466
12 Finite Element Method in Vibration Analysis
[M] e =
mab
36
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
4 0 2 0 1 0 2 0
0 4 0 2 0 1 0 2
2 0 4 0 2 0 1 0
0 2 0 4 0 2 0 1
1 0 2 0 4 0 2 0
0 1 0 2 0 4 0 2
2 0 1 0 2 0 4 0
0 2 0 1 0 2 0 4
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(12.117)
12.7 Flexural Vibration of Plates
Let us consider a rectangular plate bending element. The plate element dimensions,
the node numbering and the coordinate system have been shown in Fig. 12.10. Each
node has three degrees of freedom-one vertical displacement and two rotations along
x and y-axes. The displacements at node 1 therefore, are w 1 , θ x 1 and θ y 1 . The complete
displacement vector for this element is
{X }
T
e = { w 1 θ x 1 θ y 1 w 2 θ x 2 θ y 2 w 3 θ x 3 θ y 3 w 4 θ x 4 θ y 4 }
(12.118)
The displacement function is given by
{ f } = {w} = α 1 + α 2 x + α 3 y + α 4 x
2
+ α 5 x y + α 6 y
2
+ α 7 x
3
+ α 8 x
2 y + α 9 x y
2
+ α 10 y
3
+ α 11 x
3 y + α 12 x y
3
(12.119)
Fig. 12.10 A rectangular
plate in bending