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p
( )
t
=
P
exp(
i
λ
t
)
(29)
K
+
K
0
ss
dd
B
=
,
0
−
M
ss
then the steady state response of the structure and
the vector of internal variables can be expressed as
p
( )
t
(25)
s
( )
t
=
0
q
( )
t
=
Q
exp(
i
λ
t
),
s
s
i
λ
(30)
q
( )
t
=
Q
exp(
t
)
d
d
The solution to the homogenous state equation,
i.e., when
s
(
t
=
in (22), is assumed to be in the
0
After substituting relationships (29) and (30)
into Equations (20) and (21), the following rela-
tionships are obtained:
form:
x
( )
t
=
a
exp( )
st
(26)
D
( )
Q
( )
D
( )
Q
( )
P
,
λ
λ
+
λ
λ
=
This leads to the following linear eigenvalue
problem:
ss
s
sd
d
D
λ
Q
λ
+
D
λ
Q
λ
=
0
(31)
( )
( )
( )
( )
ds
s
dd
d
(
s
A B a
+
)
=
0
(27)
where
from which the (
2
n
+
) eigenvalues
s
i
and ei-
genvectors
a
i
can be determined. In the case of
an undercritically damped structure, the
2
n
ei-
genvalues (eigenvectors) are complex and con-
jugate numbers (vectors) while the remaining
r
eigenvalues (eigenvectors) are real numbers (vec-
tors).
The frame with VE dampers is characterized
by the natural frequencies
ω
i
and the non-dimen-
sional damping parameters
γ
i
. The above-men-
tioned quantities are defined as:
2
D
( )
λ
= −
λ
M
+
i
λ
C
+
K
,
ss
ss
ss
ss
D
( )
λ
=
i
λ
C
+
K
,
sd
sd
sd
D
( )
λ
=
i
λ
C
+
K
,
ds
ds
ds
D
(
λ
i
C
K
=
λ
+
(32)
dd
dd
dd
Finally, it is possible to write the relationships
Q
( )
λ
=
H
( ) ,
λ
P
s
ss
Q
λ
=
H
λ
P
(33)
( )
( )
d
ds
2
2
2
γ
= − /
µ
ω
ω
=
µ
+
,
η
(28)
i
i
i
i
i
i
where the frequency response functions
H
ss
(
λ
and
H
rd
(
λ
could be written in the following form:
= Im( )
. Equation (28)
refer to complex eigenvalues only.
The considered system can also be character-
ized by the frequency response functions. To
determine these functions the steady state har-
monic responses of the system are considered. If
the excitation forces vary harmonically in time,
i.e., when
where
µ
i
=
Re( )
,
η
i
s
s
i
i
−
1
−
1
H H
≡
( )
λ
=
D
( )
λ
−
D
( )
λ
D D
( )
λ
,
ss
ss
sd
dd
ds
−
1
H
( )
λ
= −
D
( )
λ
D
( )
λ
H
( )
λ
ds
dd
ds
ss
−
1
= −
D
−
1
( )
λ
D
( )
λ
D
( )
λ
−
D
( )
λ
D
−
1
D
(
λ
dd
ds
ss
sd
dd
ds
(34)
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