Identification of the dynamic characteristics of nonlinear structures

NUMERICAL CASE STUDIES
A 30DOF mass-spring ‘chain’ model shown in Fig.5.5 is used in the numerical case
study. Cubic stiffness nonlinearity is introduced between 
and 
and 40% analytical
stiffness modelling errors are introduced between coordinates 
and 
When
the excitation coordinate is chosen to be 
mode 2 is found to be the most sensitive
mode 
and 
based on (5-14) with 1%
proportional damping) to the thus-introduced nonlinearity and therefore is used
in the location process. The point receptances of coordinate 
with a 25% stiffness
change in the nonlinear stiffness element are shown in Fig.5.6, demonstrating the
sensitivity of mode 2 to the localised stiffness nonlinearity (clear shift of the natural
frequency of mode 2).
40% stiffness modelling errors
localised stiffness nonlinearity
40% stiffness modelling errors
. . . . . .
x1
x2
x 3
x15
x16
x29
x30
Fig.5.5 A Mass-spring Model Used in Numerical Case Studies

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 Location of Structural Nonlinearity
153
Fig.56
of 
with 25% Stiffness Change of 
When the measured coordinates are complete, the location results are as shown in
(3-dimensional plots showing the absolute values of the elements of a
matrix against its two dimensions) using the modal data of the second mode ‘measured’ at
two different response levels (the stiffness change of the nonlinear stiffness element
corresponding to the lower and higher response levels is 25% of its original value).
Fig.5.7 shows the stiffness modelling errors [AK] 
calculated based on (5-4)
and Fig.5.8 shows the stiffness modelling errors [AK] and stiffness change 
due to
nonlinearity 
calculated based on (5-3). By taking the difference
of these two error matrices, shown in figure 5.7 and Figure 5.8 based on 
the
location of the nonlinearity becomes clear, as shown in Fig.5.9.
Fig.57 Stiffness Error [AK] 
Fig.5.8 Stiffness Error 

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5
Location of 
Nonlinearity
154
Fin.5.9 Stiffness 
Due to Nonlinearity 
However, as mentioned, it is unlikely that all the coordinates (which are specified in the
analytical model) will be measured in practice. To simulate coordinate incompleteness of
practical measurements, only the odd numbered coordinates are included as ‘measured
and the unmeasured (even-numbered) coordinates are interpolated based on 10) using
the analytical model itself. The location results for this case are shown in Figs.5