508
13 Finite Difference Method for the Vibration Analysis …
⎡
⎣
−[N
]
T
0
0
0
−[N ]
T
0
0
0 2[N
]
T
⎤
⎦
⎡
⎣
D x D 1 0
D 1 D y 0
0 0 D xy
⎤
⎦
⎡
⎣
−[N
]
T
0
0
0
−[N ]
T
0
0
0 2[N
]
T
⎤
⎦
⎡
⎣
[]
[
]
[
]
⎤
⎦ d(area)
=
a
0
[]
T
, [
]
T
, [
]
T
⎡
⎣
[C 1 ] [C 3 ] 0
[C 2 ] [C 4 ] 0
0
0 [C 5 ]
⎤
⎦
⎡
⎣
[]
[
]
[
]
⎤
⎦ dy
=
a
0
[]
T
[C 1 ][] + []
T
[C 3 ][
] + [
]
T
[C 2 ][]
+[
]
T
[C 4 ][
] + [
]
T
[C 5 ][
]
dy
(13.54)
[C 1 ] =
b
0
[N
]
T D x [N
]dx, [C 2 ] =
b
0
[N
]
T D 1 [N ]dx,
[C 3 ] =
b
0
[N ]
T D 1 [N
]dx, [C 4 ] =
b
0
[N ]
T D y [N ]dx,
[C 5 ] =
b
0
[N
]
T D xy [N
]dx
(13.55)
D xy , D y , etc., being the usual orthotropic plate constants.
The integrations in Eq. (13.55) can be worked out explicitly since only
simple polynomials are involved while the integrations for the coupling matrices
a
0 φ
(k)
i
φ
(l)
j dy in Eq. (13.54) may be reduced to summation operations of the integrals
φ
(k)
i φ
(l)
j dy over a typical segment. Such segment integrals have also been
tabulated explicitly.
They are given in Tables 13.3, 13.4, 13.5, 13.6 and 13.7.
The explicit value of stiffness matrix is given in Table 13.8 [15]
13.7.5 Consistent Mass Matrix of the Plate Strip
A consistent mass matrix is formulated on the basis of lateral displacement w. The
displacement is given by Eq. (13.45)
{ f } = [N ] [] {X ] e = [N mw ] {X } e
(13.56)