Identification of the dynamic characteristics of nonlinear structures

Location of Structural Nonlinearity
171
Resp. Level
Low resp. level
High resp. level
(0. 
Nat. Freq. (Hz.)
92.05
96.03
mode
shapes
0.715
0.630
0.325
0.276
-0.266
-0.249
-0.776
-0.688
-0.283
-0.229
0.211
0.227
0.725
0.691
0.345
0.27 1
-0.169
-0.251
-0.523
-0.605
-0.520
-0.602
-0.167
-0.247
0.338
0.265
0.739
0.705
0.211
0.226
-0.285
-0.230
-0.772
-0.684
-0.269
-0.253
0.303
0.258
0.721
0.635
Table 5.2 Measured Modal Parameters of Mode 2 at Different Response 

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5
Location of Structural Nonlinearity
172
Mode No.
1
2
3
4
6
Nat. Freq. (Hz)
50.36
51.41
94.85
152.40
222.88
255.55
mode
shapes
0.1624
0.6826
0.7189
0.3553
0.4543
0.4306
0.3168
0.7605
0.6611
-0.0144
-0.2832
0.4958
0.7697
-0.5566
-0.7878
-0.2796
0.2756
-0.6030
-0.2690
-0.5225
-0.2756
-0.6030
0.2690
-0.5225
-0.7697
-0.5566
0.7878
-0.2796
-0.66 11
-0.0144
0.2832
0.4958
-0.4543
0.4306
-0.3168
0.7605
-0.1624
0.6826
-0.7 189
0.3553
0.1624
0.6826
-0.7 189
-0.3553
0.4543
0.4306
-0.3168
-0.7605
0.6611
-0.0144
0.2832
-0.4958
0.7697
-0.5566
0.7878
0.2796
0.2756
-0.6030
0.2690
0.5225
-0.2756
-0.6030
-0.2690
0.5225
-0.7697
-0.5566
-0.7878
0.2796
-0.6611
-0.0144
-0.2832
-0.4958
0.4543
0.4306
0.3168
-0.7605
-0.1624
0.6826
0.7189
-0.3553
0.4250
0.7680
0.1755
-0.7259
-0.2913
0.2913
0.7259
-0.1755
-0.7680
-0.4250
0.4250
0.7680
0.1755
-0.7259
-0.2913
‘0.2913
0.7259
-0.1755
-0.7680
-0.4250
0.4528
-0.1704
-0.6282
-0.4528
0.7256
0.7256
-0.4528
-0.6282
-0.1704
0.4528
0.4528
-0.1704
-0.6282
-0.4528
0.7256
0.7256
-0.4528
-0.6282
-0.1704
0.4528
Table 5.3 First 6 Modes of the FE Model

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CHAPTER 
IDENTIFICATION OF MATHEMATICAL
MODELS OF DYNAMIC STRUCTURES
6.1 PRELIMINARIES
As with the identification of the dynamic characteristics of a linear structure, the ultimate
target involved in the identification of a nonlinear structure is obviously to establish its
nonlinear mathematical model which is a function of vibration amplitude 
and
or 
As will be shown, the establishment of such a nonlinear mathematical
model becomes possible only when, on the one hand, an accurate linear mathematical
model (corresponding to very low vibration amplitude and therefore, can be regarded as
linear model of the nonlinear structure) is available and, on the other, the location
information of the 
nonlinearity is given. Such requirements in the modelling of a
nonlinear structure, which are different from those for the modelling of a linear structure,
are due to the fact that the mathematical model of a nonlinear structure has to be
established on a mode by mode basis as will be explained later in this chapter. Since an
accurate linear model is an essential pre-requisite for the modelling of a nonlinear
structure, most of the effort will be devoted in this chapter to discussions of how an
accurate linear model can be be established.
As discussed in Chapter 5, techniques for locating the nonlinearities (which are usually
of practical structures have been developed and these techniques are further
employed in the modelling of nonlinear structures in this Chapter.

