Page 94 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 77

the kinetic and potential energies of the system entirely ignored this aspect, proceeding as
if the system were closed, yet fortuitously ended up with the correct equation of motion.
Benjamin (1961a,b) was the first to derive a proper statement for Hamilton’s principle,
in his work related to articulated and continuously flexible cantilevered pipes. Benjamin
rightly maintained that Housner’s derivation was erroneous, since the proper statement
of Hamilton’s principle was not used; thus, although the correct equation of motion was
spuriously obtained for pipes with supported ends through a fortuitous error in the kinetic
energy expression (Benjamin 1961a), there is no question that Housner’s derivation would
fail if applied to cantilevered pipes. The controversy was resolved by McIver (1973) with
the aid of a more general form of Hamilton’s principle for open systems, concluding
that Benjamin’s argument was correct, but Housner’s derivation was also ‘correct’, in a
sense, though for unexpected reasons. Hence, in this section Hamilton’s principle will
be reproduced as per McIver’s work, and then the form obtained by Benjamin and the
equations of motion will be derived therefrom: finally, Housner’s derivation for pipes
with supported ends will be considered.
Let us first rewrite the principle of virtual work for a system of N particles, each of
mass mi and subjected to a force Fi. By d’Alembert’s principle,

(3.43)

where ri is the position vector of each particle and Sri the associated virtual displacement
compatible with the system constraints. It is first noted that

6
5 6ri = 6w - 6v, (3.44)
i= 1
is the virtual work by the applied forces, part of which has been expressed in terms of
the potential energy V. Then, by re-writing

where T is the kinetic energy of the system, equations (3.43)-(3.45) lead to

(3.46)

Consider next the closed system of Figure 3.7(a) associated with the closed control
volume y(t), bounded by the surface Yc(t), containing a collection of particles of density
p, each with position vector r and velocity u. The principle of virtual work in the form
just derived may be written as

(3.47)

where Zc = T, - 1! is the Lagrangian of the closed system, 6W is the virtual work by
the generalized forces, and D/Dt is the material derivative following a particle; hence,