Civil Engineering Reference
In-Depth Information
The smallest eigenvalue is found by iterating on the equation [
A
−1
][
x
]=
l
−1
[
x
] in the same manner as the largest value. A common structural prob-
lem is the modal node analysis of a multi-story frame. The general steps
are as follows:
[
K
][
x
] − λ[
M
][
x
] = 0
The value
l
=
w
2
, where
w
is the frequency of the building. The equation
can be rewritten as follows:
1
l
[][[]
=
[]
BMx
x
The values of the matrices are [
K
] for stiffness, [
M
] for mass, and [
B
] for
flexibility.
Example 2.17
Power method
Determine the first mode shape (lowest eigenvector) for the following set
of equations using the power method with
e
= 0.001 and assuming [
x
] =
[1] as an initial value. The [
B
] and [
M
] matrices are for a four-story single
mass structural model.
10
6
−
10
3
112112 112112
1121160 160 160
112116037 120 37 120
1
/ / / /
/ / / /
/ / / /
/12 11 60 37 120 57 120
6000
0500
0040
0003
0
0
0
0
=
/
/
/
The [
B
] and [
M
] matrices can be combined by matrix multiplication.
0 000500 0 000417 0 000333 0 000250
0 000500 0 000917 0 000733 0
. . . .
. . . .000550
0 000500 0 000917 0 001233 0 000925
0 000500 0 000917 00
0
0
0
0
=
.
.
.
.
.
.
.01233 0 001425
.
The iteration process and the final solution are shown in Table 2.20. The
x
vector was normalized to the top value.
