Page 81 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

64 SLENDER STRUCTURES AND AXIAL FLOW
any influence on subsequent research, except in an important way on a set of nonlinear
studies to be discussed in Chapter 5.
The next study, some 20 years later, was Benjamin’s (1961a,b), mainly on the dynamics
of articulated cantilevers conveying fluid [Figures 2.l(b) and 3.1(d)], but with an author-
itative discussion of the continuous system [Figure 3.l(c)].’ One of the principal accom-
plishment, among many, of this work was the establishment of the appropriate form of the
Lagrangian equations for-this ‘open’ system (open, in the sense that momentum constantly
flows in one end and out the other), namely

(3.10)

in which T and V are the total kinetic and potential energies of the system, RL is the
position vector of the free end and t~ the unit vector tangent to the free end [Figure 3.1 (d)];
qk are the generalized coordinates, typically the angles made by each of the rigid pipes
of the system with the undeformed line of equilibrium. The corresponding statement
of Hamilton’s principle was also obtained, from which the equations of motion of the
continuous system (and the articulated one, if so desired) may be derived.
The equation of motion of the continuous cantilevered system is the same as that of
a pipe with supported ends, equation (3.1); this will be derived in Section 3.3, and there
are subtle differences in the derivation for these two cases (Section 3.3.3). However,
physically, it seems reasonable that the same equation should hold. Similarly, the same
expression, equation (3.3, holds true for the work done by the fluid on the pipe over a
period T of periodic oscillation, but in this case it is equal to

where (&/at), and (aW/ax)L are, respectively, the lateral velocity and slope of the free
end. In Ziegler’s (1968) classification, since some of the forces associated with AW # 0
are not velocity-dependent [the MU2(a2w/ax2) follower load leading to the second term
in (3.11)], this is a circulatory system. The dynamics of this system was elucidated by
means of this expression by Benjamin (1961a) and elaborated by PaYdoussis (1970).
For U > 0 and sufficiently small for the second term within the square brackets to
be much smaller than the first, it is clear that AW < 0, and free motions of the pipe
are damped - an effect due to the Coriolis forces, which, unlike the case of supported
ends, in this case do do work. If, however, U is sufficiently large, while over most of
the cycle (aW/ax)L and (awlat), have opposite signs, then AW > 0; i.e. the pipe will
gain energy from the flow, and free motions will be amplified. The requirement that
(aW/ax)~ (&/at), < 0 suggests that, in the course of flutter, the pipe must execute a sort
of ‘dragging’, lagging motion that one would obtain when laterally oscillating a long
flexible blade or baton in dense fluid. This, indeed, is what is observed, as remarked by
Bounihres (19391, Benjamin (1961b) and Gregory & Pai‘doussis (1966b).

+‘A continuous system’ will henceforth denote the distributed parameter system involving a continuously
flexible pipe.