Identification of the dynamic characteristics of nonlinear structures

CHAPTER
LOCATION OF
STRUCTURAL NONLINEARITIES
5.1 PRELIMINARIES
So far, different types of dynamic phenomenon observed in nonlinear structures have
been discussed and techniques for analysing them have been presented. For practical
structures whose nonlinearities are such that the measured first-order 
using
sinusoidal excitation (when the amplitude of the excitation force is kept constant) display
the nonlinear behaviour (most practical nonlinearities are of this nature), a new
identification method has been developed in Chapter 2 which can not only quantify the
extent of the nonlinearity, but also identify its type in some cases. On the other hand, for
some nonsymmetric nonlinearities such as the quadratic and bilinear stiffness
nonlinearities, as discussed in Chapter 3, the measured first-order 
are effectively
linear and analysis of the higher-order 
becomes necessary so that such
nonlinearities can be identified in practice. However, both first-order and higher-order
analyses are largely based on the assumption of periodic-input periodic-output and
for some nonlinear structural systems (chaotic systems), this assumption is no longer
valid. Based on a system with backlash stiffness nonlinearity, qualitative as well as
quantitative ways of identifying chaotic vibrational systems have been presented in
Chapter 4. Some important engineering applications of the techniques developed have
been discussed.

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5
Location of Structural Nonlinearity
145
In this and later Chapters, we shall confine ourselves to the analysis of first-order 
only although we accept that such analysis could, in some cases, be very approximate. It
will be shown how measured first-order FRF data (or their derived modal data), together
with an analytical model of the structure (usually an FE model), can be used to locate the
structure’s localised nonlinearity (Chapter 5) and later how an accurate mathematical
model of a dynamic structure can be established by correlating an analytical model and
measured dynamic test data (Chapters 6 7).
5.2 
NECESSITIES AND REQUIREMENTS FOR NONLINEARITY
LOCATION
It is usually believed that, if they exist, structural nonlinearities are localised in terms of
spatial coordinates as a result of the nonlinear dynamic characteristics of structural joints,
nonlinear boundary conditions and nonlinear material properties such as plasticity. The
ability to locate a structure’s localised nonlinearity 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 based on 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, as will be discussed in detail in Chapters 
location information is essential if a nonlinear mathematical model of the structure is to be
established.
In practical measurements, the data measured are usually quite limited (both measured
modes and coordinates are incomplete) and this is especially true when a nonlinear
structure is considered, as will be discussed in some detail in Chapter 6. It is therefore
believed that the task of locating a structure’s localised nonlinearity can only become
possible by correlating an analytical model, which may contain 
errors but can
represent the structure to some accuracy, and the results from dynamic test of the
structure. To illustrate the above argument, consider a typical nonlinear structure (two
linear components connected by a nonlinear joint) as shown in Fig.5.1. Mathematically,
the structure possesses a mass matrix 
which is constant, and a stiffness matrix
which is a function of of response amplitude, if stiffness nonlinearity is

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 Location of Structural 
146
considered (for the convenience of discussion, the structure is assumed to be undamped).
Clearly, if the impedance matrix [Z(o)] of the structure can be measured, then the
nonlinearity location becomes straightforward as shown in Fig.5.2. However, what can
be measured in practice is the the receptance matrix 
which is the inverse of
[Z(
O
)], and in this inverse format the 
stiffness change due to the nonlinearity at
different response amplitudes is spread over the whole matrix, as shown schematically in
Fig.5.3. Of course, one may obtain [Z(o)] by inverting the measured [a(o)] but,
unfortunately, such a process is found to be extremely sensitive to measurement noise
and, therefore, is often not implementable in practice. Hence, it becomes clear that in
order to locate the nonlinearity based on measured data only, 
the coordinates and