Identification of the dynamic characteristics of nonlinear structures

 
 
+ 
= f(t)
(3-47)
 


 
+ 
= 0
(3-48)
 
 + 
+ 
= 
0
Fig.3.10 A Three Degrees-of-freedom Nonlinear System

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3
of Nonlinearity Using Higher-order 
86
The first-order Volterra kernel transforms of the system are the frequency response
functions of the linear system 
here only the second-order Volterra kernel
transforms are going to be calculated. Similarly to the SDOF case, let 
then
N =
(N+k)! k!
+ 
(3-37)
where 
are the transfer Volterra kernel transforms between
coordinates 
and 
Substitute x,(t) and its derivatives 
and 
into 
and let the coefficients of 
on both sides of the equations be equal respectively,
so that the following algebraic equations are obtained:
I 
 
 
+
+ k] 
=
 
 
(3-51)
 
 
 
 
 
+
+ 
+
+ k] 
=
(3-52)
+ 
[ 
 
 + 
k] 
=
( 3 - 5 3 )
From 
the three unknowns 
can be calculated. The
analytical second-order point frequency response function 
(excitation and
response are at the same coordinate) is shown in 
In this case, the appearance of
combinational resonances (peaks which do not lie on the two diagonals) is clearly
demonstrated.

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Identification of 
Nonlinearity Using Higher-order 
87
Analytical Second Order Frequency Response Function of a 3DOF Nonlinear
System (Modulus Linear Scale, x-axis -275 275, y-axis 
275 
Since the above calculations are based on the definition of 
the calculated
are of receptance-like frequency response functions. If x(t) is changed
into 
in 
then the calculated Volterra kernel transforms are the mobility-like
frequency response functions and it can be easily seen that the receptance-like and
mobility-like Volterra kernel transforms are related by
. . . .
= 
+ 
. . .+ 
(3-54)
3 . 3 . 3
MEASUREMENT OF HIGHER-ORDER FREQUENCY RESPONSE
FUNCTIONS USING HARMONIC PROBING METHOD
As in the case of the measurement of frequency response functions of a linear system,
different measurement techniques can be employed to measure the frequency response
functions of a nonlinear system, such as (i) multi-impulse technique 
(ii) harmonic
probing method 
(iii) correlation analysis using random input 
and (iv)
time series modelling technique 
etc.. Among these different methods, the

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