Vibrations of the Mass Variable Systems
35
Differentiating (53) 2 , it is
¨
x = −
2
˙
A(t)Ω(τ ) + A(t) ˙
Ω(τ )
α + 1
sa(1, α, ψ(t)) −
2 A(t)Ω
2
(τ )
α + 1
ca
α+1
(α, 1, ψ(t))
−
2 A(t)Ω(τ ) ˙
θ (t)
α + 1
ca
α+1
(α, 1, ψ(t))
(56)
Substituting (53) and (53) into (47) and after some modification, it is
−
2
˙
A(t)Ω(τ ) + A(t) ˙
Ω(τ )
α + 1
sa(1, α, ψ(t)) −
2 A(t)Ω(τ ) ˙
θ(t)
α + 1
ca
α+1
(α, 1, ψ(t))
=
ε
M(τ )
2 A(t)Ω(τ )
α + 1
dM
dτ
sa(1, α, ψ(t))
(57)
(55) and (57) are two first-order equations which correspond to (47).
After uncoupling equations for ˙
A and ˙
θ and using the relations (48) and (54), it
follows
˙
A
sa
2
(1, α, ψ) + ca
α+1
(α, 1, ψ)
= −
ε
M
A
α + 1
dM
dτ
sa
2
(1, α, ψ)
(58)
A ˙
θ
sa
2
(1, α, ψ) + ca
α+1
(α, 1, ψ)
= −
ε A
M
dM
dτ
sa(1, α, ψ)ca(α, 1, ψ) (59)
Finally, for the identity [18]
sa
2
(1, α, ψ) + ca
α+1
(α, 1, ψ) = 1
( 6 0 )
it is
˙
A = −
ε
M
A
α + 1
dM
dτ
sa
2
(1, α, ψ), A ˙
θ = −
ε A
M
dM
dτ
sa(1, α, ψ)ca(α, 1, ψ) (61)
Equations (61) are the rewritten version of (47) into new variables: A and θ.
Bearing in mind that the Ateb functions are time periodic, the averaging of the
equations over the period T =
2Π α
Ω
where 2Π α = 2B
1
α+1
,
1
2
and B is the complete
beta function is done. For
1
T
T
0
sa
2
(1, α, ψ)dψ =
α + 1
α + 3
,
1
T
T
0
sa(1, α, ψ)ca(α, 1, ψ)dψ = 0,
(62)