Page 367 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 347

The system is then further simplified by the method of normal forms, following Gucken-
heimer & Holmes (1983; exercise 7.4.1) and Sethna & Shaw (1987) - see Appendices F
and H. Suffice it to say that the final form of the perturbation equations on the centre
manifold is given by

2
i. = pir - (r2 + bz2)r + h.o.t., i = pzz + (cr2 + z )z + h.0.t.; (5.127)
h.0.t. stands for higher order terms. For 6 = 0.2, b = 1.518 and c = 3.954.
A local bifurcation analysis shows that, generally, there are four equilibrium points:

(5.128)

for the values of b and c just given, the third equilibrium exists only for p1 + bp2 < 0 and
cpl + p:! > 0. Topological features of the system near these equilibria can be determined
by the eigenvalues of the matrix of the linearized autonomous version of (5.127). For an
equilibrium point (ro, zo), this matrix is

(5.129)

Figure 5.29 shows phase flows emanating from the equilibria in the fourth quadrant of
the parameter plane. Within the segment defined by the dashed lines, the equilibrium of
the third equation (5.128) emerges, and it may be a spiral sink or a source; all other
equilibria are saddles. A bifurcation curve exists, where the sink and the source collapse

Figure 5.29 Bifurcation diagram for the doubly degenerate up-standing cantilevered pipe on the
centre manifold, showing typical phase flows in the fourth quadrant (Li & Paidoussis 1994).