116
A. K. Misra and S. Cohen
4.1 Longitudinal Oscillations
The frequencies of longitudinal oscillations of the ribbon can be obtained by solving
the eigenvalue problem
M
A A
+ K
A A = 0
(33)
where the elements of the matrices M
A , K
A are given by [8]
M
A
ik = M p
1
0
exp[F(ξ)]ψ i ψ k dξ + M c ψ i1 ψ k1
(34)
and
K
A
ik = ¯
Ω 2
1
0
exp[F(ξ)]
dψ i
dξ
dψ k
dξ
dξ
− M p
1
0
exp[F(ξ)]ψ i ψ k dξ − M c ψ i1 ψ k1
− 2
β
Λ
3
M p
1
0
exp[F(ξ)]ψ i ψ k
[(1/Λ) + ξ(1 + ε 0 )] 3 dξ + M c
ψ i1 ψ k1
[(1/Λ) + (1 + ε 0 )] 3
(35)
where the indices i and k vary from 1 to N . The prime symbol appearing in Eq. (33)
denotes differentiation with respect to nondimensional time τ . The nondimensional
quantities appearing in Eqs. (33) to (35) are defined by
τ = Λt, ξ =
s
L 0
, M p = γ A m
L 0
m tot
, M c =
m c
m tot
(36)
Λt =
L 0
R E
, β =
R G
R E
, ¯
Ω =
E A m
m tot L 0
1
2
Λ
(37)
and
F(ξ) =
R
2
E
¯
h R G (1 + ε 0 )
3
2
−
R G
R E + ξ(1 + ε 0 )L 0
−
R E + ξ(1 + ε 0 )L 0
2
2R
2
G
(38)
The generalized eigenvalue problem given by Eq. (33) is solved using MATLAB.
Before solving, a set of basis functions ψ i (s) appearing in Eq. (22) must be chosen.
They must all be zero at the base, i.e., at s = 0 or ξ = 0. Also, to allow the counter-