Page 102 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 85

(a) First method

The simplest form of the equation of motion, equation (3.1), will be considered first,
which in dimensionless form becomes
+
a417 a2q a2r a2r
+
- u - 2pu +--0, (3.76)
at4 ap ~ acat at2
subject to the appropriate boundary conditions; e.g. for a pipe with simply-supported
('pinned') ends,
(3.77)

while for a cantilevered pipe,

(3.78)

Consider now solutions of the form

r~, ~WO)~'"'I, (3.79)
r)
=
where w is the dimensionless circular frequency defined by (3.73). In general, w will
be complex, and the system will be stable or unstable accordingly as the imaginary
component of w, 9m(w), is positive or negative; in the case of neutral stability w is
wholly real. Substituting (3.79) into (3.76) leads to

(3.80)

Next, we take a trial solution
Y(6) = A&, (3.81)
where A is a constant. When this is substituted into equation (3.80), the equation deter-
mining the permissible values of the exponent a is obtained, namely

a4 - u2,2 - 2pum - 02 = 0, (3.82)
and since this equation is of fourth degree, the complete solution of (3.76) is given in
general by

(3.83)

in which the four A, must be determined from the boundary conditions. This is illustrated
here for the cantilevered system. Making use of (3.78), we find