Identification of the dynamic characteristics of nonlinear structures

steady-state response. The 
steady-state response of
a system is what remains after the transient has decayed to zero and from what has been
taught in linear system theory, it can either be an equilibrium point, periodic or 
periodic solution. This basic principle of linear systems has been so deeply-rooted in the
mind of most engineers that they may subconsciously extrapolate it for nonlinear systems
as well. We know that for some nonlinear systems, there exist a wide range of parameters
for which the steady-state responses are bounded, but are 
periodic. In fact, the
response 
becomes erratic with a broad continuous frequency spectrum (rather
than discrete, as in the periodic case). Moreover, the response is so sensitive to initial
conditions that unless a computer with infinite word length is used in the simulation, no
long-term prediction of the precise waveform is possible.
For mechanical systems, the study of chaotic vibration is important for several reasons.
First, in the design of mechanical control systems, it is essential to avoid the occurrence
of chaotic oscillation at design stage because chaos means unpredictability and so
uncontrollability. Secondly, the random nature of the response to a deterministic (usually
periodic) excitation of a mechanical structure makes life prediction difficult and statistical
stress/fatigue analysis becomes necessary. Finally, from a machine monitoring point of
view, that a broad continuous response spectrum can be due a single sinusoidal input
makes the reliable diagnosis in most cases difficult and suggests that the development of
new techniques is required.
In the following section, the basic theory of chaotic vibration is introduced based the
well-known Duffing’s and van der Pol’s systems. Ingredients which are essential for
understanding chaos are presented.

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4
Identification of Chaotic Vibrational Systems