Page 308 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 289

where L(q) represents the linear terms, N1 (q) the nonlinear terms not involving integrals,
and N~(v) those that do and wherein all nonlinear inertial terms lie. The terms involving
a are omitted for simplicity in the derivation.
It is noted that L(v) - O(E) and Nl(q) and N2(q) - O(c3). After some manipulation
of L(q), starting with

one obtains

which transforms one of the nonlinear inertial terms in N2(q). Integration of (5.41) from
6 to 1 yields the other nonlinear inertial term. These two terms are replaced in N2(q) to
obtain, after some long but straightforward algebra,

Hence, the transformed equation is correct to 6(c3), as is equation (5.39). However, in
view of the additional approximations introduced, this is a more approximate equation,
albeit of the same order as the original.
(el Equations for pipes with fixed ends

For pipes with both ends fixed, some additional nondimensional quantities need be intro-
duced, as follows:

EA L2
PL2
T~L~
r=- &=- n=-, (5.44)
EI ’ EI ’ EI