Page 430 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

402 SLENDER STRUCTURES AND AXIAL FLOW

while two potential wells now exist, centred at [ltO.Ol, 0) in the phase plane. With
p # 0, period-1 motions exist only for 0 < p < 0.0255, as shown in Figure 5.62(a). For
p > 0.0256, a period-doubling sequence leads to transient chaos (near p = 0.0283); then
to stable period-2 motion, followed by another period-doubling cascade and chaos for
p = 0.04, as seen in Figure 5.62(b). The extremely low values of p in this case are noted.

-0.016 0.016 -0.02 0.02
(a) 41 (b) 41
Figure 5.62 Post-divergence phase-plane diagrams for the system of Figure 5.61 for uo = 4.7077;
(a) global view for p = 0.02, (b) p = 0.04 (Jayaraman & Narayanan 1996).

Although the results presented in this work are very interesting, it should be remarked
that, at least for the parameters chosen, the ranges of uo for chaos are extremely narrow,
in fact so narrow that their experimental realization could well be problematical.
To conclude, the nonlinear analysis of parametric resonances in pipes with supported
ends has brought to light a number of interesting features, e.g. the possible ‘subcritical’
onset of resonance and the subcritical (hardening) behaviour for its cessation as p is
increased. Furthermore, the amplitude of oscillations can be computed, at least close to
the resonance boundaries. Finally, chaotic oscillations have been found to exist in narrow
uo -ranges.

5.9.2 Cantilevered pipes

Two main studies have been conducted in this case: a complete bifurcation analysis of
the principal primary resonance in the vicinity of the flutter boundary by Bajaj (1987b)
and a combined analytical, numerical and experimental study by Semler & PaIdoussis
(1996), both for planar motions. The problem in this case is more complex than that of
pipes with supported ends, since several modes are required for accuracy in the Galerkin
expansion (a one-mode approximation being totally meaningless), and nonlinear inertial
terms create additional difficulties in numerical solutions.
The Bajaj (1987b) analysis is at once very powerful, nonstandard and difficult to
condense; hence, only an outline of the methods used will be given here. The Lund-
gren et al. (1979) form of the equation of motion is used for motions in a horizontal
plane; therefore, apart from harmonic perturbations associated with flow pulsations, the
mean flow velocity is steady. Proceeding as in Bajaj et al. (1980), the system is re-written
in the vector form
au
- = Lu + ep[L~ cos 2wt + L2 sin 2wtJu + EN(u, UO) + S(c2), (5.151)
at