Page 476 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

a

CURVED PIPES CONVEYING FLUID 447

mode
Fourth

Third mode

First mode

0 1 2 3
Dimensionless flow velocity, E*

Figure 6.14 Dimensionless eigenfrequencies w* for in-plune motion of a clamped-clamped
semi-circular pipe conveying fluid as functions of the dimensionless flow velocity ii*, for = 0.5:
__- , Chen (19724; -, conventional inextensible theory (Misra et al. 1988a); - * ~- , modified
inextensible theory (Misra et al. 1988a).

Now, as shown in Figures 6.10 and 6.12, the differences between the full extensible
theory and the version in which the extension of the centreline [Se(rf + Oq;)] is ignored
are small, especially in the first mode; and this is the mode in which the system, according
to the conventional inextensible theory, would lose stability. Hence, insofar as in-plane
motions are concerned, the effect of neglecting extensibility of the centreline is small,
and the modified inextensible theory provides a reasonable approximation to the results
obtained by the full extensible theory, at considerably smaller computational cost.