Applied Acoustics 21 (1987) 191-198

A Note on Flexural Vibrations of a Pipeline Containing Flowing Fluid

P. A. A. Laura, G. M. Ficcadenti de Iglesias and P. L. Verniere de lrassar

Institute of Applied Mechanics, Puerto Belgrano Naval Base, 81 ! 1 (Argentina)

(Received 25 February 1986; accepted 9 June 1986)

S UMMA R Y

This paper deals with the determination of natural frequencies of transverse vibration of a simply supported pipeline containing flowing fluid. No claim of originality is made but the results may be of some interest since, apparently, there is an algebraic error in the determination of the series solution of the governing differential system that affects the values of the natural frequencies already available in the technical literature. The effect is considerable for ratios of the parameter fluid mass/total mass of the system approaching unity.

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

Excellent papers, reports and books deal with the important practical problem of fluid-conveying pipes that execute transverse vibrations. 1-4

The purpose of the present paper is simply to point out an apparent algebraic error 1-4 in the determination of the solution of the governing differential equation, which has practically no effect when the ratio fluid mass/fluid plus conduit mass is very small, but carries considerable weight when the value of the previously defined parameter is close to unity.

191 Applied Acoustics 0003-682X/87/$03"50 © Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain

192 P. A. A. Laura, G. M. Ficcadenti de lglesias, P. L. Verniere de lrassar

M A T H E M A T I C A L ANALYSIS OF THE PROBLEM

In the case of free vibrations and in accordance with Housner's classical paper I the governing differential system (see Fig. 1) is given by

04y 02y 02y O2y EI-~-~x 4 + pAov2-ff~x2 + 2pAov o~-~x + M - ~ = 0 (1)

y(o, t) = y(L, t) = 0

82Y (0, t) = t~2y tL, t)=o

(2a)

(2b)

where

A o, E,I: cross-sectional area, Young's modulus and moment of inertia, respectively, of the conduit

p: density of the fluid M: total mass per unit length (M = pAo + m) m: conduit mass per unit length v: velocity of the fluid (assumed constant).

Y~

Y(x t) ~ ~ v ~x

//////

q

Fig. 1.

Section 1-1 Fluid-conveying conduit executing transverse vibrations.

Flexural vibrations of a pipeline containing flowing fluid 1 9 3

Following Refs 1 and 3, one takes

rl T~ X yj(x, t) = a. sin -~---. sin co~t . = 1 , 3 , 5 . . .

nnx + a. sin-~---'cos ogjt

, = 2 , 4 , 6 . . .

where w~ is circular frequency of the jth mode. Substituting Eqn (3) for Eqn (1), one obtains

j = 1,2,3 .... (3)

Eli ~ a,(-~-)'sin-~-sinoojt+ n = 1 , 3 , 5 . . .

a" (-~-)4 sin ~ f - c°s °gJ t ] n = 2 , 4 - , 6 . . .

+PA°v2 1 - ~ a" ( 7 ) 2 s i n ~ - sin °9~t n = 1 , 3 , 5 . . .

X (7) a. sm ~ cos e)jtJ n = 2 , 4 - , 6 . . .

I ~ n~ rlT~x + 2pAov a.coj --£- cos - - ~ cos oJjt n = 1 , 3 , 5 . . .

+ M E -

Expressing now

C O S - - - -

where

"1 r / / Z r / 7 ~ X . |

-

n = 2,4.,6...

Z n = 1 , 3 , 5 . . .

Y/7~X .

a.oJ~ sin T sm oojt

-~ a.co~ s i n - ~ cos mjt] = 0 n = 2 , 4 , 6 . . .

(4)

mrxL -

r = 1 , 2 , 3 . . .

rlzx b.r sin T (n = 1,2, 3,...) O)

2r [1 - ( - 1)" +'] b . r ~ - - r 2 - 12 2

194 P. A. A. Laura, G. M. Ficcadenti de lglesias, P. L. Verniere de Irassar

substituting for Eqn (4) and grouping terms which multiply sin cn/ and cos cojt results in

E1 a.n 4 s i n ~ - - + pAo v2 ( - a . ) n 2 s i n ~ - -

n = 1 , 3 , 5 . . . n = 1 , 3 , 5 . . .

+ 2pA°v ~ (-a")~J nrc ~ L b,r sm. rzrx

n = 2 , 4 , 6 . . . r = 1 , 2 , 3 . . .

sin ~ - = 0

n = 1 , 3 , 5 . . .

E1 a.n 4 s i n - ~ - - + PAo v2 (-- a")n 2 7r 2 s i n ~ -

n = 2 , 4 , 6 . . . n = 2 , 4 , 6 . . .

(6a)

+ 2pAov a.coj L b,, sin rrcxL

n = 1 , 3 , 5 . . . r = 1 , 2 , 3 . . .

. ? l T ~ X

2 sin ~ - = 0 (6b) + M ( - a.) coj

n = 2 , 4 . , 6 . . .

