Page 415 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 389

The system can also be perturbed by adding a small oscillatory component to the flow,
so that u = u, + u sin wt - cf. Sections 4.5 and 5.9. Prior to doing this, Li & Paldoussis
(1994) conducted a Melnikov analysis on a reduced subsystem, the details of which are
not given here; for nonautonomous systems, this can give an indication of the parameters
near which chaos can arise. The central idea is to construct a scalar function which gives a
measure of the distance between the stable and unstable manifolds when the unperturbed
heteroclinic (or homoclinic) orbit is broken by perturbations. If and when this distance
vanishes as a system parameter varies, the two manifolds intersect transversally; one
such intersection implies infinitely many, yielding horsehoes and chaos (Guckenheimer
& Holmes 1983). The analysis in this case gives u > d(w) for this to occur, where
(T = 0.1 is the viscous damping coefficient [equation (3.71)], viscous damping being the
only dissipation included, and R(w) is a function of the forcing frequency of the system,
which has a minimum at w = wo = 2.45, 00 being the frequency of oscillation at the Hopf
point. Simulations for a forcing frequency of o = 2.50 while varying v show that chaotic
oscillations arise for v > 1.22; this value of u is very much higher than the minimum
required according to the Melnikov calculation. The sequence of oscillatory states for
increasing u is (a) quasiperiodic, (b) periodic and (c) chaotic motions. The results are
similar to those discussed next, and hence no further details are given here.
The next case to be considered is that of the pipe-spring system, discussed in relatively
great detail in Section 5.7.3(a). In this case, heteroclinic orbits arise along one of the lines
shown in Figure 5.27(a). Here, simulations are conducted exclusively with a periodic
flow-velocity perturbation of the system, with parameters as given in Figure 5.55, where
a bifurcation diagram and a few phase-plane plots are shown; a fuller set, consisting of
several power spectra, time traces, phase-plane plots and Lyapunov exponent calculations
are given in Paldoussis & Sender (1993b). The sequence of oscillatory states for increasing
u is (i) periodic oscillations around one or the other of the two buckled states (both are
shown, obtained via different initial conditions), (ii) quasiperiodic oscillations around both
buckled states, (iii) periodic motions with sub-, combination, and super-harmonic content
(3 < u < 8 approximately), and (iv) chaotic oscillations. It is of interest that quasiperiodic
and chaotic oscillations in Figure 5.55(a) look not too dissimilar, but the difference in the
Lyapunov exponents is quite clear: zero in the former case and positive in the latter.
Three-dimensional motions of the same system are considered by Steindl & Troger
(1996), who determine in a map of j3 versus the location of the spring support, tS, the
regions of existence and stability of heteroclinic cycles. Physically, the heteroclinic cycle
involves the following set of transitions, as shown in Figure 5.56(a): (i) the system is
buckled in one of the two mutually perpendicular planes; (ii) oscillations develop in that
plane about the buckled state; (iii) the amplitude of these oscillations increases, while the
static deformation due to buckling diminishes, eventually leading to oscillations about the
straight equilibrium state (about the origin); (iv) oscillations develop in the perpendicular
plane, with decreasing amplitude as the amplitude of buckling increases in that plane;
(v) eventually buckling in that plane results, with no oscillation. By symmetry, this state is
fundamentally identical to the initial one, and so the sequence just described begins anew.
Steindl (1996) considers another type of heteroclinic cycles for the same system, this
time associated with Hopf-Hopf bifurcations, rather than Hopf-pitchfork ones as in the
foregoing, again obtaining a map of stable heteroclinic cycles in the {B, &} plane, as shown
in Figure 5.56(b). In this case, the oscillations in one plane develop a secondary bifurcation
at the TB boundary (corresponding to a simultaneous occurrence of a Hopf bifurcation and