Page 315 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

296 SLENDER STRUCTURES AND AXIAL FLOW

equations derived for cantilevered pipes, those by Lundgren et al. (1979) and Bajaj et al.
(1980) are found to be absolutely correct, while those by Rousselet & Herrmann (1981)
are found to be correct except for a small order-of-magnitude inconsistency. Furthermore,
both sets contain a distinct refinement vis-&vis those derived here: the flow velocity is
not assumed to be constant; instead, the upstream pressure is taken to be constant, while
the flow velocity generally varies with deformation. Of the equations derived for pipes
withJixed ends, the set derived here is considered to be the only one available, correct to
the same order as that for the cantilevered pipes. On the other hand, the simple equation
derived originally by Holmes (1977) is correct as far as it goes and may be preferred in
some cases because of its simplicity. It is of interest that the origin of the terms in the
equations - even some of the linear terms - as well as the structure of the equations are
distinctly different for pipes with both ends fixed as compared to cantilevered ones.

5.3 EQUATIONS FOR ARTICULATED SYSTEMS
Traditionally, articulated models of columns subjected to axial loading have been widely
used as an aid in the study of their continuous, distributed parameter counterparts
(Herrmann 1967). The same has occurred with pipes conveying fluid. For nonlinear
dynamics this is particularly attractive, since many of the methods of nonlinear dynamics
are best suited to low-dimensional discrete systems; with articulated systems, no questions
need arise as to the adequacy of the Galerkin discretization of a continuous system: the
physical system is discrete and may be low-dimensional to start with.
Most of the interesting dynamics is associated with cantilevered systems, and hence
most of the research has been devoted to such systems. Furthermore, virtually all of that
work has been confined to two-degree-of-freedom (N = 2) systems (Figure 5.2), and this
despite the caveat (Section 3.8) that the dynamical behaviour of the N = 2 system is not
generic with respect to N (Paidoussis & Deksnis 1970).
Two representative sets of equations are given here, both for cantilevered two-degree-
of-freedom systems. The first set was derived by Rousselet & Herrmann (1977) by
straightforward Newtonian methods via free-body diagrams and moment balances on
the two segments of the pipe, yielding

(5.64)

where M and m are the mass of the fluid and of the pipe per unit length, and U the
flow velocity; ZI and 12 are the lengths of the upper and lower segments of the pipe
(Figure 5.2), and kl and kz are the stiffnesses of the interconnecting springs. The flow