Identification of the dynamic characteristics of nonlinear structures

TIME DOMAIN ANALYSIS
As in the identification of linear structures for which time domain methods such as
Ibrahim’s method (ITD) 
can be used to obtain modal parameters of a linear structure,
structural nonlinearities can be identified by analysing the measured time force and
response data directly. The simplest method of nonlinearity analysis in time domain is the
phase-space or, more specifically, the phase-plane approach 
which gives both local
and global behaviour of a nonlinear system and provides an exact topological account of
all possible system motions under various operating conditions. In the case of sinusoidal
excitation, the 
map which is the discrete phase-space trajectory of the motion,
can be calculated to detect the existence of harmonic components and so the existence of
nonlinearity.
The force state mapping technique for the identification of nonlinearity was initiated by
Masri and Caughey 1 and independently developed by Crawley and 
12-
13]. The technique aims to establish the relationship between the restoring force and the
vibration displacement and velocity 
For an SDOF nonlinear system described by
m + 
= f(t)
where 
is the restoring force, if, by some means, the mass m of the system is
known and quantities x, and f(t) are measured, then the restoring force 
at
given state (x, can be calculated. If the system is linear, then the restoring force surface
is a plane as shown in Fig. 
If the system is nonlinear, then some surface distortion
is expected, as shown in Fig.l.l(b) for the case of cubic stiffness nonlinearity. The
extension of this technique to certain nonlinear MDOF systems has been investigated [ 1 
and the practical application of the technique has been discussed

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1 Introduction
7
(a) linear SDOF system
nonlinear 
SDOF system
Fig. 1.1 Restoring Force Surfaces of Linear and Nonlinear Systems
Time series analysis techniques [ 
have been widely used in the modelling of linear
systems and have recently been extended to the identification of nonlinear systems 16-
17]. In general, a linear time-invariant system can be represented by higher-order (more
than the second) differential equation with constant coefficients. Such a differential
equation can, in theory, be approximated by a difference equation whose accuracy
depends on the time interval of sampled data points. The time series analysis seeks to
calculate the coefficients of the difference equation model based on the measured 
output time series data. In the case of a nonlinear system, extra coefficients have to be
identified which represent the effect of the nonlinear behaviour 
After the difference
equation model, which describes the dynamic characteristics of a nonlinear system, has
been identified, the first-order and higher-order frequency response functions of the
system can be calculated [ 
1.3.3 
AMPLITUDE DOMAIN ANALYSIS
Nonlinearities can also be identified by calculating the amplitude probability density
function of the response due to random excitation. For a linear system, if the input force
is a random signal with its amplitude probability density function 
being Gaussian,
then the amplitude pdf of the response will also be Gaussian. For a nonlinear system
however, this simple relationship no longer holds and some distortion in the response
amplitude pdf from Gaussian distribution is expected and from this distortion, the
existence of nonlinearity can be identified 
The method is developed based on the
Fokker-Planck-Kolmogorov 
equation of a nonlinear system which is described in
in 
To illustrate the idea, consider an SDOF nonlinear system as

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where f(t) is a Gaussian noise signal. The corresponding Fokker-Planck-Kolmogorov
equation of (1-5) is described as 
+ 
+ 
= 
where 
is the power spectrum of the input force and 
and 
are the probability
density functions of the displacement and velocity, respectively. Solving 
can
be obtained as 
(l-7)
where is a constant which can be determined by the normalisation condition. From 
it can be seen that only when 
is linear does 
have a Gaussian distribution.
On the other hand, 
can be calculated based on the measured time response data and,
therefore, nonlinearity can be easily identified experimentally based on the distortion of
measured 
from the standard Gaussian distribution.
In the research described in this thesis, we will concentrate mainly on the development of
analysis techniques in the frequency domain although time domain techniques such as
phase-plane and 
map approaches will be used in the 
of chaotic
vibrational systems.
1.4 
MODELLING OF NONLINEAR STRUCTURES
For many engineering applications, accurate mathematical models (spatial models in terms
of mass and stiffness matrices) of nonlinear structures are required. So far, much
progress has been made in the mathematical modelling of linear structures 
As
mentioned above, a mathematical model of a linear structure can be established either
using analytical FE analysis (an analytical model) or based on measured dynamic test data
(an experimental model). Due to the existence of modelling errors in most practical cases,
the analytical model needs to be validated using measured test data so that an accurate
mathematical model can be established. In the case when a structure to be 
is
nonlinear, its mathematical model becomes a function of response amplitude 

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 Introduction
9
for the case of stiffness 
and what needs to be established is a series of
linearised models corresponding to different vibration amplitudes by correlating the
analytical model and measured first-order 
data.
In fact, since structural nonlinearity cannot be foreseen and so cannot generally be
analytically predicted, measurement is crucially important in the modelling of
nonlinearity. However, measurement alone cannot, in general, establish a practically
realistic model because measured data are usually very limited (as will be shown, this is
especially true for the case of nonlinear structures). It is believed that a reasonably
accurate linear model of a nonlinear structure (corresponding to very low vibration
amplitude) and location information of the 
nonlinearity are necessary in order to
establish the mathematical model of a nonlinear structure.
In the present research, we shall focus on the development of techniques for both the
location and the mathematical modelling of structural nonlinearities. The procedure is as
proposed below. First, an analytical model is updated using vibration test data measured
at very low response amplitude to obtain an accurate linear model of the nonlinear
structure. Then, the nonlinearity is located, based on this linear model and measured data
at higher response amplitudes. With this location information available, modelling of the
nonlinearity can be concentrated on the region where the structural nonlinearity is and
then by correlating the linear model and measured FRF data at different response
amplitudes, a mathematical model of the nonlinear structure can be established.