0d0,

(9

0

where p(x, r, 8, t) is the fluid pressure, which is given by p=-P,,,

ay

and p is the fluid density. Equations (1) to (10) are the complete mathematical statement of the system. The objective is to find the dynamic characteristics of this fluid/structural system.

3. FREQUENCY EQUATION For free vibration, let y(x, t) = Q(x) efSdr, jj(x, t) = G(x) elRr, 1

(11)

P(x, t) = Q(x) eisar,

(12)

P(x, t) = Q(x) ei”‘, 1

where Sz is the circular frequency. Substitution of equations (11) and (12) into equations (l), (2), (3) and (4) yields (13)

@2!?=0

at

x = 0,

@2%0

at

x = 1,

dx2 dx2

I

(14)

I (15)

$cd2=() dx2 &=*=O dx2

at

x = 0,

at

x = 1.

(16) I

Q(x) and 6(x) are modal functions of the rod and shell with fluid coupling. In general they are different from those without fluid coupling. In this analysis, G(x) and 6(x) are represented in series form as superpositions of the uncoupled rod and shell modal functions : i.e., @(x) = *z, =i Mx),

390

S.-S.

CHEN

Here, the #I(x)‘s are orthonormal modal functions of the rod and the &(x)‘s are orthonormal modal functions of the shell in VCICUO. In other words, &(x) and $ i (x ) are solutions of equations (13) and (14), and (15) and (16), with Q = Q = 0, respectively. Therefore,

where wI and 8, are natural frequencies of the rod and shell in vacua, and & is the Kronecker delta. Next, consider the fluid field. The solution of equation (6) is taken as Y = X(x) Y(r) O(0) e*ot,

(19)

where X, Y, and 0 are solutions of the following equations :

d20

-@+q28

=o.

(20)

Here, s and q are constants to be determined. Solving equations (20) gives X(x) = C sin (sx) + D cos (sx), Y(r) = Al&r)

+ BK,(sr),

O(e) = Esin(q0) + Fcos(qe),

(21)

where Z4and K4 are modified Bessel functions of order q, and A, B, C, D, E and Fare arbitrary constants to be determined from boundary conditions. To satisfy equations (8), c = 0, s = m/l,

n = 0, 1,2, 3, . . ..

(22)

Introducing the axial modal function for the fluid,&(x), such that 1

1 7

s f;(x)

dx = 1,

n = 0, 1,2, 3,.

. .,

(23)

0

for the present case, we have n = 0, n> 1,

f”(X) = 1, f” = 2/2COS(S”X),

s, = ml/l.

(24)

Therefore, equation (19) may be written as Y = nzo D&(x)

Y(r) g(e) elRt.

(25)

COUPLED

FLUID/STRUCTURAL

SYSTEM

391

To satisfy equations (7), E=O, 4= 1,

with respect to r. From equations (21), (25), and (26),

The prime denotes differentiation Y can be written as

where G, and H,, are determined from the following equations :

Multiplying equations (28) byf&), tions (17), gives

integrating with respect to x from 0 to I, and using equa-

GkZ;(skR) + HkK;(skR) = isZ f

aRiai,

J=l

GkZ;(skR)

+ HkK;(skR)

= if2 5 cikJdj, J=l

(29

where

(30) Solving equations (29) yields G

=

iQ

Jz,

IakJaJ

Hk

=

iQ

Jgl

[-%JUJZ;(QR)

Ki(&

1)

-

%J

i-

&J

K;<&

R)l/&

&JdJZ;($R)]/dk,

d,=z;(s,R)K;(s,R)-z;(s,R)K;(s,R).

(31)

The fluid pressure is obtained from equations (10) and (27): i.e., (32) Substituting equation (32) into equations (9), carrying out the integration, and using equations (12) and (31), then yield

S.-S. CHEN

392

where Qn=

[-~,hlR)G(&lR +-G(%~)K(~"R)ll&

0”=

[-I x4l R)JG(%IRI + 1,(S”RI K; (4l ml/d”,

7, =

Fn =

HX~“~)~,(%~) + ~,(hz~)fG(%~)ll& Hd~n&I%R) + ~;(~nR)M~n~)WL.

(34)

With equations (18), substitution of equations (33) into equations (13) and (15) gives

Multiplying the first equation by 4,(x) and the second equation by &(x) in equations (35) and integrating with respect to x from 0 to I yields

(36) where

M = prR2/m, ii2 = pd+i,

Q=-

lrn c R n=O

__ %k

OII %I.

(37)

Equations (36) consist of an infinite number of ordinary equations. However, only a finite number of equations are taken in any particular case according to the desired accuracy. The frequency equation is obtained by requiring the determinant of the coefficient matrix of the unknowns a, and a1 to be equal to zero. It is a functional relation between the frequency Q and system parameters, such as mass ratio, uncoupled frequencies, and geometrical ratios. Therefore, the frequency may be written as or

F(Q, cot, c&, M, i@, l/R,it/R) = 0

(38)

F(SZ,EI, i?i, m, 6, R, R, I, p) = 0.

