Identification of the dynamic characteristics of nonlinear structures

1.2 STRUCTURAL NONLINEARITY
Most of the theories upon which structural dynamic analysis is founded rely heavily on
the assumption that the dynamic behaviour of the structure to be analysed is linear. By
this is meant that (i) if a given loading is doubled, the resulting deflections are doubled
and (ii) the deflection due to two (or more) simultaneously applied loads is equal to the
sum of the deflections caused when the loads are applied one at each time. This
superposition principle of linear systems can be expressed mathematically as
= 
k = l
k = l
where x is the deflection, f(t) is the loading force and a is a constant. Linear
mathematical models of engineering structures based on this superposition principle have
proven to be very useful in numerous engineering applications. From general theoretical
considerations based on the superposition principle, successful methods have been
developed and applied to the dynamic analysis of linear structures.
Failure to obey the superposition principle implies that the 
is nonlinear. fact,
most practical engineering structures exhibit a certain degree of nonlinearity due to
nonlinear dynamic characteristics of structural joints, nonlinear boundary conditions and
nonlinear material properties. For practical purposes, they are in many cases regarded as
linear structures because the degree of nonlinearity is small and therefore insignificant in
the response range of interest. For other cases, the effect of nonlinearity may become so
significant that it has to be taken into account in the analysis of dynamic characteristics of
the structure.
It is often supposed that unless a real measurement is taken, the existence of a
nonlinearity in a practical structure cannot be foreseen based on analytical prediction nor
the degree of nonlinearity can be analytically quantified. Experimental investigation
becomes essential in the identification of dynamic characteristics of nonlinear structures.
The present research focuses on the identification, location and mathematical modelling of
practical nonlinearities based on measurement of the input/output dynamic characteristics
of nonlinear structures. Although there have been several efforts directed towards the

---

1 Introduction
4
identification of nonlinearity, there has not been a complete and systematic development
of identification techniques which are required in order to analyse the numerous different
nonlinear phenomena that occur in engineering practice. Further, the location of
nonlinearity (information about where the nonlinearity is located in a structure), which
has important engineering applications has not been investigated to date. Also, as an
ultimate target of nonlinear system analysis, the establishment of a nonlinear spatial
mathematical model which is a function of the response amplitude, has not been
investigated in spite of its practical relevance to numerous engineering applications.
1.3
IDENTIFICATION OF DYNAMIC CHARACTERISTICS OF
NONLINEAR STRUCTURES
System identification, which is generally considered as the inverse problem of system
dynamics, is in the scope of various fields such as structural and control engineering.
Although mathematicians and engineers have developed a number of approaches to
address the identification problem, most of the work to date has been restricted to linear
systems. Nonlinear systems are, however, often assumed to be linearisable in some
manner, and the resulting linear model is then used to analyse the behaviour of the
system. Significant inaccuracy arises when conditions and/or assumptions required for
the linearisation are violated.
The identification of linear time-invariant system is relatively well understood and
theoretically well developed. The same is not true for the case of a nonlinear system.
Nevertheless, over the past years, some progress has been made in the development of
both theories and techniques in the identification of nonlinear systems. A very brief
review is presented here in terms of frequency, time and amplitude domains and a more
detailed discussion will be given in some later chapters when specific topics are described
or referred.
1.3.1 FREQUENCY DOMAIN TECHNIQUES
Techniques developed for the identification of nonlinearirities in the frequency domain
are, in general, based on the comparison of different characteristics of the measured
frequency response functions of linear and nonlinear structures (nonlinearity detection)
and extension/modification of classical linear analysis methods to nonlinear structures
(nonlinearity quantification). As the first task of nonlinearity analysis, the detection of the
existence of nonlinearity is believed to be relatively easy. For most practical nonlinear
structures, frequency response functions 
measured using sinusoidal excitation

---

1 Introduction
with constant forcing amplitude will show certain form of distortion as compared with
those of a linear structure. Distortions of measured 
when they are displayed in the
form of a Bode plot 
or reciprocal receptance 
have been employed to detect the
existence of nonlinearities. Also, as discussed in 
when a structure is nonlinear, the
isometric damping plot calculated based on the measured FRF data will show systematic
variation (surface distortion) and this variation is an indication of nonlinearity. These
detection techniques are simple and easy to implement in practice.
As a more sophisticated method, the Hilbert transform technique can be used to detect,
and to some degree, to quantify structural nonlinearities. The theory of the Hilbert
transform, which is an integral transform, is described in detail in 
The basis that the
Hilbert transform technique can be used to identify nonlinearity is due to the fact that for a
linear structure, the real and imaginary parts of a measured FRF constitute a Hilbert
transform pair (that is: 
and vice versa), while for the FRF of a
nonlinear structure, these Hilbert transform relationships do not hold. By calculating the
Hilbert transform of the real part (or the imaginary part) of a measured FRF and
comparing it with the corresponding imaginary part (or real part), the existence of
nonlinearity can be identified based on the difference of the transform pair 
For most practical applications, not only does the nonlinearity need to be detected, but
more importantly, it needs to be quantified. The Hilbert transform approach seeks to
quantify the nonlinearity by measuring the degree to which the Hilbert transform pair
differ from each other. As a more practical way of quantifying structural nonlinearity, the
Inverse Receptance method was developed 
which aims to establish the relationship
between the natural frequency and the vibration amplitude of a nonlinear structure.
However, the method is restricted to the case in which the mode to be analysed is real.
All the above-mentioned techniques are formulated for the identification of nonlinearity
based on the measured first-order 
which are obtained by considering only
the fundamental frequency component of the response signal, as will be defined later).
For some nonlinear structures, the measured first-order 
are effectively linear and
for some practical vibration problems in which the harmonic components of the response
become as important as the fundamental component, the measurement and analysis of
higher-order 
becomes necessary. The theoretical basis of higher-order 
is the
Volterra series and its extended Wiener series theory 
However, research activities
had been restricted in electrical and control engineering since Wiener’s early work 
and
until recently that the theory has been applied to the identification of nonlinear mechanical
structures 
and found to be quite useful.

---

 Introduction
6
1.3.2