On stability of a nonlinear pendulum
Методом Ляпунова знайдено достатню умову стiйкостi руху нелiнiйного маятника.
Gespeichert in:
| Datum: | 2003 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2003
|
| Schriftenreihe: | Нелінійні коливання |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/176933 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On stability of a nonlinear pendulum / E. Kengne // Нелінійні коливання. — 2002. — Т. 5, № 4. — С. 191-196. — Бібліогр.: 16 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-176933 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1769332021-02-11T01:29:03Z On stability of a nonlinear pendulum Kengne, E. Методом Ляпунова знайдено достатню умову стiйкостi руху нелiнiйного маятника. By means of Liapounov’s method, we establish a sufficient condition for stability of a nonlinear pendulum. 2003 Article On stability of a nonlinear pendulum / E. Kengne // Нелінійні коливання. — 2002. — Т. 5, № 4. — С. 191-196. — Бібліогр.: 16 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/176933 517.929.51 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Методом Ляпунова знайдено достатню умову стiйкостi руху нелiнiйного маятника. |
| format |
Article |
| author |
Kengne, E. |
| spellingShingle |
Kengne, E. On stability of a nonlinear pendulum Нелінійні коливання |
| author_facet |
Kengne, E. |
| author_sort |
Kengne, E. |
| title |
On stability of a nonlinear pendulum |
| title_short |
On stability of a nonlinear pendulum |
| title_full |
On stability of a nonlinear pendulum |
| title_fullStr |
On stability of a nonlinear pendulum |
| title_full_unstemmed |
On stability of a nonlinear pendulum |
| title_sort |
on stability of a nonlinear pendulum |
| publisher |
Інститут математики НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/176933 |
| citation_txt |
On stability of a nonlinear pendulum / E. Kengne // Нелінійні коливання. — 2002. — Т. 5, № 4. — С. 191-196. — Бібліогр.: 16 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT kengnee onstabilityofanonlinearpendulum |
| first_indexed |
2025-07-15T14:53:11Z |
| last_indexed |
2025-07-15T14:53:11Z |
| _version_ |
1837725061320015872 |
| fulltext |
UDC 517.929.51
ON STABILITY OF A NONLINEAR PENDULUM
ПРО СТIЙКIСТЬ НЕЛIНIЙНОГО МАЯТНИКА
E. Kengne
Univ. Dschang
P.O. Box 173, Dschang, Cameroon
e-mail: ekengne6@yahoo.fr.
By means of Liapounov’s method, we establish a sufficient condition for stability of a nonlinear pendulum.
Методом Ляпунова знайдено достатню умову стiйкостi руху нелiнiйного маятника.
1. Introduction. Various interesting dynamical processes described by ordinary differential
equations with many-dimensional rapid and slow variables have been studied by several authors
[1 – 4]. Using the methods of works [5 – 8], Matviitchuk [9, 10] established and proved sufficient
criteria of stability for the class of the mentioned dynamical processes.
The purpose of this paper is to establish by the means of Liapounov’s method [11] a suffici-
ent criterion for stability of a nonlinear pendulum. In Section 2, we obtain a differential system
for oscillations of a nonlinear pendulum. The obtained system is a dynamic system with two
rapid variables and one slow variable. In Section 3, we find the interval of stability of nonlinear
pendulum described in Section 2.
