466
M. Dutkiewicz and M. R. Machado
G(ω) =
⎡
⎢
⎢
⎣
1
1
e
−ik L
e
−kL
−ik
−k
ike
−ik L ke
−kL
e
−ik L
e
−kL
1
1
−ike
−ik L
−ke
−kL ik
k
⎤
⎥
⎥
⎦
(10)
The dynamic stiffness matrix for the spectral beam element under axial tension
can be determined as:
S(ω) = K (ω) − ω
2 M(ω)
(11)
where
K (ω) =
L
0
E I g
(x)
T g
(x) + T g
(x)
T g
(x)
dx
(12)
M(ω) = ρ A
L
0
g(x)
T g(x)dx
(13)
where
express the spatial partial derivative. By solving the integral, the dynamic
stiffness matrix is:
S(ω) =
EI
⎡
⎢
⎢
⎣
s 11
sym
s 12
s 22
s 13
s 23
s 33
s 14
s 24
s 34
s 44
⎤
⎥
⎥
⎦
(14)
where = cos(k L) cosh(k L) − 1 and the components of element matrix (14) are
given as
s 11 = −k
3
(cos(k L) sinh(k L) + sin(k L) cosh(k L))
s 12 = −k
2 sin(k L) sinh(k L)
s 13 = k
3
(sin(k L) + sinh(k L))
s 14 = k
2
(cos(k L) − cosh(k L))
s 22 = k(cos(k L) sinh(k L) − sin(k L) cosh(k L))
s 23 = k
2
(cosh(k L) − cos(k L))
s 24 = k(sin(k L) − sinh(k L))
s 33 = −k
3
(cos(k L) sinh(k L) + sin(k L) cosh(k L))
s 34 = k
2 sin(k L) sinh(k L)