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(2-3)
(2-4)
When different combinations of points
are used, a flat plane which is the surface
plot of the estimated damping ratio
against its two variables
and
can be
obtained in the case of linear FRF data
On the other hand, if the measured FRF data are from a nonlinear structure, distortion of
the isometric damping plot (no longer a flat plane) calculated based on equation (2-5) will,
in general, be expected as shown in Fig.2.6 for the data measured from a Beam/Absorber
structure shown in figure 2.1.
2
Identification of Nonlinearity Using First-order
28
Fig.26 Isometric Damping Plot for Data Measured from Beam-absorber Structure
R e s
The reason why this distortion occurs is discussed below. Suppose the FRF data are
measured from a system with stiffness nonlinearity, then the natural frequency
of
mode which is sensitive to nonlinearity becomes a function of response amplitudes
(where is the response amplitude of certain coordinate) and since different
frequency points have different response amplitudes for data measured with constant
force, the effective natural frequencies at different data points are therefore, different.
With this in mind, equation (2-3) and (2-4) become:
=
_
(2-6)
(2-7)
The exact damping loss factor can be calculated as:
In the case when damping is linear, the calculated damping loss factors based on equation
(2-8) will be constant and therefore, it is not difficult to see that the distortion of the
isometric damping plot obtained based on equation (2-5) is determined by the the
Identification of Nonlinearity Using First-order
2 9
difference between the estimates of equation (2-5) and (2-8) which, to the first-order
approximation, becomes:
From equation
it can be seen clearly that in the case of stiffness nonlinearity, the
distortion of the damping plot calculated based on equation (2-5) is caused by the
different response amplitudes and therefore
frequency points chosen.
different natural frequencies of various
The isometric damping plot technique, as has been demonstrated, can be used to detect
the existence of nonlinearity. However, as in the case of nonlinearity detection based on
Bode and Nyquist plot techniques, anything beyond detection in the identification of
nonlinearity will be truly difficult because the method is of a qualitative nature rather than
quantitative, although it has been suggested that, by comparing the different distortion
patterns of damping plot of commonly encountered nonlinearities, the identification of the
type of nonlinearity may become possible. With a more ambitious objective of
quantifying nonlinearity, the Inverse Receptance method was developed
The method
will be presented next and its limitations when applied to FRF data measured from
practical nonlinear structures will be discussed.
2.3.3 INVERSE RECEPTANCE METHOD
As discussed before, nonlinearities in FRF data will cause distortion of a plot of the
inverse receptance data and such characteristics as plot distortion have been employed for
the detection of the existence of nonlinearity. These inverse receptance data can be further
employed for the purpose of quantifying nonlinearity. In this section, the Inverse
Receptance method
is introduced. The limitations of the method for the analysis of
practical nonlinear structures will be discussed and further possible improvements will be
pointed out.
For a nonlinear SDOF system, the natural frequency
are, in general, response amplitude dependent. With this
receptance can be expressed as:
and damping loss factor
in mind, the reciprocal of the
2
Identification of Nonlinearity Using First-order
1
+
A
A
where A is the modal constant which is assumed
its real and imaginary parts, so that
Re( l/a(o)) =
A
Im( l/a(o)) =
A
(2-10)
to be real. Separate equation (2-10) into
(2-l 1)
(2-12)
Suppose that the input force signal F(o) is also recorded during the measurement, so that
the response amplitude at each frequency can be easily calculated as:
(2-13)
It becomes clear that if the modal constant A can be estimated by some means, then the
relationships of
and
can be calculated based on equations
11)
and (2-12) as:
=
+ A
(2-14)
(2-15)
where is the response amplitude at frequency and can be calculated from equation (2-
13). The calculation of the modal constant A in this Inverse Receptance method is based
on a trial-and-error approach and the criterion for the correct estimation of A is based on
the fact in which satisfactory results have been obtained.
Based on this method, FRF data measured from analogue computer circuit with cubic
hardening stiffness nonlinearity have been analysed and the results are shown in Fig.2.7.
The effect of a hardening stiffness nonlinearity is clearly demonstrated.
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