Identification of the dynamic characteristics of nonlinear structures

MEASUREMENT OF FIRST-ORDER FREQUENCY RESPONSE
FUNCTIONS
A brief introduction has been made so far of the theoretical analysis of nonlinear systems
based on the known differential equations. However, the primary target which is sought
in this study is the identification of the unknown mathematical models of nonlinear
structures based on measured input/output dynamic characteristics. Therefore, it becomes
necessary before the introduction of any identification techniques to discuss how the
dynamic characteristics of a structure (linear or nonlinear) can be measured.
First of all, it is necessary to explain what is meant by the 
response functions 
of a nonlinear structure. In concept, first-order frequency response
functions (first-order 
are the extension of frequency response functions 
of
linear systems to nonlinear systems. Similar to the measurement of 
of a linear
structure, in the case of sinusoidal excitation (the excitation is a pure sinusoid), the 
order FRF 
of a nonlinear structure is defined as the spectral ratio of the response
X(o) and the force F(o) at the excitation frequency: 
During the
estimation of 
all the harmonic components (subharmonics, superharmonics and
combinational resonances) are ignored and only the fundamental frequency component of
the response is retained. Similarly, in the case of random excitation (the excitation is
wide-band random signal), first-order frequency response function is defined as the
spectral ratio of cross-spectrum of the force and response and the auto-spectrum of the
force: 
(or its equivalent form 
H,(o) 
The measured
first-order 
of a nonlinear structure based on sinusoidal and random excitations are
in general different and their relationship will be discussed in Chapter 3.
For linear structures, the first-order frequency response functions (often referred simply
as frequency response functions) are unique and, therefore, will not vary according to
different excitation techniques and conditions. For nonlinear structures, however, the
measured first-order frequency response functions are, in general, not unique. They
depend not only on the excitation conditions (input force levels), but also on the different

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2 Identification of Nonlinearity Using First-order 
15
excitation signals used to measure them. Therefore, the first problem of nonlinearity
investigation will necessarily be to decide a proper means of excitation so that nonlinearity
can easily be revealed and then identified. There are three types of excitation method
widely used in vibration study practice sinusoidal, random and transient and each of
them is discussed below.
2.2.1