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where r denotes the r th story.
t,), P(t 2 ), .-, P(t L )] r (2.9)
P(t,) = [P,(t,), P,(t,),..., P N (t,)f ( 2 - 10 )
p r (t,) = -m r x r (t,)-m r k< rt (t,) r = l,...,N (2.11)
Step3: From Eqn.2.11, at any sample time instant t,, the ground acceleration x g (t,) can be estimated
from p r (t,) for each story as
^r) (t l ) = x r (t 1 )-p r (t,)/m r r = l,.",N, i = l,-,L (2.12)
According to the structural dynamics, all the N ground acceleration Xg r) (t,) obtained from Eqn.2.12
should be the same at the time t, no matter which story it is estimated from. However, they may not be
the same because of the difference between the assumed initial values and the actual values of the
structural parameters. Therefore, a statistical average process is introduced to compute the average of
x g r) (t,) to force the N ground accelerations to be the same.
*.(t,) = Z*(O i = V",L (2.13)
r«l
Step4: Now, reconstruct the vector P in Eq. (2,7) using the averaged value x g (t,)and the measured
structural responses, leading to
p r (t I ) = -m r x r (t I )-m r I g (t 1 ) r = l,-,N (2.14)
where the symbol '-'denotes the modified value.
PW-faW, P 2 (U -, P N (t,)F (2-15)
p^lSco, f(t a ), »., p(tjf (2.16)
StepS: Eqns. 2.14 to 2.16 are different from Eqns. 2.9 to 2.10 in that after Step 4 the estimated ground
acceleration time histories at each story are forced to be identical in Eqns. 2.14 to 2.16. Therefore, the
improved estimation of the unknown structural parameters can be obtained by
^ = [H T H]" I H T P (2.1)
Step6: Replace # 0 by 0 { , and repeat Step 2 to 6 until the following convergence criteria are satisfied.
max (2-18)
( 2 - 19 )
where the subscript tier stands for the current iterative step, / means the /th element of vector 0; s e
and Eg are the predetermined index for the structural parameter and the input vector, respectively.
They are generally a small number of the magnitude between 10" 4 to 10" 6 . Clearly, the first criterion is
imposed on the estimated structural parameter while the second criterion is imposed on the estimated
ground motion. Once the iterations converge, the vector 0 obtained in Step 5 gives the final