Page 29 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

12 SLENDER STRUCTURES AND AXIAL FLOW

for i = 1, . . . , N are those for i = N + 1,. . . ,2N, multiplied by A;. Since the A; are
complex, so are the f2; - the real part of R being associated with the frequency of
oscillation and the imaginary part with damping (see Section 2.3); recall that A; = in;.
A modal matrix, [A], is then constructed, and the transformation {z) = [A]{y) intro-
duced. In view of the weighted orthogonality of the [A);, for a set of distinct eigenvalues,
one obtains
[f'lIyl+ [SIIy) = [AIT[@) = {*I), (2.19)
where [PI = [AIT[B][A] and [SI = [AIT[E][A] are diagonal. Hence, each row reads y; -
A;y; = ai*;, i = 1, 2, . . . , 2N, which is easily solvable. As before, the solution in terms
of [q), and redundantly in terms of [q}, is obtained by (z) = [A][y).
In fluidelastic systems [C] and [K] are often nonsymmetric, and the foregoing decou-
pling procedure then needs to be modified (Meirovitch 1967). To that end, the adjoint of
eigenvalue problem (2.18) is defined,

(WI - WIT) {A) = IO), (2.20)
the eigenvalues of which are the same as those of (2.18), but the eigenvectors, [A];,
are different. Then, the original system may be decoupled by introducing in (2.17) the
transformation [z) = [AIIy), and (ii) making use of the biorthogonality properties

{A]T{A)j = 0, {A)T[Y]{A), = [0), for i # j, (2.21)
which lead to a decoupled equation, similar, in form at least, to (2.19).

2.1.3 The Galerkin method via a simple example

As already mentioned, it is advantageous to analyse distributed parameter (or continuous)
systems by transforming them into discrete ones by the Galerkin method (or, for that
matter, by collocation or finite element techniques), and then utilizing the methods outlined
in Section 2.1.2. The Galerkin method will be reviewed here by means of an example.
Consider a uniform cantilevered pipe of length L, mass per unit length m, and flexural
rigidity EZ. The simplest equation describing its flexural motion is
a4w a2w
EZ- + m- = 0, (2.22)
ax4 at2
where w(x, t) is the lateral deflection - according to the Euler-Bernoulli beam theory,
as opposed to the Timoshenko or other higher order theories. The boundary conditions
are

The solution of this problem is well known [e.g. Bishop & Johnson (1960)l. After sepa-
ration of variables, with separation constant A:, the spatial equation admits a solution
consiting of exponentials of &A,. and &A,.i. Substitution into (2.23) gives a system of
four homogeneous equations, the condition for nontrivial solution of which leads to the
characteristic equation,
COS A,.L cosh A,.L + 1 = 0. (2.24)