Page 181 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 163

development of divergence, makes it impossible. Finally, (iii) if K is increased, so that
K = 40 and K* = 10, then the system loses stability by divergence once more. Yet, (iv) if
K = 100, the behaviour with K* = 0 and 10 is qualitatively similar: the system loses
stability by divergence and at higher flow by flutter in both cases. Hence, K and K* do not
act synergistically; the dynamics of the system is affected not only by the values of the
individual spring constants, but also by their relative magnitudes. Equivalent dynamical
behaviour is found in aeroelasticity (Dowel1 et al. 1995; Section 3.6).
The foregoing peculiar stability behaviour follows the same pattern as in Figure 3.66,
obtained by Lin & Chen (1976) for = 0 - thus for a column subjected to a follower

60'
I
(a) I
I
I
50'-
40 -
Divergence
30 -
20 - .I , I
.N
I
I
I
I
i K1
I K2 I
10 d I ,

K
Figure 3.66 Stability of a cantilevered column subjected to a follower load 9 (or equivalently a
pipe conveying fluid with j3 = 0, where u2 = 8), supported at the free end by a translational and a
rotational spring of dimensionless stiffness K and K*, respectively: (a) for K* = 0; (b) for K* = 10;
(c) for K* = 30. At K = K, and ~4 the divergence and flutter bounds coincide (Lin & Chen 1976).