Identification of the dynamic characteristics of nonlinear structures

CHAPTER 
IDENTIFICATION OF
CHAOTIC VIBRATIONAL SYSTEMS
4.1 
PRELIMINARIES
So far, techniques for identifying structural nonlinearity based on the measured first- and
higher- order frequency response functions have been developed and discussed in some
detail in Chapter 
If the first-order and higher-order (usually second-order)
frequency response functions of a nonlinear structure are measured in the ways as have
been discussed in previous Chapters then, in most cases, the structural nonlinearity can
be detected, quantified and identified by analysing these measured 
However, for
some nonlinear systems (chaotic systems, which are discussed in this Chapter), the
behaviour is so complex that the first- and higher-order FRF analyses, which are
largely based on the assumption of periodic input periodic output, become inadequate. In
order to analyse these systems, the development of yet more analysis techniques becomes
necessary.
In the case of a single sinusoidal input, the possible behaviour of the response spectrum
of a nonlinear dynamic system can be illustrated in 
The first-order frequency
response function analysis only considers the fundamental frequency components, while
the higher-order frequency response function analysis takes the harmonic (both

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4
Identification of Chaotic Vibrational Systems
107
subharmonic and superharmonic) and combinational frequency components into account.
However, except for these fundamental and harmonic frequency components, the
subharmonics of some nonlinear systems bifurcate in such a way that the response
spectrum due to a single sinusoid input changes from discrete (periodic) to continuous
(nonperiodic) distribution. Such newly-discovered strange behaviour of nonlinear system
a deterministic system exhibiting apparently random behaviour is called chaos and is
one of the most exciting research topics in nonlinear systems research.
subharrnonics
I component
superharmonics
approaching chaos
energy transfer to higher frequencies
Response Frequency Components of a Nonlinear System
In the last fifteen years, clues to the emergence of randomlike motion in deterministic
dynamic systems have been uncovered by new topological methods in mathematics. At
the same time, experimental measurements and numerical simulations have provided
supporting evidence to the mathematical analysis which shows that many physical
systems may exhibit chaotic behaviour without random inputs. Research on chaos has
become an interdisciplinary subject and applications of the study have been found in
almost all engineering subjects. In mechanical engineering, it has been well known that
Duffing’s system 
with negative stiffness, such as that which represents mechanical
structure of pre-stressed buckled beams, and, some impact mechanical oscillators 
exhibit chaotic behaviour under certain excitation and initial conditions. These systems
represent very special types of mechanical nonlinear structures which are not commonly
encountered in practice. In order to investigate the possible chaotic behaviour of practical
mechanical structures with more commonly-encountered nonlinearities, a mechanical
system with backlash stiffness nonlinearity is considered in this Chapter. Extensive
numerical as well as experimental research work has been carried out and apparently, it is
the first time in literature that the chaotic nature of mechanical system with backlash
stiffness nonlinearity under realistic system parameters has been revealed. Such a
nonlinear mechanism as backlash stiffness represents an important and extensive group of
mechanical structures with manufacturing clearances such as gearing systems. Based on

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 Identification of Chaotic Vibrational Systems
108
the mechanical backlash system, the basic theory of chaotic vibration is introduced and
qualitative as well as quantitative ways of identifying chaotic behaviour of a nonlinear
system are presented. Possible engineering applications of the study presented in this
Chapter are suggested.
4.2 INTRODUCTION TO CHAOTIC VIBRATION THEORY
Chaos is an nonlinear phenomenon which permeates all fields of science. Although
identified as an important research area only recently, chaos has existed from time
immemorial. It is now known that chaos can readily occur not only in man-made
systems, but in all natural and living systems where nonlinearity is present.
Roughly speaking, chaos is an exotic