Identification of the dynamic characteristics of nonlinear structures

7 Possibilities and Limitations of Analytical 
Model 
Improvement
238
V
l 
measured nodes
0 
unmeasured nodes
Fig.7.21 The Fixed-free GARTEUR Structure
Fig.7.22 Exact Stiffness Error Matrix

---

7 Possibilities and Limitations of Analytical Model Improvement
Y
<
X
Fig.7.23 First Iteration Results
Y
<
X
Fig.7.24 Second Iteration Results

---

7 Possibilities and Limitations of Analytical Model Improvement
240
Fig.7.25 Third Iteration Results
some cases, it may not be appropriate to assume that the mass and stiffness modelling
errors can be simply expressed as that of (7-22) and more independent physical design
variables need to be considered. To illustrate this point, the free-free GARTEUR
structure (figure 7.1) is also investigated. Every element shown in figure 7.1 is a
superposition of an axial bar element and a bending beam element and these pairs of
elements are considered to be independent of each other. Physically, this means that
during the modelling, the cross-section area of the axial bar element and the second
moment of area of the bending beam element are treated independently. As a result, the
stiffness error matrix can in general be expressed as:
= 
 
 
1
(7-24)
where 
is the expanded extensional element stiffness matrix (which is proportional to
the cross-section area) and 
is the expanded bending element stiffness matrix (which

---

7
Possibilities and Limitations of Analytical Model Improvement
241
is proportional to the second moment of inertia). Since, in this exercise, the number of
elements is 
166 unknowns need to be calculated in each iteration when only
stiffness errors are considered. The stiffness errors are introduced by overestimating the
1 
and 77th bending element stiffness matrices and the 
and 
extensional element stiffness matrices by 100%. The exact stiffness
error matrix is shown in Fig.7.26 (the stiffness errors for the bending and extension are
of 100 times different in magnitudes and therefore they are plotted separately). Some of
the ‘experimental’ and analytical receptance 
are shown in Fig.7.27. For
comparison, the method developed in Chapter 6 is employed in this case. One incomplete
column all the hatched nodes with their translational degrees of freedom 
of the
receptance matrix of the ‘experimental’ model with data covering the first 5 modes is
supposed to have been measured. In each iteration, 20 FRF data points which are
randomly chosen in the ‘measured’ frequency range are used and the iteration results are
shown in 

---

q