Page 336 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 3 17

Here, Thompson’s (1982b) magic black box (Section 3.2.2, Figure 3.3) is discussed
in detail. The system in the black box in Figure 5.11(a) is generally nonconservative. It
consists of an articulated pipe, the downstream end of which is constrained by a string,
supporting a weight Eg. Assuming equal spring stiffnesses, k, and lengths, I, neglecting
gravity effects in the pipe system, and assuming small angular deflections, 61 and 02,
Thompson (1982b) conducted an interesting static analysis of the system. In terms of
statics, the fluid acts as a follower compressive load of magnitude MU2 (Section 3.2.1).
Talung moments about the joints, one obtains

?fig1 + k(O2 - 61 1 = 0, 2Egl + k61 + MU21(02 - 61) = 0. (5.100)

The deflection at the end of the pipe system is x = -1 (01 + 02), which, from the solution
of (5.100), may be re-written as
x = -5igl’ (T 2MU21 - 5) lk.

(5.101)

The flexibility may be defined as x/E (more usually x/Eg); its inverse, E/x, is the
stiffness of the system. It is clear that the stiffness is positive for small values of U,
becomes infinite at the point of divergence (2MU21 = 5k), and then negative for larger
values of U. This dynamical behaviour is illustrated in Figure 5.1 l(b) from experiments
with Lunn’s (1982) articulated pipes, involving Perspex or copper tubing and rubber
joints, as described in Section 5.6.2. The observed behaviour is a little more complex
than the linear relation between x and MU2 in (5.101), but essentially the dynamics is as
predicted. In particular, in the region of negative stiffness, when the weight ?fi is doubled,
x is halved, approximately; i.e. as the weight is increased, it goes up (Figure 3.3) - a
graphic demonstration of ‘paradoxial’ mechanics due to negative stiffness.
The other interesting observation made by Thompson is this. For conventional structural
systems (e.g. the inverted pendulum, loaded arches), the equilibrium path in the negative
stiffness region is unstable under ‘dead’ load and, hence, can only be studied experimen-
tally by using a suitable ‘rigid’ load, e.g. via a screw loading device (Thompson 1979).
Here, however, we have a system in which the complete, stable load-deflection curve
can be obtained, covering also the region of negative stiffness, by using dead weight
loading - precisely because the system is nonconservative.

5.6.2 Unconstrained cantilevers

The main objective of virtually all nonlinear studies in this area is related to the character-
ization of the nature of the Hopf bifurcations leading to flutter. In the case of 2-D motions,
this distinguishes subcritical from supercritical bifurcations in the parameter space. In the
case of 3-D motions, however, the nonlinear analysis also defines whether the resulting
flutter is planar or rotary. Of special interest is another set of studies, concerned with the
dynamics of systems in the vicinity of a double degeneracy, characterized by two coinci-
dent bifurcations via which a rich variety of dynamical states may emerge, as discussed
in part (c) of this section.