Page 103 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

86 SLENDER STRUCTURES AND AXIAL FLOW

For nontrivial solution, the determinant of the Aj must vanish, yielding

1 1

(3.84)

Since the roots of (3.82) cannot be expressed in simple explicit form in terms of u, w
and j3, and in view of the complexity of (3.84), it is not possible to obtain solutions by
direct methods, Three methods of solution were given by Gregory & Paidoussis (1966a):
(i) a rather ingenious method of transforming the original problem into one easier to
solve numerically in 1966;+ (ii) a straightforward numerical method; and (iii) a Galerkin
solution. Of these, only (ii) will be outlined here, as follows: (a) starting with a small
value of u, say u = 0.1, and trial values of %e (w) and 9m(w), say those for u = 0, a
minimizing procedure (e.g. a secant method) finds the appropriate values of %e(@) and
$m(w) which result in %e(A) = 9nr(A) = 0 to within desired accuracy; (b) the value
of u is increased by 6u, say by 0.1, and using the %e(w) and 9m(w) found in (a) as first
approximations, the minimizing procedure determines the complex frequency for u = 0.2;
and so on.
Clearly, this method has to be applied for each mode separately (for a given value of
#?), the locus to be followed depending on the initial trial value for %e(w).

fbl Second method

The fuller equation of motion (3.70) is nonhomogeneous, since the coefficients of deriva-
tives of are explicit functions of 6 and/or implicit ones of T, because u = u(r); hence,
the foregoing method of solution is inapplicable. A solution for u = const. is, however,
readily possible via the Galerkin method and will be given here; the case for u =
u(r) is considered in Chapter 4. This is approximate, not only in the strict numerical
sense, but also because of the finite number of terms utilized in the Galerkin expansion
(Section 2.1).
Let
00
rl(6, = 4rO) 41- (TI, (3.85)
r= 1

where qr(t) are the generalized coordinates of the discretized system and &(c) are the
dimensionless eigenfunctions of a beam with the same boundary conditions as the pipe
under consideration, and hence they are appropriate comparison functions (Section 2.1.3).
It is presumed that the series (3.85) may be truncated at a suitably high value of r, r = N.
Substitution of (3.85) into (3.70) with iC = 0, followed by multiplication by and

+Computers were then new and slow, and 6 ntvh tixvcrs Kampy&<6mi; i.e. poverty (necessity) develops
ingenuity!