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Identification of Mathematical Model of Dynamic 
174
On the other hand, for the modelling of linear structures (in fact, as will be shown, the
problem of modelling a nonlinear structure is essentially the same as that for a linear
structure except that in the former case, a series of linearised models need to be
established), a new method is presented in this chapter which tackles the modelling
problem by using the measured frequency response data directly. The method is then
extended to the modelling of nonlinear structures by combining the updated linear model
(corresponding to very low vibration amplitude) and the nonlinearity location results.
6.2 MODELLING OF LINEAR AND NONLINEAR STRUCTURES
Mathematical models of practical continuous structures, both linear and nonlinear, play an
important role in dynamic analysis. They are frequently used in response and load
prediction, modification/sensitivity and stability analysis, structural coupling etc.. Due to
the development of mathematical and physical sciences, the closed-form modelling of
some basic mechanical components such as uniform beams and plates has now become
possible. However, for complicated practical structures, there do not in general exist
analytical solutions (or if they do, it is extremely difficult to find such analytical solutions)
and therefore some discrete approximate models which can well represent the structure
under given conditions have to be sought so that the dynamic characteristics of the
structure can be analysed mathematically. According to Berman 
such a discrete
model can be considered 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 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. It is the establishment of such a physical model of the
structure which is discussed in this chapter.
Basically, there are two ways of establishing a discrete mathematical model of a practical
structure and they are (i) experimental modelling and (ii) theoretical modelling (Finite
Element Modelling). Due to the advances made in measurement instrumentation and
techniques, nowadays it is usually agreed that measured data should be considered as the
true representation of the structure while the FE model because of the idealisation
involved, lack of knowledge about the structure and difficulty of modelling of boundary
conditions is usually considered to be inaccurate and therefore should be updated using
measured data if possible. Based on the assumption that the measured data are correct,
two approaches to modelling exist: (i) using experimental data only to establish a
mathematical model in terms of measured coordinates of interest and (ii) correlating the

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Identification of Mathematical Model of 
Structures
175
FE model and dynamic testing data to update the FE model. Both modelling activities will
be reviewed and further possible development will be discussed.
As for the mathematical modelling of nonlinear structures, correlation between analytical
model and dynamic testing data becomes essential because, in general, structural
nonlinearities cannot be foreseen and, therefore, cannot be analytically 
but can
be measured. It is believed that the modelling of a nonlinear structure becomes possible
only when an accurate linear mathematical model of the nonlinear structure is available. In
addition to the availability of an accurate linear model, location information on the
structural nonlinearity is also essential in most cases so that the number of unknowns
involved in the modelling process can be reduced because, unlike the modelling of a linear
structure, in which the model to be sought is unique and so all the data measured are
consistent and can be used at the same time,
structure has to be established based on a mode
mode is available each time as illustrated below.
the mathematical model of a nonlinear
by mode basis so that only one measured
Consider the system shown in Fig.6.1. In measurement, in order to linearise the
structure, the response amplitude of a chosen point 
for example) is kept constant.
Then, when the excitation frequency 
(around the mode), the displacements of all
coordinates are determined by the 
modeshape and therefore, the relative vibration
amplitude of the nonlinear stiffness, which determines the value of equivalent 
stiffness, can be considered to be proportional to the vibration amplitude of 
This is
especially true when the mode to be analysed is well separated from its neighbours.
However, when the excitation frequency is around the 
mode, 
for the same
vibration amplitude of (assuming this to be possible), the relative vibration amplitude
of the nonlinear stiffness will also be proportional to the amplitude of 
but will be
different in magnitude from that of the 
mode because these two modeshapes are
different. Therefore, even when the response amplitude of a certain coordinate 
in this
case) is constant, the data measured around different modes could be the data from
different 
systems and this means that only one measured mode can be used each
time in the modelling of a nonlinear structure due to the inconsistency of measured data.

