Identification of the dynamic characteristics of nonlinear structures

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FREQUENCY HZ 
Fig.3.3 Response Spectrum of an SDOF System with Input 
 3
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0
2 2
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F R E Q U E N C Y
Response Spectrum of an SDOF System with Input 
+ 

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3
Identification of Nonlinearity Using Higher-order 
7 2
3.2.2 THE 
 SERIES REPRESENTATION
Volterra series have been described as “power series with memory” which express the
output of a nonlinear system in “powers” of the input. A wide class of nonlinear systems
encountered in engineering can be represented as Volterra series. Given an input f(t), the
output x(t) of a time invariant system can, in general be expressed as
 
. . .
. . . .
) 
n = l
r = l
(3-4)
where the kernels 
. . . .
are the Volterra kernels which describe the system. It
should be noted that the first-order kernel 
is the impulse response due to the linear
part of the nonlinear system and the higher-order kernels can thus be viewed as 
order impluse responses which serve to 
the various orders of nonlinearity. In
the special case when the system is linear, all the higher-order kernels except 
are
zero. The Volterra series representation (3-4) of a nonlinear system is homogeneous. In
order to illustrate this, rewrite equation (3-4) as x(t) 
+ 
+ . . . + 
+ . . .
where
x,(t) =
. . .
. . . .
. . . 
. . . 
(3-5)
From 
it is easy to see that when the input changes from f(t) to 
then the 
component of the output becomes 
and the total output x(t) becomes
x(t)= 
s = l
This homogeneous property of Volterra series representation has been
applied to the measurement of Volterra kernels of electrical nonlinear circuits by repeating
the measurements using different input levels of the same signal 
Since almost all
physical systems, whether they are linear or nonlinear, are causal (a system is said to be
causal if, for any input, the output at any instant of time does not depend upon the future
input), all the kernels have to satisfy
. . . . 
= 0
for any 
0
s = 1, n n = 1, 

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3
Identification of Nonlinearity Using Higher-order 
7 3
Like a Taylor series representation of a nonlinear function, the Volterra series
representation of a general nonlinear system is theoretically infinite and, as will be
discussed later, the effort of computing the n*-order kernel increases exponentially as n
increases so that one has to be satisfied with the first few kernels only (usually, up to the
third kernel). Fortunately, good approximations can be obtained for most engineering
problems by just considering these first few kernels and this is why this theory has been
widely applied to the 
of practical nonlinear systems.
3.3