Fig. 6.26 displays typical solutions as obtained by numerical integration of
(6.60). Phase planes, Poincaré maps and frequency spectra for the antisymmetric
amplitude f(t) are shown for four values of the excitation frequency X. Note how
quasiperiodic motion (Fig. 6.26(c) resembles chaotic motion (Fig. 6.26(d)) in the
phase plane, whereas in the Poincaré map they are easily distinguished. The routes
to chaos for this system, as was discussed in Sect. 6.4, involve transient chaos,
period-doublings, quasiperiodic oscillations, and intermittency.
Fig. 6.27(a) depicts regions of chaotic motions in the plane of the loading
parameters X and q, as obtained by evaluating the largest Lyapunov exponents on a
100 Â 100 grid. Grid-points corresponding to chaotic motion ð ^ k 1 [ 0Þ are shown in
dark gray, whereas quasiperiodic motion ð ^ k 1 % 0Þ is shown in lighter gray.
A nonlinear stability diagram for the zero solution (Fig. 4.10, obtained by local
perturbation analysis) has been superimposed. Chaos is seen to prevail in the
knife-shaped region C. In this region, according to the perturbation analysis of
Sect. 4.3.3, all small-amplitude solutions are unstable. Since a simple analytical
expression exists for the boundary curve of region C we here have, seemingly, an
approximate sufficient condition for chaos to occur.
Fig. 6.27(b) shows the magnitudes of the largest Lyapunov exponents on a grid
similar to that of Fig. 6.27(a). To expound the numbers, we note that if ^ k 1 ¼ 0:1
and initial conditions are specified to within one part per million (20 bits), then after
20/0.1 = 200 nondimensional seconds of time all information about the state of the
system is lost. This period of time corresponds to merely 48–89 forcing periods
when X is in the range 1.5–2.8. For chaotic motions, it appears from the figure,
higher magnitudes of the loading q cause ^ k 1 to increase, and thus lowers the
time-horizon of predictability.
The Autoparametric Vibration Absorber This system, shown in Fig. 4.1, also
obeys coupled equations having quadratic nonlinearities:
€ x þ 2b 1 _
x þ x
2
1 x þ c 1 h
2
¼ q cosðXtÞ;
€ h þ 2b 2
_
h þ x
2
2 h þ c 2 xh ¼ 0:
ð6:61Þ
The vibration absorber is especially designed so as to utilize internal two-to-one
resonance, and is thus inherently autoparametric. The equations of motion are
similar in structure to (6.60) for the non-shallow arch. As we saw in Chap. 4 a local
perturbation analysis yields similar results for the two systems (compare, e.g.,
Fig. 4.3 for the absorber to Fig. 4.8 for the arch). Hence, what has been said above
on chaotic motions of the non-shallow arch is likely to hold as well for the
autoparametric vibration absorber.
Other Autoparametric Systems Among the large number of nonlinear studies by
Prof. Ali H. Nayfeh and co-workers at Virginia Polytechnic Institute and State
University, many treat autoparametric systems, and some of these consider chaos.
A system consisting of two orthogonally clamped beams with a two-to-one internal
6.6 Mechanical Systems and Chaos
375
(6.60). Phase planes, Poincaré maps and frequency spectra for the antisymmetric
amplitude f(t) are shown for four values of the excitation frequency X. Note how
quasiperiodic motion (Fig. 6.26(c) resembles chaotic motion (Fig. 6.26(d)) in the
phase plane, whereas in the Poincaré map they are easily distinguished. The routes
to chaos for this system, as was discussed in Sect. 6.4, involve transient chaos,
period-doublings, quasiperiodic oscillations, and intermittency.
Fig. 6.27(a) depicts regions of chaotic motions in the plane of the loading
parameters X and q, as obtained by evaluating the largest Lyapunov exponents on a
100 Â 100 grid. Grid-points corresponding to chaotic motion ð ^ k 1 [ 0Þ are shown in
dark gray, whereas quasiperiodic motion ð ^ k 1 % 0Þ is shown in lighter gray.
A nonlinear stability diagram for the zero solution (Fig. 4.10, obtained by local
perturbation analysis) has been superimposed. Chaos is seen to prevail in the
knife-shaped region C. In this region, according to the perturbation analysis of
Sect. 4.3.3, all small-amplitude solutions are unstable. Since a simple analytical
expression exists for the boundary curve of region C we here have, seemingly, an
approximate sufficient condition for chaos to occur.
Fig. 6.27(b) shows the magnitudes of the largest Lyapunov exponents on a grid
similar to that of Fig. 6.27(a). To expound the numbers, we note that if ^ k 1 ¼ 0:1
and initial conditions are specified to within one part per million (20 bits), then after
20/0.1 = 200 nondimensional seconds of time all information about the state of the
system is lost. This period of time corresponds to merely 48–89 forcing periods
when X is in the range 1.5–2.8. For chaotic motions, it appears from the figure,
higher magnitudes of the loading q cause ^ k 1 to increase, and thus lowers the
time-horizon of predictability.
The Autoparametric Vibration Absorber This system, shown in Fig. 4.1, also
obeys coupled equations having quadratic nonlinearities:
€ x þ 2b 1 _
x þ x
2
1 x þ c 1 h
2
¼ q cosðXtÞ;
€ h þ 2b 2
_
h þ x
2
2 h þ c 2 xh ¼ 0:
ð6:61Þ
The vibration absorber is especially designed so as to utilize internal two-to-one
resonance, and is thus inherently autoparametric. The equations of motion are
similar in structure to (6.60) for the non-shallow arch. As we saw in Chap. 4 a local
perturbation analysis yields similar results for the two systems (compare, e.g.,
Fig. 4.3 for the absorber to Fig. 4.8 for the arch). Hence, what has been said above
on chaotic motions of the non-shallow arch is likely to hold as well for the
autoparametric vibration absorber.
Other Autoparametric Systems Among the large number of nonlinear studies by
Prof. Ali H. Nayfeh and co-workers at Virginia Polytechnic Institute and State
University, many treat autoparametric systems, and some of these consider chaos.
A system consisting of two orthogonally clamped beams with a two-to-one internal
6.6 Mechanical Systems and Chaos
375