Page 530 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

500 SLENDER STRUCTURES AND AXIAL FLOW

i.e. (I - P>f(C) = C cos(n.$), where C is a constant to be determined. Therefore, the
projection P is defined by

Pf(8 = f(C> - c cos(n0. (F.90)

The constant C may be calculated from the condition that each element in the range of
L is orthogonal to ker L* ,

(F.9 1 )

which leads to C = 2 Jd f(() cos(?rc) de. Thus, the projection of any function f(4) onto
kerL* is given by

(F.92)

Now we decompose the solution according to (F.83), $ = $c + qS, where

qC = q cos(7tC) E ker L and $s = h(q, u, 6) E range L'. (F.93)

In (F.93), q is the amplitude of the buckling mode. The bifurcation equation is obtained
by projecting the original equation (F.88) onto the kernel of L* through (F.92), which
leads to

I' [$" + u($ - + . . .)] cos(rC)dt = 0. (F.94)

Equation (F.94) is solved by making use of (F.93), and by introducing the unfolding
parameter ,u = u - u,. From the symmetry of the problem, it can be seen that it is not
necessary to find h explicitly if one wants to find a bifurcation equation to the third-order
only. After some manipulations, this leads to

(F.95)

The unfolding parameter p represents the small deviation from the bifurcation point,
and from equation (F.95), it is obvious that p = O(q2). Therefore the term pq3 can be
neglected in comparison to u,q3, so that the final bifurcation equation then becomes
(8p/n2>q - q3 = 0. (F.96)

F.6.3 The method of alternate problems

Although the method of alternate problems is usually applied to finite dimensional
problems, it is introduced very briefly here because of its similarity with the
Lyapunov-Schmidt method presented in Section F.6.2. A detailed presentation may be
found in Hale (1969) and Bajaj (1982). The method is particularly useful if the scaling
relationship between the small parameters and the amplitude is a priori unknown, this
scaling being suggested ultimately by the bifurcation equation, as in the example of
the previous section. In spirit, the method of alternate problems is very similar to the