The author has observed hours of transient chaos with a harmonically forced
laboratory pendulum, before the pendulum settled down into regular motion.
Bishop and Clifford (1996) report on a seemingly intermittent type of chaos for
the pendulum, which they call tumbling chaos. In this case the pendulum completes
an apparently random number of full rotations in one direction and then changes
direction of rotation, sometimes after a number of oscillations about the hanging
position.
With an external periodically varying torque we obtain another chaotic pendulum system (e.g., Schuster 1989):
€ h þ b _
h þ x
2
0 sin h ¼ Q cosðXtÞ:
ð6:52Þ
Blackburn et al. (1987) considered, experimentally and numerically, the chaotic
dynamics of a pendulum subjected to combined constant and periodic torque:
€ h þ b _
h þ x
2
0 sin h ¼ Q 0 þ Q 1 cosðXtÞ:
ð6:53Þ
Finally we mention a chaotic pendulum system studied experimentally and
numerically by Moon, Cusumano and Holmes (1987):
€ h þ b _
h þ x
2
0 sin h þ q 1 cosðXtÞ cos h ¼ 0;
ð6:54Þ
for which a Melnikov criterion was provided. This parametric equation models the
rotation of a magnetic dipole in a time-periodic magnetic field.
6.6.4 Piecewise Linear Systems (D ! 3)
Piecewise linear systems behave linearly within certain intervals of the state variables. Many mechanical systems having free play or stops, i.e. impact systems
belong to this class (Babitsky 1998; Burton 1968; Kobrinskii 1969). So do systems
with bi-linear or multilinear elastic restoring forces, see, e.g., Shaw and Holmes
(1983). Fig. 6.21 shows some examples.
The beam in Fig. 6.21(a) is similar to the Moon beam, though with the magnets
replaced by a rigid stop. The chaotic dynamics of this system have been examined
numerically and experimentally in a number of studies, in particular by Moon and
Shaw (e.g., 1983). The partial differential equations of motion are linear with linear
boundary conditions at the clamped end, w 0; t
ð Þ ¼ w
0 0; t
ð Þ ¼ 0. However, at
x = l the beam is considered to be free ðw
00 l; t
ð Þ ¼ w
000 l; t
ð Þ ¼ 0Þ when w(l, t) < D,
while simply supported ðw; ðl; tÞ ¼ w
00
ðl; tÞ ¼ 0Þ when w(l, t) = D and the beam
moves to the right ð _
wðl; tÞ [ 0Þ. Thus, the beam performs linear vibrations between
the times of impacting with the stop. A single-mode approximation for the equation
of motion takes the nondimensional form (Moon and Shaw 1983):
366
6 Chaotic Vibrations
laboratory pendulum, before the pendulum settled down into regular motion.
Bishop and Clifford (1996) report on a seemingly intermittent type of chaos for
the pendulum, which they call tumbling chaos. In this case the pendulum completes
an apparently random number of full rotations in one direction and then changes
direction of rotation, sometimes after a number of oscillations about the hanging
position.
With an external periodically varying torque we obtain another chaotic pendulum system (e.g., Schuster 1989):
€ h þ b _
h þ x
2
0 sin h ¼ Q cosðXtÞ:
ð6:52Þ
Blackburn et al. (1987) considered, experimentally and numerically, the chaotic
dynamics of a pendulum subjected to combined constant and periodic torque:
€ h þ b _
h þ x
2
0 sin h ¼ Q 0 þ Q 1 cosðXtÞ:
ð6:53Þ
Finally we mention a chaotic pendulum system studied experimentally and
numerically by Moon, Cusumano and Holmes (1987):
€ h þ b _
h þ x
2
0 sin h þ q 1 cosðXtÞ cos h ¼ 0;
ð6:54Þ
for which a Melnikov criterion was provided. This parametric equation models the
rotation of a magnetic dipole in a time-periodic magnetic field.
6.6.4 Piecewise Linear Systems (D ! 3)
Piecewise linear systems behave linearly within certain intervals of the state variables. Many mechanical systems having free play or stops, i.e. impact systems
belong to this class (Babitsky 1998; Burton 1968; Kobrinskii 1969). So do systems
with bi-linear or multilinear elastic restoring forces, see, e.g., Shaw and Holmes
(1983). Fig. 6.21 shows some examples.
The beam in Fig. 6.21(a) is similar to the Moon beam, though with the magnets
replaced by a rigid stop. The chaotic dynamics of this system have been examined
numerically and experimentally in a number of studies, in particular by Moon and
Shaw (e.g., 1983). The partial differential equations of motion are linear with linear
boundary conditions at the clamped end, w 0; t
ð Þ ¼ w
0 0; t
ð Þ ¼ 0. However, at
x = l the beam is considered to be free ðw
00 l; t
ð Þ ¼ w
000 l; t
ð Þ ¼ 0Þ when w(l, t) < D,
while simply supported ðw; ðl; tÞ ¼ w
00
ðl; tÞ ¼ 0Þ when w(l, t) = D and the beam
moves to the right ð _
wðl; tÞ [ 0Þ. Thus, the beam performs linear vibrations between
the times of impacting with the stop. A single-mode approximation for the equation
of motion takes the nondimensional form (Moon and Shaw 1983):
366
6 Chaotic Vibrations