Information Technology Reference
In-Depth Information
where
N
1
N
1
N
2
N
1
N
2
N
2
Symmetric
N
x
N
x
N
y
N
y
=
=
N
3
N
1
N
3
N
2
N
3
N
3
N
4
N
1
N
4
N
2
N
4
N
3
N
4
N
4
Simple integration of (3) leads to the consistent mass matrix with
1
/
91
/
18
1
/
36
1
/
18
1
/
91
/
18
1
/
36
m
∗
=
1
/
91
/
18
Symmetric
1
/
9
To form a lumped mass matrix, choose the four nodes
(
k
=
1
,
2
,
3
,
4
)
as integration points.
Since the shape functions exhibit the property
1
i
=
k
N
i
(
x
k
,y
k
)
=
(
i
and
k
=
1
,
2
,
3
,
4
)
0
i
=
k
as utilized in Eq. (10.12a), upon integration (3) becomes a diagonal matrix,
m
i
=
diag
(
m
11
m
22
m
33
m
44
|
m
11
m
22
m
33
m
44
)
(4)
where
b
a
4
N
i
(
W
(
4
)
k
N
i
(
W
(
4
)
i
m
ii
=
t
γ
x, y
)
dx dy
=
t
γ
x
k
,y
k
)
=
t
γ
i
=
1
,
2
,
3
,
4
0
0
k
To calculate
m
ii
(
,
the integration rule of Eq. (6.125), Chapter 6, is used to find
the weights
W
(
4
i
. Because the four nodes of the element are spaced equally in the
x
and
y
coordinate directions, two in each direction, Newton-Cotes quadrature can be used. From
Eq. (6.125), the entries of the mass matrix of (4) are
i
=
1
,
2
,
3
,
4
)
b
a
2
2
W
(
2
)
β
N
i
(
W
(
2
)
α
N
i
(
m
ii
=
t
γ
x, y
)
dx dy
=
t
γ
x
α
,y
β
)
(5)
0
0
α
β
and
W
(
2
)
β
The
W
(
2
)
α
can be obtained from Table 6.6. In the
x
direction
W
(
2
α
=
(
−
)
C
(
2
α
=
a
0
;inthe
y
direction
W
(
2
β
=
(
C
(
2
β
=
1
1
2
a
(α
=
1
,
2
)
b
−
0
)
2
b
(β
=
1
,
2
)
. By comparison of the
expressions for
m
ii
in (4) and (5),
1
2
a
1
1
4
ab
W
(
4
)
k
W
(
2
β
=
W
(
2
)
α
=
2
b
=
(
k
=
1
,
2
,
3
,
4
,
α
and
β
=
1
,
2
)
so that
1
4
abt
W
(
4
)
i
m
ii
=
t
γ
=
γ(
i
=
1
,
2
,
3
,
4
)
(6)
and the element lumped mass matrix becomes
1
1
1
1
4
abt
1
m
i
=
γ
(7)
1
1
1
1
Search WWH ::
Custom Search