Page 390 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

3 66 SLENDER STRUCTURES AND AXIAL FLOW

reasonable quantitative agreement for the key bifurcations - though with unrealistically
large pipe displacements. As a result of fractal dimension calculations, higher-N models
were then employed, and convergent solutions were obtained with the correct K and (b
and a better impacting model. Inevitably, this was interpreted to mean that the use of
N = 2 was the main factor responsible for the original numerical difficulties. However,
this was disproved, when it was found that, if the nonlinear equation of motion is used,
then N = 2 with the proper K, &, and impact model gives perfectly reasonable results
(albeit not as accurate as N = 4); and, moreover it gives the correct order of magni-
tude of displacements and retrieves forgotten features of physical behaviour, such as the
‘sticking’ to the constraints.
These studies, therefore, may be looked upon as providing particular views of the
dynamics through different ‘sections’ of the multidimensional parameter space of this
system. Thus, the N = 2 model of Paldoussis & Moon (1988) must be judged as so fragile
(nonrobust), as to make one wonder if the agreement with experiment were not fortuitous.
The Paldoussis et al. (199 1 a) model with N > 2 was decidedly more successful and more
robust; yet, it too failed for N = 2. Finally, the most successful model (Pa‘idoussis &
Semler 1993a) is also the most robust: small excursions in the parameter subspace of this
model have little effect on the dynamics. It is at this stage only that it can be concluded
that the excellent agreement with experiment cannot be fortuitous.
Another of the experimental cases of Pai’doussis & Moon - pipe 9 (Table 5.3), but
with softer, leaf-spring-supported bars, closer to the pipe (wb/L = 0.025) - was studied
numerically by Makrides & Edelstein (1992), with the Lundgren et al. (1979) nonlinear
equations of motion and a finite element and penalty function solution approach - see
Section 5.7.1. It is found that the onset of chaos is dependent on the stiffness of the motion
constraints (modelled as trilinear springs), in agreement with observations. Furthermore,
in this case, the route to chaos is via quasiperiodicity (see Section 5.8.3), although in the
experiments a period-doubling route appears to be followed. Still, the predicted threshold
to chaos, u,/~ 2: 8.05 for an assumed K = IO3, is not too far from the experimental one,
u,h 2: 9.1. One weakness here is that in the theoretical model, as per the original form
of the Lundgren et al. equations, gravity effects are neglected, whereas in this particular
experiment they are not negligible (y = 26.8); a nonzero y would nevertheless raise uH
and hence u,h. One observation that ought to be made here, in view of the foregoing
discussion, is that through the use of the nonlinear equations of motion, similarly to
Pai‘doussis & Semler (1993a), no difficulties are reported in obtaining convergent solutions
with amplitudes of the correct order of magnitude. In conclusion, Makrides & Edelstein’s
work shows that, with different motion constraints (and perhaps other parameters), it is
possible that a different route to chaos may be followed - cf. Paldoussis & Botez (1995)
for a system discussed in Volume 2 - which adds to the interest in this system.
The interested reader is also referred to Miles et al. (1992), who demonstrate the power
of bispectral analysis techniques to isolate nonlinear phase coupling and energy exchange
of the various Fourier components of motion, using this particular system as an example.

5.8.2 Magnetically buckled pipes

Chaotic oscillations of a conservative system with passive damping added, such as a
buckled beam subjected to deterministic forced excitations, have been studied by Moon
(1980), Holmes & Moon (1983) and Dowel1 & Pezeshki (1986), among others; rarer