3 Identification of Nonlinearity Using Higher-order
6 7
symmetric counterparts and the harmonic components which are usually present in the
response signal of nonlinear systems are filtered out. This symmetrisation of
nonsymmetrical nonlinearities and the elimination of harmonic components mean that the
first-order frequency response function analysis is not very appropriate for the analysis of
structures with nonsymmetrical nonlinearities. In fact, it will be shown that for some
specific nonlinear systems, such as
quadratic and bilinear systems, the analysis technique
is incapable of analysing them at all. From the response prediction point of view,
calculations made using the first-order frequency response functions only can be quite
inaccurate in some cases as described in the application conditions of harmonic balance
analysis in Chapter 2 because, mathematically, this means that only the linear term of the
Taylor expression of a nonlinear function at certain point has been retained. These
limitations of first-order frequency response analysis are illustrated next.
As mentioned, systems with nonsymmetrical nonlinearities such
as quadratic and bilinear
systems cannot be identified based on the first-order frequency response function analysis
because these nonlinearities are such that the measured first-order frequency response
functions based on sinusoidal excitation are effectively linear. Suppose the nonlinearities
are of a stiffness type and their force-displacement relationships are shown in
then the equivalent stiffness value corresponding to specific response amplitude can be
calculated based on harmonic balance theory as discussed in Chapter 2. Assume the
vibration to be sinusoidal as
then the describing
function coefficients
(equivalent stiffnesses)
for the case of quadratic stiffness and
for the case of
bilinear stiffness can be calculated as:
=
+
sinot dot = k
0
(3-l)
=
sinot dot +
dot =
(3-2)
0
Identification of
Using Higher-order
6 8
1
Fig. 3.1 Force
Relationshins of Quadratic and Bilinear Stiffness
1
Since both
and
which are the equivalent stiffnesses, are constant
(independent of the response amplitude), the measured first-order frequency response
functions of these systems are linear with equivalent linear constant stiffnesses of k for
the quadratic stiffness case and
for the bilinear stiffness case.
On the other hand, the existence of nonlinear phenomena such as, sub-, super- and
combinational resonances in nature is well known. Nayfeh
mentioned that Lefschetz
described a commercial airplane in which propellors induced
a subharmonic vibration in
the wing which in turn induced subharmonic vibration in the rudder. The oscillations
were violent enough to cause tragic consequences. Also, reports have been found in the
literature that excessive vibrations were caused by superharmonic excitation and
combinational resonances. In those cases, the analysis of the harmonic components
becomes as important as that of the fundamental frequency component and the response
predicted using first-order FRF data in such circumstances could be very inaccurate. To
illustrate this point, superharmonic excitation is considered
for the case of an SDOF
system with cubic stiffness nonlinearity. When the external excitation frequency is far
from one third of the natural frequency of the system (linear natural frequency, as if the
cubic term were not introduced), the response prediction based on the first-order FRF is
very accurate. When the excitation frequency is close to one third of the natural
frequency, then the structural resonance will be excited by the third harmonic component
generated by the cubic nonlinearity and as a result, the response
prediction based on the
first-order FRF in this case becomes very inaccurate. Comparisons of the true responses
and the responses predicted using first-order frequency response function data when the
excitation frequencies are of
and
the natural frequency are shown in Fig.3.2.
