Page 180 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

162 SLENDER STRUCTURES AND AXIAL FLOW

Two interesting features of the theoretical results are that: (i) the effect of K is not
dramatic until the critical value of K for which divergence becomes possible is approached
(K, = 35 for c3 = 1; K, = 280 for ts = 0.5); (ii) as K, is approached, ucf can be decreased
with increasing K if .$ = 1, which is surprising, but generally increases for tS = 0.5.
Furthermore, in the experiments for K E K= it was found that after the onset of divergence,
if the flow was increased slightly and the pipe was straightened by hand, it would remain
straight upon release, so that the theoretical restabilization was actually observed; at higher
flow, again as predicted, stability was lost once more by flutter.
This work has been extended to the case of several spring supports by Sugiyama
er al. (1991).
Finally, Lin & Chen (1976) and Noah & Hopkins (1980) consider the case where the
downstream end is supported simultaneously by a translational and a rotational spring
[Figure 3.61(c)]. In this case the two boundary conditions at 6 = 1 are

where C is the rotational and K the translational spring stiffness. The solution of the
equation of motion, equation (3.76), subject to the boundary conditions 11 = ar/ac = 0
at 6 = 0 and equations (3.120), was obtained by Galerkin’s method. The comparison
functions, however, are obtained by solving the beam equation subject to these boundary
conditions, yielding the following eigenfunctions (Noah 62 Hopkins 1980):
$j(t) = cash (hjc) - COS (hjt) - aj(sinh (hit) - sin (hit)), (3.121a)
with
(K*/h,)(sinh hi + sin h,) + cosh hj +cos hj
0- - (3.121 b)
’ - (K*/hj)(cosh h, - cos hj) + sinh hj + sin hi ’
which were shown to be orthogonal. The eigenvalues h, are solutions of
K K*
-(tan hj - tanh hj) + - (tan h, + tanh hj)
1; hj
1 1
- I) (COS hj cosh h,
In this way convergence, as the number of comparison functions is increased, is quite
rapid (see Section 2.1).
If only a rotational spring is present and it is sufficiently stiff, then the system
approaches a clamped-sliding systemt and loses stability by divergence. However, the
nonconservativeness of the system generates unexpected results when both K and K* are
present. Consider the following set of results obtained by Noah & Hopkins (1980) for
,8 = 0.125, a = lop3. (i) With K = K* = 0 the system loses stability by flutter, but with
K* = 10 it does so by divergence. (ii) If K = 25 and K* = 10, however, the system loses
stability by flutter once again, the divergence not occurring at all (not even at higher
flow velocities). Thus, the addition of a translational spring, instead of aiding in the

+Sliding in the transverse direction, corresponding to the standard sliding support condition for a beam, with
boundary conditions w’(L) = 0, Elw”’(L) = 0 (Bishop & Johnson 1960); not to be confused to axial sliding at
an otherwise clamped or pinned end, as discussed in the foregoing.