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leads to
51
15
1
2
(
m
lumped
)
=
ρ
12
m
high
-
order
=
m
consistent
+
(10.15)
where, from Eq. (10.10b),
21
12
m
consistent
=
ρ
6
and, from Eq. (10.11b),
10
01
m
lumped
=
ρ
2
2
5
3
5
A study [Kim, 1993] contends that
α
=
and
β
=
gives the best results for the axial
1
9
8
9
vibration of a bar, while
α
=
and
β
=
leads to the best results for a cantilevered beam.
The literature contains various proposals for using numerical integration schemes for
generating mass matrices for two and three dimensional elements. As mentioned earlier,
the most common approach is simply to use the numerical integration scheme employed
for the integration of
k
i
to perform the integration to form
m
i
See the references cited above
for further discussions concerning the establishment of approximate mass matrices.
.
10.2
Reduction of Degrees of Freedom
For a complex vibrating system, some of the DOF may have little influence on the dynamic
behavior of the system. For example, when one mass in a two DOF system is much smaller
than the other mass, the system can often be treated as a single-DOF system. Dynamic
response problems tend to be so complicated numerically that it is accepted as a basic
premise that any plausible reduction in the DOF should be implemented. Some DOF can
complicate a computational procedure, as is often the case for a lumped mass matrix with
zero diagonal elements. These zero elements have little effect on the dynamic response of
the system. Certainly, it is reasonable to simply neglect the small and zero masses and then
deal with the remainder of the mass matrix. A more accurate method of achieving this
is called
kinematic condensation.
This is similar to static condensation procedures (Chapter
5), which are used in structural analysis to reduce the size of the stiffness matrix. Use the
two-DOF system of Fig. 10.4 to illustrate this procedure.
The governing equation for this system is
M V
+
KV
=
0
(10.16)
FIGURE 10.4
A two-DOF system.
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