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all
the
modes of the
should be measured so that the mass and stiffness matrices of
the structure at different response amplitudes, which are necessary to the location of
nonlinearity, can be reconstructed. This demonstrates the difficulties of using measured
data only to do the location task.
A typical Nonlinear Structure
at resp.
[Z(o)] at resp.
stiffness change
Fig.5.2 Schematical Illustration of Location Process
Location
of Structural Nonlinearitv
147
at
resp.
at
error receptance
Fig.5.3 Schematical Illustration of Inverse Process
Fortunately, due to the development of analytical modelling techniques, an analytical
model of a structure can be employed. Although it may contain modelling errors, these
are usually of second order when compared with the analytical model itself in the sense of
the Euclidean norm. With such an analytical model available, it will be shown in this
Chapter that by correlating the analytical model and the measured dynamic test data,
location of nonlinearity can be achieved.
5.3
TECHNIQUE FOR THE LOCATION OF STRUCTURAL
NONLINEARITY
5.3.1 LOCATION USING MEASURED MODAL DATA
The location method developed in this Chapter is based on the correlation between an
analytical model which contains modelling errors and dynamic test data which are
measured at different response levels. A nonlinearity location method based on the use of
measured modal data is discussed first. This method is then extended to the case of using
measured
data.
Before discussion, it is necessary to mention that the ‘modes’ of a nonlinear structure are
difficult to define (if indeed they exist at all) in an exact mathematical sense
because of the existence of harmonic response components, and so the term ‘modes of a
nonlinear structure’ is used in this Chapter to mean the natural frequencies and
modeshapes which are derived from the analysis of measured first-order
in which
only the fundamental frequency component of the response is of interest. For most
nonlinear mechanical structures, the thus-obtained natural frequencies and modeshapes
5 Location of Structural Nonlinearity
148
are response levtl dependent. As far as stiffness nonlinearity is concerned, the stiffness
matrix of the structure corresponding to different response levels will be different and,
therefore, if this difference in stiffness matrix can be calculated in some way, the problem
of nonlinearity location can be resolved.
Suppose that the eigenvalues and eigenvectors of the
mode (which is sensitive to the
nonlinearity) corresponding to a lower response level,
are
and those
corresponding to a higher response level,
are
and that these have been
obtained from the analysis of measured first-order
(either based on the new
nonlinear modal analysis method discussed in Chapter 2 or based on standard linear
modal analysis methods by linearising the structure using response control). Suppose
also that the analytical model which contains second-order modelling errors
(corresponding to lower response level) is available. Then, from the eigendynamic
equations, the following relationship can be established:
[M,] +
) +
+
=
( (
+ [AM]
+
+
+
(5-2)
Post-multiply (S-2) by
then
[AM] + [AK] + [AK,])
= (-
+
Post-multiply (5-l) by
we have
[AM] + [AK])
= (-
+
(5-4)
Subtract (5-4) from (5-3) and rearrange, then
+
[M,] +
)
(5-5)
Since
is a perturbed modeshape of
due to the stiffness change of
is of second order compared with
in the sense of the
Euclidean norm (one can notice that all the diagonal
of’
Location of Structural
149
are zero), as a result, if the modelling errors [AK],
and stiffness change due to
nonlinearity [AK,,] are of the same order of magnitude (also in the sense of the Euclidean
norm), then to the first order approximation, (5-5) becomes
+
+
As a special case in which [AK]=[O] and [AM]=[O] (no modelling errors), then (5-6)
becomes an exact statement for
The principle of the nonlinearity location process
based on equation (5-6) is illustrated in Fig.5.4. If the nonlinearity is localised, then
will be a very sparse matrix (only those elements where the structural nonlinearity
is located are
and, as shown in figure 5.4, the dominant
elements of the
resultant matrix after the matrix multiplication will indicate the location of localised
nonlinearity. Also, it should be noticed that during the location process, only one
measured mode is required and it is recommended that the mode which is the most
sensitive to nonlinearity in the measured frequency range should be used. Extra modes
can be used to check the consistency and reliability of the location results.
full matrix
resultant matrix
Fig.5.4 Illustration of Nonlinearity Location Process
5.3.2 EXPANSION OF UNMEASURED COORDINATES
In the theoretical development of the location method, it is assumed that the measured
coordinates are complete. In practice, however, this is very difficult to achieve because
certain coordinates are physically inaccessible, such as internal
and the rotational
coordinates are very difficult to measure and so the unmeasured coordinates have to be
interpolated first before the location process can be carried out. This interpolation of
5
Location of Structural Nonlinearity
150
unmeasured coordinates can be achieved
using the analytical model itself based on
Kidder’s expansion method
Although the analytical model contains modelling errors, in order to interpolate the
unmeasured coordinates, it is assumed that the following relationship between the
analytical model and the
measured and unmeasured sub-modes holds:
are the
measured and unmeasured sub-modes. Upon multiplying
out
the following two equations are established
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