Page 348 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

328 SLENDER STRUCTURES AND AXIAL FLOW

5.7 CANTILEVERED PIPES
A rapid scan of publication dates will convince the reader that most of the activity on
the dynamics of pipes conveying fluid has in recent years concentrated on the nonlinear
dynamics of continuous cantilevered pipes, to be discussed here, or articulated cantilevers,
covered in Section 5.6; this is more striking if one includes the work on chaotic dynamics,
presented in Section 5.8. The reason is that these systems display as varied and fascinating
nonlinear dynamical behaviour as the cornucopia in the linear dynamics domain already
discussed in Chapters 3 and 4.
In keeping with the rest of this chapter, most important findings are discussed to a
greater or lesser extent, but the mathematical methods and analytical details are only
skimmed; only in one case, in Section 5.7.3, are they presented in fair detail.
As for articulated cantilevers, the character of the Hopf bifurcations leading to flutter is
a question of considerable interest. For 2-D motions, this defines whether the bifurcation
is sub- or supercritical, as discussed in Section 5.7.1. For 3-D motions, this additionally
decides whether the flutter is planar or three-dimensional, as discussed in Section 5.7.2.
Again as for articulated cantilevers, it is of interest to examine the dynamics near different
types of double degeneracy conditions; this is done in Section 5.7.3.

5.7.1 2-D limit-cycle motions
Planar limit-cycle motions of a vertical cantilever in the vicinity of the critical flow
velocity where the Hopf bifurcation arises were studied by Rousselet & Herrmann (1981).
A constant-pressure tank is assumed to feed the flow into the pipe [see Figure 5.2(a)],
while the flow velocity is pipe-deformation dependent, as discussed in Section 5.6.2 for the
articulated counterpart of this system. Hence, there are two coupled governing equations,
which are solved iteratively, in a manner similar to that described in Section 5.6.2: (i) the
homogeneous, linear solution to the equation of motions is obtained first; (ii) the solution
is substituted into the ‘flow equation’ which yields Au and AM, the flow velocity and
acceleration due to pipe deforniation; (iii) the homogeneous solution together with Au
and AU are substituted into the full nonlinear equation of motion, which is solved by the
Krylov -Bogoliubov averaging method, to first order, yielding the averaged amplitude, 2,
and the corresponding phase. Of course, at the very threshold of the Hopf bifurcation
there is zero damping. A small amount of positive or negative external viscous damping,
as
a
is
added
h~~, control parameter, used to achieve purely real eigenfrequencies for
u = u, f Au,, where u, is the critical value; an equal and opposite amount of ‘damping’,
hi, is added to the nonlinear part of the equation, thus cancelling the total added
damping. Also added to the nonlinear part of the equation are the gravity terms, which
are small. These addenda to the nonlinear part of the equation are significant in the
discussion of the results.
Unfortunately, not all parameters are defined in the results obtained. It is mentioned
that, ‘for a steel or plastic pipe in. in diameter and 2 ft in length’ and j3 = 0.5, the
gravity parameter is y = 0.01 and 0.3, respectively; so, presumably y is in that range for
the calculations in Figure 5.18. The dimensions also give a clue as to the likely value for
the fluid friction constant a in equation (5.53), which can play an important role in the
dynamics (Bajaj et al. 1980).