Page 149 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 131

now textbook material (Hughes 1986; Chapters 5 and 7). Nevertheless, for fluid-structure
interaction phenomena, destabilization by dissipation is sufficiently perplexing to deserve
further attention.
Several attempts have been made to understand the mechanism of destabilization. Of
these, Benjamin’s (1963) work, applying to all fluid-structure interaction systems, will be
discussed first, followed by that of Bolotin & Zhinzher (1969) and Semler et nl. (1998).
An attempt to explain the phenomenon in simple terms was made by Benjamin (1963)
in connection with the stability of compliant surfaces in fluid flow. Specifically, consid-
ering a one-degree-of-freedom mechanical system, rnq + cq + kq = Q, where Q = Mq +
CG + Kq is associated with the fluid forces, and introducing the concept of an ‘activation
energy’, Benjamin shows that (i) if rn > M and k > K, dissipation stabilizes the system
(class B instability), while (ii) if m < M and k < K, dissipation destabilizes it (class A
instability). Since -M is the added mass, M < 0 must hold for a physically meaningful
system, and hence the condition rn < M is nonphysical. Benjamin recognized this and
so considered next an infinitely long compliant surface, disturbed by a sinusoidal wave
travelling along it. In this case, physically meaningful conditions are obtained for the exis-
tence of class A and B instabilities, once again with the aid of the activation energy [see
also Ye0 & Dawling (1987)l; as before, these conditions are dependent on the fluidsolid
mass and stiffness ratios. This work is discussed in greater detail in Appendix C.
It was initially thought (Paidoussis 1969) that Benjamin’s work could explain both
dissipative destabilization and the stability curve jumps in the pipe problem. Certainly,
for B < Bsl. where is the value for the first discontinuity, dissipation is stabilizing
(Figure 3.35) and for j3 > it is destabilizing. However, as seen in Figure 3.35,
dissipation continues to be destabilizing across the second discontinuity at &. Hence,
Benjamin’s work can only explain the destabilizing effect of damping for j3 > PSI, but
cannot explain the jumps themselves.
Another point of view was expressed by Bolotin & Zhinzher (1969), whose thesis
may be summarized as follows: the very statement that ‘damping is destabilizing’ in a
nonconservative system is flawed in that the analysis with zero damping gives a false
indication of the stability region, a portion of which, if the analysis is properly conducted
with some (even infinitesimally small) damping, is really unstable. Thus, the presence
of purely imaginary eigenvalues on the imaginary axis merely indicates ‘quasi-stability’
rather than stability. This work is very important and it can explain the dynamics for
= 0 and j3 = O+ discussed at the end of Section 3.5.3; see also Section 3.7. However,
it applies to nongyroscopic nonconservative systems and hence cannot help us, since the
instability here is via a classical Hopf rather than a Hamiltonian Hopf bifurcation. For the
pipe system, one not only obtains that nonzero dissipative forces are destabilizing vis-&vis
the undamped system, but also that in some cases (e.g. Figure 3.35 for o = 0.23 and 1.42
and also Figure 3.43) increased dissipation further destabilizes the system. In this regard
the dynamical behaviour is more closely related to Benjamin’s system. Under conditions
where dissipation-induced destabilization occurs (class A instability), the system must be
allowed to do work against the external forces providing the excitation; i.e. the absolute
energy level of the whole system must be reduced in the process of creating a free
oscillation. The interested reader is also referred to Craik (1985) and Triantafyllou (1992)
for a discussion of ‘negative energy modes’, requiring an energy sink in order to be
excited.