Identification of the dynamic characteristics of nonlinear structures

6 Identification of Mathematical Model of Dynamic 
191
to these different response levels. 
cases of both complete and incomplete
measured coordinates were investigated but only the results of the incomplete coordinates
case are given here.
The analytical model used is that shown in Tables 
and the ‘experimental’ model
corresponding to the lowest response level is the same as the ‘experimental’ model used
in the numerical case study of incomplete coordinates in 
with stiffness and
damping error matrices previously shown in Fig.6.5. Both cubic stiffness and quadratic
damping nonlinearities are introduced between 
and are studied separately. Four
coordinates are supposed to be measured 
and and the point receptances of
coordinate corresponding to different response levels for the case of stiffness and
damping nonlinearities are illustrated in Fig.6.9 in the frequency range of mode 3. First,
the analytical model is updated using the measured frequency response functions
corresponding to the lowest response level to obtain an accurate base-level (‘zero’
amplitude) linear model of the nonlinear structure. Then, based on this linear model and
the measured FRF data at higher response amplitudes, the nonlinearity can be located in
the ways discussed in Chapter 5. After the location has been made, only those unknowns
corresponding to the nonlinear region are retained and, therefore, only the FRF data
around one mode are necessary (mode 3 in this case study) to identify the mathematical
model of the nonlinear structure (in fact, the only data available due to the inconsistency).
The calculated stiffness and damping changes versus nondimensionalised response levels
of coordinate are plotted in Fig.6.10. Clearly, the cubic stiffness and quadratic
damping features of the nonlinearities are demonstrated.
l 
a.m
stiffness nonlinearity
damping nonlinearity
Fig.6.9
of a Nonlinear Structure with Stiffness and Damping Nonlinearity

---

6 Identification of Mathematical Model of Dynamic Structures
192
0
0.1
0.3
0.4
0.6
0.1
Od
OS
0.1 02 
0.4 
0.6 0.7 
0.9 
1.0
Calculated Stiffness and Damping Changes Versus Response Levels
6.4.4
APPLICATION OF THE METHOD TO THE
GARTEUR STRUCTURE
In the previous section, numerical case studies based on an 8DOF mass-spring system
have been carried out to verify the new method and here an FE model of a more complex
structure is considered. In general, as for the global mass and stiffness matrices, the mass
and stiffness error matrices [AM] and [AK] can be expressed as linear combinations of
element mass and element stiffness matrices 
and 
which have been appropriately
(according to their positions in the global mass and stiffness matrices) expanded to the
global dimension of the system, respectively, as:
= 
and [AK] =
(6-14)