Civil Engineering Reference
In-Depth Information
to give
} −
ω
2
[
M
m
]
[
K
m
]
{
A
{
A
} = {
0
}
(3.123)
Solution of this eigenproblem by the techniques previously described results in
neq
eigenvalues
ω
2
,where
neq
(in program terminology) is the total
number of degrees of freedom in the finite element mesh. These eigenvectors or “mode
shapes” can be considered to be columns of a modal matrix [
P
], where
and eigenvectors
{
A
}
=
{
A
nmodes
+
}
[
P
]
A
1
}{
A
2
}
..
{
(3.124)
where
nmodes
is the number of modes that are contributing to the time response. Often it
is not necessary to include the higher frequency components in an analysis, so that
nmodes
≤
neq
.
Because of the properties of eigenproblems that mode shapes possess orthogonality,
one to the other, such that
T
[
M
m
]
A
j
=
{
A
i
}
0
T
[
K
m
]
A
j
=
i
=
j
(3.125)
{
A
i
}
0
T
[
M
m
]
A
j
=
m
ii
{
A
i
}
i
=
j
(3.126)
T
[
K
m
]
A
j
=
k
ii
{
A
i
}
where
m
ii
and
k
ii
are the diagonal terms of the diagonal global “principal” mass and
stiffness matrices, [
M
] and [
K
] respectively. Use of these relationships in (3.121) has the
effect of uncoupling the equations in terms of the principal or “normal” coordinates
U
{
}
,
thus
[
M
]
d
2
U
d
t
2
[
K
]
U
+
= {
0
}
(3.127)
The effect of uncoupling has been to reduce the vibration problem to a set of
nmodes
independent second order equations (3.127).
The actual displacements can be retrieved from the normal coordinates by a final super-
position process given by
[
P
]
U
{
U
} =
(3.128)
Inclusion of damping
Free damped vibrations, governed by
[
C
m
]
d
U
d
t
[
M
m
]
d
2
U
d
t
2
[
K
m
]
{
U
} +
+
= {
0
}
(3.129)
can be handled by the above technique if it is assumed that the undamped mode shapes are
also orthogonal with respect to the damping matrix [
C
m
] in the way described by (3.125)
