Page 77 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

60 SLENDER STRUCTURES AND AXIAL FLOW

of the rotating shaft, and thus complements them both as a tool for the development of
new dynamical theory and methods of analysis (Pa’idoussis 1987; PaTdoussis & Li 1993);
(iv) it belongs to a broader class of dynamical systems involving momentum transport:
that of axially moving continua, such as high speed magnetic and paper tapes, band-saw
blades, transmission chains and belts (Mote 1968, 1972; Wickert & Mote 1990), in paper,
fibre and plastic film winding, as well as in extrusion processes.
In terms of the topics covered in this book, all of which deal with axial flow along
slender structures, the pipe conveying fluid constitutes the main paradigm, on the basis of
which the qualitative dynamics of other systems are explained. This is one of the reasons
why so much emphasis is placed on this topic.
This chapter together with Chapter 4 deal with the linear dynamics of initially straight
pipes conveying fluid. The nonlinear dynamics of the same physical system is the subject
of Chapter 5. The dynamics of curved pipes conveying fluid is presented in Chapter 6,
and that of shells containing flow in Chapter 7 (Volume 2).
The dynamics of pipes with steady mean axialflow is presented first, starting with
a discussion of the fundamentals and the derivation of the equations of motion, in
Sections 3.2 and 3.3. The dynamics of pipes with supported ends, which is an inherently
conservative system (Le. a conservative system in the absence of dissipative forces), is
treated next (Section 3.4), followed by cantilevered pipes, an inherently nonconservative
system (Section 3.3, and then hybrid and articulated pipe systems. Other, more complex
systems and applications are the subject of Chapter 4.

3.2 THE FUNDAMENTALS

3.2.1 Pipes with supported ends

After Bounikres (1939), the study of pipes conveying fluid was re-initiated by Ashley &
Haviland (1950) in an attempt to explain the vibrations observed in the Trans-Arabian
Pipeline. Feodos’ev (1951), Housner (1952) and Niordson (1953) were the first to study
the dynamics of pipes supported at both ends, obtaining the correct linear equations of
motion in different ways, and reaching the correct conclusions regarding stability.
If gravity, internal damping, externally imposed tension and pressurization effects are
either absent or neglected, the equation of motion of the pipe in Figure 3.l(a-c) takes the
particularly simple form

where EZ is the flexural rigidity of the pipe, M is the mass of fluid per unit length, flowing
with a steady flow velocity U, rn is the mass of the pipe per unit length, and w is the
lateral deflection of the pipe; x and t are the axial coordinate and time, respectively. The
fluid forces are modelled in terms of a plug flow model, which is the simplest possible
form of the slender body approximation for the problem at hand. This equation will be
derived in various ways and forms in Section 3.3. Suffice it to point out here, however,
that if one uses the slender body approximation (2.83), together with (2.69) and u = O,$

‘As will be seen later, the equation of motion is independent of fluid frictional effects, and equation (3.1)
holds true if pressure drop in the pipe is taken into account, i.e. for u # 0.