Page 187 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 169

subject to the boundary and compatibility conditions

(3.126)
a2w2
w2 = 0, EI(=)+C2(2)=0 at x2=b,

10.
In
In
a2w2
a3w2
(S) = (z) (S) = (q)
lo.
The system, once rendered nondimensional, may be solved by straightforward means (cf.
Section 3.6.1). Its dynamics is governed by the following parameters:
tS = a/L, 9 = PL2/EI, K; = ClL/EI, K; = C2L/EI. (3.127)
It is noted that here @ = 0, and hence there are no Coriolis terms, and 9 = -r, f' being
the nondimensional tension, while 9 is a compression. By assigning to KT and K; the
value of zero or infinity, a pinned or clamped end condition may be obtained at either
end, or both, without change in the basic formulation.
Some typical and interesting results are presented in Figure 3.70(b,c). The stability
boundary for a pinned-clamped system (KT = 0, K; = 00) is shown in Figure 3.70(b). It
is seen that the system loses stability by divergence throughout, with no coupled-mode
flutter for higher values of 9 as would be the case for a pipe. The eigenfrequencies
remain real until Ycr is reached, when they become imaginary; but they do not coalesce
on either the real- or imaginary-frequency axis. Physically, it is clear that the follower
force, once the beam is flexed as in Figure 3.70(a), cannot resist the moment generated
by 9 and hence flutter cannot develop. Note also that in the absence of Coriolis forces
there cannot be post-divergence restabilization. Hence, although the system is inherently
nonconservative, it is effectively conservative, as is the case when both ends are pinned.
The behaviour of the clamped-pinned system (K; = 00, K; = 0) is quite different, as
seen in Figure 3.70(c); the conservative results (where the force remains parallel to the
undeformed axis) are also shown. For these boundary conditions, for 0.2 < tS < 0.45
approximately, the system loses stability by coupled-mode flutter rather than divergence;
this comes about through coalescence of two eigenfrequencies while on the real axis, either
the first and the second or the second and third [cf. Figure 3.4(c)]. For lower tS, progres-
sively higher modes would be involved. For eS = 1, the system becomes conservative.
Experiments were conducted by using a long aluminium blade (L 2: 1.2m, 50.8 mm
wide in the vertical plane, and 5mm thick), clamped at one end, and simply-supported
and free to slide axially at the other, so as to oscillate in the horizontal direction. The