Civil Engineering Reference
In-Depth Information
where
S
is the diagonalized singular values matrix,
U
is the left-singular vector matrix, and
V
is the right-singular vector
matrix. Therefore, the extended observability matrix
Γ
i
and the state sequence
X
i
are equal to:
U
1
S
1
1
=
2
Γ
i
¼
;
(8.18)
X
i
¼
S
1
1
=
2
V
1
T
(8.19)
Another similar oblique projection
P
i
1
is defined as:
Y
f
Y
p
þ
¼ Γ
i
1
X
iþ
1
P
i
1
¼
(8.20)
Γ
i
without the last low. Therefore
X
i
þ
1
can be calculated as:
where
Γ
i
1
is defined as the matrix
X
i
þ
1
¼ Γ
i
1
y
P
i
1
(8.21)
The system equation is finally stated as:
¼
X
i
X
iþ
1
Y
iji
A
C
ρ
w
ρ
v
þ
(8.22)
X
iþ
1
and
X
i
are extracted from the SVD of
P
i
1
(
8.20
) and
P
i
(
8.17
)
where
ρ
w
and
ρ
v
are the noise residues. Since
respectively, a least square method is used to find A and C:
¼
X
i
y
X
iþ
1
Y
iji
A
C
(8.23)
The damping ratio and natural frequency of the
ith
mode can be calculated by converting the discrete-time eigenvalues of
A to the continuous-time domain:
¼ ΨΛΨ
1
A
(8.24)
where
Λ
is the diagonal matrix containing the eigenvalues of A and
Ψ
contains eigenvectors in each column. The mode shape
vectors can be calculated as:
Φ ¼½Φ
1
Φ
2
s
Φ
n
¼
C
Ψ
(8.25)
As the measurements have noise and signal aliasing, not all the eigenvalues obtained are the corresponding eigenvalues of
the modal frequencies of the structure. In addition, for low energy modes (e.g., torsional modes), the corresponding
eigenvalues of these modes may not be able to be obtained from SSI. To autonomously extract the correct modal frequencies
directly from the data collected from the sensor network, examination of the corresponding damping ratios of each mode are
used to eliminate pure mathematical results. As for regular civil engineering structures, the damping ratio is typically lower
than 5 %. If damping results from SSI is more than 10 %, it is highly likely that the corresponding mode has not resulted from
the structural response and therefore such results should be discarded as a mathematical artifact.
8.4 Modal Analysis Results of New Carquinez Bridge
Modal properties of an invariant system are fixed in theory. However, for operational bridges, the operational environment
changes thereby introducing some variance into the system; therefore the observed modal properties are not fixed. The New
Carquinez Suspension Bridge, as one of the largest bridges in the Bay Area, experiences continual heavy traffic loads and
large temperature variations. With the implementation of the long-term structural monitoring system, automated modal
