2
Identification of Nonlinearity Using First-order
6 4
plot method achieves its nonlinearity detection by calculating the damping matrix based on
linear modal analysis theory and then examining the distortion of the damping plot due to
the variation in response amplitude and so in the effective
frequency differences of
data points around Nyquist circle. These methods are convenient for the nonlinearity
detection stage but not so applicable for nonlinearity quantification and identification.
With a more ambitious objective of nonlinearity quantification, the Inverse Receptance
method as discussed was developed by analysing stiffness
and damping nonlinearity
separately based on the real and imaginary parts of the inverse receptance data. However,
as demonstrated in some detail in this Chapter, some assumptions have been made during
the development of the method which are, in general, not valid for data measured on
practical structures with nonlinearity and so the method is limited in terms of its practical
applications.
The theoretical aspects of the existence of complex modes have been discussed. The
necessary and sufficient condition for the existence of complex modes is that the damping
distribution of the system is nonproportional. The effect of natural frequency spacing on
the degree of complexity has been illustrated. Numerical as well as experimental examples
are given.
The harmonic balance theory, which is the mathematical basis
of the new nonlinear modal
analysis method proposed in this Chapter, is presented together with its practical
application conditions. The relationship between the analytical analysis of a
nonlinear
system based on harmonic balance theory and the experimental measurement of FRF data
of a practical nonlinear structure has been discussed. Based on harmonic balance theory,
a dynamic system having stiffness nonlinearity will take a different equivalent linearised
stiffness values (describing function coefficient
for different response amplitudes,
so that each FRF data point from a measurement with constant force actually relates to a
specific FRF data curve measured with constant response, thereby containing all the
information of the latter curves. Due to this specific characteristic
of FRF data from
constant force measurement, thus measured data can be analysed to quantify and identify
the nonlinearity of the test structure if the force level is appropriately chosen.
With a theoretical basis of the harmonic balance analysis, a new method has been
proposed to analyse nonlinearity from measured first-order FRF data. In addition to
deriving an indication of the nonlinearity, the method aims at establishing the
relationships between the modal parameters of interest and response amplitude from the
FRF data measured using sinusoidal excitation. The final results of the analysis are the
response-amplitude-dependent eigenvalues
and eigenvectors
of nonlinear
2 Identification of Nonlinearity Using First-order
6 5
systems. These
identified modal data
can be used subsequently to derive a linearised
spatial model
and
or
matrices) of the structure. Also, it is
necessary to mention that the condition of constant force is not necessary when measuring
the FRF data for the subsequent modal analysis using this new method. In fact,
satisfactory analysis can be carried out as long as the response amplitude varies
sufficiently to expose the nonlinearity and embraces the range of displacements which
must be described by the model.
The method has been extended to the analysis of nonlinear MDOF systems and the effect
of residual on the analysis accuracy has been discussed. By linearising neighbouring
modes at the measurement stage and applying the SIM
method at the analysis stage, the
residual of the mode to be analysed can be removed to its minimum level and so accurate
analysis results can be obtained.
Although the quantification of nonlinearity in modal space has been achieved using the
proposed method, identification of the describing function coefficient
and so of the
physical characteristics K(x) and/or C(x) from the identified modal parameters
and
which will be
in later chapters, is by no means straightforward when
MDOF systems are considered. Moreover, it has to be pointed out that,
during sinusoidal
excitation measurement, since the DC component of the response signal of a
nonsymmetric nonlinear system has been filtered out, the nonsymmetric nonlinearity has
been made symmetrical and the FRF data measured are the data from an equivalent
symmetric nonlinear system. Due to this symmetrisation, for some nonsymmetric
nonlinear systems, such as bilinear systems, the existing nonlinearity cannot be revealed
from measured first-order FRF data which are effectively linear. Furthermore, except for
the fundamental
frequency component, the response of a nonlinear system usually
contains super-, sub- and combinations of harmonics. However, in the first-order FRF
analysis, all these harmonics, which are in some cases as important as the fundamental
component in vibration analysis, are filtered out. Therefore, first-order FRF analysis is
limited in the sense that it reveals the nature of the nonlinearity and in order to identify
nonsymmetric nonlinear systems and take into account these super-, sub- and
combinations of harmonics, higher-order FRF analysis becomes necessary and is to be
introduced in next chapter.
