Optimal Design and Practical Considerations of Tuned Mass Dampers for Structural Control - Design Optimization of Active and Passive Structural Control Systems - page 130

Figure 2. A multi-story building equipped with multiple tuned mass dampers (Lin et al., 2010)

are n × n mass, damping and stiffness matrices of
the primary building, respectively.
M s

respectively. The superscript ( i ) indicates the
position of vector 1 in matrix u .
x = T T T

= diag m k C s = diag c k is the displacement vector.

x p ( ) and v s ( ) denote the vector of displacement
of building relative to the ground and the vector
of MTMD's displacement relative to the i ith floor
(called stroke), respectively. r is ( n+p ) × 1 influ-
ence vector with each element equal to -1.  x t

( ) t [ ( ) t ( )] t

.[ ], .[ ], x v

s s p s

C u T

K s = diag k k ], M

= , C

M u = − ( ) ,

.[

s sp ps s

K K u T

= − ( ) where M s , C s , and K s are p × p
diagonal matrices; m s k , c s k , and k s k are mass,
damping coefficient and stiffness coefficient of
the k th unit of MTMD k

ps s

g ( )

( )

= 1 2 ,  , , p ;

represents the ground acceleration.
Let Φ be the n × n mode shape matrix of the
building and η ( t be the n × 1 modal displacement

u [ 0 0  1

( ) i

 0 ]

p n where 0 and 1 are
p × 1 vectors with each element equal to 0 and 1,

=

×