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Free Damped Vibrations of Sandwich Shells of Revolution 183<br />
Figure 5. Kinematical assumptions for sandwich finite element in the thickness direction.<br />
are five degrees of freedom: three displacements (u, v, w) and two rotations (γ x , γ ϕ ),<br />
where u, v and w are displacements in the directions of reference axes x 1 , x 2 and x 3<br />
of the shell. Stiffness and mass matrices of this finite element have been derived in<br />
detail in Reference [38]. Using the displacement continuity conditions between<br />
layers, the sandwich finite element was developed. A similar approach was used<br />
for developing the sandwich plate finite element PLW54 [12].<br />
The sandwich shell finite element has 54 independent degrees of freedom. For<br />
the bottom layer (six nodal points) in every node there are three displacements<br />
( 1) ( 1) ( 1)<br />
( ) ( )<br />
( u0<br />
, v0<br />
, w0<br />
) and two rotations ( γ<br />
1 , γ<br />
2 ),<br />
x1 x2<br />
while for the top layer there are<br />
two displacements ( ( 3 ) , ( 3<br />
u v<br />
) ) and two rotations ( γ<br />
( 3) , γ<br />
( 3)<br />
) only.<br />
0<br />
0<br />
x1 x2<br />
DISPLACEMENT REPRESENTATION<br />
In each layer of the sandwich structure the kinematics, Equation (4), are assumed.<br />
As the reference surface the midsurface of the bottom layer is used and displacements<br />
and rotations of all layers are defined in the coordinates of midsurface<br />
of the bottom surface. For the first (bottom) layer the displacements can be ex-
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