Page 331 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

3 12 SLENDER STRUCTURES AND AXIAL FLOW

where E(t), comprising the first four terms of (5.96), is the Hamiltonian (conservative)
energy, i.e. the kinetic plus the potential energy
I
I
V17) = ;{I17 112- u 2 117 I2 + :sQlr’141; (5.98)
thus, the integration constant, CO, is equal to the initial energy supplied to the pipe. For
17 = vf, with the aid of the first of equations (5.87), it is shown that 1v”l2 = n21$I2, and
hence, by utilizing the second of (5.87), it is found that
min”Ir(q) = -k(u2 - n2)2/~. (5.99)

The integral term in (5.97) is strictly increasing with time and, since Co is constant,
E(r) must decrease unless t = 0. However, rj = 0 only at the equilibrium points; i.e. for
u > n, at the saddle points where the minima of T(q) given by (5.99) occur. Therefore,
these are also the minima of E(+ Thus, the pipe will always approach an equilibrium
point as t + 03. Consequently, by infinite dimensional analysis it has definitively been
shown that no limit-cycle oscillation can exist in this system.
This completes the presentation of Holmes’ work on this system, proving that ‘pipes
supported at both ends cannot flutter’. Or does it? The question of whether even this
unequivocal statement has to be qualified is discussed next.

IC) Flutter in the Hamiltonian system
Lunn (1982) examined the equivalent problem to Holmes’: a pin-ended pipe with one end
free to slide, but constrained by an axially-disposed spring, so yielding equation (5.82)
directly, with no approximation. On the other hand, the system was generalized by intro-
ducing an elastic foundation; hence, a term kq appears in the equation of motion, where
k is the dimensionless foundation stiffness - cf. equations (3.70) and (3.71). A two-
mode Galerkin discretization is considered and, with the aid of centre manifold theory,
similar conclusions to Holmes’ are reached; but, in the process, a number of important
contributions are made, as follows.
It is first observed that the region of gyroscopic stabilization, occurring for high enough
j3 (Section 3.4; Figures 3.1 1 and 3.12) between the first and second critical flow velocities,
is ‘of purely “academic” concern’ since, on first buckling, the deflections of the pipe would
grow sufficiently to make the study of higher stability regions ‘inapt’. Therefore, a system
is sought which would remain stable up to the point of onset of linear coupled-mode
flutter. This is achieved by a judicious choice of the elastic foundation (Section 3.4.3). For
k = 4n4, it is found that with zero dissipation the two eigenvalues reach zero in the Argand
diagram at the same value of u, namely u = fin - cf. Figure 3.20; however, diver-
gence does not develop thereafter, because of gyroscopic stabilization, and the eigenvalues
remain purely imaginary up to ucf = 7.66, where the system loses stability by Hamilto-
nian coupled-mode flutter. However, if even infinitesimally small dissipative forces are
included, the gyroscopic stabilization is destroyed and hence coupled-mode flutter ceases
to be the first instability to occur, divergence developing instead at u = An; which
leads to qualitatively the same dynamics as discussed heretofore. This, however, raises
the following question: Is it possible that the nonexistence of coupled-mode flutter in the
nonlinear analysis is primarily due not to nonlinear effects but to dissipation?
To answer this question, Lunn reconsiders the nonlinear system, without any founda-
tion but also without any dissipation. The startling result is illustrated in Figure 5.9(a),