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For node b ,
1
2 ρ
1
2 ρ
m b =
element1 +
element2 =
43
.
231
+
37
.
44
=
80
.
67 kg
r y +
r y +
2
2
= ρ 2
element1 + ρ 2
m rb
(2)
12
12
element2
m 2
=
.
+
.
=
.
·
43
547
28
356
71
90 kg
m 2
Similarly, for node c, m c =
114
.
66 kg, m rc =
86
.
88 kg
·
and for node d, m d =
77
.
22 kg,
m 2 .
The global displacement vector is
m rd =
58
.
529 kg
·
d ] T
V
=
[ U Xa
U Za
U Xb U Zb
b U Xc U Zc
c U Xd U Zd
(3)
a
and the global mass matrix is
M
=
diagonal
(
m a
m a
m ra m b m b m rb m c m c m rc m d
m d
m rd )
(4)
After the boundary conditions are imposed, the displacement vector is
[ U Xb U Zb b U Xc U Zc c ] T
V
=
(5)
and the corresponding global mass matrix is
M
=
diagonal
(
m b m b m rb m c m c m rc )
(6)
=
diagonal
(
80
.
67 80
.
67 71
.
90 114
.
66 114
.
66 86
.
88
)
10.1.3
Alternatives for the Formation of the Mass Matrix
Many schemes have been proposed for the formation of mass matrices. For example, a
lumped mass matrix can be formed by using a diagonal mass matrix approach. This kind
of matrix is usually constructed from the consistent mass matrix. One method for accom-
plishing this is [Cook, 1981]:
1. Compute only the diagonal coefficients of the consistent mass matrix.
2. Compute m , the total mass of the element.
3. Compute s as the sum of the diagonal coefficients m ii associated with translation (but
not rotation).
4. Scale the diagonal coefficients m ii by multiplying them by the ratio m
/
s
.
This kind of element is also particularly applicable to structures whose translational DOF
are mutually parallel, such as occurs for beam and plate elements.
Another alternative form of the mass matrix is a linear combination of the consistent
and the lumped mass matrices
m
= α
m consistent + β
m lumped
(10.14)
This m is referred to as a non-consistent or high-order mass matrix. Sometimes [Hughes,
1987], a simple average is employed so that
α = β =
1
2
.
For a two-node bar element, this
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