Geology Reference
In-Depth Information
MTMD with identical stiffness and damping
coefficient involves (
p
+ 1) independent param-
eters,
r
are considered. An alternative strategy is to select
only one of them as a representative.
From Equation (7c), it can be seen that the
first MTMD unit (
m
s
1
, called Unit1) has the
heaviest weight among the MTMD units if
r
f
1
is
the smallest frequency ratio and
r
f
p
is the largest
one. Since each MTMD unit has the same stiffness
and damping coefficients, Unit 1 would probably
experience the largest stroke. Consequently, the
stroke of Unit1 is selected to be reduced.
Now, a modified performance index,
R
j
α
with
the consideration of stroke, is defined as
, , ,
and
ξ
s
0
.
Theoretically, with
given
ω
j
,
ξ
j
,
and
ϕ
ij
,
the optimal MTMD param-
eters,
(
r
r
f
f
f
p
1
2
r
f
1
opt
)
,
(
r
f
2
opt
…,
(
)
,
r
f
p
opt
and
(
)
,
)
ξ
s
0
opt
can
be obtained by solving the following system of
equations which are established by differentiating
R
j
with respect to the (
p
+ 1) parameters and
equating to zero, respectively, to minimize
R
j
.
∂
∂
R
r
∂
∂
R
r
∂
∂
R
r
∂
∂
R
j
j
j
j
=
0
,
=
0
,
...,
=
0
,
=
0
ξ
f
f
f
s
1
2
p
0
(9)
α
R
= −
(
1
α
)
R
+
α
R
(11)
j
j
v
s
1
Then, the optimum stiffness coefficient,
(
k
s
0
opt
optimum damping coefficient,
(
)
,
c
s
0
opt
)
,
where
α
is the stroke weighting factor ranging
from 0 to 1.0.
R
v
s
1
means the stroke ratio of Unit1.
When
α
= 0,
R
j
α
reduces to the conventional
performance index with unlimited movement of
MTMD, while
α
= 1.0 indicates that the MTMD
are locked. To associate the parameters of Unit1
with the rest of MTMD units, the frequency dif-
ferences between Unit1 and the other MTMD
units (
r
and optimum mass for
k
th MTMD unit,
(
m
s
k
opt
can be obtained. The optimization process can
also be performed by numerical seaching tech-
niques which can be found in mathematical
software packages, such as MATLAB.
)
A New Optimization Criterion:
Considering Stroke Limitation
−
,
r
r
−
, …,
r
r
r
−
1
) from the
first-stage design are assumed unchanged in the
second-stage. With the new performance index
R
j
α
which includes the first-stage optimization
result, the second-stage optimization can be pro-
ceeded in the same manner as Equation (9) as
f
f
f
f
f
f
2
1
3
1
p
To consider the stroke of each MTMD unit in
design stage, the
k
th stroke ratio,
R
v
s
k
,
which is
used to measure the reduction of stroke, is defined
as
2
E v
[
]
α
α
∂
∂
R
r
∂
∂
R
s
with MTMD
R
=
(10)
k
j
j
=
0
,
=
0
(12)
v
2
E v
[
]
s
k
ξ
s
with MTMD
0
f
s
k
1
0
with MTMD
0
represents the mean square
stroke of the
k
th MTMD unit based on the optimum
parameters obtained by the first-stage optimization
procedure. Because there are
k
units of MTMDs,
it can be a complicated problem if all units' strokes
where
E v
s
k
[
2
]
By solving Equation (12), the optimum damp-
ing ratio,
(
ξ
α
s
1
opt
ξ
α
α
s
p
opt
and optimum
)
,
(
s
2
opt
…,
(
)
,
ξ
)
,
α
can be
obtained with a prior selection of MTMD mass
ratio and the weighting factor,
α
.
α
α
frequency ratio,
(
r
f
1
)
,
(
r
f
2
)
,
…,
(
r
f
p
)
opt
opt
opt
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