Identification of the dynamic characteristics of nonlinear structures

 
 
 
+ 
 + . . . + 
(3-42)
where 
. . . .
is defined in such a way that
= 
[ 
+ 
+ 
 
+ 
(3-43)
It should be noted here that higher nonlinear terms 
have no influence on the
lower-order Volterra kernel transforms
while the lower nonlinear
terms 
have an influence on all the higher-order Volterra kernel transforms,
Therefore, a system with cubic stiffness nonlinearity does not possess 
order Volterra kernels while a system with quadratic stiffness nonlinearity has, in general,
all the Volterra kernels. The second- and third-order Volterra kernel transforms of the
SDOF nonlinear system described by equation (3-30) are calculated and are shown in
Figs.3.53.7.

---

3 Identification of Nonlinearity Using Higher-or&r 
ig.3.5 Analytical Second Order Frequency Response Function of an SDOF Nonlinear
System (Modulus Linear Scale, x axis 
-275 275, y axis 
0 
275 
Fig.3.6 Analytical Second Order Frequency Response Function of an SDOF Nonlinear
System (Phase Linear Scale, x axis 
-275 275, y axis 
0 275 

---

3 Identification of 
Nonlinearity Using Higher-or&r 
ig.3.7 Analytical 
Third 
Order Frequency Response Function of an SDOF Nonlinear System
(Modulus Log. Scale, x axis 
-275 275, y axis 
0 275 
To illustrate 
physical interpretation of higher-order frequency response functions, the
second-order frequency response function shown in figure 3.5 is discussed. As shown in
Fig.3.8, the components of the second-order frequency response function near both
frequency axes represent the fundamental frequency components of the response. While
components along both diagonals are the static components 
and second
harmonic components 
of the response respectively. All the other components
defined on the 
plane are the combinational frequency components of the
response which, as will be discussed later, are important in cases of nonlinear 
systems because these combinational frequency components can excite the system into its
resonances when they coincide with some of the natural frequencies of the system.

---

3
Identification of Nonlinearity Using Higher-order 
8 4
Fig.3.8 Physical Interpretation of Second-Order FRF
Also, one special case which has been treated in the literature 
and is discussed here is
the square-law system given as:
+ + kz = 
f(t)
(3-44)
(3-45)
This is an ideal Volterra system for which all the kernels except the second one are zero
and it is easy to prove that its second-order Volterra kernel transform is
(3-46)
where HI(o) is given by (3-36). This second-order frequency response function of the
square-law system is shown in Fig.3.9. Because of its purely quadratic nature, the
system’s response is dominated by the second harmonic and static components.

---

3 Identification of 
Using Higher-order 
85
ig.3.9 Analytical Second Order Frequency Response Function of the Square-law System
(Modulus Linear Scale, x axis 
-275 
275, y axis 
0 
275 
The same argument holds for MDOF nonlinear systems, although the analytical
calculation becomes a bit complicated. To see this, the 3DOF nonlinear system as shown
in Fig.3.10 is considered. The governing differential equations of the motion are given as
+ 
+