Page 397 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 373

the pipe-strip bond and even the pipe as a whole. More successful was the use of plate
guides confining the motion to a plane. Under these circumstances the motion did become
chaotic in any case.
The route to chaos is not clear from the experimental data. It could be via
quasiperiodicity - see Section 5.8.3(b); however, in at least one case, a sequence of
two period doublings was observed to precede chaos.
Finally, the case of p = 0 does appear to be singular, in that no secondary bifurcation
arises. The motion remains periodic to the maximum flow available - a conclusion also
reached by Copeland & Moon (1992).
In the theoretical study of the system, equation (5.28) or (5.39) is used, rather than
(5.42) and (5.43), so that inertial nonlinearities are left intact and no restriction on their
magnitude needs be imposed. Of course, equation (5.28) needs be modified to include
the effect of the end-mass, which is twofold: (i) the inertial terms now involve m + M +
Ah(x - L), where A is the end-mass; (ii) the gravity-induced tension terms are similarly
modified. Hence, with U = 0, the dimensionless equations are

f y[l + ph(6 - l)]ql + 2ufili’ + q”ll + N(q) = 0 (5.140)

where

1111 12
+ rl q + 4q’q’’q’’’ + q”3 + q’[l + pS(6 - l)]

(5.141)

p is defined by (5.139). Thus, via the use of the Dirac delta function, the effect of p is
incorporated in the equation of motion, while the boundary conditions remain the same
as for p = 0.’ This facilitates the discretization of the system, via Galerkin’s method.
As discussed in Section 5.4, care has to be exercised in selecting appropriate numerical
methods for the solution of the resultant N second-order ordinary differential equations,
because of the presence of the nonlinear inertial terms - methods which should give
accurate, convergent solutions. The finite difference (FDM) and the incremental harmonic
balance (IHB) methods have been found to be particularly efficient and complementary
(Semler et al. 1996).
Typical results are shown in Figure 5.46; they are seen to be sensibly the same whether
computed with N = 3 or 4 in the Galerkin discretization. In (a) it is seen that, for p = 0,
the theory also finds no bifurcation beyond the Hopf bifurcation at u 2 6.2. The Hopf
bifurcation is clearly supercritical, in contrast to the experimental results.

‘The appropriateness of this formulation and method of solution is demonstrated at the end of Section 4.6.2.