(39)

The frequency equation is expressed in terms of the physical properties of the structures and

those of the fluid.

COUPLED FLUID/STRUCTURAL

SYSTEM

393

4. ONE-MODE APPROXIMATION

Frequently, the fundamental frequency is of interest. Its value can be estimated by using a one-mode approximation. For the first mode of the rod and shell, from equations (36), the following frequency equation is obtained : 0: - P( 1 + Mb, *) -MC,, i2* A-22 CG:- JP(1 + lr;i6, ,) = O.

-nc,,

w

The coefficients b, , ,6,, , cl ,, and El1 are given by equations (37). By neglecting all other terms except the one for n = 0 in equations (37), and using equations (34), the following approximate expressions are obtained : 61, = [(a* + R*)/(R* - R2)] a:,, b,, = [(R* + R*)/(R* - R*)] t5&, cl1 = -[2R*/(R* - R*)] sol E,,, E, , =

-[2R*/(R* - R*)] sol iio,.

(41)

Substituting equations (41) into equation (40) and solving for Sz gives 522= &J:(l

+ /3& J8) + a:(1 + /3cY& M) ??([0&l + j%;, Kf)

- f_i.$(l+ #&& M)]2 + 4(/3*- 1) a& E;‘ kffl0J: G’:)“2} + (1 +

/3(a& M + iif1 W)

-t a& Gil MD}

(42)

where /I = (R* + R*)/(& - R*). The frequencies given by equation (42) are the first two frequencies of the coupled system; the fundamental frequency is the upper bound of the exact value. Equations (42) are useful in the cases when an estimate of the fundamental frequency is desired. When the two structures are relatively far apart, the coupling becomes small. As R/R becomes infinite, /3 = 1 and the frequencies become Q* = (&I( 1 + IR* = (c$)/(

Ma&),

1 + +J&&).

(43)

These results also can be obtained for a rod immersed in an infinite fluid or a shell containing fluid. 5. NUMERICAL RESULTS AND DISCUSSIONS Numerical results are presented for simply-supported structures. The orthonormal modal functions for the rod and shell are, respectively, fjI = 1/Z sin @x/Z), & = fi

sin (im/J).

(W

The corresponding rod and shell frequencies without fluid coupling are w1 = (At/J*) (EZ/m)“*, cii, = (At/J*) (l?flFi)“*,

(45)

where hl = (in)*. It is more convenient to use dimensionless terms, and thus introduce (1= (m/EZ)“* 52J*.

(46)

394

S.-S. CHEN

Similarly, the uncoupled structure frequencies w1 and wi can also be expressed in dimensionless form : hi = (m/EZ)“‘Oi 1’~ xi = (m/EZ)i’2 w, 12.

(47)

From equations (45) and (47), it is seen that A*= Z&

(48)

where ZL= (i?~m/EZKi)‘12. With these non-dimensional parameters, for simply-supported structures, equation (38) may be written as F(A.7 Ai, ~3 My AY, 113, R/R) = 0.

(49)

The frequency equation is solved numerically. Table I shows the values of (1 obtained from equation (49) for various approximations where, for an n-mode approximation, n terms are included in equations (37). For an n-mode approximation, the frequency equation is a polynomial equation of 2n order; therefore, 2n frequencies can be obtained. From Table 1, it is seen that the rate of convergence is very fast. In general, if n frequencies are of interest, an n-mode approximation will yield results with sufficient accuracy. In particular, if only the fundamental frequency is needed, a one-mode approximation will be acceptable. This can be seen from the values in the first row of Table 1; the difference between the fundamental frequency obtained from a one-mode approximation and that from a five-mode approximation is less than eight percent. TABLE1

Frequencies obtainedfrom

various approximations

Approximations Frequencies

One-mode

Two-mode

Three-mode

Four-mode

Five-mode

(a) l/R = 10, i?/R = 2.0, M = 0.5, A? = 1.0, p = 2.0 (3, (1, fL

7.364 14443

;:

7.052 14.023 30.581

7.052 14-023 28.933

7.044 14,012 28,933

7.044 14.012 28.872

59.178

56.762 71.194

67.345 56.760

67.345 56660

136.201

130.684 128.620

124.072 129.962

248.020

210.160 236.188 397.176

:; ia A:, (b) 118 = 10, R/R = 1.2, M = 0.5, M = 1.0, p = 2.0 i1

14.125 4.854

13.973 4.509

13.973 4.509

13.969 4.499

13.969 4.499

A:

20.579

18.573

18.573

18.503

zi; 2:

56.953

48.742 56.032 129.033

43.621 56.030 126.542 91.326

43.621 55.991 126.542 81.697

231.117

150584 226.096 364.248 -

395

COUPLED FLUID/STRUCTURAL SYSTEM

Figures 2 and 3 show the first five frequencies as functions of the uncoupled frequency ratio CL.The mode shapes for five frequencies at p = 2, those circled on Figure 2, are shown in Figure 4. There exist in-phase modes and out-of-phase modes. When the motions of the two

60

0

I

3 2 /z b;i,/A,)

4

5

Figure 2. The dimensionless natural frequency as a function of uncoupled frequency ratio p for M - 05, M= 1.0,1/R= 10,and R/R = 2.

