Page 100 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 83

plane) to make the argument simplest:

aP aw aT aw
-A - -qS+ F - = 0, - +qS - F - = 0, (3.65)
ax ax ax ax
which, when added give
a
-(T - PA) = 0. (3.66)
ax
Thus, the frictional force qS is replaced by its twin effects: (i) as a tension on the pipe
and (ii) as a pressure drop in the fluid. Equation (3.66), when integrated from x to L gives
(T - PA), = (T - ~A)L, the equivalent of equation (3.36). Ignoring externally imposed
tensioning and pressurization, which do not enter the argument (and which are discussed
in Section 3.4.2), and thus considering for simplicity the fluid to discharge to atmosphere,
both p and T vanish at x = L, and hence

T - PA = 0 for x E [O. L]. (3.67)

It follows that the term related to T and p in equation (3.34), the precursor to the final
equation of motion, vanishes, i.e.
a
- [(T - PA):] = 0,
ax (3.68)

because of (3.66) and (3.67). Therefore, the two effects of friction - tensioning and
pressure drop - cancel each other entirely and vanish from the equation of motion, to
the order of the linear approximation (Benjamin 1961a; Gregory & Pdidoussis 1966a).
This has been verified experimentally (see Sections 3.4.4 and 3.5.6), and also numeri-
cally in calculations with shell theory for beam-mode vibrations (n = 1) in Chapter 7.

3.3.5 Nondimensional equation of motion

Consider the most general form of the equation of motion derived so far, equation (3.38).
It will help further discussion if this equation is generalized a little by considering the
possibility that the pipe may be supported all along its length by a Winkler-type elastic
foundation, which involves distributed springs of stiffness K per unit length: thus, a term
Kw is added to the equation of motion.
The resultant equation may be rendered dimensionless through the use of

(3.69)

The dimensionless equation is