Issue 54
F. Brandão et alii, Frattura ed Integrità Strutturale, 54 (2020) 66-87; DOI: 10.3221/IGF-ESIS.54.05
m
0
0 0 0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
1
m
0
2
m
0 0 0 0 0 0 0 0 0 0 0 0
0
0 0
0 0 0
0 0 0
0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
n
m
0
d
1,1
m
0 0
d
2,1
m
0 0 0 0 0 0
0 0 0
0
0 0
dm
,1
m
0 0
M
d
1,2
m
0
d
2,2
,2 dm m
0 0
0 0
0
0
0
0
0
0
0
m
0 0 0 0
0 0 0 0
0 0
0 0
0 0
0 0
0 0
0
0 0
1, d n
m
0
2, d n
(4)
m
0 0
0 0
0
0
0
0
0
0
0
, dm n
where m n , c n, and k n represent the mass, damping coefficient, and stiffness of the n th story of the structure, respectively. Also, m di , c di , k di (i = 1, 2, 3, …, m) where m is the last TMD in each floor considering horizontal distribution, are the mass, damping coefficient and stiffness of i th TMD, respectively. To solve Eq. 1, a numerical integration method can be applied.
st Floor
1
m N
TMD
1 2 k k
k
k
1,1 k k d d
k
0
0 0
0
0 0
0
di
dm
,1
2
2,1
,1
i
1
nd Floor
2
m N
TMD
k
2 3 k k
k
1,2 k k d d
,2 dm k
0
0 0
0
0 0
0
di
2
,2
2,2
i
1
thn Floor
m N
TMD
k
k
1, d n k k
k
0
0
0 0
0 0 0
0
n
, di n
2, d n
, dmn
i
1
k k
k
0 0
0 0
0
0 0 0 0 0 0
0 0
0 0 0 0
0 0
d
d
1,1
1,1
k
0
d
d
2,1
2,1
K
k
k
0
0 0 0
0 0 0 0 0 0
0 0
0 0 0
0 0 0 0 0 0
0 0 0
dm
0 dm
,1
,1
k k
k
0 0
0
d
d
1,2
1,2
k
0 0
d
d
2,2
2,2
k
,2 dm k
0
0
0 0
0 0 0
0 0
0
dm
,2
(5)
k k
k
0 0
0 0
0 0 0 0
0 0 0 0 0 0
0 0
0
0 0
1, d n
1, d n
k
0
2, d n
2, d n
k
k
0
0
0 0
0 0 0
0
0 0
, dmn
, dmn
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