Page 426 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

398 SLENDER STRUCTURES AND AXIAL FLOW

in which

The quantities wi, wj, Si, i = 1, 2, are related to the eigenvalues of the unperturbed system,
p1,2 = a6; f i(w1 + awl) and p3.4 = as; zt i(w2 +am;).
The next step is to introduce the new time t = wt and the detuning parameter 6
through w = wo(1 - 6), and to further transform the problem into polar coordinates:
u1 = a1 sin @,, u2 = a1 cos @I, @I = (q/Wg)t + 41; u3 = a2 sin @2, u4 = a2 cos @2,
@2 = (w2/wo)t + $2, leading to first-order ODES in al, a2, $1 and 42. These are then
reduced by the method of averaging to a set of simpler equations in the averaged Lii and
$i (Appendix F) of the type

d& 1
- = -[iiias’ + p(~i sin 24 + vi COS 24);i + fi?~il,
dt wo
(5.148)
d$i 1
=
& - + -[iiiwia + p(ui cos 24 - vi sin 24)ii + ii?si],
dt 00
in which Ui, V,, Ri and Si are simply constants. This equation is very important, since
it represents a parametrically perturbed one-degree-of-freedom oscillator. Most of the
nonlinear studies of parametrically excited pipes conveying fluid, whether supported
at both ends or cantilevered (but near the Hopf bifurcation point), end up with equa-
tion (5.148) or a variant thereof. Therefore, the methodology followed thereafter in most
studies is the same: (i) the stability of the origin (trivial solution) is investigated using the
linearized version of (5.148) around the origin; (ii) nontrivial solutions or fixed points are
sought, determined via (5.148), and their stability is examined. From a physical point of
view, a stable (or unstable) nontrivial fixed point in the reduced system (5.148) represents
a stable (or, respectively, unstable) periodic solution of the original system; the loss of
stability of a fixed point via a Hopf bifurcation signifies the possibility of quasiperiodic
motions. The complete study of a periodically perturbed Hopf bifurcation may be found
in Bajaj (1986) and Namachchivaya & Ariaratnam (1987).
Namachchivaya (1989) and Namachchivaya & Tien (1989a,b) analyse the averaged
equations in the case of the principal primary resonance, wo = 2w,, Y = 1,2, and combi-
nation resonance wo = 01 + w2.
Typical results are shown in Figure 5.60(a,b) for the principal first-mode resonance of a
clamped-clamped pipe. As w is increased from the left at a fixed p, the stable trivial equi-
librium point of the averaged system becomes unstable through one eigenvalue crossing
the origin in the complex plane at point S if dissipation is not zero, or through a double
crossing of the origin at SD if dissipation is zero (see Figures 2.10 and 3.4). At this point
the trivial solution bifurcates into a stable nonzero fixed point, the bifurcation diagram for
which is traced in Figure 5.60(b); therefore, the solutions of the original system (5.145)
are periodic, of period 2n/w,. It is recalled that averaging provides a solution valid only
in the vicinity of w/w, = 2 and, since the whole bifurcation diagram cannot be traced
with high accuracy, its upper part is not given. On the other hand, if w is reduced from the
right, the trivial solution loses stability subcritically, the bifurcating solutions in this case