Page 527 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

SOME OF THE BASIC METHODS OF NONLINEAR DYNAMICS 497

pair of purely imaginary eigenvalues,
*iwov = L(u,)v, (F.72)

and the other eigenvalues (infinitely many!) have negative real parts. At uc, the steady-state
solution, which lies in the centre manifold, may be expressed as
yo = r(vei@ + Vepi@), (F.73)

with r and ,9 = $ - mot being arbitrary, and V being the complex conjugate of v. Similarly
to the case of ordinary differential equations, both r and j3 will be slowly varying in time
as ~i is adjusted slightly away from u,. At u = u, + EP, expressing y(x, t) = y~(x, t) +

EYI (x, 0, and
dr
dlCI
- = CA and - = wo + EB,
dt dt
and equating coefficients of equal powers of E, we obtain

aY0
wo - -
allr

Expression (F.73) is the solution for (F.74a). Unlike equations (F.67a,b), the left-hand
side of the above equations involves both the cyclic variable $ and the spatial variable
x. Introducing a linear operator E defined by

(F.75)

and its adjoint
E*y* = 0, (F.76)

then the necessary and sufficient condition for the existence of a solution for (F.75) is

1' F(y)y* d$dw = 0. (F.77)
27r

Comparing (F.74b) with (F.75), the necessary and sufficient condition for finding a peri-
odic solution of u1 can be derived from (F.77). By substituting uo of (F.73) into (F.74b)
and applying condition (F.77), one obtains

A + Bri = pbr + cr3, (F.78)

where f(y0, u,) has been assumed to be homogeneous and cubic, and

b = L 2n 12= 1' (u,)(vei@ + Ve-'@). v*ei@ d+ dw,