Page 345 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 325

for a special set of parameter values. It has already been mentioned previously that for
different values of y and p (or B) the system may lose stability by divergence (pitchfork
bifurcation) or flutter (Hopf bifurcation); for the right combinations of (y, p), these two
bifurcations may occur simultaneously, i.e. at the same u. Codimension-three refers to
three parameters being used to ‘unfold’ the bifurcations in the vicinity of this double
degeneracy - i.e. to develop the evolution of the bifurcations gradually as one or more
parameters are varied. This is normally a codimension-two problem (Guckenheimer &
Holmes 1983), but here a third parameter corresponding to imperfection-related asymme-
tries is added.
The system is first transformed into Jordan canonical form, and then through centre
manifold reduction and averaging reduced to the deceptively simple set of equations

I- = pl r + r(-r2 - bz2), i = p4 + p2 z + z(--cr2 + 2’1, (5.110)
where pi, i = I, 2 and 4, are the unfolding parameters which are related to variations
of the original system parameters from the critical point. By comparing these to the
original nonlinear equations [cf. the planar version of equations (5.74)-(5.77)] involving
more than 10-15 terms each, it is clear that a very dramatic simplification has been
achieved. Yet, equations (5.110) are capable of capturing the essential dynamics of the
system, as is illustrated, for instance, by comparison with simulations from the full form
of the equations. The r equation is similar to the averaged equation for the classical
van der Pol oscillator, with r representing the amplitude of oscillatory response, in this
case due to the Hopf bifurcation, while the z equation represents pipe response due to
the pitchfork bifurcation. The results are illustrated in Figure 5.16 for the case of no
asymmetries (p4 = 0), which in fact corresponds to codimension-two bifurcations - cf.
Section 5.7.3(d) where, for a similar system, the analysis is outlined in greater detail. The
parameter p~, for p1 > 0, gives rise to a pitchfork bifurcation (only one side of which is
shown) and to a new equilibrium point q1 (on the r-axis) for the averaged system (5.1 10);
in the original system this corresponds to the amplitude of periodic motions. For p2 < 0,
we additionally have a static subcritical pitchfork bifurcation, and the point q2 (on the
z-axis) is unstable. Of particular importance is line 2, on which there exists a heteroclinic
cycle, across which the character of the solutions and the stability of the new fixed point
q3 change. According to Smale’s horseshoe theory (Guckenheimer & Holmes 1983; Moon
1992), it is known that homoclinic and heteroclinic tangles may lead to complex dynamics
and chaos.
The three-parameter, codimension-three case is very much more complex and will not
be discussed here, even in outline. In addition to periodic motions, amplitude-modulated
oscillations, i.e. motions on a ‘two-torus’ in four-dimensional space, are generally also
possible. These manifest themselves as periodic orbits in r - only on line 2 for the system
of Figure 5.16, but more widely for the asymmetric system - cf. Section 5.7.3(d). In
total, in this remarkable study, 23 distinct open sets are found in the three-dimensional
(p 1, p2, p4) parameter space, each corresponding to qualitatively different dynamics!
Yet another type of double degeneracy in articulated pipes was studied by Langthjem
(1995): the case of two Hopf bifurcations occurring simultaneously. This, though impos-
sible for N = 2 and 3 systems, can occur for the N = 4 system. Hence, to study this
system, Langthjem derives the nonlinear equations for the four-degree-of-freedom system,
considering planar motions and a deflection-independent flow velocity; the connecting
springs are taken to be nonlinear, with a linear and a cubic component.