Page 520 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

490 SLENDER STRUCTURES AND AXL4L FLOW

(F.29)
Consequently, the inverse operator zil exists if and only if Ak.i # 0. Introducing the
subset xk of Hk representing the eigenvectors of YA with nonzero eigenvalue, it can be
said that the components of fk(y) lying in Xk can be eliminated by a proper choice of
hl(y), while the components of fk(y) not lying in Xk cannot be eliminated. In other words,
the terms of fk(y) lying in the range of 2~ can be eliminated, while those lying in the
kernel of YA have to stay.
From the analysis, three important characteristics become apparent: (i) the normal form
method is local, since the coordinate transformation is generated in the neighbourhood of a
known solution (usually a fixed point for vector fields); (ii) the coordinate transformation
is a nonlinear function of the dependent variables, but it is found by solving a linear
problem; (iii) the structure of the normal form is determined entirely by A, since the
transformation depends only on the eigenvalues of A - see equation (F.29).
Examples of how to find normal forms may be found in many books, e.g. Guckenheimer
& Holmes (1983), Wiggins (1990), Arrowsmith & Place (1990). Because of its importance,
here we consider the case of a two-dimensional system with purely imaginary eigenvalues:

(F.30)

and, because of its simplicity, we follow the methodology developed by Nayfeh (1993).
Equation (F.30) is first transformed into a single complex-valued equation using the trans-
formation
x1 = c - 4, x2 = i(C - f), (F.31)

to obtain
4 = it + ic [(a, - ia5)(< + 5)’ + i(a2 - ia6>(< + f)’(~ - 5)

-(a3 - ia7)(~ + - 5)’ - i(a4 - ia8)(< - 513] . (F.32)

Using the methodology described previously, we assume
2‘ = 9 + ~h(q, 3) and q = ir] + cg(q, 7). (F.33)

Substituting (F.33) into (F.32) and equating the coefficients of E on both sides yields

Next, the function h must be chosen so as to eliminate the nonresonance terms, Le. all
nonlinear terms that do not produce secular terms (which implies that resonance terms
are those producing secular terms). The form of equation (F.34) suggests choosing h in
the form
h = r,q3 + r2q27 + r3qij2 + r4ij3. (F.35)