Page 428 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

400 SLENDER STRUCTURES AND AXIAL FLOW

regimes. Equation (5.143) with u = ug( 1 + p sin or) is discretized into one- and two-
mode Galerkin approximations for a pinned-pinned pipe, the former of which is simply
;il + (an4 + o)ql + {n4 + n2[r - ui + iy - 24p sin wt

- i,9/2~op~ wt1}ql + $n4q? + +in4q:ql = 0, (5.149)
COS
which, because no sliding of the ends is permitted and hence (5.143) used, is much
simpler than that analysed by Yoshizawa et al. (1986).+ This equation and its two-mode
counterpart are studied numerically, while (5.149) is also studied analytically by the
multiple time-scale method, in both cases for the fundamental resonance (w 2 wl) only.
The solution of (5.149) is approximated by

qi(r, p) = qio(To, Ti, T2) + p~~qli(T0, Ti, T2) + p2q12(To, Ti, T2) +. . . , (5.150)
where To = t is the fast time-scale associated with frequencies w and 00, and TI =
pr, T2 = p2r, etc. are slow time-scales associated with modulations in amplitude
and phase resulting from the nonlinearities and parametric excitation. Assuming q10 =
A(T1, T2) exp(iwoT0) TI, T2) exp(-iwoTo), where is the complex conjugate of A,
equations of like powers of p are solved sequentially while eliminating secular terms, so
that equations similar to (5.148) are obtained.
Typical numerical results with the one-mode approximation for wo = 8.875 and uo <
U,-dr i.e. before the loss of stability with steady flow, show the following. (i) For 0 5
p 5 0.259, the trivial solution converges to zero, while for p > 0.295 it loses stability.
(ii) For 0.229 < p < 0.610, a nontrivial stable periodic solution exists. This implies that
for 0.229 < w -= 0.259, two solutions coexist: the trivial one and a finite-amplitude peri-
odic one, as confirmed by analytical results by the multiple time-scale method for wo = 8.8
shown in Figure 5.61(a). (iii) For p > 0.61, a period-doubling sequence ensues, a period-
8 phase-plane diagram being shown in Figure 5.61(c), leading to chaos at p = 0.7123.
The associated bifurcation diagram is shown in Figure 5.61(b). The extent of the chaotic
region is very limited, 0.7123 5 p 5 0.7162. For higher k, transient chaos is observed
(Moon 1992): initially, the motion is chaotic with two separate patches in the PoincarC
map of the motion; but, as time progresses, these two patches grow and eventually come
into contact, whereupon chaotic motion is destroyed and is succeeded by period-1 motion.
(iv) As p is decreased from 0.7164, period-1 motion is found to coexist with the period-
8, period-4, and so on, motions found in the foregoing, down to /L = 0.66, as seen in
Figure 5.61(b). It should be remarked that values of p larger than 0.5 ought to be judged
as being too large, from both the physical and mathematical viewpoints.
The dynamics obtained with the two-mode approximation is qualitatively similar to
that just described. Here it ought to be said that the one-mode approximation is rather
hazardous since, as seen in equation (5.149), no Coriolis terms are present because the
gyroscopic matrix is skew-symmetric.
For ug > u,d = 4.196, the system in steady flow becomes a ‘buckled beam’. It is not
too surprising, therefore, that its dynamical behaviour with pulsating flow is qualita-
tively similar to a harmonically excited buckled beam, represented by Duffing’s equation
(Dowel1 & Pezeshki 1986). For uo = 4.7077 and w = 0, the trivial equilibrium is a saddle,

+The unusual factor in some of the terms, e.g. y, is due to suppressing the fi in the beam eigenfunctions
dr = &! sin(rn<) used in the Galerkin scheme.