Identification of the dynamic characteristics of nonlinear structures

 
Rewriting equation 
1) in its reciprocal form and assume modal constant 
to be real:
1
+
 
 
= 
=
+
From equation 
it can be seen that if FRF data are expressed in their reciprocal
form, 
(related to stiffness nonlinearity) and 
(related to damping nonlinearity) can be
identified separately from the real and imaginary parts of the reciprocal FRF.
When FRF data are obtained for linear 
the relationships of R(o) 
and
are straight lines for the case of hysteretic damping as shown in Fig.2.2. Any
distortion from a straight line gives indication of the existence of nonlinearity. When FRF
data from nonlinear structures are to be analysed, the effect of a stiffness nonlinearity will
show up in the real part of the reciprocal of receptance data while the effect of a 
nonlinearity will appear only in the imaginary part of the data. To illustrate this point,
FRF data with stiffness and damping nonlinearity measured from simulated analogue
circuits are analysed and the real part (R(o)
and imaginary part (I(o)
of the
reciprocal FRF are shown in Figs.2.3 
2.4. In the case of stiffness nonlinearity,
distortion of the reciprocal FRF data only appears in the real part, as shown in Fig.2.3.
On the other hand, as shown in Fig.2.4, when damping nonlinearity is considered, its
effect is clearly observed to be confined to the imaginary part of the reciprocal FRF data.

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2
Identification of Nonlinearity 
Using 
First-order 
2 5
 
Real part
Imaginary Part
Fig.2.2 Real and Imaginary Parts of Linear Reciprocal 
Data
R e a l P a r t
P a r t .
1
1 5 . 7 0