Identification of the dynamic characteristics of nonlinear structures

2 5 3
achieve (if not impossible) based on measured data alone because they are in general quite
limited and correlation between an analytical model and dynamic test data becomes
essential. Based on such understanding, a nonlinearity location method was developed
based on the correlation between an analytical model which contains modelling errors and
modal test data which are measured at different response levels. The method was then
extended to the practical case where measured coordinates are incomplete by interpolating
the unmeasured coordinates based on the analytical model. Further, it has been shown
that the method can be generalised when measured first-order 
data are used. As
illustrated in numerical case studies, a successful location can be achieved provided that
the modelling errors are of second order in the sense of the Euclidean norm when
compared with the analytical model itself and some coordinates around where the
nonlinearities are located are measured.
The location technique has been demonstrated by a practical application. The structure
used is a frame structure with a 
stiffness nonlinearity simulated using an 
dynamic shaker and an analogue computer circuit based on feed-back system control
theory. Only a third of the coordinates specified in the analytical (FE) model were
measured. The second bending mode was found to the most sensitive one to nonlinearity
and so was used in the location process. The unmeasured coordinates were first
interpolated using the FE model and then the complete modeshape was used to locate the
nonlinearity. The location process successfully indicates the location of the introduced
nonlinearity.
8.3 MODELLING OF LINEAR AND NONLINEAR STRUCTURES
In the identification of dynamic characteristics of practical structures, analytical prediction
and experimental modal testing are involved because both have their own advantages and
shortcomings. Using modal testing results to improve an analytical model effectively
makes use of the advantages of both while at the same time overcoming their
shortcomings. It is therefore believed that by doing so, the most reliable mathematical
model can be established.
Since practical structures are continuous, it is apparent that any constant coefficient
models derived to represent their dynamic characteristics are approximate with limits on
the frequency range of applicability. A constant coefficient model can be regarded as a
good 
model if it will not only predict responses over the frequency range of interest, but
will also be representative of the physical characteristics of the structure. Thus it must

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 Conclusions
254
have the capability to predict the effects of changes in physical parameters and to
represent correctly the structure when it is treated as a component of a large system.
In effort to establish such a good model, many different methods have been developed in
recent years to improve analytical models using modal testing results. Review and
discussion of those recently-developed methods have suggested their limited practical
applicability but at the same time, have revealed some of the existing problems. The full
matrix updating methods such as the Berman’s method and the Error Matrix method
tackle the problem of analytical model improvement from a purely mathematical
viewpoint, such as an optimisation one, rather than to consider structural characteristics
as well, such as physical connectivity. As a result, the ‘improved’ model could be optimal
in a mathematical sense, but physically unrealistic. Also, the completeness of measured
coordinates is in general assumed in those methods and is critical to their success in most
cases in spite of the fact that it is extremely difficult in practice to measure all the
coordinates which specified in the analytical model. Furthermore, modal data are assumed
to be used during analytical model improvement and since the number of measured modes
are often quite limited. Consequently, the updating problem is usually mathematically
underdetermined. Nevertheless, it is understood that the measured FRF data contain, in
theory, the necessary information of all the modes of a structure. Based on such
understanding, a new generalised model updating method has been developed which
tackles the updating problem by directly using measured FRF data. The method allows
the physical connectivity of the analytical model to be preserved and deals with the
practical case in which measured coordinates are incomplete. It has been shown that
model updating methods based on modal data are, in a broad sense, discrete versions of
the present generalised method. Based on this method, the uniqueness of the updating
problem has been discussed in some mathematical 
Numerical studies demonstrate
the marked advantages of the new method as compared with other existing methods.
When the connectivity information of the analytical model is available, the model updating
problem can in general be turned into an overdetermined problem even when, as in
practice the measured data are limited. Therefore, it becomes possible and necessary to
develop criteria concerning just how much data (modes and coordinates) need to
measured in order to solve the updating problem uniquely. Such criteria are believed to be
practically important because they enable the analyst to judge whether a set of measured
data have the potential to solve the updating problem so that blind tries can be avoided.
The final target in the analysis of a nonlinear structure is perhaps to establish its nonlinear
mathematical model. It is argued that such a mathematical model of a nonlinear structure

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 Conclusions
2 5 5
becomes possible only when, on the one hand, an accurate linear mathematical model
(corresponding to very low response amplitude) is available and on the other, the location
information of the 
nonlinearity is given because, unlike the 
of a linear
structure in which measured data are consistent, a nonlinear mathematical model has to be
established based on mode by mode basis. Mathematically, structural nonlinearity can be
considered as modelling errors and as a result, the problem of modelling nonlinear
structures becomes the same as that of analytical model updating of linear structures
except that in the former case, a series of 
models are to be established. In the
same way, the proposed method is ideally suited for the application of nonlinear
structures for which 
data can be measured while modal data are sometimes difficult
to obtain.
8.4 SUGGESTION FOR FURTHER STUDIES
Whereas extensive research work on the identification of the dynamic characteristics of
nonlinear structures has been carried out in this thesis, the study undertaken has revealed
that some further development in the field may be necessary and of interest. Some general
suggestions for possible further studies are summarised below.
Although extensive numerical simulations have been carried out in order to assess the
practical applicability of the measurement and analysis of higher-order 
and it can be
anticipated that similar results can be obtained in practical measurement as those of
numerical simulation, no real measurement has been undertaken in this study due to the
limited period of time available. Further research on the measurement of clean, consistent
higher-order 
of practical engineering structures is recommended.
As discussed in Chapter 4, it can be envisaged that the existence of chaotic behaviour of
such a simple nonlinear mechanical system with backlash stiffness nonlinearity will have
important engineering applications such as fatigue analysis, condition monitoring and
robotics design. Studies on these specific applications are recommended.
The nonlinearity location technique developed in Chapter 5 needs to be applied to more
sophisticated practical nonlinear structures. In this study, the application has been made to
a numerical study and a comparatively simple frame structure due to the difficulties
involved in designing a nonlinear structure with known 
nonlinearity.

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8 Conclusions
256
As for the mathematical modelling of nonlinear structure:, it has to be mentioned that the
mathematical models obtained based on the measured first-order 
are only the first
order approximation of the true models of nonlinear structures. In order to improve the
modelling accuracy, it is obvious that the measured higher-order 
should be
incorporated into the 
process. Some research work on this subject is necessary.

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