Page 39 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

22 SLENDER STRUCTURES AND AXIAL FLOW

have A* = A. In the nondimensional notation used here, 9 = [0, 11 and t = x/L. This
problem has in fact been solved by Chen (1987), but it is not difficult to reproduce the
results. One finds 2* = 2, but a new set of boundary conditions for the adjoint problem,
namely

@(o) = @'(o) = 0, $"(I) + 9$(1) = 0, @"'(l) + p$'(l) = 0. (2.58)

Solving the two eigenvalue problems, one obtains
x0) = AI sin pt + A2 cos pt + A3 sinh qt + A4 cosh &,
$(t> B1 sin pt + B2 cos pt + B3 sinh qt + B4 cosh qt,
=

AI = 1, A2 = -(p2 sin p + pq sinh q)/(p2 cos p + q2 cosh q), (2.59)
A3 = -p/q, A4 = -A2,
[(9 p2) sin p - (p/q)(B + q2> sinh 91
-
B1 = 1, B2 = - '
-
[(9 p2) cos p - (9 + q2) cosh q]
B3 = -p/q, 84 = -B2.
The characteristic equation is

P2 + 2h(l + cos pcosh q) + 9fi sin p sinh q = 0,

and it is the same for both problems; hence, so are the eigenvalues.
The essence of this method is that it achieves direct decoupling of the equations of
motion via the so-called biorthogoriality of the initial and adjoint eigenfunctions, viz.

(2.60)

By introducing I]N = xx,(t)q,(t) into equation (2.52), then muItiplying by $r(t) and
integrating over 9, the system is decoupled in a single operation, by virtue of relations
(2.60). yielding

m, q, + k,q, = J sin Wft, j = 1,2,. . . , N. (2.61)
Calculations with the same set of parameters produce virtually identical results as those
shown in Figure 2.3 for N = 4.z What is more surprising is that the rate of convergence
with N is not better with this method than with the previous one. Clearly, therefore, in this
particular case, there is no advantage in utilizing this second, more general but more labori-
ous, procedure rather than the first. Similar conclusions are reached by Anderson (1972),
who tested a very similar problem, essentially by the same two methods - although very
small differences are found in that case in the results obtained by the two methods.

tA typographical error in p and q is noted in Chen (1987, Appendix C).
The results obtained by integrating the equations numerically are also identical, although in that case it
took about one order of magnitude longer in time to obtain them.