2. Mathematical considerations. Throughout this paper we use the following notations: g is
the acceleration of gravity; α the angular deviation from vertical; x(τ) the slowly variation of
the length of the thread; τ = µt, where µ is a small positive quantity [2]. We assume that the
length of the thread changes not by means of external forces, but by means of the self energy of
the system, that is, the rate of change of the length of the thread depends only on α, α̇ and x.We
also assume that the rate of the plastic deformation of the thread is very small and proportional
to the tension of the thread, that is, ẋ = λµP, where P is the tension of the thread and λµ a
small coefficient of deformation. If we denote by x0 the initial length of the thread then we can
write x(τ) = x0 + ξ(τ), with 0 ≤ ξ ≤ K − x0, where K is a positive constant. The oscillations
of such a pendulum are governed by the dynamic system [1]
dα
dt
=
p
m(x0 + ξ)2
,
dp
dt
= −mg(x0 + ξ)α,
dξ
dt
= λµ
[
mg cosα+
p2
m(x0 + ξ)3
]
,
(1)
c© E. Kengne, 2003
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2 191
192 E. KENGNE
defined in the region
D = {α, ξ, p, µ, t : − π < α < π, 0 ≤ ξ ≤ K − x0, − p0 ≤ p ≤ p0, p0 = const > 0,
0 < µ < µ0, µ0 = cosnt > 0, 0 ≤ t ≤ µ−1
}
.
A pendulum whose oscillations are governed by system (1) is called a nonlinear pendulum.
By setting ε = λµ, system (1) under the transformation t = t/ε is reduced to the following
system with a small positive parameter along the two first derivatives:
ε
dα
dt
=
p
m(x0 + ξ)2
,
ε
dp
dt
= −mg(x0 + ξ)α,
dξ
dt
= mg cosα+
p2
m(x0 + ξ)3
.
(2)
At every point (α, p, ξ) of the phase space (or the region where system (2) is defined) system
(2) defines the vector of the phase velocity
v(α, p, ξ) =
(
1
ε
p
m(x0 + ξ)2
,
−mg(x0 + ξ)α
ε
,mg cosα+
p2
m(x0 + ξ)3
)
. (3)
As it is seen, the third component of the phase velocity (3) possesses a finite value, and, generally
speaking, the two first components are infinitely large. Therefore for the phase portrait of
system (2) we notice the presence of rapid movements and slow movements. Nevertheless the
rapid movements occur far from the region [12]
Γ = {α, p, ξ : α = 0, p = 0, 0 ≤ ξ ≤ K − x0}
almost in parallel to the plane αp, and the slow movements occur near the region Γ.
System
ε
dα
dt
=
p
m(x0 + ξ)2
,
ε
dp
dt
= −mg(x0 + ξ)α
in which ξ is considered as a parameter is called the system of rapid movements, corresponding
to system (2), and the variables α and p are called rapid variables [12] of system (2). The variable
ξ is called the slow variable [12] of system (2). Therefore, (1) is a dynamic system with many-
dimensional rapid and slow variables (two rapid variables, α and p, and one slow variable, ξ).
Because
f1(α, p, ξ) =
p
m(x0 + ξ)2
, f2(α, p, ξ) = −mg(x0 + ξ)α
and
f3(α, p, ξ) = mg cosα+
p2
m(x0 + ξ)3
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
ON STABILITY OF A NONLINEAR PENDULUM 193
are continuous functions in Ω̃ = {−π < α < π, 0 ≤ ξ ≤ K−x0,−p0 ≤ p ≤ p0, p0 = const >
> 0}, we have the following conclusion [12 – 14]: if ϕ(t, ε) = (α(t, ε), p(t, ε), ξ(t, ξ)) is a solution
of system (1) (with ε = λµ) verifying the initial condition ϕ(0, ε) = ϕ0 and ϕ0(t) a solution of
the singular system
dα
dt
=
p
m(x0 + ξ)2
,
dp
dt
= −mg(x0 + ξ)α,
dξ
dt
= 0
(4)
(this system is obtained from (1) by setting µ = 0) corresponding to system (1), defined in the
region 0 ≤ t ≤ T and verifying the same initial condition ϕ0(0) = ϕ0, then
ϕ(t, ε) = ϕ0(t) +R0(t, ε),
where R0(t, ε) tends uniformly to zero with respect to t ∈ [0, T ] as ε → 0.
If we consider α and α̇ to be very small, then cosα ≈ 1−α
2
2
. Therefore, system (1) relatively
coincides with the nonlinear system
dα
dt
=
p
m(x0 + ξ)2
,
dp
dt
= −mg(x0 + ξ)α,
dξ
dt
= λµ
[
mg − 1
2
mgα2 +
p2
m(x0 + ξ)3
]
.
