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13 Finite Difference Method for the Vibration Analysis …
Fig. 13.4 1st mode shape
13.4 Free Vibration of Rectangular Plates
The free vibration equation of the plate is given by
∂
4
w
∂ x 4 + 2
∂
4
w
∂ x 2 ∂ y 2 +
∂
4
w
∂ y 4 =
mp
2
D
w
(13.11)
The plate is divided into equally spaced square mesh, having x = y = h
(Fig. 13.5). The finite difference expressions are written with respect to the node (i,
j). ‘i’s are numbered in the x-direction and ‘j’s are in y-direction. Therefore,
∂
4
w
∂ x 4
i, j
=
1
h 4
w i+2, j − 4w i+1, j + 6w i, j − 4w i−1, j + w i−2, j
(13.12)
∂
4
w
∂ y 4
i, j
=
1
h 4 [w i, j + 2 − 4w i, j + 1 + 6w i, j − 4w i, j − 1 + w i, j − 2 ] (13.13)
∂
4
w
∂ x 2 ∂ y 2
i, j
=
∂
2
∂ y 2
∂
2
w
∂ x 2
i, j
=
1
h 2
∂
2
w
∂ x 2
i, j+1
− 2
∂
2
w
∂ x 2
i, j
+
∂
2
w
∂ x 2
i, j−1
=
1
h 4
w i+1, j+1 − 2w i, j+1 + w i−1, j+1
− 2
w i+1, j − 2w i, j + w i−1, j
Fig. 13.5 Finite difference representation of a plate node