Page 214 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

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Pipes Conveying Fluid:

Linear Dynamics II

4.1 INTRODUCTION
The linear dynamics of the basic system of a pipe conveying fluid has been considered
in detail in Chapter 3, including, in more abbreviated form, the dynamics of some impor-
tant modified systems (Section 3.6). A characteristic of all these systems, if they are
continuously flexible, is that they are all governed by equations (3.38) and (3.70) and the
dimensionless parameters of (3.7 l), or by simple variants thereof. Furthermore, solution of
these equations may generally be achieved by one of the two methods of Section 3.3.6, or
by straightforward extensions of these methods. The only ‘unusual’ system in this respect
is that of articulated pipes, dealt with in Section 3.8.
The systems considered here, on the other hand, either are governed by substantially
modified forms of the equations of motion or require different methods of solution.
Specifically, the following topics are discussed. Nonuniform pipes are pipes with nonuni-
form cross-section and axially variable flow area. Aspirating or.sucking pipes are pipes
ingesting flow at a free end, rather than expelling it. Short pipes also require special
treatment: from the solid mechanics side the use of Timoshenko beam theory, and from
the fluid mechanics side the use of potential flow theory and the introduction of so-
called ‘outflow models’ for the fluid discharging to atmosphere. Pipes with harmonically
perturbed flow velocity are subject to parametric resonances and require special methods
of solution; so does the treatment of forced vibration of pipes conveying fluid. Finally,
the section on applications presents some expected and unexpected uses of the work
discussed in Chapters 3 and 4.

4.2 NONUNIFORM PIPES
4.2.1 The equation of motion

The equation of motion will be derived for a pipe with a nonuniform flow passage
and, generally, a nonuniform external form also. Variations in the shape of the pipe are
axisymmetric, gradual and smooth with respect to the axial coordinate [see Figure 4.1(a)].
The pipe is immersed in air or water, so that hydrostatic, added-mass and damping effects
associated with the external fluid need generally be taken into account.
In this derivation (Hannoyer & Paidoussis 1979a), the lateral dimensions of the flow
passage will not a priori be considered to be negligible. However, the other assumptions

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