Page 254 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 235

Table 4.4 The eigenfrequencies of a heavy clamped-clamped
short pipe (u = 0) for y = 10, #3 = 0.5, p = 0 = 0, by Timo-
shenko plug-flow (TPF) and Timoshenko refined-flow (TRF)
theories.
A Mode w by TPF theory w by TRF theory

1 19.552 19.599
100
2 44.365 44.752
1 9.670 9.837
10
2 19.342 20.493

I 1 I I I I I I
1- -

1 I I I I I I I I I I I
0 2 4 6 8 10
@IO A
Figure 4.19 The critical dimensionless flow velocities for divergence, u,d, of a pipe clamped at
both ends, showing the effect of slenderness and related transverse shear [see equation (4.37)], for
= 0.5, y = 10, p = 0 = 0. + Timoshenko refined-flow (TRF) theory; --A--, Timoshenko
,
plug-flow (TPF) theory (PaYdoussis et al. 1986).

fluid per unit length - the ‘slender body’ approximation [cf. Section 2.2.2(e)(ii)] which
in this case reduces to the plug-flow model. According to the refined model, however,
this is smaller because of ‘end effects’ or departures from two-dimensionality [cf. equa-
tion (2.139) and the discussion of (4.58)], which are more important for short than for
long pipes. Hence, the effective total mass per unit length is m + M’, with M’ < M where
M = pAf, and the values of o [generally equal to (generalized stiffness)/(generalized
mass)] are therefore larger.
Considering the critical flow velocities for divergence next (Figure 4.19), it is seen
that the results for the TRF theory are indistinguishable from those obtained by the TPF