Page 353 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 333

experimentally by Chen (see Section 3.5.6), the limit cycle appears to be supercritical, but
this is not made absolutely clear, although in the experiments it is definitely subcritical.
Nevertheless, agreement with the experimental stable limit-cycle amplitude and flutter
frequency is reasonably good.
Before ending this section, the singular behaviour of the system in the vicinity of
B = 0.3 and 0.69 in Figures 5.18-5.20 should be remarked upon; i.e. at the same values
of fl where S-shaped discontinuities in the u, versus B plot occur (Figure 3.30), and with
which so many interesting features of linear dynamics are associated (Sections 3.5 and
3.6). Furthermore, the ‘stabilizing/destabilizing’ and in-plane/out-of-plane ranges of /3 in
Table 5.1 straddle these same S-shaped discontinuities. Referring to the work presented
in Section 35.4, this is hardly surprising: the modal content of the flutter mode and the
energy transfer mechanism both experience radical alterations about these critical values
of B. Hence, it is reasonable that the nature of limit-cycle motions should also be modified
across these same values of B.

5.7.2 3-D limit-cycle motions

As already mentioned, the main objective in the case of 3-D motions of the pipe is the
prediction of whether the flutter motions are planar or three-dimensional (orbital). The
necessary analysis, using a simplified set of equations with a deflection-independent flow
velocity, was done by Bajaj & Sethna (1984). As in Bajaj & Sethna’s (1982a.b) similar
analysis of articulated pipes (Section 5.6.2), stability in this system is lost via two pairs
of complex eigenvalues crossing the imaginary axis simultaneously, i.e. via a generalized
Hopf bifurcation. This system is said to have O(2) symmetry, possessing both rotational
(about the x-axis) and ‘reflective’ symmetry across that axis.
The original governing PDE is reduced directly to a set of ODES on the centre manifold
without introducing Galerkin expansion, similarly to Bajaj et al. (1980). The discretized
equations are then projected onto the centre manifold, and periodic solutions close to the
critical flow velocity are sought. The resulting equations are then brought into normal
form by the method of averaging. Eventually, equations similar to (5.113) are obtained,
but in this case two different amplitude parameters, al and 122, are involved; for rotary
motions a1 = a?. The final results are shown in Figure 5.21. Similarly to the articulated
system (Section 5.6.2(b)), whenever two supercritical bifurcations occur for the same
system, that with the larger amplitude is the stable one, while the other is unstable.
Once more, the singularities in this figure, namely the values of /3 where the limit-cycle
amplitude can be zero, correspond to the location of the S-shaped curves in the linear
stability diagram (Figure 3.30). However, the points where limit-cycle motions switch
from planar to circular (rotary) and vice versa do not correspond to any of the ranges in
Table 5.1.
The dynamics of the same system was also studied (Bajaj & Sethna 1991) in the
presence of small imperfections, representing different bending stiffnesses and additional
viscous damping in two mutually perpendicular directions, imposed to break the rotational
symmetry. Thus, the linear flexural rigidity term in one equation is multiplied by (1 + ~6)
,
and in the other by (1 - ea), while the damping coefficients are EC~ and ~ 2 respectively.
E being a small parameter. The rotational symmetry is therefore broken by the parameters
6 and (CI - c?). Similar analytical techniques as in Bajaj & Sethna (1984) are used. At