464
12 Finite Element Method in Vibration Analysis
{ f } =
u
v
=
1 x y xy 0 0 0 0
0 0 0 0 1 x y xy
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
α 1
α 2
. . .
α 7
α 8
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(12.104)
or,
{ f } = [C] {α}
(12.105)
Nodal displacements can be expressed in terms of nodal coordinates with the help
of Eq. (12.104).
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u 1
v 1
u 2
v 2
u 3
v 3
u 4
v 4
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
1 0 b 0 0 0 0 0
0 0 0 0 1 0 b 0
1 a 0 0 0 0 0 0
0 0 0 0 1 a 0 0
1 a b ab 0 0 0 0
0 0 0 0 1 a b ab
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
α 1
α 2
α 3
α 4
α 5
α 6
α 7
α 8
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(12.106)
or,
{X } e = [A] {α}
(12.107)
From Eq. (12.107), one can write
{α} = [A]
− 1
{X } e
(12.108)
Replacing {α} from Eqs. (12.108) with (12.105), we get
{ f } = [C] [A]
− 1
{X } e
or,
{ f } = [N ] {X } e
(12.109)
where
[N ] =
N 1 0 N 2 0 N 3 0 N 4 0
0 N 1 0 N 2 0 N 3 0 N 4
(12.110)