Vibrations of the Mass Variable Systems
37
is higher for smaller orders of nonlinearity: Increase is the fastest for linear oscillator, while amplitude remains approximately constant for extremely high order of
nonlinearity.
4.1 Reactive Force Is Omitted
In the most of published papers in vibrations of mass variable oscillators, the reactive
force is omitted. For that case, the right side of the (47) is zero and the equations of
vibration are
˙
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)
(66)
The corresponding averaged equations are
˙
A
A
=
1
α + 3
dM
M
, ˙
θ = 0
(67)
After integration, the amplitude and angle variation follow
A = A(0)
M
M(0)
1
α+3 ,
(68)
ψ = θ (0) + A(0)
α−1
2
t
∫
0
M
M(0)
α−1
2(α+3)
α + 1
2
k
M(τ )
dt
(69)
Comparing the obtained solutions (64) and (68), it is obvious that the tendency
of amplitude change is opposite for the two cases. Namely, if the mass is decreasing
and the reactive force is assumed the amplitude of vibration increases. However, for
the same mass decrease, if the reactive force is neglected the amplitude of vibration
also decreases. This result is already obtained for the mass variable linear [29] and
Duffing [30] oscillators. The result in the paper shows that it is generally the case for
any oscillator with variable mass independently on the type of nonlinearity in elastic
term.
5 Conclusion
The dynamics of the body with continual and discontinual mass variation is considered. In the rigid body which is separated into two parts, the linear and angular
momentums are varying causing the change of the velocity of the mass center and
of the angular velocity of the body. The linear and angular velocity variation is
37
is higher for smaller orders of nonlinearity: Increase is the fastest for linear oscillator, while amplitude remains approximately constant for extremely high order of
nonlinearity.
4.1 Reactive Force Is Omitted
In the most of published papers in vibrations of mass variable oscillators, the reactive
force is omitted. For that case, the right side of the (47) is zero and the equations of
vibration are
˙
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)
(66)
The corresponding averaged equations are
˙
A
A
=
1
α + 3
dM
M
, ˙
θ = 0
(67)
After integration, the amplitude and angle variation follow
A = A(0)
M
M(0)
1
α+3 ,
(68)
ψ = θ (0) + A(0)
α−1
2
t
∫
0
M
M(0)
α−1
2(α+3)
α + 1
2
k
M(τ )
dt
(69)
Comparing the obtained solutions (64) and (68), it is obvious that the tendency
of amplitude change is opposite for the two cases. Namely, if the mass is decreasing
and the reactive force is assumed the amplitude of vibration increases. However, for
the same mass decrease, if the reactive force is neglected the amplitude of vibration
also decreases. This result is already obtained for the mass variable linear [29] and
Duffing [30] oscillators. The result in the paper shows that it is generally the case for
any oscillator with variable mass independently on the type of nonlinearity in elastic
term.
5 Conclusion
The dynamics of the body with continual and discontinual mass variation is considered. In the rigid body which is separated into two parts, the linear and angular
momentums are varying causing the change of the velocity of the mass center and
of the angular velocity of the body. The linear and angular velocity variation is