Page 314 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 295

(bl Holmes‘ work
Holmes (1977) was one of the first to develop the new tools of modem dynamics, and to
introduce them into the study of fluid-structure dynamical systems; he was therefore less
concerned with the derivation of the equations so much as he was with their srructure. In
that spirit, he considered only the major nonlinear terms associated with the deflection-
induced tension in the pipe.
Starting from the linear equation obtained by Paldoussis & Issid (1974), Holmes adds
the effect of the axial extension. To a first order approximation, the axial tension induced
by lateral motions is
T=aA= (E&+E*&)A,

in which a Kelvin-Voigt viscoelastic material has been considered and where E is the
averaged axial strain defined by

l
E = 2L L (.~)’~ds.+

Thus, an axial force T is added to the linear equation, where

EA E*A
T=-- (z’~) ds + I (z’ 2’) ds. (5.63)
L
~
The addition of this extra deflection-dependent axial force leads to one equation with
two cubic nonlinear terms. This same axial force T (with 9 = 0) has also been obtained by
Ch’ng & Dowel1 (1979) and by Namachchivaya (1989) through the energy method. In this
case, however, attention must be paid to the order approximation, as already mentioned
in Section 5.2.1.
It is noted that Holmes’ version of the nonlinear equation is a single scalar one, as
compared to the two equations derived in Section 5.2.7 and also by others. The implication
is that, in Holmes’ work, axial motion of the pipe is considered to be negligibly small
and also that it is symmetric vis-&-vis the undeformed pipe shape.

5.2.1 0 Concluding remarks

The nonlinear equations of motion of a pipe conveying fluid have been derived in a simple
manner, by both the energy and, in Appendix G, by the Newtonian method, following
Semler et al. (1994). It is shown that the equations of motion of a cantilevered pipe and
of a pipe fixed at both ends are fundamentally different. In the first case the pipe may
be considered to be inextensible and nonlinearities are mainly geometric, related to the
large curvature in the course of arbitrary motions. In the case of a pipe fixed at the ends,
nonlinearities are mainly associated with stretching of the pipe and the nonlinear forces
generated thereby.
Of the other derivations, some have been found to be absolutely correct, some correct for
the purposes for which they are used, and some to contain errors or inconsistencies. Of the

‘There are some errors in sign in a few intermediate steps in Holmes’ derivation (1977); the final equation,
however. is correct.