|
-0.60 - 0 . 3 2 - 0 . 0 4 0 2 4
1 . 0 8 1 . 3 6
displacement
A = 100,
187
- 3 . 6 - 2 . 4
-1.2 0.0 1.2 2.4 3.6 4.8
-7.2
-5.4
-3.6
0.0 1.8 3.6 5.4 7.2
displacement
A =
1000,
Fig.4.29 The
Maps of Different Forcing Amplitudes
To assess the effects of damping on chaotic vibration, different damping levels were
introduced for case 1 with
As expected, an increase in damping
was found to reduce the ‘randomness’ in the chaotic motion and the
map of the
motion becomes more compact as the damping increases, as shown in Fig.4.30. The
fractal dimensions of the Poincare maps for different values of damping c were also
calculated and the results shown in Fig.4.31. Clearly, the introduction of damping is an
effective way of avoiding unsatisfactory motions of chaotic systems.
4
Identification of Chaotic Vibrational Systems
138
0.80
0 2 4
0.80
-0.60 -0.32 -0.04 0.24 0.52 0.80 1.08 1.36 1.50
-0.60 -0.32 0 . 0 4 0.24 0.52 0.80 1.08 1.36 1.50
displacement
Fig.4.30 The
Maps at Different Damping Levels
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Damping Coefficient
Fig.4.31 Fractal Dimension Versus Damping Level c
Identification of Chaotic Vibrational Systems
139
4.3.7 TWO-DEGREE-OF-FREEDOM
SYSTEM
So
far, the chaotic behaviour of an SDOF mechanical system with backlash stiffness
nonlinearity has been investigated in some detail. In order to study chaotic vibrations of a
nonlinear MDOF system, the 2DOF system with backlash stiffness nonlinearity as shown
in Fig.4.32 is considered. It is generally believed that if chaotic vibration occurs in a
MDOF system, the motion will become yet more complex than that in an SDOF system
with same type of nonlinearity because, in this case, the interactions between all the
degrees of freedom act as chaotic excitations and these chaotic excitations make the
resulting motion more complex. As in the case of the SDOF backlash system, it was
found that for very low and very high excitation levels, the motions are periodic.
However, there exists a wide range of forcing parameters in which chaotic vibrations
occur.
lkg
Ikg
=
Fig.4.32 2DOF Mechanical System with Backlash Stiffness Nonlinearity
The system parameters are as shown in figure 4.32 and it was found that when the
forcing amplitude
and forcing frequency
the motions become
chaotic. As before, the results are presented in the time-, frequency- and state-space
domains as shown in Fig.4.33. However, as the 2DOF system is a four-dimensional
system since there are four state variables
the projection of the
dimensional
map onto the
vs plane (2-dimensional) disguises the
properties of the true Poincare map.
4
Identification of Chaotic Vibrational Systems
140
time response of coordinate
response spectrum
-1.6
-1.2
-0.8
0.0 0.4
0.8
1.7.
1.6
-1.6 -1.2 -0.8 4.4 0.0 0.4 0.8 1.2 1.6
phase-plane trajectory
map in
plane
Fig.4.33 Chaotic Response of 2DOF Backlash System
4.3.8
EXPERIMENTAL INVESTIGATION
Detailed numerical studies on the chaotic vibration of backlash system have been carried
out and reported alone. However, the question remains: “do chaotic vibration exists in
real practical backlash systems To answer this question, an experiment was designed
based on a test structure comprising a simply-supported beam with a mass at its midpoint
to simulate the SDOF system, as shown in Fig.4.34. The first natural frequency of the
structure was designed to be around
with the second mode much higher so that
when the excitation is around
the structure behaves effectively like an SDOF
system. The backlash stiffness nonlinearity was introduced by providing motion
constraints on both sides of the mass (figure 4.34) so that the stiffness characteristics of
the equivalent SDOF system can be represented by that shown in figure 4.17(a). The
response was detected by a strain gauge attached on the beam near the mass such that the
measured strain is proportional to the displacement of the mass. A sinusoidal excitation
force is produced by an
shaker acting on the mass.
Identification of Chaotic Vibrational Systems
141
Response (strain gauge)
Excitation force
Fig.4.34 Experimental Model of Backlash System
As in the numerical studies, periodic as well as chaotic responses were found to exist at
different forcing amplitudes and frequencies. The time response and auto-spectrum of a
chaotic response at excitation frequency
are presented in Fig.4.35. A pseudo-
map (x(n+l) x(n), where x(n) is the sampled response signal with sampling
frequency equal to the excitation frequency which is
is shown in Fig.4.36
because in the experimental case, usually only one signal (displacement or velocity) is
available (the simultaneous measurement of displacement and velocity is practically
difficult in some cases and it has been mathematically established
that the same
amount of information about the motion of the system can be obtained from the pseudo-
map instead of the true
map). From these results, the chaotic nature of
the response is clearly demonstrated. However, it is difficult to compare these results
with those from the numerical calculations because the necessary system parameters of
the experimental rig are difficult to
Fig.4.35 Time Response and Auto-spectrum of Measured Chaotic Response
4
Identification of Chaotic Vibrational Systems
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