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

539
in which
H = [H(t l ), H(t 2 ), -.., H(t L )] T (2.3)
0 = [c 13 c 2 , -.., C N , k p k 2 , .... k N f (2.4)
P = [P(t l ), P(t 2 ), »., P(t L )f (2.5)
where H is the response matrix of velocity and displacement of the building. H is a rectangular
matrix of order (L x N) x J, where L is the number of sampling points in the measured structural
response time history and J is the number of unknown parameters. For the present case, J =2*N.
Vector 9 contains all the unknown parameters and P is a vector related to the ground acceleration
and building inertia forces.
At any sample time instant t,, one has
H(t,) =
"*,
0
X 1 -x 2 0
x 2 — x, x, — x 3
0
0
x, x,-x, 0 0
0 x,-x t x 2 -x 3 0
(2.6)
0
0 0
x n -x
0 0 0 x n »x Q
P(t 1 ) = [-m l x 1 (t l )-m l x g (t t ), -m.jXjtt.J-mjX^t,), •••, -m N x N (t l )-m N x g (t l )] r (2J)
The responses of the structure are supposed to be measured in terms of displacement, velocity and
acceleration. If the ground acceleration X g (t) is available, parameter estimation is trivial using, for
instance, the least-squares technique (Hsia, 1977) as
0 = [H T H] 1 H T P (2>8)
However, the problem here is the ground acceleration is unknown and it is expected to simultaneously
estimate with the structural parameters. In this connection, a dynamic compound inverse method is
suggested below.
Dynamic Compound Inverse Method
The dynamic compound inverse method is actually an iterative identification procedure that consists of
the least-squares technique for parameter identification and the statistical average method for forcing
the identified input excitation to comply with the dynamic equilibrium of the concerned building. The
implementation of the dynamic compound inverse method can be divided into the following steps.
Stepl: Since both the structural parameters 0 and the ground motion X g (t) are unknown, initial
values for the structural parameters have to be assumed, for instance, let 9 Q = [l, 1, •••, l] T . It will
be demonstrated later that the performance of the suggested method is not sensitive to this initial
assumption. The symbol ' A ' stands for the estimated value and the subscript 4 0* of 0 stands for the
number of iteration times.
Step2i From Eqn.2.2, the vector P can be estimated using # 0 and the measured structural responses,
resulting in P = H0 0 . Then, comparing with Eqn.2.7, one may have