Page 358 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

338 SLENDER STRUCTURES AND AXIAL FLOW

where the A; are the dimensionless eigenvalues associated with the beam eigenfunctions
$i, used here as comparison functions in the discretization. The coefficients bjj, qj and
dij are computed from the integrals of the eigenfunctions (Section 3.3.6, Table 3.1) while
aijk/, bijk-, Cijk[ and dijkr are computed numerically (Semler 1991; Li & Paidoussis 1994).
The repeated indices in (5.115) implicitly follow the summation convention. The
nonlinear terms have been multiplied by E, used here as a book-keeping device to indicate
that they are small. Equation (5.115) may be re-written in first-order form:

in which p = ;I; or
Y = [AIY + ef(y), (5.117)
where I, K, C are the identity, stiffness and damping matrices, and q, p and y are under-
stood to be vectors. Although all this is applicable to any order of discretization, all
numerical results (and the centre manifold calculations in subsection (b)) are confined to
a two-term Galerkin discretization, N = 2.
In the remainder of this subsection, a linear and then a nonlinear stability analysis is
undertaken, in the latter case supplemented by simulation. A typical Argand diagram of
the eigenvalues of [A] as u is increased is shown in Figure 5.24(a). The system loses
stability by divergence at u = 11.47 in its first mode. Then, according to linear analysis,
it is subject to flutter at u = 12.48 in its second mode. The two branches of the divergent
mode merge, and that mode regains stability, at u = 15.07.
A nonlinear analysis of the dynamics in the vicinity of the fixed points is conducted
next. The fixed points are given by
Kijqj + aijkl qyqiqe = 0, (5.1 18)
0
where the superscript 0 denotes the fixed point. Then, considering perturbations about a
fixed point, qi = qi 0 + u;, pi = v;, the perturbation equations are obtained from (5.1 15)
and (5.1 17):