Yim and Lin (1991) studied the chaotic dynamics associated with the rocking of
slender objects subjected to horizontal base excitation (Fig. 6.21(b)). A Melnikov
criterion was set up. This type of analysis is pertinent to the protection of
earthquake-excited structures, e.g., tall buildings, storage tanks, ancient towers and
nuclear reactors.
The impacting pendulum in Fig. 6.21(c) has been investigated in several studies
(e.g., Moore and Shaw 1990). Used as a vibration damper in certain types of
rotating machinery, it soaks up energy from a structure to be damped and dissipates
the energy through impacts with the rigid wall. With the above examples in mind,
you might presume correctly that this device can turn chaotic for certain values of
the system parameters.
Fig. 6.21(d) shows another implementation of an impact-type damper. Sung and
Yu (1992) examined the system numerically. Among other findings they reported a
period-doubling route to chaos.
Putting a small roller-ball on a strongly vibrating table, you may observe the ball
performing a strange random-like dance (until it leaves the plate and you cannot
find it). The chaotic dynamics of the bouncing ball at a periodically vibrating table
have been studied for many years (e.g., Guckenheimer and Holmes 1983; Moon
1987; Tufillaro et al. 1992). For one-dimensional bouncing, the ball moving only up
and down, the equations of motion may be posed as a map relating subsequent
post-impact times and velocities (Moon 1987):
T k þ 1 ¼ T k þ v k ;
v k þ 1 ¼ av k À c cosðT k þ v k Þ; k ¼ 0; 1; . . .:
ð6:56Þ
Here T k is the nondimensional time of the k’th impact and v k the post-impact
velocity. The two-dimensional variant of the problem (Fig. 6.21(e)) was studied by
Kozol and Brach (1991).
As a final example we consider the simple device in Fig. 6.21(f). Two rigid bars,
the one excited by a time-varying force, are joined by a smooth pin with play. This
system was studied by Li et al. (1990), motivated by experimental observations of
chaotic responses for a pinned truss structure. Assuming inelastic collisions and
time-harmonic forcing, the motion of the pin x(t) is governed by the nondimensional equation of motion:
€ x ¼ A sin t for x
j j\1;
_
x ! Àr _
x
for x
j j ¼ 1;
ð6:57Þ
where A is the forcing amplitude and r the coefficient of restitution.
Period-doublings and chaos were reported for large values of A. This system is
called a zero-stiffness impact oscillator, since there is no elastic restoring term.
368
6 Chaotic Vibrations
slender objects subjected to horizontal base excitation (Fig. 6.21(b)). A Melnikov
criterion was set up. This type of analysis is pertinent to the protection of
earthquake-excited structures, e.g., tall buildings, storage tanks, ancient towers and
nuclear reactors.
The impacting pendulum in Fig. 6.21(c) has been investigated in several studies
(e.g., Moore and Shaw 1990). Used as a vibration damper in certain types of
rotating machinery, it soaks up energy from a structure to be damped and dissipates
the energy through impacts with the rigid wall. With the above examples in mind,
you might presume correctly that this device can turn chaotic for certain values of
the system parameters.
Fig. 6.21(d) shows another implementation of an impact-type damper. Sung and
Yu (1992) examined the system numerically. Among other findings they reported a
period-doubling route to chaos.
Putting a small roller-ball on a strongly vibrating table, you may observe the ball
performing a strange random-like dance (until it leaves the plate and you cannot
find it). The chaotic dynamics of the bouncing ball at a periodically vibrating table
have been studied for many years (e.g., Guckenheimer and Holmes 1983; Moon
1987; Tufillaro et al. 1992). For one-dimensional bouncing, the ball moving only up
and down, the equations of motion may be posed as a map relating subsequent
post-impact times and velocities (Moon 1987):
T k þ 1 ¼ T k þ v k ;
v k þ 1 ¼ av k À c cosðT k þ v k Þ; k ¼ 0; 1; . . .:
ð6:56Þ
Here T k is the nondimensional time of the k’th impact and v k the post-impact
velocity. The two-dimensional variant of the problem (Fig. 6.21(e)) was studied by
Kozol and Brach (1991).
As a final example we consider the simple device in Fig. 6.21(f). Two rigid bars,
the one excited by a time-varying force, are joined by a smooth pin with play. This
system was studied by Li et al. (1990), motivated by experimental observations of
chaotic responses for a pinned truss structure. Assuming inelastic collisions and
time-harmonic forcing, the motion of the pin x(t) is governed by the nondimensional equation of motion:
€ x ¼ A sin t for x
j j\1;
_
x ! Àr _
x
for x
j j ¼ 1;
ð6:57Þ
where A is the forcing amplitude and r the coefficient of restitution.
Period-doublings and chaos were reported for large values of A. This system is
called a zero-stiffness impact oscillator, since there is no elastic restoring term.
368
6 Chaotic Vibrations