Identification of the dynamic characteristics of nonlinear structures

part) to the stiffness matrix has not been included. Link 
used a multi-point excitation
technique (more than one excitation forces) to establish the mass, incomplete stiffness and
incomplete damping matrices using measured force and response data. Luk 
employed the incomplete modal data set and used the pseudoinverse to calculate system
matrices which are minimum 2-norm least-squares solution in a mathematical sense. The
possibility of establishing a complete mathematical model when the measured data contain
less modes than the number of coordinates remain to be investigated.
Alongside these studies, a number of methods have been published in the literature to deal
with analytical model improvement by correlating FE models and measured data. The
philosophy behind this practice is that the analytical model, while containing modelling
errors, is assumed to represent the structure with some accuracy so that the limited
measured test data available will offer the possibility of updating it (otherwise the
modelling problem will become the same as that of using experimental data only). Based
on an inverse first-order sensitivity analysis, and considering the random nature of
measurement errors, Collins et al 
employed an iterative procedure to adjust their
analytical model so that the difference between the measured and analytical modal data is
minimised in terms of the Euclidean norm. Later, Chen/Garba 
modified this
procedure by introducing matrix perturbation concepts to avoid the need for an
eigensolution at every iteration (which is required in the formulation in 
Lallement
extended the method in 
to pinpoint first where the significant errors are located
and then to reduce the number of unknowns to improve the solution condition. On the
other hand, based on the assumption that the mass 
is correct, 
[83-
84] introduced a kind of objective function together with an orthogonality property so
that the analytical modes are optimised in such a way that they are closest to the measured
ones in a weighted Euclidean sense. These optimised analytical modes are then used to
derive the updated stiffness and flexibility matrices. Berman later extended this theory to
the case of mass matrix updating 
Having 
the mathematical difficulty of

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6 Identification of Mathematical Model of Dynamic Structures
178
whole system matrix updating, simple eigendynamic equations are used in 
to locate
the major modelling errors first and then to employ the limited measured modes to turn the
updating problem into an overdetermined one. All these above-mentioned activities are
based on the correlation between an analytical model and measured modal data and the
completeness of measured coordinates is, in most cases, critical.
Recently, there have also been publications on the identification of mass, stiffness and
damping matrices in terms of measured coordinates of a system from measured FRF data.
used an Instrumental Variable method to identify system matrices based on the
measured force and response data 
extended the time domain
invariant imbedding filter to the frequency domain to estimate system parameters 
later extended the method in 
to allow correction of reduced-order
finite element model (Guyan-reduced and so fully-populated) by minimising the
difference between the analytical and identified models 
In the following section, a new advantageous model updating method is developed which
tackles the problem by using the measured frequency response function data directly. The
new method is then extended to the case where the structure to be 
is nonlinear.
The advantages of using FRF data over modal data to update an analytical model are
demonstrated. It is shown that model updating methods based on modal data are, in a
broad sense, discrete versions of the present generalised method where only FRF data at
resonance frequencies are employed. Based on this method, the uniqueness of the
updating problem is discussed in some mathematical 
Special attention is given to
the application of the method to the case where both measured modes and coordinates are
incomplete. The practical applicability of the method is assessed based on the 
exercise which is intended to represent practical problems in terms of the incompleteness
of both measured modes and coordinates.
6.4