Identification of the dynamic characteristics of nonlinear structures

2 Identification of Nonlinearity Using First-order 
From equations 18) and 
it is clear that if cubic stiffness is introduced as shown
in figure 2.9, both natural frequency and modal constant of mode 1 are functions of
response amplitude. The relationships 
and 
are illustrated in
with 
and nondimensionalised response = 0. 1.0.
Fig. 2.10 Natural Frequencies and Modal Constants of a Nonlinear System
As far as the quantification of nonlinearity is concerned, since FRF data measured 
practical nonlinear structures usually contain mode complexity and the modal constant of
the mode to be analysed cannot be assumed to be constant, the analysis results obtained
based on the Inverse Receptance method can be erroneous and sometime misleading.
Therefore, it becomes necessary to develop more realistic techniques so that the
complexity of the mode and the variation of modal constant can be taken into account and
more accurate modal parameters of nonlinear structures can be obtained.
2.4 COMPLEXITY OF VIBRATION MODES
2.4.1 THEORETICAL BASIS
There exist two different types of mode known as real modes and complex modes in
structural vibration analysis. In real mode vibration, individual elements of a system
move exactly in or out of phase with each other while in the case of complex modes,
individual elements vibrate with different phase angles (relative to each other). The reason
for the existence of complex modes is known to be a nonproportional distribution of the
structure’s damping. However, the degree of complexity of modes when the damping is
nonproportional is largely determined by the closeness of the natural frequencies of the

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2
Identification of Nonlinearity Using First-order 
system. In what follows, the necessary and sufficient 
for the existence of
complex modes, the influence of mode spacing on the complexity of a mode and the
relationship between viscous and hysteretic damping models from mode complexity point
of view, will be discussed.
It is well known that an undamped linear dynamic system described by
(2-20)
possesses real modes when [M] is nonsingular and 
has a full set of
eigenvectors, 
Such real modes can be used to find the principal coordinates in
which the equations of motion of the system are decoupled. Suppose 
is the mass
normalised modeshape matrix and let 
then equation (2-20) becomes:
= 
(2-21)
Pre-multiply equation (2-21) by 
and since 
and 
then
equation (2-21) can be decoupled in terms of principal coordinates (p) as:
+ 
= 
(2-22)
In the presence of damping (assuming viscous damping for the convenience of analysis),
equation (2-20) is modified to become
(2-23)
In this case, the criterion for the existence of real modes of the damped system is that the
real modes of the corresponding conservative system (without damping) can be used to
decouple the equations of motion of the damped system. For damped systems, in general,
the decoupling property is violated and the modes become complex. However, certain
conditions on the form of the damping matrix have been found under which a damped
system can still possess real modes. Such damping condition have been discussed in
detail by Caughey 
who pointed out that the sufficient condition for the existence
of real modes in a damped system is that the damping matrix of the system can be
expressed as:

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2
Identification of Nonlinearity Using First-order 
3 6
= 
(2-24)
s=l
where N is the dimension of the system. In the case when 
and 
equation (2-24) becomes the familiar 
damping which is
=
+
(2-25)
To prove the sufficiency of equation 
pre-multiply both sides of equation (2-24) by
and post-multiply by 
then
= 
= 
(2-26)
From equation 
since the damping matrix is 
by the real modes of the
corresponding conservative system, these real modes are also the real modes of the
damped system. On the other hand, if the corresponding conservative system has no
repeated eigenvalues, then condition described in equation (2-24) is also the necessary
condition. To illustrate this point, rewrite
s l
= 
into linear algebraic
equations in terms of unknowns 
as:
. . 
N - l
.
. . . .
.
. . . .