Identification of the dynamic characteristics of nonlinear structures

s = l
s = 1
where 
is the number of elements, and are the design variable changes associated
with the element, 
and 
are the 
expanded element mass and stiffness
matrices. In the case when the measured coordinates are incomplete, replacing the
unmeasured 
receptance terms in 
(on the RI-IS of (6-6)) by their analytical
counterparts and substituting 
into the RHS of 
with
[AM] and [AK] being expressed in 
the RHS of (6-6) becomes a vector with each
element being a linear combination of all the unknown coefficients and
and
(6-6) becomes:

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6 Identification of Mathematical Model of Dynamic 
193
 
(6-15)
Deleting 
the rows on both sides of (6-15) where the 
terms have not been
measured, the following reduced-order linear algebraic equations are obtained:
(6-16)
where [A(o)] and 
are known for specific frequency 
Again, equation (6-16)
is based on FRF data at one frequency point. When data for a number of frequencies are
used, (6-16) becomes overdetermined and the SVD technique can be used to solve for
(P} and then to reconstruct the updated analytical model. Since some approximation has
been made during the formulation of (6-16) due to the incompleteness of measured
coordinates, the updating problem has to be solved iteratively, as discussed in 
A plane truss structure as shown in 
which is a part of the GARTEUR structure
as will be explained in next Chapter, is studied. In the formulation of FE model, 
(two translational and one rotational as shown figure 6.11) are considered. As compared
with the previous mass-spring system, this example is larger in dimension and more
typical because it is formulated based on a real structure. Modelling errors are introduced
by overestimating the mass matrix for the 18
element (nodes 17-18) and the stiffness
matrices for the 1 
and 
element (nodes 10-12) by 100% (notice that in this case, the
= 
= 
0.324). The exact mass and stiffness error matrices are
shown in 
One incomplete column of the receptance matrix of the ‘experimental
model all the odd numbered nodes with their translational degrees of freedom 
is
supposed to be measured over a frequency range covering just the first 5 modes and some
of the measured and analytical frequency response functions are shown in 
Based on 
FRF data at 20 frequency points were randomly chosen for each
iteration in the measured frequency range to construct the coefficient matrices [A(o)] and
and the iteration results are shown in 

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Identification of Mathematical Model of Dynamic Structures
194
2
5
12
11
10
9
a 
5000 mm
m e a s u r e d n o d e s
O u n m e a s u r e d n o d e s
A Free-free Frame Structure
Error Mass Matrix
Error Stiffness Matrix
Fig.6.12 Exact Mass and Stiffness 
Matrices
point receptances 
 
transfer receptances 
Fig.6.13
of Analytical and ‘Experimental’ Models
analytical, ‘experimental

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6 Identification of Mathematical 
of Dynamic Structures
195
Mass Matrix (first iteration)
Error Mass Matrix (third iteration)
Error Mass Matrix 
iteration)
Error Mass Matrix 
iteration)
Error Stiffness Matrix (first iteration)
Error Stiffness Matrix (third iteration)
Error Stiffness Matrix 
iteration)
Error Stiffness Matrix 
iteration)
Fig.6.14 Iteration Results

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Identification of Mathematical Model of Dynamic Structures