to be chaotic. For this response the value of (a, p) is in a range where, according to
perturbation analysis, the zero solution is destabilized both by linear and by nonlinear terms. This causes an explosive rise in amplitude, allowing the orbit to
explore a much larger region of phase space. In this larger region one can show that
the dynamic behavior of the double pendulum is affected by the action of no less
than twenty-seven states of equilibrium with mixed kinds of stability. Adding the
like existence of limit cycles, one is not surprised that the flow of orbits displays
extreme sensitivity to initial conditions.
In Fig. 6.22(d) the value of (a, p) corresponds to a situation where the
zero-solution is stabilized by linear terms, but destabilized by nonlinear terms. That
is, the system is on the dangerous, some may say ‘exciting’, side of a subcritical
pitchfork bifurcation for the oscillation amplitudes. Recall that in this case a stable
zero solution can be destabilized by a strong disturbance. This is confirmed by the
figure, which was computed for initial conditions corresponding to a strong velocity
impulse. The large-amplitude periodic orbit in the figure was approached only after
a very long chaotic transient (8,000 nondimensional seconds).
Figure 6.23 illustrates the different types of stationary motions of the pendulum
encountered when varying the loading parameters (a, p). The analytical stability
diagram obtained in Chap. 4 (Fig. 4.12) has been superimposed. The figure was
obtained by computing the largest Lyapunov exponent ^ k 1 on a 512 Â 330 grid
spanning the (a,p)-range, and then coloring the grid-points according to the value of
^ k 1 . Periodic motion (gray, ^ k 1 % 0) prevails in regions where, according to perturbation analysis, the only solution is a stable limit cycle. This region is defined by
a a < a < a b and p above the curve D 3 = 0 of supercritical Hopf bifurcations.
Chaotic motion (black, ^ k 1 [ 0) prevails in regions where, according to perturbation
analysis, subcritical pitchfork bifurcations have occurred, that is, where a > a b and
p is above the upper branch of H 4 = 0. On these observations one may suggest a
simple analytical and seemingly sufficient criterion for chaos to occur, though it
does not hold for the periodic window (gray band) inside the main region of chaos.
Chaos-like phenomena for the follower-loaded double pendulum with steploading have been examined in several studies by Kounadis (e.g., 1991 and references cited herein). Other variants of the double pendulum are likely to behave
chaotically as well, e.g., the double pendulum with eccentric load or load-dependent
stiffness treated in Guran and Plaut (1993).
Fluid-conveying Pipes A large number of studies consider the dynamics of fluidconveying pipes (Fig. 6.24). M. P. Païdoussis, in particular, has contributed to this
area (for chaos-related studies see, e.g., Païdoussis and Semler 1993; Païdoussis
et al. 1989). Physically, a fluid-conveying pipe is a far more complicated system
than the double pendulum discussed above. However, applying mode expansion for
the PDE of the system in Fig. 6.24, one can reduce the dynamics of the system to
the following set of approximating ODEs:
6.6 Mechanical Systems and Chaos
371
perturbation analysis, the zero solution is destabilized both by linear and by nonlinear terms. This causes an explosive rise in amplitude, allowing the orbit to
explore a much larger region of phase space. In this larger region one can show that
the dynamic behavior of the double pendulum is affected by the action of no less
than twenty-seven states of equilibrium with mixed kinds of stability. Adding the
like existence of limit cycles, one is not surprised that the flow of orbits displays
extreme sensitivity to initial conditions.
In Fig. 6.22(d) the value of (a, p) corresponds to a situation where the
zero-solution is stabilized by linear terms, but destabilized by nonlinear terms. That
is, the system is on the dangerous, some may say ‘exciting’, side of a subcritical
pitchfork bifurcation for the oscillation amplitudes. Recall that in this case a stable
zero solution can be destabilized by a strong disturbance. This is confirmed by the
figure, which was computed for initial conditions corresponding to a strong velocity
impulse. The large-amplitude periodic orbit in the figure was approached only after
a very long chaotic transient (8,000 nondimensional seconds).
Figure 6.23 illustrates the different types of stationary motions of the pendulum
encountered when varying the loading parameters (a, p). The analytical stability
diagram obtained in Chap. 4 (Fig. 4.12) has been superimposed. The figure was
obtained by computing the largest Lyapunov exponent ^ k 1 on a 512 Â 330 grid
spanning the (a,p)-range, and then coloring the grid-points according to the value of
^ k 1 . Periodic motion (gray, ^ k 1 % 0) prevails in regions where, according to perturbation analysis, the only solution is a stable limit cycle. This region is defined by
a a < a < a b and p above the curve D 3 = 0 of supercritical Hopf bifurcations.
Chaotic motion (black, ^ k 1 [ 0) prevails in regions where, according to perturbation
analysis, subcritical pitchfork bifurcations have occurred, that is, where a > a b and
p is above the upper branch of H 4 = 0. On these observations one may suggest a
simple analytical and seemingly sufficient criterion for chaos to occur, though it
does not hold for the periodic window (gray band) inside the main region of chaos.
Chaos-like phenomena for the follower-loaded double pendulum with steploading have been examined in several studies by Kounadis (e.g., 1991 and references cited herein). Other variants of the double pendulum are likely to behave
chaotically as well, e.g., the double pendulum with eccentric load or load-dependent
stiffness treated in Guran and Plaut (1993).
Fluid-conveying Pipes A large number of studies consider the dynamics of fluidconveying pipes (Fig. 6.24). M. P. Païdoussis, in particular, has contributed to this
area (for chaos-related studies see, e.g., Païdoussis and Semler 1993; Païdoussis
et al. 1989). Physically, a fluid-conveying pipe is a far more complicated system
than the double pendulum discussed above. However, applying mode expansion for
the PDE of the system in Fig. 6.24, one can reduce the dynamics of the system to
the following set of approximating ODEs:
6.6 Mechanical Systems and Chaos
371