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Identification of Mathematical Model of 
Structures
175
FE model and dynamic testing data to update the FE model. Both modelling activities will
be reviewed and further possible development will be discussed.
As for the mathematical modelling of nonlinear structures, correlation between analytical
model and dynamic testing data becomes essential because, in general, structural
nonlinearities cannot be foreseen and, therefore, cannot be analytically 
but can
be measured. It is believed that the modelling of a nonlinear structure becomes possible
only when an accurate linear mathematical model of the nonlinear structure is available. In
addition to the availability of an accurate linear model, location information on the
structural nonlinearity is also essential in most cases so that the number of unknowns
involved in the modelling process can be reduced because, unlike the modelling of a linear
structure, in which the model to be sought is unique and so all the data measured are
consistent and can be used at the same time,
structure has to be established based on a mode
mode is available each time as illustrated below.
the mathematical model of a nonlinear
by mode basis so that only one measured
Consider the system shown in Fig.6.1. In measurement, in order to linearise the
structure, the response amplitude of a chosen point 
for example) is kept constant.
Then, when the excitation frequency 
(around the mode), the displacements of all
coordinates are determined by the 
modeshape and therefore, the relative vibration
amplitude of the nonlinear stiffness, which determines the value of equivalent 
stiffness, can be considered to be proportional to the vibration amplitude of 
This is
especially true when the mode to be analysed is well separated from its neighbours.
However, when the excitation frequency is around the 
mode, 
for the same
vibration amplitude of (assuming this to be possible), the relative vibration amplitude
of the nonlinear stiffness will also be proportional to the amplitude of 
but will be
different in magnitude from that of the 
mode because these two modeshapes are
different. Therefore, even when the response amplitude of a certain coordinate 
in this
case) is constant, the data measured around different modes could be the data from
different 
systems and this means that only one measured mode can be used each
time in the modelling of a nonlinear structure due to the inconsistency of measured data.

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Identification of Mathematical Model of Dynamic Structures
176
f = F sinot
keep = 
+ 
A 
+ 
for
 
Illustration of Inconsistency of Measured Data of a Nonlinear Structure
Although the inevitable limitations of measured data could present a problem, the greatest
difficulty involved in the modelling of a nonlinear structure, as will be shown, is the
establishment of an accurate linear model. When such a linear model is available, by using
the measured frequency response data together with location results of the nonlinearity,
the modelling problem can be resolved in most cases. For this reason, most of the space
in this Chapter will be devoted to a discussion of how to obtain an accurate linear model
of a structure by correlating the analytical model and measured dynamic test data.
When the identification of structural dynamic characteristics is undertaken by both
theoretical analysis (normally FE analysis) and experimental modal testing, discrepancies
often exist between the vibration characteristics predicted by the theoretical model and
those identified experimentally. In such cases, the analytical model needs to be modified,
if practically necessary, so that it represents more accurately the dynamic characteristics of
the actual structure. When both the analytical model of a structure and experimental modal
testing data are available, analytical model improvement can be mathematically formulated
as described below.
Given the analytical mass matrix [M,] and stiffness matrix [K,] (containing modelling
errors [AM] and [AK]) and dynamic testing data, which can be either modal data 
and 
or frequency response data [a,(o)] (but in both cases incomplete in terms of
measured modes and/or coordinates), then by correlating the analytical model and
measured data, the former can be updated so that it better represents the dynamic
characteristics of the actual structure.
6.3 
REVIEW OF ANALYTICAL MODEL IMPROVEMENT METHODS
As mentioned earlier, there exist two different branches of modelling activity, these being
modelling using experimental data only and modelling based on the correlation between
an analytical model and experimental data. For mathematical modelling via direct use of

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Identification of Mathematical Model of 
Structures
177
measured modal data, there have been attempts in recent years to use results of dynamic
testing to identify the parameters in the equations of motion directly, but it is generally
believed that unless the number of measured modes is greater than or equal to the number
of coordinates of interest, a mathematically unique model cannot be obtained 
A
survey of the work on the mathematical modelling using measured modal data only was
conducted in 1969 by Young and On 
In 1971, Berman 
assumed a simple form
for the mass matrix and then used the limited measured modes to construct a mass matrix
by invoking the orthogonality equations. This mass matrix is then used together with the
measured modes to construct the so-called ‘incomplete’ stiffness matrix: ‘incomplete’
because the contribution of all the unmeasured higher modes (which is usually the major