Structural Health Monitoring Using Smart Sensors - ideals ...

F
=
– D g T
(6.15)
D g = diag
d 1 d 2 d j
is the matrix of modal normalization constant, d . If all of
j
the modes are observed and available for reconstruction, F can be estimated. When only a
part of structural modes is available, Eq. (6.19) gives an approximation of F. Gao (2005)
showed that F can be approximated well from a limited number of modes.
The normalization constants can be estimated through the following relation:
–p +
T
d j
= j
j Bqf diag m,j
T qm
–
(6.16)
where mode shapes at sensor locations m
and eigenvectors satisfies the following
relationship:
m
=
C
(6.17)
A
=
(6.18)
is the j-th row of the inverse of eigenvector matrix , and is the j-th row of the
T
transposed mode shape matrix m
. is a diagonal matrix of eigenvalues. When there is
more than one colocated sensor and actuator pair, multiple estimates of d j
will be
obtained. Bernal and Gunes (2004) suggested that d j
corresponding to the component in
vector m,j
with the largest magnitude might be used. Using the normalization
constants, the flexibility matrix is estimated as in Eq. (6.15); only a part of the flexibility
matrix corresponding to sensor locations is practically estimated by replacing the mode
shape matrix with the matrix of mode shapes at sensor location, .
j
T
T qm
– T
m,j
When the measurement of input force is impractical, output-only measurements
before and after a known mass perturbation are utilized to estimate normalization
constants for flexibility matrix reconstruction. The estimated mode shapes and
eigenvalues, as well as normalization constants, reconstruct F as follows:
m
F
=
=
– T
–
T
(6.19)
where is the matrix of mass normalized mode shapes, and is the matrix of arbitrarily
normalized mode shapes. is a diagonal matrix of mass normalization constants, i
.
Note that estimation of normalization constants from a known mass perturbation is
theoretically derived for a proportionally damped system; mode shapes should be real. An
approximate flexibility matrix is estimated from a limited number of modes.
In the mass perturbation approach (Bernal, 2004), the mode shapes and eigenvalues
are determined before and after a known mass perturbation is introduced to the structure.
The mass matrix of the modified structure, , is expressed as
M 1
M 1 = M 0 + M
(6.20)
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