Since the b.,'s are identically zero when (n + r) results in an even number, one writes now

4r b,, - rc(r 2 _ n2 ) (n + r: odd)

and interchanging the subscripts one has

4n brn = _ ~z(r 2 - n 2) (7)

(Equation (7) does not contain the negative sign in Refs 1 and 3. Apparent ly the negative sign resulting from the cross derivative was not taken into account either.)

After rearranging Eqn (6) one obtains

8pAowo j ~ rn mzx L ar r2 _ n~ sin T

r = 2 , 4 - , 6 . . .

(n = 1,3,5. . . ) (8a)

Flexural vibrations of a pipeline containing flowing fluid 195

a.[E'n4(L)4-pAov2n2(L)2-Mog~]sinn-~ x

8pA°v°gJ ~ a, nr sinnnX L r 2 - n z L

r = 1 , 3 , 5 . . .

(n = 2, 4, 6...) (8b)

Equations (8a) and (8b) lead to an infinite, linear system of homogeneous equations in the a,'s (n = 1,2, 3...). From the non-triviality condition a secular determinant in the natural frequencies of the system is generated.

NUMERICAL RESULTS

Taking the first term of each series appearing in Eqn (3) one obtains, from the determinantal equation, the following functional relation

2 ~ / = v 2 4 (m ~2 ~_x/~z 4[l_(v.)1 [ _(~)2] (9)

where

ml: fundamental frequency of the system ogN: fundamental frequency of the simply supported beam

Vc \ - L J x / - ~ o (critical velocity which leads to buckling of the pipe)

= 8 " 5 - 2-5 9n 2 M }

Table 1 indicates that the convergence of the series approach is quite adequate since a two-term solution provides excellent engineering accuracy.

Figure 2 illustrates the variation of (o~l/~N) as a function of v/vc for the limiting situations (pAo/~l) = 0 and (pAo/M)= 1. The values of ~01/o~N are lower for the latter while an opposite trend is indicated in Ref. 3.

196 P. A. A. Laura, G. M. Ficcadenti de Iglesias, P. L. Verniere de lrassar

Fig. 2.

3

0-5 -

I i I I I 1 I I I i 0 0"1 0'2 0.3 0.4 0"5 0"6 0.7 0.8 0"9 1.0

v l v c

Dimens ion less p a r a m e t e r tol/~o u versus v/v~ for different values o f pAo/M. (pAo/M) = O . - - ' - - : (pAo/M) = 0 ' 5 . - - - - : (pAo/M)= 1.

Flexural vibrations of a pipeline containing flowing fluid

T A B L E 1 Analysis of the Convergence of the Solution

v/vc Two-term solution Four-term solution

pAo M

PAo M

- - = 0 0 0-1 0.2 0.3 0.4 0-5 0-6 0.7 0.8 0-9 1

-1 0 0"1 0'2 0"3 0-4 0'5 0"6 0-7 0"8 0'9 1

1 1 0-994 987 0"994 988 0-979 795 0"979 795 0"953 939 0"953 940 0"916515 0-916515 0"866 025 0"866 025 0'800 000 0.800 000 0.714 142 0-714 142 0-600 000 0"600 000 0.435 889 0.435 889 0 0

1 1 0'994 031 0'994 031 0-976 024 0"976 024 0"945 658 0"945 658 0'902 324 0"902 324 0"844 979 0"844 979 0-771 840 0'771 840 0'679 672 0"679 672 0"561 818 0-561 819 0-400 374 0'400 379 0 0

197

C O N C L U D I N G R E M A R K S

Obviously the error has a small effect on the results (less than 5%). This can be seen in Fig. 2, which gives curves that bound the range o f the influence of the error. The solution is unaffected by the error, at both zero flow velocity and at buckling instability, because the term in which it arises is the Coriolos, component of force. This force has no effect on the instability when the ends are either pinned or clamped.

It is impor tant to point out that damping has been excluded in the simplified model studied in this paper, and this has important consequences at flow velocities away f rom the critical buckling velocity.

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

The present study has been sponsored by C O N I C E T Research and Development Program 009400-85.

198 P. A. A. Laura, G. M. Ficcadenti de lglesias, P. L. Verniere de lrassar

R E F E R E N C E S

1. G. W. Housner, Bending vibrations of a pipe line containing flowing fluid. J. Applied Mechanics, 19 (1952) 205-8.

2. H. L. Dodds and H. Runyan, Effect of High-Velocity Fluid Flow in the Bend- ing Vibrations and Static Divergence of a Simply Supported Pipe. NASA TN D-2870, National Aeronautics and Space Administration, 1965.

3. R. D. Blevins, Flow-inducedivibration, Van Nostrand Reinhold Company, New York, 1977.

4. R. W. Gregory and M. P. Paidoussis, Unstable oscillation of tubular cantilever conveying fluid--I. Theory. Proc. Roy. Soc. (London) Ser. A, 293 (1966) 512-97.