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Free Damped Vibrations of Sandwich Shells of Revolution 185
u= N = [ N , N ,..., N ] =∑ N
() e 1 2 6 () e i i
i= 1
6
(32)
where
T
() e = 1 2 6
{ , ,..., }
(33)
are generalised displacements of the finite element and
T (1) (1) (1) (1) (1) (3) (3) (3) (3)
i = u0 v
i 0
w
i 0 γ i x γ i ϕ
u
i 0
v
i 0
γ i x
γ
i ϕi
{ , , , , , , , , }
(34)
is the vector of the displacement parameters at node i. Here
N
i
= Ni
I
9
(35)
where N i are the shape functions for node i expressed in terms of local triangular
coordinate system (L 1 ,L 2 ,L 3 ) of the finite element and I 9 istheunit9×9matrix.
UNDAMPED VIBRATION
The Hamilton’s principle given by Equation (21) is discretised as
∫
N
t e
2
t1
∑( δUl −δTl −δ Wel) dt = 0
l=
1
(36)
where N e is the total number of finite elements and the subscript l represents the
contribution of the lth element. Equation (36) can be expressed in terms of the displacements
and rotation angles (u 0 ,v 0 ,w 0 ,γ x ,γ ϕ ). Following the standard finite element
technique, we obtain the discretised equations in terms of nodal degrees of
freedom x:
Mẋ̇ + Kx = F
(37)
where M, K, x, F are the global mass and stiffness matrices, the displacement and
load vector, respectively. For free vibration problems F = 0 and Equation (37) becomes
2
( K− M)
=
0
(38)
where is the natural frequency and is the corresponding mode shape of the
structure.