Page 202 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

184 SLENDER STRUCTURES AND AXIAL FLOW

subject was done by Paidoussis & Deksnis (1970), Bohn & Herrmann (1974a,b), Sugiyama
& Noda (1981), Bajaj & Sethna (1982a), Sugiyama & Pafdoussis (1982), Lunn (1982),
Sugiyama (1984) and Sugiyama et al. (1986a,b) on linear aspects; a considerable amount
of work was also done on the nonlinear dynamics of the system, which is discussed in
Chapter 5.
The dynamics of the articulated system mirror those of the continuous one (which is
treated first in this book), with the following difference: the cantilevered articulated system
is not only subject to flutter but also to divergence, unlike the continuous system. The
importance of this discrepancy should be viewed in the context of the popularity of low-
dimensional (low-N) models for studying the dynamics of continuous systems (Henmann
1967; Herrmann & Bungay 1964; Herrmann & Jong 1965, 1966). For columns subjected
to axial loading, the dynamics is qualitatively the same in the discrete and continuous
systems, and hence low4 models may be used without worry; however, this is not the
case for pipes conveying fluid, as discussed in Section 3.8.2.

3.8.1 The basic dynamics

Consider the articulated system shown in Figure 3.l(d), oscillating in a vertical plane.
The mass of the pipe per unit length is rn and that of the fluid M, the length of the upper
pipe 11 and of the lower one 12; the corresponding spring constants are kl and k2, while
the generalized coordinates are q1 = 8, q2 = 4. The equations of motion can be derived
with the aid of (3.10) from the expressions for the kinetic and potential energies, correct
to second order,

+ {const. + iM [($Zi + Q2 + llZ:6$ + $Z:$’ + 211Z2U(+ - 8)8]}, (3.156)

V = ;{k1O2 + k2(8 - 4)2 + $(m +M)g[(l: + 21112)8’ + Z;qj2]],

and R = (l,O+l2+)k- i(Z102 +12+2)i and r = @k+i, where k and i are the unit
vectors, respectively in the lateral z-direction and the axial x-direction. The equations
of motion are rendered dimensionless by defining a dimensionless time t = [3k2/(M +
rn)Z;]’/*t and the parameters a = Zl/lz, = 38 = 3M/(M + m), K = kl/k2, u = [(M +
m)12/3k~]”’CJ and y = (M + rn)g1$/2k2.
For the system defined by a = K = 1, 8 = and y = 0 it is found that (i) the first
mode remains stable, receding to even larger 9m(w) as u is increased, w being the
eigenfrequency; (ii) the system loses stability in its second mode at u = 1.733 by flutter;
(iii) thereafter the second-mode locus reaches the 9m (w)-axis and remains thereon,
tending to o + 0 as u + 00. For y # 0, however, stability may be lost by divergence.
No attempt was made by Benjamin to draw the map showing where divergence and
where flutter would occur; this was done later, e.g. by Lunn (1982) - see Figure 5.13.
Nevertheless, in the case where the restoring forces are due to gravity alone (kl = k2 = 0),
Benjamin shows that the system loses stability by divergence if p > i, and by flutter for
lower p. Hence, in a typical experimental system, if the fluid conveyed is water the
instability first observed will typically be divergence; if it is air, however, it will be
flutter.