2 Identification of Nonlinearity Using
Coefficients
Frequencies
. .
Amplltudc
Amplitude
of
C o n s t
.
.
8
8
Resp. Amplltudc
Amplitude
Fig.2.23 Analysis Results of Data with Complexity
It also needs to be mentioned that although the phase angle of a mode is a measure of a
linear system’s complexity, linear modal analysis of the data from a nonlinear system can
produce an erroneous phase angle which could be misinterpreted as complex mode. This
is illustrated in Fig.2.24 where the estimated phase angle using
the classical circle-fit is
29” while the true phase angle is
Identification of Nonlinearity Using First-order
54
o f
Fig.2.24 Misestimation of Phase Angle Using Linear Modal Analysis Method
2.5.4 EXTENSION OF THE METHOD TO NONLINEAR
MDOF SYSTEMS
In order to assess the applicability of the above method to MDOF systems, constant force
of 2DOF systems with cubic stiffness nonlinearity have been analytically generated
with (a) only one mode (the second mode is made linear by controlling the input force to
be very low) and (b) both modes are nonlinear respectively. In this case,
in order to
analyse the mode accurately, the residual must be subtracted. The removal of the residual
can be accomplished by the method called SIM
which analyses the neighbouring
modes first and then subtracts the influence of these analysed modes from the one to be
analysed. For the case of only one nonlinear mode, the residual can be removed almost
completely as shown in Fig.2.25 in which the Nyquist circle passes through the origin
(the Nyquist circle looks the same as those of SDOF systems). After the residual has been
removed, the mode can be analysed accurately and the analysis results are shown in
Fig.2.26. For the case
of both modes being nonlinear, however, it becomes very difficult
to remove the residual completely as shown in Fig.2.27 for the case of the first mode (the
circle does not pass through the origin, the data points are not exactly on the circle and are
not symmetrical with the imaginary axis) and therefore, the analysis results obtained
could be in error. The main difficulty of removing the residual in this case lies in the
wrong estimation of the phase angle of the neighbouring mode obtained by the linear
modal analysis. This difficulty can be overcome in practical analysis by linearising the
neighbouring modes in the measurement (by controlling the response amplitude) so that
their modal parameters can be accurately estimated based on linear modal analysis
method(s) and the residual can therefore be correctly subtracted.
2
Identification of Nonlinearity
Using First-order
Fig.2.25 FRF of 2DOF Nonlinear System
D a m p i n g C o c f f
N a t u r a l
R e c e p t .
M o d u l u s o f M o d a l C o n s t
R e c e p t .
R e c e p t .
P h a s e
o f M o d a l C o n s t
R e c e p t .
l t u d e
Fig.2.26 Analvsis Results of 2DOF Nonlinear Svstem
q
2 Identification of Nonlinearity Using First-order
56
Fig. 2.27 FRF of 2DOF System with both
Modes
Nonlinear
Although some measurement and analysis techniques can be employed to remove the
residual effect, as mentioned above, when a structure is nonlinear, it is not possible for
the data to be analysed to become completely residual free because
in this case the residual
is a function of response amplitude. Therefore, it is very important for a nonlinear modal
analysis method based on the SDOF assumption to obtain satisfactory results even when
a small amount of residual exists and it is necessary to undertake a residual analysis. For
convenience, an assumption is made that the residual for the mode to be analysed is a
complex constant (in fact, this is quite accurate for the
separated modes). The
mathematical expressions of the modal parameters obtained based on the proposed
method are as
follows:
=
R,)
+
+
+
+
(2-50)
(2-5 1)
+
(2-52)
+ (I,
+
(2-53)
Where
A, and are the natural frequency, damping coefficient, real and
imaginary parts of the modal
constant respectively while
and are the
2
Identification of Nonlinearity Using First-order
5 7
frequencies, real parts of the receptances and imaginary parts of the receptances of the
two points at either side of the resonance chosen.
Because of the similarity of these equations, equation (2-50) can be regarded as their
representative for the residual analysis. For convenience, suppose that the receptances of
the two points chosen satisfy:
(this is the case for a real mode
with constant force input) and the complex constant for the residual is
Then the
percentage error for the estimation of
based on equation (2-50) is:
+
+
If
we denote
) =
then
becomes:
AR
(2-54)
(2-55)
From equation
it can be seen that the percentage error for the estimation of
is
proportional to the real residual ratio AR/R and the imaginary part of the residual has no
effect on the estimate. Although this is true only for the case of natural frequency
estimate, the percentage errors for estimation of the other parameters
A, and
are
more or less at the same level of
+
Also, from equation
it can be
seen that the accuracy of the estimation of
can be improved if the frequency difference
between
the two selected points
is small so that becomes large. Therefore,
when the mode to be analysed is influenced by other modes, some measures can be taken
at both measurement and analysis stages in order to obtain satisfactory results. At the
measurement stage, (i) it is possible to linearise the neighbouring modes so that they can
be analysed accurately using linear modal analysis method(s) and (ii) the response levels
can be controlled so that it is possible to obtain enough points just around the resonance
and so the values of R, I and can be increased (in fact, it is possible to quantify
structural nonlinearity by analysing FRF data measured at different response amplitudes
at only two frequency points around resonance).
At the analysis stage, on the other hand,
the SIM method can be used to subtract the residual until it is at its minimum level.
To see how residual effects influence the analysis results, analogue computer FRF data
representing dry friction damping nonlinearity and with a 1% artificially-added residual
(here 1% residual means that the complex constant of the residual is
