Structural Health Monitoring Using Smart Sensors - ideals ...

where R xxref
is a vector consisting of R xi x ref
. When the sensor node at location i has a
time synchronization error of t xi
relative to the reference node, the correlation function,
can be written as
R xi x ref
R xi x ref
= Ex i t – t xi
+ x ref t = R xi x ref
t – t xi
(5.11)
where E is the expectation operator. If t xi
is positive, the correlation function for the
interval (0, t xi
) does not have the same characteristics as the succeeding signal. This
portion of the cross correlation function will correspond to negative damping; therefore,
the beginning portion of the signal needs to be removed. When t xi
is unknown, a
segment corresponding to the maximum possible time synchronization error, t xmax
, is
truncated from the correlation function.
The correlation functions, each having independent time synchronization errors, do
not satisfy the equation of motion, Eq. (5.10). The correlation functions after the
truncation, R xi x ref
' , however, can be decomposed into modal components as follows:
R xi x ref
'
=
'A
'
=
2n
=
m
– t x1
–
t x1
2n –
2n t x1
–
t x
–
t x
2n2 –
2n t x
–
t xm
m
–
t xm
2nm
–
2n
t xm
(5.12)
= diag(
2n
)
i
= h i
i
+ j i
– h i
A = diag( a
a
a n
)
where ij
is j-th element of i-th mode shape, h i
and i
are i-th modal damping ratio and
modal natural frequency, respectively, and a i
is a factor accounting for the relative
contribution of the i-th mode in the correlation function matrix. Modal analysis techniques
such as ERA can be used to identify these modal parameters. As Eq. (5.12) shows, the
natural frequencies and damping ratios remain the same. The observed mode shapes '
are, however, different from the original mode shapes; changes in mode shape amplitude
are negligible due to small h i
and t xi
; however, phase shifts in the mode shapes can be
substantial. Mode shape phases can indicate structural damage and are important modal
characteristics from an SHM perspective. The requirements on time synchronization
accuracy for modal analysis need to be assessed mainly from the viewpoint of the mode
shape phase.
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