8.14 Collocation Method for Obtaining Normal Modes …
353
Fig. 8.19 Example 8.7
Numerical integration is done by the trapezoidal rule. According to trape-zoidal
rule, the weighting numbers are
[w] =
L
3
⎡
⎣
1 0 0
0 1 0
0 0 1/2
⎤
⎦
Since the beam is uniform, we can write
[A] = A
⎡
⎣
1 0 0
0 1 0
0 0 1
⎤
⎦
Therefore, from Eq. (8.188)
⎧
⎨
⎩
Y 1
Y 2
Y 3
⎫
⎬
⎭
= p
2
ρ A
L
3
162 E I
⎡
⎣
2 5 8
5 16 28
8 28 54
⎤
⎦
⎡
⎣
1 0 0
0 1 0
0 0 1
⎤
⎦ L
3
⎡
⎣
1 0 0
0 1 0
0 0 1/2
⎤
⎦
⎧
⎨
⎩
Y 1
Y 2
Y 3
⎫
⎬
⎭
or,
1
p 2
⎧
⎨
⎩
Y 1
Y 2
Y 3
⎫
⎬
⎭
=
ρ A L
4
486 E I
⎡
⎣
2 5 4
5 16 14
8 28 27
⎤
⎦
⎧
⎨
⎩
Y 1
Y 2
Y 3
⎫
⎬
⎭
The eigenvalues and eigenvectors can be solved by any suitable procedure. For
determining the fundamental frequency, we adopt matrix iteration procedure.
ρ A L
4
486 E I
⎡
⎣
2 5 4
5 16 14
8 28 27
⎤
⎦
⎧
⎨
⎩
1
1
1
⎫
⎬
⎭
=
ρ A L
4
486 E I
⎧
⎨
⎩
11
35
63
⎫
⎬
⎭
=
11ρ A L
4
486 E I
⎧
⎨
⎩
1.000
3.182
5.727
⎫
⎬
⎭
ρ A L
4
486 E I
⎡
⎣
2 5 4
5 16 14
8 28 27
⎤
⎦
⎧
⎨
⎩
1.000
3.182
5.727
⎫
⎬
⎭
=
ρ A L
4
486 E I
⎧
⎨
⎩
40.818
136.090
251.725
⎫
⎬
⎭
=
40.81ρ A L
4
486 E I
⎧
⎨
⎩
1.000
3.334
6.167
⎫
⎬
⎭