Page 435 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 407

only one nontrivial solution and it is stable. Beyond q = 0, the trivial solution is unstable
and there are no nontrivial periodic solutions. On physical grounds, however, the response
should remain bounded; although the methods used preclude finding the solutions existing
in this area, by analogy to the results obtained for the continuous cantilevered system,
they are expected to be amplitude-modulated motions. As q is increased to 6 = -0.1
in (b), a portion of the nontrivial response for q > 0 becomes unstable, similarly to the
behaviour displayed in Figure 5.63(b). For 6 = 0.4, we see in (c) that the nontrivial
periodic solutions ‘pinch-off‘ the trivial one, giving rise to an isolated solution.
Similar results are shown in Figure 5.65(d-f) in the A versus 6 plane. For q = -0.075
we see in (d) the usual jump phenomenon and the associated hysteresis for high enough
6. In (e) and (0 we see that the trivial solution is unstable and there are segments of (7
over which the nontrivial solution also is unstable. In (f) we again see an isolated solution.
Figure 5.65(g-i) shows plots for B = x where the Hopf bifurcation is subcrit-
ical. In (g) and (h) it is seen that for large enough negative detuning one or two nontrivial
solutions generally exist, depending on 6 and q; the lower branch is unstable, while
for high enough q a portion of the upper branch becomes unstable also. In (i) the solu-
tion is isolated, and both branches are unstable. The remarks already made regarding
amplitude-modulated motions apply here too.
It is therefore seen that the dynamics of the system, articulated or continuous, for para-
meters such that self- and parametrically excited oscillations are close, is very interesting.
In most cases the Hopf bifurcation is suppressed and the dynamics is dominated by the
parametric resonance. Jump phenomena, subcritical onset of resonance, isolated resonance
branches, and amplitude-modulated motions are all possible for given combinations of b,
q, 6 and p.
Semler & Paidoussis’ (1996) contribution is an extension of Bajaj’s (1987b) analytical
work, but numerical solutions of the full nonlinear equations are also presented, as well
as some experiments. Equation (5.39) is utilized, thus retaining the inertial nonlinearities
intact, and is discretized by Galerkin’s method into

where u = uo(1 + p sin ut) and
+
+
c;j = a~g~;j 2~”~u0(1 p sin wt)bjj,
(5.156)
2
~ i ; = A:S;~ + ui(1 + p sin ut) c;; + B 112 puOw cos ut(dij - cij) + y(bij - clj + dij),
the A; being the dimensionless cantilevered beam eigenvalues and bij, cij and d;j are
as given in Table 3.1, while aijkl, B;jkl and yijkl are similar to the aijkl-dijkl given in
Section 5.7.3(a), but different since the inertial nonlinearities here are intact.
For thc analytical solution of the problem, we confine ourselves to the vicinity of the
Hopf bifurcation in steady flow, i.e. to u 2: u,f, as in Bajaj (1987b), but proceed in a
more standard manner, as follows. The system is transformed to first order, such that
y = (q, q}T, and then into standard form via y = PX, where P is a modified modal matrix
evaluated at uCf, thus yielding an equation of the form
+
X = Ax + p(w cos cot B1 + sin ut B2)x + p2 sin2 wt B~x F(x, x), (5.157)