Structural Health Monitoring Using Smart Sensors - ideals ...

The eigenproblem of the system matrix is then solved:
D – 1/2 P T H rs kQD – 1/2 i = z i i
(6.11)
The natural frequencies f i
and damping ratios i
are then calculated from the eigenvalues,
z i
as
i z i kt
=
f i
= i
i
=
Re( i
) i
(6.12)
where z i
is the principal value of natural logarithm of z i
. t is the sampling period.
Re takes the real part of a complex number. The mode shape, i
, corresponding to z i
is calculated as
i
= PD 1/2 i
. (6.13)
E p
T
The initial modal amplitudes, , which can be utilized to calculate modal amplitude
coherence, an indicator to quantitatively distinguish the system and noise modes, is
estimated as follows:
=
-1 D 1/2 Q T E m
(6.14)
6.3 Damage Locating Vector method
One of the flexibility-based SHM approaches is the Damage Locating Vector (DLV)
method, which is briefly described here. The DLV method, first developed by Bernal
(2002), is based on the determination of a special set of vectors, the so-called damage
locating vectors (DLVs). These DLVs have the property that when they are applied to the
structure as static forces at the sensor locations, no stress is induced in the damaged
elements. This unique characteristic can be employed to localize structural damage (Gao,
2005).
For a linear structure, the flexibility matrix, F , at the sensor locations is constructed
from measured data. Gao (2005) explained two approaches to estimate the flexibility
matrix. Both of them reconstruct the flexibility matrix using mode shapes and
normalization constants. One approach estimates the normalization constants assuming
that the input force can be measured, while the other utilizes output-only measurements
before and after a known mass perturbation.
Formulation of the flexibility matrix based on input force measurements is
summarized by the following equation involving complex conjugate pairs of arbitrary
normalized mode shapes, = .
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