Page 521 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

SOME OF THE BASIC METHODS OF NONLINEAR DYNAMICS 49 1

Substituting (F.35) into (F.34) leads to

where fi(c) represent functions of C, i = 1, 3,4 (not given here for brevity) and of the
coefficients cyj, and

cz = (3a1 f a3 + + 3a8)/8, b = (a2 + 3a4 - 3a5 - a7)/8. (F.37)

Equation (F.36) is independent of r2, indicating that v27j is a resonance term. This can be
explained by using a ‘multiple scales’ approach: carrying out a straighforward expansion
to order E in (F.32) by putting < = A exp(it), one finds that the term proportional to q2C
produces a secular term proportional to A2A t exp(it), whereas the remaining ones do not.
It is therefore possible to choose r1, r3 and r4 in (F.36) to eliminate the nonresonance
terms, thereby reducing g to the form

g = 4(a + ib)q2q, (F.38)

and the normal form, to first order, is

rj = iq + 4c(a + ib)q2q. (F.39)
The normal form (F.39) can now be expressed in polar coordinates, using

I
q = ?r exp(iB). (F.40)
Substituting (F.40) into (F.39) and separating real and imaginary parts yields

i- = car 3 , B = 1 +cbr2. (F.41)

The analysis in which equation (F.30) is treated with real rather than complex variables
can be found in Nayfeh (1993). The final result is of course the same, but the algebra
involved is much more complicated.

F.4 THE METHOD OF AVERAGING
The method of averaging, originally due to Krylov & Bogoliubov (1947) is particu-
larly useful for determining periodic solutions of weakly nonlinear problems or small
perturbations of a linear oscillator. In contrast to the treatment of normal forms presented
previously, the method of averaging can be extended easily to the case of nonautonomous
systems, i.e. systems in which time appears explicitly [see, e.g. Semler & Paidoussis
(1996), for the treatment of a nonautonomous system with normal form theory]. To begin
with, let us consider the nonlinear harmonic oscillator

i + mix = Ef(X, i). (F.42)

When E = 0, the solution of (F.42) can be written as
X(l) = r cos(w0t + ,B), (F.43)