Page 184 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

166 SLENDER STRUCTURES AND AXIAL FLOW

First, consider the lowest three curves in the figure. Comparing the cases with a single
mass at the downstream end (J = 1, ,$I = l), it is seen that increasing the magnitude
of the mass from pl = 0.2 to 0.3 destabilizes the system further, which agrees with the
statement made just above. If, however, a second mass is added at mid-point (J = 2;
p1 = 0.2 at (I = 1; p2 = 0.2 at ,$z = OS), the effect is to stabilize the system slightly,
even though the combined additional mass is now higher than in either of the other two
cases. On the other hand, if one starts with a mass at mid-point (top curve, other than
G-P), then the addition of a second mass at the end (the next, lower curve) severely
destabilizes the system.
Second, based on the foregoing, one might conclude that adding a mass at mid-point
is always stabilizing. If, however, the mid-point mass is the only one added, as in the
uppermost curve (J = 1; p1 = 0.2 at = OS), the effect is stabilizing for p I
0.27
and destabilizing for larger p. This is another instance where the qualitative dynamical
behaviour of the system is radically different on either side of the S-shaped bend in
the stability curve - or, as is the case here, close to that bend (note that the transition
at = 0.27 is half-way between B = 0.295 for the system without an added mass and
Bequiv = M/[m + M + rnl/L] = 1/(1 + p1) = 0.246 for the system with one.)
Another point of interest in Figure 3.68 is the sharpness of the S-shaped bends, more
like kinks here, in the lower stability curves. This is explained by Hill & Swanson as
being due to sudden switches of the system from losing stability in one mode just below
the B concerned, and in another mode just above it (and for a critical B two modes losing
stability at the same u) - instead of the behaviour as in Figures 3.27 and 3.28 involving
destabilization, stabilization and destabilization once more; thus, in this case there are
real discontinuities in the values of w,f, as shown by Hill & Swanson, but not here for
brevity.
Finally, the various data points (0, 0, etc.) correspond to experimental points obtained
by Hill & Swanson, utilizing surgical rubber pipes conveying water in an apparatus similar
to that of Gregory & Paidoussis (1966b). The agreement with theory is excellent, although
if dissipation had been taken into account in the theory, it might have been less so.
Further studies on this problem have been made by Chen & Jendrzejczyk (1985)
and Jendrzejczyk & Chen (1985) for a mass at the free end, and by Sugiyama et al.
(1988a), who consider an additional mass together with a spring at some point x < L.
Sugiyama et al. find that the u versus K curve displays S-shaped discontinuities for selected
combinations of K and p1 = rnl/[(rn + M)L]. This means that there exists a region of
restabilization between two critical values of ucf for loss of stability (cf. Figure 3.28).
It is of interest that the flutter mode is quite different at these two critical values, as
shown in Figure 3.69: in both cases there are very strong travelling-wave components in
the motion; however, in (a) the presence of the spring and mass at = 0.25 is hardly
manifest, while in (b) there is a quasi-nodal point not far from cl.
In another study, Silva (1979, 1981) examines the stability of pipes with attached
valves - which can be quite massive relative to the pipe. Masses centred on the pipe or
eccentric [overhanging, as shown in Figure 3.67(b)] are considered for both cantilevered
and simply-supported pipes, but ignoring out-of-plane motions and possible coupling with
torsional modes. As in the foregoing, the dynamical behaviour is affected by the value of
p, (in this case j = 1 always) and also h = h/L, where h is the distance of the point mass
from the pipe centreline. As expected, the effect of the additional mass on the stability
of simply-supported pipes is on coupled-mode flutter, rather than divergence which is a