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CHAPTER
CONCLUSIONS
8.1 IDENTIFICATION OF STRUCTURAL NONLINEARITY
It is believed that most practical engineering structures possess a degree of nonlinearity.
In some cases, they are treated as linear structures because the degree of nonlinearity is
small and therefore insignificant in the response range of interest. In other cases, the
effect of nonlinearity may become so significant that it has to be taken into account in the
analysis of the structure’s dynamic characteristics. In fact, for many engineering
applications, structural nonlinearities need to be identified and, subsequently, nonlinear
mathematical models must be established.
Unlike theoretical studies, where the vibration characteristics of nonlinear systems can be
described by differential equations, a major problem in the identification of nonlinearity is
to study unknown types of nonlinearity. In practice, not only the existence of nonlinearity
needs to be detected, but more importantly, the degree of nonlinearity must be quantified
and then the physical characteristics of the nonlinearity identified. A review and
discussion of those methods currently used to investigate nonlinearity have shown their
practical applicability. However, conclusive identification of practical structural
nonlinearities is still problematic due to the existence of various different types of
nonlinearity and numerous qualitatively different nonlinear phenomena.
8 Conclusions
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For those structures whose nonlinearities are vibration amplitude dependent, the dynamic
characteristics can be identified based on the analysis of measured first-order
It is
found that one first-order FRF curve measured with a constant force level contains all the
information of a series of FRF data with constant response controls. Therefore, a study
of nonlinearity can be carried out using a single FRF curve measured with constant force.
Based on this observation, a new nonlinearity analysis method has been proposed in this
work. Instead of assuming that the mode to be analysed should be real, as in the case of
inverse receptance method, this new method deals with the practical situation in which
measured modes contain complexity due to the nonproportional distribution of structural
damping. The final results of the analysis based on the proposed method are the
response-amplitude-dependent eigenvalues
and eigenvectors
which can be used
not only to quantify the degree of nonlinearity but also to derive
spatial models
of a nonlinear structure. The method has been effectively applied to the data measured
from practical nonlinear structures even when the modes to be analysed are markedly
complex. Also, it is found that the method does not require the condition of using
constant force to measure the first-order FRF data as long as the force level is large
enough to expose nonlinearity.
On the other hand, for structures whose nonlinearities are such that the measured
order
are apparently linear (the nonlinearity is of an nonsymmetrical type), a
higher-order FRF analysis becomes necessary for the identification of such nonlinear
structures. Fundamentally different from the analysis of first-order
(in which only
the fundamental frequency component is of interest), the analysis of higher-order
takes into account the super-, sub- and combinational frequency components which, in
some practical applications, are as important as the fundamental frequency component.
By extending first-order FRF analysis to higher-order FRF analysis, it can be seen that
linear system theory has been extended in a natural way to cover nonlinear systems. It has
been found that many phenomena exhibited by nonlinear systems cannot be explained
based on classical first-order
but can be interpreted with a series of
of
different orders. Furthermore, measured higher-order
provide valuable information
about the nature of system nonlinearities and can be used not only to identify structural
nonlinearities, but also together with the first-order
to improve the response
prediction of a nonlinear system due to known input.
Both first-order and higher-order FRF analysis techniques are essentially based on the
classical assumption that the output of a nonlinear structure is periodic if the input is
periodic. However, it has been
that for some nonlinear systems (chaotic
8 Conclusions
252
systtms), a periodic input will result in an output of a random nature. Such a
discovered phenomenon is called chaos and is the most complicated dynamic behaviour
of nonlinear systems. It is believed that the ability to identify such nonlinear behaviour is
of practical importance. In Chapter 4, for the first time, the hidden chaotic behaviour of a
mechanical backlash system with realistic system parameters has been revealed and,
based on this system, qualitative as well as quantitative ways of identifying chaotic
structures are presented. Both numerical studies and experimental investigations have
been carried out. Indeed, there exist wide parameter regions, both in the system
parameters and external forcing conditions, for which chaotic vibrations occur. Such
nonlinear mechanisms as backlash stiffness nonlinearity represent an extensive group of
engineering structures and it can be expected that many structural systems will exhibit
chaotic behaviour under certain operating conditions. The anticipated engineering
applications of the research work presented include, (i) design of mechanical control
systems, (ii) statistical stress/fatigue analysis and, (iii) condition monitoring and
diagnosis of machinery.
8.2
LOCATION OF STRUCTURAL NONLINEARITY
It is believed that structural nonlinearities, when they exist, are generally
in
terms of spatial coordinates as a result of the nonlinear dynamic characteristics of
structural joints, nonlinear boundary conditions and nonlinear material properties. The
ability to pinpoint a structure’s localised nonlinearity(ies) thus has some important
engineering applications. First, the information about where the structural nonlinearity is
may offer opportunities to separate the structure into linear and nonlinear subsystems so
that these can be analysed separately and efficiently using a nonlinear substructuring
analysis. Second, since nonlinearity is often caused by the improper connection of
structural joints, its location may give an indication of a malfunction or of poor assembly
of the system. Third, from a materials property point of view, the stress at certain parts of
the structure during vibration can become so high that the deformation of that part
becomes plastic and the dynamic behaviour becomes nonlinear. In this case, location of
the nonlinearity may offer the possibility of failure detection. Finally, the location
information is essential if a nonlinear mathematical model of the structure is to be
established.
Since structural nonlinearities cannot be foreseen and so cannot be analytically predicted,
measurement becomes crucially important in locating and identifying them. Nevertheless,
it is understood that such a task as to locate structure’s localised nonlinearity is difficult to
8 Conclusions
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