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Fig.5.19 Calibration Results of Measurement
- 6 8
Hz.
299.95
Typical Measured FRF Data
Location of
161
from
51.08
53.50
Fig.521 The First 2 Close Modes of the Frame
From figure 5.20, it can be seen that mode 2
97 Hz) is the strongest mode in the
measurement frequency range and is the most sensitive one to the stiffness nonlinearity
introduced between
and
because of its particular modeshape (see table 5.1).
Accordingly, mode 2 was chosen to study the nonlinear effects of the simulated nonlinear
structure. Frequency response functions in the vicinity of mode 2 corresponding to
different response levels (linearised) are measured (in fact,
at different constant
force levels are also measured and analysed as shown in
at all 20 translational
coordinates around the structure and some of these are shown in Fig.5.22. It can be seen
that a natural frequency shift of approximately
is caused by the stiffness change of
the system when it vibrates at different response amplitudes. The measured modal data of
mode 2 corresponding to two different response levels (the response signals were set to
be
for low vibration level and
for high vibration level) are analysed and
tabulated in Table 5.2. From Table 5.2, it can be seen that in addition to the
shift in
the the natural frequency, approximately 10% changes in the modeshapes are observed
due to stiffness change of the system.
Location
of Structural Nonlinearity
162
Hz.
Fig.522 Measured
at Different Response Amplitudes
5.4.3
LOCATION OF NONLINEARITY
A Finite Element analysis of the structure was performed using the PAFEC package and
the modal parameters of the first 6 modes of the FE model are shown in Table 5.3. A
correlation between the measured modal data and those from the FE model of the
is
carried out and the MAC value matrix is shown in Table.5.4. From Table 5.4, it can be
seen’that good correlation has been obtained for modes 2, 3 and 4. However, it is
surprising that good correlation has also been obtained for the first mode (in fact, 2 close
modes) even when it is treated as a single mode in the experimental
analysis.
Table 5.4 MAC Value Matrix
The mass and stiffness matrices of the frame were generated using PAFEC. However,
before these matrices are used to correlate with the measured modal data to locate the
nonlinearity, the shaker characteristics have to be compensated in the FE model. Since
what is of interest is mode 2, the compensation of the shaker properties for this mode can
be illustrated in Fig.5.23. An effective mass
and stiffness k,, can be calculated based
5 Location of Structural Nonlinearity
163
on equivalent kinetic and potential energy as
and
(m, and are
the table mass and suspension stiffness of the shaker and they are
and
Fig.523 Illustration of Shaker Property Compensation
On the other hand, since in the FE analysis, 3
(one translational and two rotational
bending and torsion) are considered at each point while in the measurement, only the
translational degree of freedom is measured, the measured modeshapes have to be
interpolated first using Kidder’s method as mentioned earlier before they can be used to
correlate with the FE model to locate the structural nonlinearity. The calculated location
results are shown in Fig.5.23. Theoretically, the errors should be contained between two
translational coordinates (x
right in the middle of the plot. However, the errors are
distributed so that almost a third of the coordinates have been contaminated. One reason
for this is that since the exact coordinate where the
stiffness nonlinearity is
introduced (between
and
has neither been measured nor included in the FE
model, this missing coordinate is expected to cause spatial leakage. Another reason is that
the measured modal data contain measurement errors and these errors may cause this
spread of location results. To check this later possibility, 2% random errors were added
to the measured modal data and the location results (Fig.5.24) show that although the
location results have not been much affected, measurement errors do have the effect of
spreading the location results.
Location
of Structural Nonlinearity
Fig.5.24 Located Stiffness Nonlinearity
Fig.5.25 Located Stiffness Nonlinearity (2% Noise on Measured Data)
5.5 EXTENSION OF THE TECHNIQUE TO MEASURED FRF DATA
The above developed location technique can be generalised when measured
data are
used. Suppose that the i column
of the receptance matrix (corresponding to
lower response level
and
(corresponding to higher response level
around the
mode (which is sensitive to nonlinearity) have been measured and, again,
the analytical model which contains second order modelling errors (corresponding to
Location of Structural Nonlinearity
165
lower response level
is available. Since the impedance and receptance matrices of a
system satisfy
=
(5-15)
by taking the column of both sides of
following equations can be established
(5-16)
(5-17)
where
is a vector with its element equal to unity and all the others zero and and
are the measured FRF data points chosen. Post-multiply (5-17) by
(
then
(5-18)
Post-multiply (5-16) by
and we have
(5-19)
Subtract (5-18) by (5-19) and rearrange, then
Unlike the case of location using modal data, here we have the chance to choose
and
properly in the measured frequency range as shown in Fig.5.26 so that following
function is minimised
Location of Structural Nonlinearity
166
(5-21)
If the modelling errors [AK],
and stiffness change due to nonlinearity
are
of the same order of magnitude in the sense of the Euclidean norm, then to a first-order
approximation,
becomes
We state here that when
natural frequency of the structure corresponding to
the lower response level) and
(the
natural frequency corresponding to the
higher response level), (5-22) will degenerate to (5-6) but we shall leave the mathematical
proof of this relationship between (5-18) and (5-6) to Chapter 6. The principle of the
nonlinearity location process based on equation (5-22) is the same as that of (5-6).
Frequency
Fig.5.26 Illustration of the Choice of Frequency Points
When the measured coordinates are incomplete, the receptance terms of the unmeasured
coordinates can be interpolated based on following equation which is derived from (5-15)
(the column of (5-15)):
(5-23)
5
Location
of Structural Nonlinearity
167
where
and (a,(o)} are the measured and unmeasured receptances of the
column of the receptance matrix. Upon multiplying out
can be calculated
as
(5-24)
5.6
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