Figure 3. The dimensionless natural frequency as a function of uncoupled frequency ratio p for M - 0.5, iiT= 1.0, I/R = 10,and RIR = 1.2.

structures are out of phase, the fluid in between has to be displaced; thus, the fluid inertia’s effect is very large. On the other hand, when the two structures are in phase, the coupling effect of the fluid inertia is reduced. Figure 5 shows the variation of frequencies with R/R. As R/R approaches 1, the frequencies of out-of-phase modes approach zero, while those of in-phase modes reach certain limits. 25

396

S.-S. CHEN

As B/R becomes large, the fluid coupling becomes weak and the frequencies approach those of a rod vibrating in an infinite fluid and those of a shell containing fluid. The beam-like motion of a coupled fluid/structural system has been analyzed. The fluid in the annular region has the effects of(i) reducing the natural frequency because of the added

Figure 4, Mode shapes of the structures for M = 0.5, h?i = 1.0, I/R = 10, RIR = 2.0and p = 2.

’

(b)

’ ’ ’ ’ ’” ,C----

___---llzd /

30( ,/-

------

20

:

IO

__-----

i

/

/--------

I

9

IO

I

I1/1111~

IO

R/R

Figure 5. The dimensionless natural frequency as a function of R/R for: (a) M = 03,M = 1*0,1/R= 10 and In-phase modes; --- out-of-phase modes. ~=2;and(b)M=1~0,~=05,I/R=10,and~=15.--,

mass and (ii) introducing the coupling between the two structures. The natural frequencies of the coupled system, in general, are much lower than those of the rod and shell without fluid. Finally, two remarks should be made. First, the theory presented is applicable for small oscillations and for the cases in which Euler-Bernoulli beam theory is valid. Second, the simply-supported end condition has been taken as a vehicle to illustrate the method of analysis.

COUPLEDFLUID/STRUCTURALSYSTEM

397

However, the results can be applied to other end conditions provided that the corresponding axial modal function of fluid and structural orthonormal functions are used. ACKNOWLEDGMENTS I wish to thank G. S. Rosenberg for suggesting this problem. This work was performed under the sponsorship of the Special Technology Branch, Division of Reactor Development and Technology, U.S. Atomic Energy Commission.

REFERENCES 1. S.-S. CHEN 1971 American Society of h4echanical Engineers Paper No. 71- Vibr.-39. Flow-induced instability of an elastic tube. 2. E. H. DOWELL 1970 American Institute of Aeronautics and Astronautics Journal 8,385-399. Panel flutter: a review of the aeroelastic stability of plates and shells. 3. H. N. ABRAMSON and D. D. KANA 1967 Proceedings of Symposium on the Theory of Shells to Honor Lloyd Hamilton Donnell, 255-310. Some recent research on the vibrations of elastic shells containing liquids. 4. H. A. NELMSand C. L. SEGA~ER1969 USAEC Report ORNL-4399. Survey of nuclear reactor system primary circuit heat exchangers. 5. N. N. MOISEYEVand V. V. RUMYANTSEV1968 Dynamic Stability of Bodies Containing Fluid. New York: Springer-Verlag (New York) Inc. 6. S.-S. CHEN and G. S. ROSENBERG1971 USAEC Report ANL-7762. Vibrations and stability of a tube conveying fluid. APPENDIX LIST OF SYMBOLS undetermined coefficients flexural rigidity of the rod flexural rigidity of the shell axial modal function of fluid modified Bessel functions length of the rod and shell mass per unit length of the rod mass per unit length of the shell = pnRZlm = p&/rii fluid pressure resultant force per unit length of fluid acting on the rod resultant force per unit length of fluid acting on the shell radial coordinate radius of the rod inside radius of the shell &I time fluid velocity component in x direction fluid velocity component in r direction fluid velocity component in 8 direction fluid velocity vector axial coordinate transverse displacement of the rod transverse displacement of the shell Rl-+RZ =p R=-RZ frequency parameter of the rod frequency parameter of the shell

398

S.-S. CHEN &77

=

l/2

(-1 FiEI

frequency parameter of the coupled system fluid density orthonormal modal function of the rod orthonormal modal function of the shell modal function of the rod modal function of the shell fluid velocity potential natural frequency of the rod natural frequency of the shell circular frequency of the coupled system