Page 79 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

62 SLENDER STRUCTURES AND AXIAL FLOW

i.e. equation (2.47). It is clear that the centrifugal force in (3.1) acts in the same manner
as a compressive load. In this way, it is easy to see and to understand physically that,
with increasing U, the effective stiffness of the pipe is diminished; for sufficiently large
U, the destabilizing centrifugal force may overcome the restoring flexural force, resulting
in divergence, vulgarly known as buckling and, in the nonlinear dynamics milieu, as a
pitchfork bifurcation.
In the foregoing argument, it was implicitly assumed that the Coriolis forces do no
work in the course of free motions of the pipe, which is true. The rate of work done
on the pipe by the fluid-dynamic forces, the only possible source of energy input, in the
course of periodic motions is

dW aw
dt (3.4)
and hence the work done by the fluid forces over a cycle of periodic oscillation of period
AW=-MULT [($)2+U(g) (;)]I L (3.5)
T is
0 dt.

Clearly if the ends of the pipe are positively supported, then (awlat) = 0 at both ends, and
AW=O. (3.6)

Nonworking velocity-dependent loads are called gyroscopic by Ziegler (1 968) and hence
this system is classified as a gyroscopic conservative system. In Galerkin discretizations
of this system, the Coriolis-related velocity-dependent matrix is purely skew-symmetric
(antisymmetric) [see, e.g. Done & Simpson (1977) and Section 3.4.1 here].
Because divergence is a static rather than dynamic form of instability, the dynamics
of the system may be examined by considering only the time-independent terms in equa-
tion (3.1), so effectively equation (3.3) with the inertia term put to zero; whereby, for a
simply-supported pipe, the particularly simple result is obtained (Section 3.4.1) for the
critical flow velocity U,, namely that the dimensionless critical flow velocity is

u, = IT, (3.7)
where u is defined as
u = (M/Et)"2UL, (3.8)
in which L is the length of the pipe. Similarly, for a simply-supported column (Ziegler
1968),
Yc = n2, 9 = PL2/Et; (3.9)

it is clear from equations (3.1) and (3.3) that the equivalent of is u2, rather than u. As
expected, the dynamical behaviour of pipes with one or both ends clamped, rather than
simply supported, is similar.
The analogy between equations (3.1) and (3.3) and the discussion just made show
also how the natural frequencies of the system should develop with increasing U. It is
physically obvious in the column problem that, as the compressive load is increased, the
effective rigidity (or stiffness) of the system is eroded, to the point where it vanishes;