Civil Engineering Reference
In-Depth Information
apparent, we introduce the change of variables
{
(10.112)
where
{
p
}
is the column matrix of
generalized displacements,
which are linear
combinations of the actual nodal displacements
{
q
}
, and [
A
] is the normalized
modal matrix. Equation 10.95 then becomes
[
M
][
A
]
q
}=
[
A
]
{
p
}
{ ¨
p
} +
[
K
][
A
]
{
p
}={
F
}
(10.113)
Premultiplying by
[
A
]
T
,
we obtain
[
A
]
T
[
M
][
A
]
[
A
]
T
[
K
][
A
]
[
A
]
T
{¨
p
}+
{
p
}=
{
F
}
(10.114)
Utilizing the orthogonality principle, Equation 10.114 is
[
I
]
[
A
]
T
[
k
][
A
]
[
A
]
T
(10.115)
Now we must examine the stiffness effects as represented by
[
A
]
T
[
K
][
A
]
. Given
that [K] is a symmetric matrix, the triple product
[
A
]
T
[
K
][
A
]
is also a symmet-
ric matrix. Following the previous development of orthogonality of the principal
modes, the triple product
[
A
]
T
[
K
][
A
]
is also easily shown to be a diagonal ma-
trix. The values of the diagonal terms are found by multiplying Equation 10.100
{¨
p
}+
{
p
}=
{
F
}
by
A
(
i
)
T
to obtain
i
A
(
i
)
T
[
M
]
A
(
i
)
+
A
(
i
)
T
[
K
]
A
(
i
)
=
2
−
0
i
=
1,
P
(10.116)
If the modal amplitude vectors have been normalized as described previously,
Equation 10.116 is
A
(
i
)
T
[
K
]
A
(
i
)
=
2
i
1,
P
(10.117)
hence, the matrix triple product
[
A
]
T
[
K
][
A
]
produces a diagonal matrix having
diagonal terms equal to the squares of the natural circular frequencies of the prin-
cipal modes of vibration; that is,
i
=
1
0
···
0
2
0
.
.
.
.
[
A
]
T
[
K
][
A
]
=
(10.118)
.
.
.
.
.
.
·
·
·
P
0
Finally, Equation 10.115 becomes
[
I
]
2
]
[
A
]
T
{¨
p
}+
[
{
p
}=
{
F
}
(10.119)
2
]
representing the diagonal matrix defined in Equation 10.118.
with matrix
[
EXAMPLE 10.7
Using the data of Example 10.3, normalize the modal matrix and verify that
[
A
]
T
[
M
][
A
]
=
[
I
]
and
[
A
]
T
[
K
][
A
]
=
[
2
]
.
