Civil Engineering Reference
In-Depth Information
The mass matrix of a discrete
n
-DOF system is usually lumped into an
n
×
n
diagonal matrix
of the form:
2
4
3
5
M
1
0
0
.
.
.
.
0
M
2
M
=
ð3
:
5Þ
.
.
.
.
.
.
.
0
0
0
M
n
However, when certain degrees of freedom are representing the rotation of joints, the mass
moment of inertia at these joints may typically be ignored by setting the value equal to zero.
In this case, the mass matrix becomes
M
0
0
0
0
1
0
0
0
0
M
0
0
M
0
d
dd
M
=
=
ð3
:
6Þ
0
0
0
0
0
0
0
0
0
0
where the subscript '
d
' represents the total number of translational degrees of freedom with
mass,
d
≤
n
, and
M
dd
is an invertible mass matrix in its condensed form with only the transla-
tional degrees of freedom.
The damping matrix for an
n
-DOF system is typically a fully populated
n
×
n
matrix of
the form:
2
4
3
5
C
11
C
12
C
1
n
.
.
.
.
C
21
C
22
C
=
ð3
:
7Þ
.
.
.
.
.
.
.
C
n
−1
,
n
C
n
1
C
n
,
n
−1
C
nn
The exact formulation of the damping matrix has never been developed in theory, so it is often
assumed to be proportional to the mass matrix
M
and stiffness matrix
K
of the form:
C
=
a
1
M
+
a
2
K
ð3
:
8Þ
where
a
1
and
a
2
are the proportional damping constants. This type of proportional damping is
called Rayleigh damping. However, if some degrees of freedom correspond to the rotation of
joints with the mass moment of inertia being ignored, the damping matrix can also be con-
densed in the form:
C
0
dd
C
=
ð3
:
9Þ
0
0
