Page 356 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

336 SLENDER STRUCTURES AND AXIAL FLOW

(iii) The B = 0.173 pipe with flats behaves essentially as in (i); this, also agrees with
Bajaj & Sethna’s theoretical predictions.
(iv) The most interesting case is the j3 = 0.231 pipe with flats. The initial limit-cycle
motion is again planar (SW). This motion persists to a certain flow rate, Uh, qualita-
tively corresponding to hh in Figure 5.22, whereupon the pipe motion suddenly develops
‘complex spatial transients’ and then settles down to a large elliptical limit cycle (TW). As
the flow rate is increased, the ellipse becomes more nearly a circle. Changes in the initial
conditions result in circular motion in the opposite direction. On reducing the flow rate,
circular motion (TW) persists to below u~, with reduced amplitudes and a more sharply
elliptical shape (large ratio of major to minor axis). At a lower flow still, corresponding
to Amin in Figure 5.22, spatial transients develop again, and then motion settles down to
a planar limit cycle (SW). These observations agree qualitatively remarkably well with
theory (Figure 5.22). In particular, for flow rates between Amin and hh, both planar (SW)
and rotary (TW) stable motions are found to exist and, by carefully controlling the initial
conditions, either can be achieved.

5.7.3 Dynamics under double degeneracy conditions

The main reason for studying doubly or multiply degenerate systems lies in the fact that,
in the presence of ‘competing attractors’, the dynamics becomes particularly interesting.
In ‘unfolding the bifurcations’ a variety of different dynamical behaviour is found, and
regions where chaos may arise can be identified.
Some double degeneracies, e.g. those associated with 3-D motions and generalized Hopf
bifurcations (which are discussed in Section 5.7.2, but which could equally well have been
covered here), are inherent in the system - whatever the system parameters - provided
that it is perfectly symmetric. Some others, such as those treated here, involving different
types of coincident bifurcations (pitchfork and Hopf) arise for particular sets of system
parameters, irrespective of symmetry. Studies of this type were conducted by PaYdoussis &
Semler (1993b), Li & PaYdoussis (1994) and Steindl & Troger (1988, 1995). In all cases,
these studies build upon Sethna & Shaw’s (1987) pioneering work on articulated pipes.
Specifically, three particular studies into the dynamics of cantilevered pipes near a
point of double degeneracy are presented in this section: (a, b) the 2-D and (c) the 3-D
dynamics of a pipe with an intermediate spring support, and (d) the 2-D dynamics of an
‘up-standing’ cantilever in a gravity field. The case of 2-D dynamics of the pipe-spring
system is the only one in this book where the analysis is presented in reasonable detail,
starting with the nonlinear equations of Section 5.2.7; hence, the presentation is broken
into two: (a) the general analysis, and (b) the analysis under double degeneracy conditions.

fa) 2-0 motions of pipe-spring system; general analysis
The planar dynamics of a vertical cantilever with an intermediate, linear spring support
(Figure 5.23) has been studied by Pai’doussis & Semler (1993b). Equations (5.42) and
(5.43) constitute the equation of motion used, but with the linear spring term

(5.1 14)
KVW - t.7)
added, where K = kL3/EI is the dimensionless spring constant and es = x,/L is the
location of the spring. The linear dynamics of the system has been discussed in