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COMPLEX MODE FROM MEASUREMENTS
It is believed that for practical structures, most of the damping comes from joints
Therefore, practical structures possess very nonproportional damping distribution and
genuine complex modes exist. To demonstrate this, modal testing of a simple
Beam/Absorber structure as shown in Fig.2.12 was carried out. The structure was found
to be slightly nonlinear, as will be discussed again later on in this Chapter. However,
during the test, the vibration amplitude of the structure was controlled to be constant at
different sinusoidal excitation frequencies and, as a result, the FRF measured is exactly
the
of a linear structure. One of the measured point
(response and excitation at
the same point) was analysed and, as shown in
a mode complexity of about 15”
is clearly demonstrated.
Excitation force
F=Asin
Fig.2.12
The Beam/Absorber Structure
2
of Nonlinearity Using First-order
42
20
Data from
Frequency Hz.
Fig.2.13 Measured Point FRF with Mode Complexity
2.5
A NEW METHOD FOR NONLINEAR MODAL ANALYSIS OF
COMPLEX MODES
So far, some of the most commonly-used nonlinear modal analysis methods have been
reviewed and their limitations when applied to practically-measured data have been
examined. In what follows, a new analysis method which avoids the aforementioned
limitations will be proposed. The harmonic balance theory, on which the present new
method is based, will be introduced together with its application conditions. In order to
extend the method to
systems, the residual effect (of other modes) on the analysis
accuracy will be examined and the practical applicability of the method will be assessed
by analysing data measured on practical nonlinear structures. Finally, the possibility of
identifying physical characteristics of nonlinearity from analysed response
dependent modal data based on the new method, when an
system is considered,
will be discussed.
3.5.1 HARMONIC BALANCE THEORY
In the analysis of nonlinear systems, the harmonic balance method is frequently used
where sustained oscillations exist. The theoretical basis of the harmonic balance analysis
lies in the equivalent linearisation theory proposed by Krylov and Bogoliubov
for
solving certain problems of nonlinear mechanics. To explain the concept of the harmonic
balance method, an SDOF system with nonlinear restoring force
driven by a
sinusoidal excitation is considered:
=
(2-35)
Identification of Nonlinearity Using First-order
4 3
To solve the above problem by the harmonic balance approach, it is necessary to make a
basic assumption that the variable x=x(t), appearing in the nonlinear function
is
sufficiently close to a sinusoidal oscillation; that is,
x
(2-36)
where the amplitude A, frequency
and phase lag are constant. Therefore, the
harmonic balance analysis belongs to those approximate methods of solving nonlinear
differential equations which are based upon an assumed solution. As such, it requires that
conditions for the assumed solution exist. Such an assumption is quite realistic since a
nonlinear system may well exhibit periodic oscillations arbitrarily close to a pure
sinusoid. If the variable x in the nonlinear function
has the sinusoidal form of (2-
36), then the variable
is generally complex, but is also a periodic function of
time. As such, it can be developed in a Fourier series:
y =
+
+
i
+ harmonics
(2-37)
When only the fundamental component is considered, the first three terms are
1
(2-38)
0
1
(2-39)
0
2x
(2-40)
where
=
+ Coefficients
are often referred as describing function
coefficients.
If we consider the case where the nonlinear function
is symmetrical about the
origin (although the analysis is equally applicable for the case of nonsymmetrical
nonlinearities), the constant term
in the Fourier series (2-37) is
0. The quantities
defined in (2-39) and (2-40) are the coefficients of the describing function
To discuss the physical meaning of the describing function N defined above, suppose
describes a backlash stiffness nonlinearity, as shown in Fig.2.14. Then, if
2
Identification of Nonlinearity Using First-order
4 4
the input x (the response of the system) is a sinusoid,
the output
will
not be a pure sinusoid (assuming that is big enough to exceed the system’s linear
regime). Expressing the output signal y in a Fourier series, the fundamental component
will be
Backlash Stiffness Nonlinearity
= sinot
(2-41)
where is the amplitude of the fundamental component which, according to the Fourier
series theory, can be calculated as:
=
0
According to equation
has the form
1
F
(2-42)
(2-43)
Therefore, the describing function
is defined as the ratio between the amplitude
of the input signal x and the amplitude of the fundamental component contained in
the output; that is
Compared with the definition of static stiffness, it can be seen that
can be interpreted
as the equivalent ‘dynamic stiffness’ of the nonlinear stiffness element corresponding to
vibration amplitude
If the integral on the right hand side of equation (2-43) is
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