Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 591

c ] T

have been imposed, is V The global matrix, after the

columns corresponding to constrained DOF have been set equal to zero and the rows
corresponding to the unknown reactions are ignored, is given by

= [ U Xb U Zb

b U Xc U Zc

.

56 . 29856 1 . 45381 13 . 60231 12 . 48000 0 . 00000 0 . 00000

1 . 45381 57 . 54593 − 3 . 93193 0 . 00000 9 . 55391 6 . 93476

13 . 60231 − 3 . 93193 16 . 45741 0 . 00000 − 6 . 93476 − 4 . 83209

M = (9)

12 . 48000 0 . 00000 0 . 00000 82 . 48709 0 . 00000 24 . 31006

0 . 00000 9 . 55391 − 6 . 93476 0 . 00000 79 . 36609 11 . 78523

0 . 00000 6 . 93476 − 4 . 83209 24 . 31006 11 . 78523 19 . 89318

10.1.2 Lumped Mass Matrix
An alternative to the consistent mass matrix approach is to establish the mass matrix of a
structure by forming the lumped mass matrix, i.e., to consider the mass of each element to
be concentrated at its nodes. For the structural elements, such as beams, plates, and shells,
the lumped mass matrix can be formed by moving the mass surrounding a node to that
node. For example, for the beam element shown in Fig. 10.3, half of the mass of the beam
is lumped at node a and the other half is lumped at node b . Then,

m a = m b = ρ/ 2 (10.11a)

Often, only the mass associated with the translational DOF is considered.
If axial motion of the beam element is included, the mass matrix, with the corresponding
displacement vector v

= [ u a w θ

a u b

w θ

b ] T , is given by a b

m a 1

m a 1

= ρ 2

0 0

m i = (10.11b)

m b 1

m b 1

0 0

FIGURE 10.3
Lumping the mass at the ends of a beam element.