Page 365 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 345

Gravity in this case is neglected (or inoperative) and the spring is positively fixed (not as
in Figure 5.23), so that associated geometric nonlinearities have to be taken into account.
In a way, these studies represent an extension to Bajaj & Sethna's (1984, 1991) work to
the case when there is an additional, intermediate spring support. This is a very complex
problem, and unfolding the various bifurcations is an arduous task.
The wholly symmetric system is considered first. To achieve this, so far as the spring
force is concerned, it is found that either the springs in the two planes should be very
long or an array of many springs radiating from the support should be used. Although not
much information is given on the method of discretization, centre manifold and normal
form theory are used as in the foregoing to simplify the system in the vicinity of the
principal bifurcation associated with loss of stability: (i) by a zero root, (ii) a zero and a
purely imaginary pair, (iii) two purely imaginary pairs, in all cases with multiplicity two,
because of the symmetry. Two unfolding parameters are used: p, associated with u - u,,
and u, associated with K - K,. A typical case of a system with a double pair of zero roots
and K slightly above K, is shown in Figure 5.28(a). The system starts from the trivial
stable state (TS), then becomes subject to divergence (static buckling, SB) and, after a
secondary bifurcation, develops planar oscillation about the buckled state (SW3) - cf.
Figure 5.24(b). With increasing flow, the amplitude grows so that the oscillation crosses
the origin and changes into planar oscillation about the origin (SW2). All solution branches
associated with a positive real part of an eigenvalue of the locally linearized system,
marked with a + and drawn as dashed lines, are unstable. In this case, TW solutions,
corresponding to rotary pipe motions, and MW solutions, corresponding to rotary pipe
motions with a superposed radial oscillation, are unstable.
Another case of a system with a zero root and a purely imaginary pair with multiplicity
of two is shown in Figure 5.28(b) - cf. Golubitsky & Stewart (1986). As seen in the
figure, there are eight solution branches associated with: (i) the trivial equilibrium state,
TS; (ii) the statically buckled state, SB; (iii) planar oscillations, SW2, about TS; (iv) planar
oscillations, SW3, about SB in the plane of SB; (v) the same, but SW = SW4, perpendicular
to SB; (vi) rotary pipe oscillations, TW; (vii) modulated motion, MW; (viii) SB with
superposed TW (SB/TW). As shown in the figure, most of these solution branches are
unstable. The system, after buckling, develops planar oscillatory motions (SW4) about the
buckled state, in a plane perpendicular to that of buckling. More complicated motions are
possible in the case of two pairs of purely imaginary roots.
The case of broken symmetry is considered next, in three ways: (i) via a small
geometric imperfection, yielding a constant term in the bifurcation equations - cf.
Figure 5.8(b); (ii) imperfect springs breaking rotational symmetry; (iii) imperfect loading
breaking reflectional symmetry (e.g. by immersing the end of the pipe in a swirling fluid).
Dynamical behaviour similar to that of Figure 5.22, but richer, is predicted.

(d) Planar motions of doubly degenerate up-standing cantilever
The planar dynamics of another type of system, that of the 'up-standing cantilever'
(Section 3.5.2) where the free end is located above the clamped one, without any
intermediate support, is considered next; this was studied under double degeneracy
conditions by Li & Pa'idoussis (1 994).+ It is recalled that, generally, this system buckles

'This study though published later, was actually conducted prior to that by PaTdoussis & Semler (1993b),
which owes a great deal to the Li & Pa'idoussis study.