Page 98 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 81

The kinetic energy of the pipe and the fluid may be evaluated by making use of (3.23)
and (3.26),

L
Tp = im 1 (x2 + i2) ds, q = &M .IL [(i + Ux’I2 + (i + UZ’)~] ds, (3.60)
0
in which m and M have been defined in Section 3.3.2; again, the subscripts p and f
stand for the pipe and fluid, respectively. The integrands in $ and Tf may be simplified
by noting that X - 6(e2), x’ 2: 1 - ;w’~, and x’~ + z’~ = 1 from inextensibility condition
(3.14). Hence, recalling also that X = U and z = w, the expressions for $ and Tf become
L L
$= lml W2ds, Tf = iMl [U2+W2+2Uww’+2UU]ds. (3.61)

It is noted that (3.61) could have been obtained directly with the aid of Figure 3.8(b); the
various terms are obtained from Cartesian components of (3.24), which may be expressed
as (W + U sin x) and (U cos x + U) with sin x 2 w’ and cos x 2: 1 - ;w’~, neglecting
terms smaller than 6(e2).
The potential energy is given by
I’
v = vp + v, = ;EI lL ds + &(m + M)g lL w’~ ds ds. (3.62)
w”’

The component of V associated with gravity may be simplified via integration by parts,
as follows:
:

ds}
wI2
i(m + M)g/‘ 1’ w’~ dsds = ;(m + M)g { [s 1’ ds] 1 - 1‘ ~w’~
0 0
L
ds.
= T(m + M)g / (L - s)~’~ (3.63)
I
0
Finally, substituting (3.61)-(3.63) into (3.59) and making use of the standard vari-
ational techniques and of the boundary conditions for a cantilever, after considerable
manipulation, this reduces to

+ (M + m)h) Swds dt = 0. (3.64)

Two items should be remarked upon in the derivation of (3.64). Firstly, the terms in
the second integral of (3.59) cancelled out with identical ones originating from the first
integral after integration by parts. For instance,
6 1 MU2uLdt = MU26 6’ lL $d2dsdt = MU2 h” w’(6w)’ ds dt

=MU2[ w‘Sw1 L dt-MU2 1” .IL W” 6w ds dt,
0