Earthquake Engineering Research - HKU Libraries - The University ...

533
cation corresponds to a node of one of the modes, that mode will not be observed. This
method allows the cross spectral density (CSD) functions to be averaged over a number of
samples to increase the accuracy in the CSDs. Windowing should be used to minimize the
effects of leakage.
Identification of Modal Parameters
Once the time domain free response data is obtained, there are numerous techniques available
for identifying the modal parameters Here the ERA (Juang and Pappa, 1985) is
adopted because it is quite effective for identification of lightly damped structures and is
applicable to multi-input/multi-output systems. In the ERA, the Hankel matrix is formed
y(k)
(5)
where y(k) is the response vector at the £th time step. The parameters s and r correspond
to the number of columns and rows (of response vectors) in the matrix. This matrix is
evaluated for H(0) and a singular value decomposition is performed,
H(0) = PDQ 7 (6)
Relatively small singular values along the diagonal of D correspond to computational
modes and the associated rows and columns are eliminated to form the condensed D^v,
P jV , and Q N matrices. The state matrix for the resulting discrete time system and the associated
matrix in the output equation are found using is found using
_! _1 _!
A = D^ 2P/H(l)QtfDtf 2 C - K T P N D N
2
where E r = [I 0. .0]. Because this is a discrete time system, it is then transformed to
the corresponding continuous time system. The natural frequencies are found by determining
the eigenvalues of the continuous time matrix A, and C is used to transform the computed
eigenvectors of the state matrix corresponding to the non-physical states in the
identified model, to the values of the mode shapes at the floors of the structure. The ERA
method was implemented in MATLAB (1997).
NUMERICAL RESULTS
The approach described in the previous section was implemented to identify the modal parameters
of the structure from simulated acceleration records. Uncorrelated stationary
broadband forces were used as the excitation in the lateral, longitudinal and transverse directions.
The excitations were applied along the deck of the bridge. Thirty minute acceleration
record with a sampling rate of 12.5 Hz was used for the identification procedure.
Three different sensor configurations were considered to study the capabilities of the
methods when the number of sensors is varied. Figure 3 shows the identification models
used and the location of the sensors on the bridge for each case. Three sensors in each tower
measure accelerations in the transverse and longitudinal directions. All deck sensors
measure acceleration in the transverse and vertical directions. Case 1 uses a total of 128
(7)