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γ)
of the element at its four nodes for
u
x
and
u
y
. If nodes are added to the edges (between the
current nodes) of the element, the lumped mass will not be this equally distributed mass
matrix.
It is evident that this matrix can be obtained by distributing equally the total mass
(
abt
After the element lumped mass matrix is formed, the global lumped mass matrix can be
assembled by the process described in Chapter 5 for the stiffness matrix. See Example 10.1.
The global mass matrix can also be formulated directly. The total mass associated with a
global node of the structure is the summation of all the masses (or rotary inertias) lumped
from the adjacent elements connected to that node. The advantage of lumped mass is that
the resulting mass matrix is diagonal, and, hence, there is no dynamic coupling between
the masses, i.e.,
m
1
0
0
0
0
m
2
0
0
.
.
.
00
M
=
(10.13)
m
i
0
.
.
.
00
0
m
n
d
where
m
i
is the mass (or rotary inertia) at DOF
i
, and
n
d
is the number of DOF. It should
be noted that often some of the diagonal elements of this mass matrix may be zero, and
the remaining diagonal elements are all positive. Hence, the mass matrix is positive semi-
definite. Although the “lumped mass” model is only an approximation to the real mass
distribution of the structure, normally, it leads to quite satisfactory results.
The lumped mass matrix is positive semi-definite when zeros occur on the diagonal,
whereas the consistent mass (element and global) matrices are positive definite. The zeroes
on the diagonal can complicate certain numerical algorithms. It is clear that a lumped mass
matrix would require less storage space than a consistent mass matrix, which on the element
level is full. It is also more economical to form and manipulate.
EXAMPLE 10.3 Lumped Mass Matrices for a Frame
Find the global lumped mass matrix of the frame of Fig. 10.2 and Example 10.1. The con-
sistent mass matrices were derived in Example 10.1.
In general, the global lumped mass matrix can be computed using the transformations of
Example 10.1 by replacing the element consistent mass matrices with the element lumped
mass matrices. Alternatively, the global lumped mass matrix for the frame can be formed by
lumping the mass at the ends of each element of the frame. In this alternative approach the
lumped mass associated with each (translational or rotary) global DOF can be calculated by
summing the mass of the adjoining beam elements at each node. If the global mass matrix
is based on lumping the mass at only nodes
a, b, c,
and
d
of the frame of Fig. 10.2, a rather
crude approximation may be anticipated. For node
a
, from Eq. (10.11)
1
2
ρ
=
1
2
×
m
a
=
24
.
96
×
3
.
464
=
43
.
231 kg
r
y
+
464
2
464
2
12
m
ra
=
ρ
2
1
2
×
3
.
10
−
2
2
=
24
.
96
×
3
.
(
8
.
581
×
)
+
(1)
12
m
2
=
.
·
43
547 kg
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