Page 436 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

408 SLENDER STRUCTURES AND AXIAL FLOW

where A is a matrix, with J and M nonzero submatrices on its diagonal and zero elsewhere;
J corresponds to the purely imaginary pair of eigenvalues, and M to the 2N - 2 eigen-
values with negative real parts, so having sub-elements R,, given as follows:

(5.158)

The system is then projected onto a centre manifold, where special care must be exer-
cised because the system is nonautonomous. To transform the system to an equivalent
autonomous one, p sin wt and @ cos WT are replaced by two new variables,

(5.159)

".Is 0 0 --o 0 0
and I*. is included in the system of equations as a trivial dependent variable, = 0. Thus,
J in (5.158) is replaced by
0
0
0
0 p2
0
0
0 0 P2 0 + j j (5.160)
ow0
the associated vector being y = {p, vl, 712, XI, ~2)~. The system is then transformed
by defining x' =EX, p' = ep, uo - ucf = eq, and the method of normal forms
(Appendix F.3) is applied: (a) to find all possible parametric resonances to O(E) and
O(e2), and (b) to determine the simplest set of equations defining these resonances. Three
separate sets of normal forms are determined: (i) for w away from both 2w0 and WO,
(ii) for w near 00, and (iii) for w near 2~0. In case (ii) one obtains the fundamental
secondary resonance, where the harmonic perturbation in u appears only at the second
order, p2; in the last case, the principal resonance is obtained, where these terms appear
to first order, p. The results for the principal resonance are identical to those obtained by
Bajaj (1986b).
Before presenting any results, the second major component of this study is briefly
discussed, namely the solution of the full equations by numerical techniques. In this case
no restrictions apply as to u0 being close to ucf. Three such methods are used - see
Section 5.4. (a) The nonlinear inertial terms are transformed into equivalent stiffness
and velocity-dependent terms [Section 5.2.7(b)], and the resulting equation can then
be integrated by a Runge-Kutta method; AUTO may also be used in this scheme,
once the equations are transformed into those of an equivalent autonomous system,
as in equation (5.160). Solutions are also obtained by (b) the Jinite difference method
(FDM) based on Houbolt's fourth-order scheme and (c) the incremental harmonic balance
(IHB) method, in both cases with nonlinear inertial terms intact; a complete expos6 of
the application of the IHB method to the problem at hand may be found in Semler
et al. (1996).
Typical results for the principal resonance are shown in Figures 5.66 and 5.67. Several
observations may be made, as follows: (i) it is seen that there is good agreement between
the normal-form and numerical solutions for the resonance boundary in Figure 5.66, but
less so for the amplitude in Figure 5.67(b), especially away from the resonance boundaries;