Identification of the dynamic characteristics of nonlinear structures

 two 
trajectories at time 
Z
. . . . . .
Fig.4.26 Trajectories Starting with Close Initial Conditions
Based on 
the Lyapunov exponents for case 1 with 
and case
2 with 
are calculated to be 0.532 and 0.624. During the calculation,

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 Identification of Chaotic Vibrational 
135
an 
time of 
cycles was used and, as shown in Fig.4.27, the calculated
value can be considered to be reliable because by that time, virtually does not change.
These positive Lyapunov exponents give quantitative measure that the trajectories
diverge, on average, at an exponential rate of = 0.532 for case 1 and 
for case
2.
 ,
,
, ,
181 
 
 
Fig.4.27 Calculated Lyapunov Exponents Versus Integration Time
4.3.6 EFFECT OF FORCING PARAMETER
ON CHAOS
AND DAMPING
Once the chaotic nature of a nonlinear system has been established, what is then of
interest is to know under what forcing conditions chaotic vibrations will occur because if
the necessary conditions for chaos have been determined, then it is possible to avoid them
or to employ them if there are some advantages of doing so. At present, the determination
of the forcing parameter field of a nonlinear system in which chaotic vibrations occur is
generally achieved by experiment although analytical predictions for some specific chaotic
systems such as 
system have been undertaken. In the present study, the forcing
parameter field for the existence of chaotic vibration of the system described by equation
(4-29) (case 1) was determined by numerical experiment, results of which are shown in
Fig.4.28. It has been found as expected that chaotic vibrations occur when the forcing
amplitudes are of moderate values for all the excitation frequencies tried. For the higher
excitation frequencies, although it cannot be proven because of limited calculation
capacity, it was found that no chaotic vibrations occurred when was greater than 85

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 Identification of Chaotic Vibrational 
136
2700 
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Excitation Frequency 
Fig.4.28 Forcing Parameter Field for the Existence of Chaotic Vibration
To see how chaotic motion changes when the forcing amplitude increases, the 
maps of different forcing amplitudes at an excitation frequency 
are calculated
and are shown in Fig.4.29. The calculated fractal dimensions show that although all the
motions are chaotic, they become more ‘regular’ as forcing amplitude increases.

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4
Identification of Chaotic Vibrational Systems
137