Page 384 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

360 SLENDER STRUCTURES AND AXIAL FLOW

another trajectory corresponding to different initial conditions. Then, defining the vari-
ational vector function u(t) = &(t) - @(t) such that IIu(I << 1, the stability of @(t) is
governed by the equation
u = Df(@)u, (5.133)

where Df(@) is the Jacobian matrix function for the vector field f(y) along @(t). Clearly,
if &(t) approaches @(t), then u(t) will tend to zero; while if it diverges away from it,
then u(s) will tend to grow. This may be expressed as
Ilu(t)ll - eUT. (5.134)

As the system is bounded, however, the exponential behaviour indicated by (5.134) cannot
continue indefinitely. Hence, in the computations the vector function u(t) is renormalized
from time to time and the calculation process reinitialized in that sense. The so-called
Lyapunov exponent may be defined as

(5.135)

Hence, the two trajectories in question, @I (t) and #(t), may be considered to converge
or diverge exponentially on the average, according as CJ is negative or positive, with
CJ = 0 corresponding to neutral orbital stability (the case of a stable periodic orbit). Note
that in an n-dimensional space there exist n Lyapunov exponents. However, the largest
one dominates the dynamics of the system. Given an arbitrary initial condition u(O), the
solution u(t) will converge to the direction of most rapid growth, which is associated
with the largest Lyapunov exponent. A chaotic trajectory is defined as one with at least
one positive Lyapunov exponent (Parker & Chua 1989; Moon 1992).
The problem at hand being represented by a fourth-order system, there will be four
Lyapunov exponents, only the largest of which is computed for the purposes of defining
the dynamical behaviour of the system. The largest one for the case corresponding
to Figures 5.34 and 5.35 is shown in Figure 5.36. It is seen that, for u i 8.027,
CJ,, 2: 0, indicating stable periodic orbits, while for u > 8.027, a,, > 0, indicating
chaotic behaviour. (For u < u~, i.e. below the Hopf bifurcation, amax < 0 is obtained.)
It is also noted that there are so-called ‘periodic windows’ at u 2r: 8.18 and 8.19, where
periodic motion is once more obtained over small ranges of u.
Simultaneously to all of the foregoing, the fractal dimension of the experimental
system was determined (Paldoussis et al. 1992). This will help answer whether the N = 2
discretization is adequate or not. There are several measures of the fractal dimension
(Moon 1992), of which the correlation dimension, as developed by Grassberger & Proc-
cacia (1983a,b) is used here. The method is outlined in Appendix I, with experimental
data from another run of the same system as in Figures 5.30-5.32.
The final results for three flow velocities, showing the correlation dimension, d,, as
a function of the embedding dimension, m, are shown in Figure 5.37, while the prelim-
inary work leading to these figures is given in Figures 1.1-1.3. In Figure 5.37(a,b,c),
corresponding to Figures 1.1, 1.2 and 1.3, the oscillation is periodic, ‘fuzzy period-2’ and
chaotic, whereby ‘fuzzy’ is signified a periodic oscillation with a small chaotic compo-
nent. The value of d, = 1.03 in Figure 5.37(a) is sufficiently close to the ideal d, = 1
for periodic oscillation; d, = 1.53 in Figure 5.37(b) is substantially different from d, = 1
because of the fuzzy nature of the oscillation. Finally, Figure 5.32(c) gives the most