Identification of the dynamic characteristics of nonlinear structures

142
Fig.4.36 Pseudo-Poincare Map of Experimental Chaotic Response
4.4 
CONCLUSIONS
In this chapter, the basic theory of chaotic vibration has been summarised and ingredients
which are required in order to understand chaotic behaviour of dynamic systems have
been illustrated. Together with Chapters 2 3, a complete picture of all the probable
nonlinear phenomena in structural dynamics and the analysis techniques for identifying
them have been presented.
For the first time, the hidden chaotic behaviour of nonlinear mechanical systems with
backlash stiffness nonlinearity has been studied in some detail both numerically and
experimentally. Particular attention has been paid to the identification of chaotic vibration
in such nonlinear systems. Indeed, as shown in the numerical simulations, there exist
wide parameter regions, both in the system parameters and the external forcing
conditions, for which chaotic vibrations occur. Qualitative as well as quantitative ways of
identifying chaotic vibration in nonautonomous nonlinear systems are presented.
The chaotic behaviour is explained in time-, frequency- and state-space domains. For
detecting the existence of chaotic vibrations, the response spectrum or, more rigorously,
the 
map of the motion, is employed. The fractal dimensions of strange attractors
are calculated and serve the purpose of quantifying the complexity of the motion. The
sensitivity of chaotic motions to initial conditions is examined and the Lyapunov
exponents are calculated, giving further indication of the existence of chaotic vibration.

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4
Identification of Chaotic Vibrational Systems
143
The system studied is a singularly simple system whose equation of motion is very easy
to understand physically. Also, as shown in this paper, an experimental model can readily
be constructed to demonstrate the predicted behaviour. Such a system is likely to become
a paradigm for further research into chaos in nonlinear dynamical systems. In mechanical
structures, such nonlinear mechanisms represent the intermittent contact between
components due to manufacturing clearances, and therefore it is expected that many
mechanical systems might exhibit chaotic behaviour under appropriate operating
conditions. Since one of the major consequences of chaos is unpredictability of the
response, it is therefore recommended that statistical methods should be applied to
stress/fatigue analysis when such conditions are anticipated. Furthermore, from a
condition monitoring view point, if a broad-band response can be caused by a purely
sinusoidal excitation (e.g., due to the eccentricity of rotational components), this makes
reliable diagnosis in most cases difficult and creates the necessity of a new understanding
of such nonlinear systems and the development of new techniques so that reliable
diagnosis can be achieved. Further, in the design of mechanical control systems such as
robots, where such backlash stiffness nonlinearity is very likely to exist, care must clearly
be taken at the design stage so that under normal service conditions, undesirable or
unpredictable chaotic motion will not occur.

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