Page 525 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

SOME OF THE BASIC METHODS OF NONLINEAR DYNAMICS 495

hence, the matrix A in equation (F.53) becomes

Following the methodology developed previously, the bifurcation parameters can be
related to the unfolding parameters pl, p2 and p3, by requiring that

(F.61)

Computing the derivatives in equation (F.61) numerically and eliminating p3 leads to a
linear relationship of the form

where a,, i = 1,. . . , 4, are real constants.

F.6 PARTIAL DIFFERENTIAL EQUATIONS
F.6.1 The method of averaging revisited

In order to adapt the averaging method to PDEs, an alternative form of averaging for a
system of ODES is first given. Thus, consider the ODE of the form

X = AOX + EAIX + Ef(X), x E R2, (F.63)

where
Ao= yo -3 AI = [p2 -w2
w1
2
When E = 0, the solution to the above system has the form

XO = r cos(w0t + B> }=r{ cos @ }-fr{ '.}ei*+;r{ (F.64)
sin(oot + B> sin @ -1