Page 335 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

316 SLENDER STRUCTURES AND AXIAL FLOW

(D = 30 mm, L = 10 m) are subjected to periodic blast waves transmitted through the gas
in the reaction chamber.
The equations of motion are similar to Ch’ng & Dowell’s and Thurman & Mote’s
(Sections 5.2.8 and 5.2.9) but more complete: three equations of motion are obtained for
motion in two mutually perpendicular planes, and the flow velocity is generally harmon-
ically perturbed as in equation (4.69) to account for pump-induced pulsations. Before
analysis, however, the tension-gravity term is considerably simplified and the axial equa-
tion of motion is eliminated, so that each of the remaining equations becomes similar to
Holmes’ (Section 5.5.2); these two equations are coupled via the nonlinear tension terms.
Furthermore, because tension-gravity effects are so large, flexural terms are neglected,
so that the system becomes a pipe-string [cf. Copeland’s work in Section 5.8.3(b)]. In the
calculations presented, the flow velocity is steady and dissipation is taken into account.
The equations of motion are discretized and then integrated numerically.
In the calculations, the pipe is excited by periodic impulses, introduced as initial condi-
tions all along the length in the plane of the blast wave, and motion is monitored in both
planes. With increasing frequency of impulses, the classical jump (down) phenomenon in
the frequency response curve is obtained, characteristic of hardening nonlinear systems,
and the associated jump (up) when the frequency is decreased.
If the pipe is perturbed in the plane perpendicular to that being excited, the oscillation
either dies out or builds up to a steady limit-cycle motion, depending on the periodic
impulse frequency; in the latter case, a generally oval whirling motion ensues with slow
precession, which would be unacceptable in actual ICF operation.

5.6 ARTICULATED CANTILEVERED PIPES

Many of the methods for analysing nonlinear systems apply to ordinary differential equa-
tions, so that continuous systems must be discretized first before these methods can
be applied. Furthermore, since most of these methods are practicable only for low-
dimensional (low-D) systems (Le. systems of only a few degrees of freedom), there
is a natural tendency to study low-D discretizations of the continuous systems. This then
opens the question, often left unanswered, of whether the low-D discretized model really
captures adequately the essential dynamical features of the continuous system. This in
turn provides the main impetus in the study of articulated systems: the very physical
system is discrete and it can be chosen a priori to be low-D.
Most of the work on the nonlinear dynamics of articulated pipes conveying fluid has
been done on the nonconservative cantilevered system (Figure 5.2). In many cases this
serves as a preamble to the study of the same aspects of the continuous cantilevered
system, discussed in Section 5.7; this is the reason for this section being where it is.
Before discussing cantilevered articulated pipes in Section 5.6.2, the case of a pipe
with a constrained downstream end is treated first.

5.6.1 Cantilever with constrained end
No systematic study has been published on the nonlinear dynamics of the conserva-
tive system of articulated pipes with supported ends - perhaps because the work in
Section 5.5 is considered to have settled all important issues.