120
E. Kormanikova et al.
I =
ρ m h 3
12
1
3
N
k=1
ρ k (z
(k)
)
3
− (z
(k−1)
)
3
,
where ρ k is the mass density of the kth layer.
For the simply supported plate on each edge let:
w 0 (x, y, t) = w
◦
0 e
iω mn t
,
Φ x (x, y, t) = Φ
◦
x e
iω mn t
,
Φ y (x, y, t) = Φ
◦
y e
iω mn t
,
(14)
where m, n are integers only, ω mn is natural angular velocity.
After substituting of Eqs. (14) into the Eqs. (12) we get:
⎡
⎣
L
11 L 12 L 13
L 12 L
22 L 23
L 13 L 23 L
33
⎤
⎦
⎧
⎨
⎩
A
mn
B
mn
C
mn
⎫
⎬
⎭
=
⎧
⎨
⎩
0
0
0
⎫
⎬
⎭
,
(15)
where:
L
11 = L 11 −
ρ m h 3
12
ω
2
mn ,
L
22 = L 22 −
ρ m h 3
12
ω
2
mn ,
L
33 = L 33 − ρ m hω
2
mn ,
L 11 = D 11 λ
2
m + D 66 λ
2
n + k
s A 55 ,
L 12 = (D 12 + D 66 )λ m λ n ,
L 13 = k
s A 55 λ m ,
L 22 = D 66 λ
2
m + D 22 λ
2
n + k
s A 44 ,
L 23 = k
s A 44 λ n ,
L 33 = k
s A 55 λ
2
m + k
s A 44 λ
2
n ,
(16)
and
λ m =
mπ
a
,
λ n =
nπ
b
.
(17)