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Free Damped Vibrations of Sandwich Shells of Revolution 181
sandwich shell can be split into two parts:
= D e , = D e
NM NM NM Q Q Q
(17)
with
e
T
NM xx ϕϕ xϕ xx ϕϕ xϕ
= { N , N , N , M , M , M }
T
Q
= { Q , Q }
x
T
NM xx ϕϕ xϕ xx ϕϕ xϕ
= { Ω , Ω ,2 Ω , χ , χ ,2 χ }
T
Q
e
ϕ
= {2 Ω , 2 Ω }
x3 ϕ3
(18)
and
D
NM
⎡Q11 Q12 Q16 B11 B12 B16
⎤
⎢
Q12 Q22 Q26 B12 B22 B
⎥
⎢
26 ⎥
⎢Q16 Q26 Q66 B16 B26 B66
⎥
= ⎢ ⎥
⎢B11 B12 B16 D11 D12 D16
⎥
⎢B 12 B22 B26 D12 D22 D ⎥
26
⎢
⎥
⎢⎣B16 B26 B66 D16 D26 D66
⎥⎦
(19)
D
Q
⎡
2
kQ 5 55 kkQ ⎤
4 5 45
= ⎢ ⎥
2
⎢⎣kkQ
4 5 45 kQ 4 44⎥⎦
(20)
The components of the in-plane (membrane), the coupling and the out-of-plane
plate (bending and torsion) stiffness matrices Q ij , B ij , D ij , i, j = 1,2,6 are defined in
Reference [26]. The shear correction coefficients k4 2 , k5 2 can be calculated by difference
approaches [47]. Here the approach based on averaging of transverse shear
strain complementary energy and outlined in Reference [40] is used.
VARIATIONAL FORMULATION
The governing differential equations of motion can be derived by using the
Hamilton’s principle [39]:
∫
t2
t1
( δU−δT −δ W ) dt = 0
e
(21)
where δU, δT, δW e are the variations of the strain energy, the kinetic energy and the
work of external forces, respectively, t denotes the time. The strain energy for the