Geology Reference
In-Depth Information
When the structure is subjected to base ac-
celeration
u t
modelled by the fractional derivative Maxwell
model. In this case, the vector of state variables
and the vectors of state variables' derivatives are
defined as:
g
( )
, the excitation vector is written
= −
, where
r
is the influence
vector with values 0 or 1. For harmonic external
forces, we have
as
p
t
M r
u t
g
( )
( )
=
λ
, where
U
g
is the amplitude of base acceleration. The displace-
ment response of the structure is given by relation-
ship (30), and
Q
s
(
λ
is determined from:
u t
( )
U
exp(
i t
)
1
T
z
( )
t
=
[
q
( ),
t
q
( ),
t
D
q
( )] ,
t
g
g
d
s
t
s
1
1
1
2
T
D t
z
( )
=
[
D
q
( ),
t
D
q
( ),
t
D
q
( )] ,
t
t
t
d
t
s
t
s
α
α
α
α
+
1
T
D t
z
( )
=
[
D
q
( ),
t
D
q
( ),
t
D
q
( )]
t
(38)
t
t
d
t
s
t
s
=
Q
λ
H
λ
U
(35)
( )
( )
s
g
The equation of motion written in state space
takes the form:
where the vector
H
= −
will be called
the vector of frequency transfer functions of
displacements caused by kinematic excitation.
λ
H Mr
λ
( )
( )
1
α
A
D t
z
( )
+
A
D t
z
( )
+
Bz
( )
t
=
p
( )
t
(39)
t
1
t
Equation of Motion for a Structure
with VE Dampers Modelled Using
Fractional Rheological Models
where
d
d
C
C
0
0
0
0
dd
ds
d
d
A
=
0 C M
0 M 0
,
A
1
=
C
C
0
,
If the dampers are modelled using the fractional
derivative Maxwell model, then the equation of
motion of structures with dampers could be written
in the form (see also the paper by Lewandowski
and Pawlak 2010):
ss
ss
sd
ss
0
0
0
ss
d
d
K
−
K
0
0
p
dd
ds
d
d
,
p
B
=
−
K
K
+
K
0
( )
t
=
( )
t
sd
sd
ss
0
0
−
M
0
sd
2
1
M q
D
( )
t
+
C
D
q
( )
t
(40)
ss
t
s
ss
t
s
+
C
d
D
α
q
( )
t
+
(
K
+
K q
d
)
( )
t
(36)
ss
t
s
ss
ss
s
In the case of a structure with dampers modelled
using the fractional-derivative Kelvin model the
equation of motion can be written in the form (see
also the paper by Lewandowski and Pawlak 2010):
C
d
D
α
q
( )
t
K q
d
( )
t
p
( )
t
+
−
=
sd
t
d
sd
d
d
α
d
α
C
D
q
t
+
C
D
q
t
( )
( )
ds
t
s
dd
t
d
d
d
−
K q
t
+
K q
t
=
0
( )
( )
(37)
ds
s
dd
d
2
1
M q
D
( )
t
+
C
D
q
( )
t
+
C
D
α
q
( )
t
ss
t
s
ss
t
s
dd
t
s
2
q
=
and
where the symbols
D
( )
t
q
( )
t
+
(
K
+
K q
)
( )
t
=
p
( )
t
t
s
s
ss
dd
s
s
(41)
=
are used in order to be consistent
with the notation. Here, it is assumed that for all
dampers the values of the parameter
α
are identi-
cal.
The equations of motion in the state space
can also be derived for a structure with dampers
D
1
q
( )
t
q
( )
t
t
s
s
Thus, the state equation is:
1
α
A
D t
z
( )
+
A
D t
z
( )
+
Bz
( )
t
=
s
( )
t
(42)
t
1
t
Search WWH ::
Custom Search