Identification of the dynamic characteristics of nonlinear structures

 [K,,]) 
 
+ 
(5-9)
Theoretically, 
can be calculated from either (5-8) or (5-9). However, when the
number of measured coordinates is less than that of the unmeasured ones, which is quite
usual in practice, (5-8) becomes underdetermined in terms of the solution of 
(the
coefficient matrix is rank deficient), and it is therefore recommended that (5-9) should be
used to interpolate 
as follows:
= (- 
+ 
10)
It has been found that the interpolation of unmeasured coordinates based on (5-10) is
quite accurate for the lower modes of vibration (this will be further discussed in Chapter
7) and from the nonlinearity location point of view, if some coordinates have been
measured where the structural nonlinearity is located, the thus-interpolated modeshapes
can be used to achieve a successful nonlinearity location. Also, it can be shown
mathematically that the located errors in the above mentioned nonlinearity location process
will only occur in the measured coordinates if the unmeasured coordinates are interpolated
based on 
This is briefly illustrated below.
+ 
and similarly 
+ [K,]
on the 
of 
can be re-written as:

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5
Location 
of Structural Nonlinearity
151
+ 
(- 
+ 
+ 
+ 
+ (- 
+ 
(5-11)
where parameters with are the modal parameters corresponding to lower response level.
When 
is interpolated based on 
then it is easy to see that in (5-1 
the
elements corresponding to the unmeasured coordinates (the lower part) are zero and 
11) becomes
(- 
+ 
+ (- 
+ 
(- 
+ 
+ (- 
+ 
Upon substitution of 
becomes
[AK,] 
= {‘PO;‘} 
[ 
(5-12)
(5-13)
5 . 3 . 3
SENSITIVITY OF MODAL PROPERTIES TO
LOCALISED NONLINEARITY
In order to make the location more reliable, it is recommended that a mode which is
sensitive to the localised nonlinearity should be used in the location process. In order to
determine which mode is the most sensitive one in the measurement frequency range
(corresponding to specific excitation point), first-order constant-force 
can be
measured and, as discussed in Chapter 2, the degree of distortion of these measured 
data around each mode can be used to give an indication of which is the most sensitive to
the nonlinearity. Accordingly, the mode sensitivity to localised nonlinearity can be
established theoretically. Suppose that a stiffness nonlinearity is introduced between
coordinates 
and and a unit (harmonic) force is applied at 
then to a first-order
approximation, the maximum relative displacement between and for the 
mode, 
can be expressed as

---

 Location of Structural Nonlineaitv
152
(5-14)
Since nonlinearities of practical structures are usually displacement dependent, can be
used to quantify the sensitivity of different modes to localised structural nonlinearity.
From 
it can be seen that if the structure is nonlinear, some of the lower modes
will appear to be nonlinear due to their particular modeshapes while the higher modes are
likely to appear linear.
5.3.4