Page 526 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

496 SLENDER STRUCTURES AND AXIAL FLOW

where r and fi are integration constants; xo is 2n-periodic in +. When E # 0, but E << 1,
it is expected that both r and fi are slowly varying with time, such that

i- = EA and $r =oo+EB. (F.65)
The purpose here is to find A and B in terms of other system parameters.
Recall that system (F.63) is autonomous. Therefore, the time variable may be eliminated
with the use of the following chain rule:

dx ax ax
- (F.66)
= EA - + (00 + EB)-.
dt ar allr
Let x = xo + 6x1; then, expanding and collecting coefficients of equal powers of E,

(F.67a)

(F.67b)

are obtained. Obviously, expression (F.64) is the solution of (F.67a). In order to get a
periodic solution of XI, secular terms on the right-hand side of (F.67b) must be eliminated.
This requirement is guaranteed if

2n ] { f}e&"d@=O. (F.68)

Substituting xo and AI into the above and integrating, yields

(F.69)

A and B are obtained by separating the real and imaginary parts of (F.69). Thus, (F.65)
becomes
[ llr+f2 (F.70)
r=c rp1+- cos 1
iEr(fl sin +Id+ ,
1
(f2cos+- fl sin+)d+ .

Revisiting system (F.42), it is noted that it can be transformed into two first-order equa-
tions: X = --ooy, y = wox - Ef /q, where it is seen that f 1 = 0, f 2 = - f /ma and
p1 = p2 = 0. Thus, the result in (F.70) is identical to (F.47). Note also the similarity
to equation (F.48).
We now consider the method of averaging for PDEs of the form

(F.71)
where both L and f are differential operators of x E (0, 1); u once again represents the
varied system parameter. It is further assumed that at u = u, the linear system contains a