(5)
Systems (4) and (5) play an important role in the investigation of the stability of nonlinear
pendulum whose oscillations are governed by system (1).
3. Stability of nonlinear pendulum. The solutions of singular system (4) read
α(t) = a cos(ωt+ β), p(t) = −ma(x0 + ξ)2ω sin(ωt+ β), ξ(t) = const, (6)
where a is the amplitude of oscillations of the pendulum; ω =
√
g/(x0 + ξ), β = const. With
the help of the singular solution (6), we write system (1) in the form [15]
dξ
dt
= µ̃Z(t, ξ, ϕ(t, ξ, a, β)) +R1(t, ξ, ϕ(t, ξ, a, β)),
dα
dt
= µ̃Z1(t, ξ, ϕ(t, ξ, a, β)) +R2(t, ξ, ϕ(t, ξ, a, β)),
dp
dt
= µ̃Z2(t, ξ, ϕ(t, ξ, a, β)) +R3(t, ξ, ϕ(t, ξ, a, β)),
(7)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
194 E. KENGNE
where
ϕ(t, ξ, a, β) = (a cos(ωt+ β),−ma(x0 + ξ)2ω sin(ωt+ β)),
Z(t, ξ, ϕ(t, ξ, a, β)) = λm{g cos[a cos(ωt+ β)] + (x0 + ξ)a2ω2 sin2(ωt+ β)},
Z1(t, ξ, ϕ(t, ξ, a, β)) =
−3a
2
√
x0 + ξ
sin2(ωt+ β)Z(t, ξ, ϕ(t, ξ, a, β)),
Z2(t, ξ, ϕ(t, ξ, a, β)) =
ωt− 3
2 sin(ωt+ β)
2(x0 + ξ)
Z(t, ξ, ϕ(t, ξ, a, β)),
R1(t, ξ, ϕ(t, ξ, a, β)) = λµ
(
mg cosα+
p2
m(x0 + ξ)3
)
− µ̃Z(t, ξ, ϕ(t, ξ, a, β)),
R2(t, ξ, ϕ(t, ξ, a, β)) =
p
m(x0 + ξ)2
+
3aµ
2(x0 + ξ)2
sin2(ωt+ β)Z(t, ξ, ϕ(t, ξ, a, β)),
R3(t, ξ, ϕ(t, ξ, a, β)) = −
{
mg(x0 + ξ)α+
µ̃
2(x0 + ξ)
(
ωt− 3 sin(ωt+ β)
2
)
×
×Z(t, ξ, ϕ(t, ξ, a, β))
}
.
Let us group, together with system
dξ
dt
= µ̃Z(t, ξ, ϕ(t, ξ, a, β)),
dα
dt
= µ̃Z1(t, ξ, ϕ(t, ξ, a, β)),
dp
dt
= µ̃Z2(t, ξ, ϕ(t, ξ, a, β)),
(8)
Liapounov’s function
V (t, ξ, α, p, µ̃) = e− sin(e−t)
[
2−1ξ2 + µ̃(sin2 α+ sin2 p)
]
in the domain D for which the derivative along the system (6), under the conditions
|µ− µ̃| < µ1, µ1 ≤ λmK
γ − µ̃K0
g +Kp3
0
, K0 = mK[g(π + 3) +Kp2
0], γ = const > 0,
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
ON STABILITY OF A NONLINEAR PENDULUM 195
has the estimate
dV
dt
=
∂V
∂t
+
∂V
∂ξ
dξ
dt
+
∂V
∂α
dα
dt
+
∂V
∂p
dp
dt
≤
≤ e−t cos(e−t)V (t, ξ, α, p, µ̃) + e− sin(e−t)(µK1 + γ),
K1 = λmg(2 + a2)
[
K + (2x0)−1 (2
√
g + 5µ̃)
]
+
p0
mx0
.
