Page 190 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

172 SLENDER STRUCTURES AND AXIAL FLOW

Equations (3.128) and (3.129) may be rendered dimensionless by means of the
following:

(3.130)

The solution of the equations of motion may be achieved by the method of
Section 3.3.6(a). Typical results are shown in Figure 3.71(b), in terms of the parameter
2u2/n2 - 1 for various values of B’ and Zr2/Cw = 1.5, for both transverse and torsional
motions. The full lines correspond to results obtained for B = 0 and hence to a
system without flow subjected to a follower force 9 = u2; however, since divergence
is independent of #?, the stability curve for B = 0 applies equally to cases with flow.
It is seen that three types of instability are possible: (i) torsional divergence for small
enough E; (ii) torsional flutter (dashed curves) for intermediate Z; (iii) transverse flutter
(horizontal dashed lines) for high E. Thus, a system with #?’ = 0.2 would lose stability by
divergence if a! = 0.5, by torsional flutter if a! = 1.5, and by transverse flutter if E = 2.5.
Of course, according to linear theory, in the case of E = 1.5, transverse flutter would arise
at higher flow (the horizontal lines for transverse flutter really extend across the figure),
and so on.
It is of special interest that torsional divergence is possible, whereas transverse diver-
gence is not. Equations (3.128) and (3.129) are similar in structure to those for transverse
motion, with the torsional terms (proportional to GJ) playing the role of a conservative
tensile load. However, it is known that tension does not induce divergence (Section 3.5.8).
Hence, torsional divergence probably arises via the M U2-related term in the boundary
conditions (largest at small E) - cf. Chapter 8.
Another plate-pipe system, used for marine propulsion, is discussed in Section 4.7.

3.6.7 Concluding remarks
The main purpose of Section 3.6 is (i) to briefly document all these interesting studies
in one place, and (ii) to show the veritable cornucopia of interesting dynamical prob-
lems that may be obtained with simple modifications to the basic system of a pipe
conveying fluid - particularly the nonconservative case of a cantilevered pipe. This,
despite the early scepticism on the practical value in studying the stability of such
systems, as expressed for instance by Timoshenko & Gere (1961; section 2.21), regarding
the critical load for a cantilevered column subjected to a tangential follower load: ‘No
definite conclusion can be made (as yet) regarding the practical value of the result,
since no method has been devised for applying a tangential force to a column during
bending’. Although a method has now been found, this is not really the important point.
What is important in the study of these systems will emerge from the chapters that
follow, and what is practically important from the pertinent sections on applications
therein.