Page 431 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 403

in which u = ((4 + 2~j?'/~q'), v)*, L, L1 and L2 are linear differential operators involving
UO, /3 and w, and N contains all nonlinear terms; u = uo + E,U cos 2wt, reflecting that the
main interest is in the principal resonance, and it is clear that, in contrast with the
foregoing, ,U here need not be small - cf. equation (5.144). It has already been shown in
Section 4.5.1 that, for the cantilevered system, resonances arise only when uo is reasonably
close to uCf, where the system loses stability by flutter in steady flow. Hence, solutions
are sought close to this flutter boundary, namely for uo = ucf + q, w = 00 - €8, where
q denotes the mean-flow velocity variations, and 8 is a detuning parameter. Then, defining
ut = t, equation (5.151) is re-written as
+
au €8 aL0
00 - = Lou + - Lou + rq - ~~(LIo 2t + L~o sin 2t) u + rN(u, ucf) + S(r2),
cos
at w0 au0
(5.152)
in which L = LO + q(aLo/auo) + S(r2), LI = LIO + O(E), LZ = L20 + S(r), N(u, UO) =
N(u, u,f) + O(E). For E = 0, equation (5.152) reduces to q(au/at) = LOU. The operator
LO (at u = u,f) has two pure imaginary eigenvalues, while the others are in the left-hand
plane. Hence, the steady-state solutions corresponding to these two eigenvalues may be
written as uo = A{w(') exp[i(t + 4)] + W(l) exp[-i(t +@)I}, where A and 4 correspond
to amplitude and phase, relative to the parametric excitation of the periodic solutions w(')
and W(') associated with the two critical eigenmodes. For E # 0, the solution is expanded
in powers of E as
u = UO(A 4, t) + GUi(A, 4, t, + ~~~264,4, + 6(c2), (5.153)
q)
t,
where A and 4 satisfy

(5.154)

Then, substituting (5.153) into (5.152) and collecting coefficients of equal power of 6,
a set of new equations with terms which are functions of t and the spatial variable 6 is
obtained. They are expanded by Fourier series in time and pertinent comparison functions
in 6. These equations are then averaged, leading to equations of similar form to (5.148).
The results are discussed in terms of modijied flow-variation, detuning and harmonic
flow-perturbation parameters: 7j, Z and ii. Eventually, the master bifurcation diagram of
Figure 5.63(a) is obtained showing, in the (5, r}-plane, curves across which the trivial
equilibrium of the averaged system undergoes a pitchfork or Hopf bifurcation, or the
nontrivial fixed points undergo a saddle-node or Hopf bifurcation.
The dynamics is illustrated in three cases by the amplitude(A)-flow(?j) bifurcation
diagrams in Figure 5.63(b-d), each for a constant value of p. In (b), as the mean flow, i.e.
-
q, is increased, at some flow less than ucf (7j < 0) the trivial position of the pipe becomes
unstable and the pipe performs periodic oscillation at half the excitation frequency. As 7 is
increased, the amplitude of the oscillation increases, reaches a maximum and then begins
to decrease. For 7j = 17; the periodic solution becomes unstable, and for 17 > 7j: there is
no stable limit cycle; it is shown that the motion thereafter is amplitude-modulated. The
Hopf solution of the unperturbed system is also shown, starting at 17 = 0, but in this case,
since the trivial solution has already become unstable prior to 17 = 0, this is not a realizable
solution. In (c), there is a small region on the left of the figure where both the trivial