Page 475 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

446 SLENDER STRUCTURES AND AXIAL FLOW

which the steady fluid loads are not neglected) are identical. Hence, the main body of the
results will be presented in Section 6.4.3. However, a comparison with Hill & Davis’ and
Doll & Mote’s extensible theories for out-of-plane motions is presented in Figure 6.13.
It is clear that the results from the three theories are even closer in this case than for
in-plane motion. It is also clear that no divergence occurs for out-of-plane motions either.

5.0 I I I

-
4.0

-
3.0
*
h
3
2.0 - -- - - - - - - -<- -

-
1.0

0.0
0.0 0.5 1 .o
ii* = iiln
Figure 6.13 Dimensionless eigenfrequencies w* versus Us for out-ofplane motion of a
clamped-clamped semi-circular pipe conveying fluid according to extensible theory: - - -, Hill &
Davis (1974); - . - , Doll & Mote (1974, 1976); -, Misra et aE. (1988b), for the parameters
as in Figure 6.11 (Van 1986).

6.4.3 Modified inextensible theory

Figures 6.14 and 6.15 show the in-plane eigenfrequencies of clamped-clamped,
pinned-pinned and clamped-pinned semi-circular pipes conveying fluid, as functions
of the flow velocity, obtained by both the modified and the conventional inextensible
theories. It is obvious that, according to the modijied inextensible theory, the effect of fluid
flow on the eigenfrequencies is not very pronounced. Flow tends to reduce the first-mode
eigenfrequency, but does not cause divergence in the flow range investigated (as high as
-
u = 6n). It is also interesting to observe that the eigenfrequencies of some of the higher
modes actually increase with flow velocity. Thus, whether the axial force Q, (or combined
force n) is taken into account or not is very important. In the conventional inextensible
theory, where I7 is neglected, the effect of internal flow on the eigenfrequencies manifests
itself via the centrifugal and Coriolis forces, whereas in the modified inextensible theory,
where I7 is taken into account, the internal flow exerts only a Coriolis force. This is
because when both ends are supported, no is a constant equal to -Z2 [equation (6.99)],
and thus in equations (6.47) and (6.49) governing in-plane motion the terms associated
with the initial forces cancel out those arising from the centrifugal force. It is recalled
that it is the centrifugal forces that are responsible for the divergence instability obtained
in the case of pipes with both ends supported (Section 3.2.1).