Identification of the dynamic characteristics of nonlinear structures

N a t u r a l
1. 
Rasp.
 
Fig. 2.34 Analysis of Nonlinearity of NASTRAN Tower Structure
Although one can deduce the type of nonlinearity in some cases by examining the 
and 
relationships, as will be discussed next, the exact identification of
the type of nonlinearity, will be difficult when most practical structures are considered.
2.5.6 
IDENTIFICATION OF NONLINEAR PHYSICAL
CHARACTERISTICS
Structural nonlinearities can now be analysed using the proposed method and
relationships between modal parameters and response amplitudes can be established. The
quantification of nonlinearity in modal space has thus been accomplished. When
nonlinear SDOF systems are considered, according to harmonic balance theory, the
describing function coefficients (linearised equivalent stiffness or damping) can be
directly calculated from the identified modal data. For example, a nonlinear SDOF
system’s 
equivalent stiffness (describing function coefficient) 
can be
calculated from the identified 
as 
 
(m is the mass of the system
which can be calculated from the identified modal constant). Although there exists another
step from 
to the system’s true stiffness K(x) (the physical characteristics of the
nonlinearity), by comparing with known types of nonlinearity, K(x) can be conclusively
identified in most cases from the calculated 
For nonlinear MDOF systems, however, the identification of 
and thus of K(x) is
not so straightforward. Considering an MDOF system with localised stiffness
nonlinearity as shown in Fig.2.16, and supposing the mode (which is sensitive to the
thus introduced localised nonlinearity) is analysed and the relationship between the natural

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Identification of Nonlinearity Using First-order 
6 3
frequency and response amplitude at certain reference coordinate is established, the
describing function coefficient 
of the nonlinear stiffness element cannot be
calculated from these analysis results alone. Therefore, an identification of K(x) which is
based on 
will not be possible. If, on the other hand, the analysed modal data are
interpreted as being from an SDOF system when identifying 
then misleading
results can be obtained because in this case the changes of measured modal parameters
depend not only on the stiffness (or damping) changes due to nonlinearity, but also on the
modification sensitivity where the nonlinear elements are located. Take the identified
natural frequency as an example. The natural frequency change of a certain mode can be
mathematically described by
=
= 
(2-56)
Since 
is unknown in the identification process and is a function of response
amplitude 
except in the case of SDOF systems in which, 
is known to be the
identified modal constant l/m, 
cannot be calculated from the identified 
and, as a result, the identification of K(x) is out of the question.
In fact, as will be discussed later on, in order to identify the describing function
coefficients and thus the physical characteristics of nonlinear element(s) of a practical
nonlinear structure, the nonlinearities have to be located first and then the linearised
equivalent stiffness matrix [K(s)] can be established by correlating the analytical model
and measured dynamic testing data.
2.6