Page 312 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 293

At first glance these equations seem wrong (as no nonlinear velocity-dependent terms are
present); however, if further simplification is carried out, equation (5.54) yields the correct
form of governing equation in terms of z. The U and U2 terms are actually ‘hidden’ in
the nonlinear inertial term. Indeed, eliminating x through the condition of inextensibility
leads to
(rn + M) Z( 1 - 1”) + 2 M U i’ + M U2 z” + EZ (z”” + 3 z’ z” z’” + z”~)

+z’~’(rn+M”i’’+z’i’)ds

L
- i” ( lL (rn + M)(i” + z’i’) ds ds - 1 (m + M) Z Z’ ds) = 0. (5.55)

By multiplying by (1 + z’~) throughout, keeping cubic nonlinear terms and replacing
nonlinear inertial terms [cf. Section 5.2.7(b)], one may bring equation (5.55) with z = w
into the same form as (5.28).
Hence, this equation of motion is irreproachable. No nonlinear terms are missing, except
for the gravity terms, since gravity has been neglected. However, the different steps from
one equation to another are not very clear in the original derivation; also, Bajaj et al.
(1980) use some implicit relationships of the curvature (Semler 1991), and the procedure
for eliminating nonlinear inertial terms is not fully explained. Hence, verification is not
easy.
Finally, similarly to Rousselet & Herrmann, Bajaj et al. also establish an equation for
the flow, by considering a force balance on a fluid element, yielding

(5.56)

where UO is the constant flow velocity when the pipe is not in motion, a! represents the
resistance to the fluid motion (proportional to a friction factor) and aMUi represents
the constant pressure force at the fixed end s = 0 of the tube. It is found that a! plays an
important role in the dynamics, as discussed in Section 5.7.1.

(d) Ch ‘ng and Dowell‘s work

Ch’ng & Dowell (1979) obtained nonlinear equations of motion of a pipe conveying fluid
by the energy method based on Hamilton’s principle. An Eulerian approach is used to
describe the dynamics of the system, and the flow is assumed to be steady. Using first
only linear relationships, the well-known linear equation is found:

EZ z”” + 2 M U i’ + M U2 z” - (M + m)g[(L - x)~’]’ + (m + M) Z = 0. (5.57)
Ch’ng and Dowell then consider the nonlinear effects due to tension associated with
the axial elongation of the pipe,

(5.58)

This relationship implies that the cantilevered pipe is extensible, which is an unusual
but by no means erroneous assumption. By assuming the tube to be Hookean, an axial