Identification of the dynamic characteristics of nonlinear structures

all the 
modes of the

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Identification of Mathematical Model of Dynamic Structures
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and, in general, the data points measured are plentiful. Thus, the updating
problem, as discussed below, is in most cases uniquely defined while in the case of
model updating using modal data, the updating problem, in most cases, is underdefined
since on the one hand, the data points are so badly chosen 
data at resonance
frequencies) that the contribution of other modes has been deliberately eliminated and, on
the other, the number of measured modes is usually quite limited.
6.6 UNIQUENESS OF UPDATING PROBLEM
The uniqueness of the model updating problem that is, how much data should be
measured so that the model updating problem can be solved uniquely (the true solution of
problem can be obtained) needs to be discussed when a new method is developed. Since
the present method seeks to obtain an updated model which will reproduce the measured
frequency response function data continuously on the frequency axis, as shown in
numerical case studies, the measured data required (the number of measured modes) in
order to solve the updating problem uniquely are much less as compared with other
existing methods. In fact, it can be proven mathematically that when the measured
coordinates are complete, a unique solution always exists regardless of the number of
modes which are included in the measured frequency range. On the other hand, when the
measured coordinates are 
incomplete, 
a unique solution can, in most cases, be obtained
even when just few modes are included in the measured frequency range if the physical
connectivity of the analytical model is employed as additional constraints.
Suppose one complete column 
of the receptance matrix is measured and its
updated analytical counterpart 
is available (calculated based on the updated
analytical model). Since the updated model should reproduce the measured data, it
follows that 
because, otherwise, it will still be possible to improve
the model. Thus: