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the Euclidean norm of all objective functions and
X axis is specified for each objective function;
therefore, each objective function has its own
graphical representation whilst Y axis of each
graph would be the same. The Pareto fronts for
Kobe earthquake are shown in Figures 8, 9, and
10 for STMD system.
It is obvious from Figure 8 that the point with
the lowest value of J 1 has high value of objective
function J 2 (Figure 9), this issue is true about the
point with the lowest value of J 2 in comparison
with its value for J 1 , so there is a conflict between
J 1 and J 2 . Likewise, there is conflict between J 1
and J 3 , also J 2 and J 3 ; as a result, three objective
functions are in conflict with each other. This
subject shows the Pareto concept. Therefore,
selected point with the lowest value of ||J|| 2 is a
good compromise point, because it has intermedi-
ate value of three objective functions. The maxi-
mum values of the displacement, velocity, and
acceleration of each floor of the building under
Kobe earthquake with lowest ||J|| 2 for the TMD
and STMD systems are shown in Figures 11, 12,
and 13.
The results of the maximum displacement,
velocity, and acceleration of each floor of the
building for the lowest J 1 , J 2 and J 3 are the same
as the results of the other point (min ||J|| 2 ). It can
be concluded form Figures 11-13 that the STMD
device has better performance and is more reliable
in comparison with the TMD device. Moreover,
Figures 14, 15, and 16 compare the uncontrolled
time history responses of top floor of the building
under Kobe earthquake with those controlled by
STMD system for the design point with lowest
value of the ||J|| 2 . It can be seen from the figures
that STMD system significantly reduces the vi-
brational responses of the building.
The values of the design parameters of the
TMD and STMD devices and the corresponding
values of the objective functions for optimum
point with the lowest value of ||J|| 2 for Kobe
Figure 8. 2-norm level diagrams of Pareto front of the STMD for Kobe earthquake (J 1 )
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