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2 Identification of Nonlinearity Using First-order
45
calculated with the explicit F(x) as shown in figure 2.14, we obtain the describing
function coefficient
for the backlash stiffness case as:
+
(2-45)
When the vibration amplitude is given, the describing function coefficient
can be
determined. In the case of nonlinear SDOF systems, since the mass property is usually
linear, the vibration-amplitude dependent natural frequency
can be calculated from
the describing function coefficient
as
as shown in Fig.2.15 for
the above-mentioned system with backlash stiffness nonlinearity
m=lkg and
In fact, the describing function
and the identified natural frequency
and damping coefficient
satisfy:
=
m
(2-46)
(2-47)
where m is the mass of the system which can be calculated from the identified modal
constant A as
Equivalent Stiffness and Natural Frequency of an SDOF Backlash System
To see how this harmonic balance theory can be applied to the measurement and analysis
of nonlinear structures, the system shown in Fig.2.16 will be considered. When the
system is excited by a sinusoidal force
then after the transient dies away, the
response of the system at any coordinate will be very close in its waveform to a sinusoid,
as will be shown to be especially true when the excitation frequency is close to one of the
resonance frequencies of the system. Therefore, corresponding to this specified excitation
2
Identification of Nonlinearity Using First-order
4 6
condition, the equivalent stiffness which the nonlinear stiffness element exhibits can be
calculated based on the harmonic balance theory. If, for example, the force amplitude F is
kept constant as the excitation frequency varies, then corresponding to different
frequencies, the vibration amplitude, and therefore the equivalent stiffness value of the
nonlinear stiffness element, is different. The information on these different stiffness
values is recorded in the measured first-order FRF data and by analysing these measured
FRF data, the nonlinearity can be identified.
U
+
A Nonlinear MDOF system
It should be noted that such a result is achieved only under the condition that it is not
necessary to include the harmonics and combinational resonances in the response signal.
For this condition to be valid, certain criteria should be satisfied by the linear part of a
nonlinear system as well as the nature of the nonlinearity. These conditions are
summarised here without mathematical proof, which can be found in
(i)
the system to be analysed should have a narrow-band filter property so that the
and super-harmonic components will be heavily attenuated;
(ii)
(k is an integer and is the excitation frequency) should not coincide with any of
the natural frequencies of the system; and
(iii) the nonlinear function
should have finite partial derivatives with respect to x
and
Condition (i) can usually be satisfied because in the analysis of structural nonlinear
systems, only the data points around the resonances are of interest and, therefore, the
system acts as a very good narrow-band filter. As for condition (iii), most practically
encountered nonlinearities, even for nonlinearities having relay (discontinuous)
characteristics, have finite partial derivatives. However, condition (ii) is sometimes
difficult to achieve because it depends on how the natural frequencies of the system are
situated along the frequency axis and is thus the major source of analysis errors.
2
Identification of Nonlinearity Using First-order
4 7
DESCRIPTION OF A NEW METHOD
Most practical nonlinearities (in both stiffness and damping elements) are response
amplitude dependent and so if, in measurement, the response amplitudes at different
frequencies are varied, then the effect of nonlinearities on the measured FRF data will be
recorded. The main target of nonlinear modal analysis is to identify the nature of any
nonlinearity by analysing thus-measured FRF data. As discussed, many different
methods have been developed for detecting the existence of structural nonlinearities.
Taking stiffness nonlinearity as an example, the nonlinearity can be exposed by observing
the FRF data measured using different force or response control techniques, or by
analysing the FRF data and examining the isometric damping plot, or by comparing the
measured FRF data with their Hilbert transform pair
etc. With the more demanding
objective of quantifying structural nonlinearity, the Inverse Receptance method was
developed. However, the method was devised based on the assumption that the mode to
be analysed should be real and the modal constant should be real and constant. As
demonstrated, these assumptions are not usually valid when practical measured data are
concerned. In order to remove these restrictions so that nonlinearities of practical
structures can be analysed accurately, a new nonlinear modal analysis is introduced
below.
According to harmonic balance theory, in the case of sinusoidal excitation, when a
nonlinear structure vibrates at specific amplitude, there will be specific equivalent
(linearised) stiffness and damping model as far as the first-order FRF is concerned.
Therefore, measured FRF data generally contain information on a series of linear models.
What the new method seeks to do is to calculate the modal parameters of these linear
models together their corresponding response amplitudes so that the relationship between
modal parameters and response amplitude can be established. Owing to the nature of
resonance, it is always possible to find two frequency points in the measured FRF data
one on either side of the resonance which have the same (or very similar) response
amplitude. These two data points constitute a specific linear model corresponding to that
specific response amplitude in the sense that all the modal parameters necessary to
determine that linear(ised) model can be calculated just using these two receptance data
points. The thus determined modal parameters are associated with that specific response
level. Therefore, if there are many point pairs of different response amplitudes available
around that resonance, a relationship between modal parameters of the mode and
response amplitudes can be established.
2 Identifkation of Nonlinearity Using First-order
Suppose
and
are known to correspond to a certain specific response level,
one on either side of the resonance, then the following two mathematical equations can be
established (assume the residual effect is negligible or has been removed at the moment
and its influence on analysis accuracy will be discussed later)
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