Structural Health Monitoring Using Smart Sensors - ideals ...

R
A· B
=
R· AB
(6.4)
A similar relation holds for higher derivatives,
R
m
A B
=
R m
AB
(6.5)
where the superscript, m, denotes the m-th derivative. Consequently, Eq.(6.3) can be
rewritten as
MR·· xx ref
+ CR· xx ref
+ KR xxref
=
(6.6)
Thus, correlation functions for the stationary responses are shown to satisfy the equation
of motion for free vibration. This fact can be directly used for the subsequent modal
analysis.
6.2 Eigensystem Realization Algorithm
ERA (Juang & Pappa, 1985) identifies modal parameters from free vibration
responses. When measurement at p sensors are available in a measurement vector yn,
the Markov parameters Yn
from m sets of measurement are constructed as follows:
Yn
=
yyym
(6.7)
A generalized Hankel matrix, H rs k–
, is formed as a r s block matrix:
Yk Yk+j 1 Yk+j s-1
H rs
k–
=
Yh 1 + k Yh 1 + k+j Yh 1 + k+j s-1
Yh r-1
+ k Yh r-1
+ k+j Yh r-1
+ k+j
1
1
s-1
(6.8)
A SVD of this Hankel matrix yields
H rs
=
PDQ T
(6.9)
where the superscript T denotes the matrix transpose. Components in these matrices
corresponding to small singular values are considered noise and replaced by zeroes. Juang
and Pappa (1985) derived that the triple, D – 1/2 P T H rs
QD – 1/2 D 1/2 Q T E E T PD 1/2
m p
is a minimum realization of the measured system. A matrix to extract the first m columns,
E m is defined using an identity matrix of order m and a null matrix of size m.
E m
T
=
I m m m
m
(6.10)
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