Page 344 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

324 SLENDER STRUCTURES AND AXIAL, FLOW

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0 0.51 1 2 3 0 0.45 1 1.19 - 2 2.41 3
(a) P (b) P
Figure 5.15 Amplitude of periodic solutions in the vicinity of uEfr for 3-D motions of a
two-degree-of-freedom articulated cantilevered system with a = 2, K = 1 and (a) y = 0 and
-
(b) y = 0.25: -, rotary supercritical; - - - , rotary subcritical; ---, planar supercritical;
, planar subcritical (Bajaj & Sethna 1982b).

analysis having been carried out to only the first approximation level. The same applies
to the ‘infinite’ amplitude at = 1.19.
Finally, Bajaj & Sethna (1982b) discuss the effect of a small asymmetry in the system,
by making the spring stiffness in one plane slightly larger than in the other. It is found
that the circular rotary motions become elliptical, but the dynamics in the foregoing are
otherwise robust.
The foregoing analysis is restricted to solutions in the neighbourhood of the straight,
vertical equilibrium. The situation when this restriction is removed has been studied
by Sethna & Gu (1985), where the ‘limiting configurations’ as u -P ca are examined:
(i) does the system perform a rotary motion in a horizontal plane with the two pipe
segments at right angles to each other, or (ii) does it take on an S-shape in a vertical
plane, or (iii) some other configuration? The authors examine five such generic shapes,
all of the type in which the equations are invariant under rotation about the vertical
axis. The stability of these generic shapes is studied by either a linear approach or by
utilizing centre manifold theory (in the case of global circular motions). It is found that
apart from shapes (i) and (ii) above, all other shapes eventually become unstable via
a (secondary) Hopf bifurcation. The analytical results are complemented by numerical
simulations.

(e) Double degeneracy
Two studies into the dynamics of articulated cantilevers near a point of double degeneracy
are discussed here.
In the first such study, by Sethna & Shaw (1987), codimension-three bifurcations are
considered of a two-segment articulated system vibrating in a plane; the double degeneracy
in this case is associated with a pitchfork and a Hopf bifurcation occurring simultaneously