That is,
dV
dt
≤ e−t cos(e−t)V (t, ξ, α, p, µ̃) + e− sin(e−t)(µK1 + γ). (9)
Let s(t, µ̃) be a function defined by
s(t, µ̃) =
t∫
t0
esin e−t0−sin e−s(µ̃K1 + γ)ds.
Let us consider the following Cauchy problem for the congruence equation [9, 10, 16]:
dy
dt
= e−t cos(e−t)[y + s(t, µ̃)],
(10)
y(t0) = y0 ≥ V0 = V (t0, ξ(t0), α(t0), p(t0), µ̃).
Cauchy problem (10) possesses a bounded and continuous solution
ỹ(t) = esin e−t0−sin e−t [y0 + (µ̃K1 + γ)(t− t0)]− s(t, µ̃).
Using Theorem 9.5 of work [8] on the differential inequalities and by virtue of the inequality
(9), the following estimate holds for the Liapounov’s function (8) along the solution of system
(1) for the values of t ∈
[
0, µ−1
]
∩
[
0, µ̃−1
]
:
V (t, ξ(t), α(t), p(t), µ̃) ≤ ỹ(t) + s(t, µ̃).
That is,
V (t, ξ(t), α(t), p(t), µ̃) ≤ esin e−t0−sin e−t [y0 + (µ̃K1 + γ)(t− t0)] .
By virtue of the fact that s(t, µ̃) is a bounded quantity on
[
0, µ−1
]
∩
[
0, µ̃−1
]
and
y(t) →
t→+∞
y0e
sin e−t0 ,
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
196 E. KENGNE
we finally have for t ∈
[
0, µ−1
]
∩
[
0, µ̃−1
]
the estimate
0 ≤ V (t, ξ(t), α(t), p(t), µ̃) ≤ esin e−t0
[
y0 + (µ̃K1 + γ)
(
1
µ̆
− t0
)]
,
where µ̆ = max(µ, µ̃). Consequently, the processes (1), subordinate to condition (5), is stable
[10] on the segment I =
[
0, µ−1
]
∩
[
0, µ̃−1
]
. In other words, the nonlinear pendulum described
in Section 2 is stable on the region I =
[
0, µ−1
]
∩
[
0, µ̃−1
]
.
1. Bogolioubov N.N., Zubarev D.N. Method of asymptotic approximations for systems with revolutionary phase
and its application to the motion of charged particles in the magnetic field // Ukr. Math. J. — 1955. — 7, № 1.
— P. 5 – 17.
2. Volossof V.M. Averaging in the systems of ordinary differential equations // Ibid. — 1962. — 17, № 6. —
P. 3 – 126.
3. Alfven H. On the motion of charged particle in a magnetic field // Ark. mat., astron. och fys. — 1940. — 37A,
№ 22. — P. 3 – 10.
4. Mitchenko E. F., Pontriaguin L.S. Relaxation oscillations and differential equations containing a small para-
meter with the senior derivative // Calcutta Math. Soc. The Golden Jubilee Commemoration Volume (1958 –
1959). — Calcutta, 1963. — P. 605 – 626.
5. Kamke E. Differentialgleichungen reeler Funktionen. — Leipzig: Akad. Verlagsgesellschaft, 1930. — 365 S.
6. Lakshmikantham V., Leela S. Differential and integral inequalities. Theory and applications. — New York;
London: Acad. Press, 1969. — Vols 1, 2. — 569 p.
7. Rao M.R.M. A note on an integral inequality // J. Indian Math. Soc. — 1963. — 24, № 2. — P. 69 – 71.
8. Sharski J. Differential inequalities. — Warzawa: Pánst. wyd-wo nauk., 1967. — 256 p.
9. Matviitchuk K.S. Remarks on the method of comparison for systems of differential equations with rapidly
rotating phase // Ukr. Math. J. — 1982. — 34, № 4. — P. 456 – 462.
10. Matviitchuk K.S. On the question of the stability of systems connecting solids with the shock elements // Ibid.
— 1983. — 19, № 5. — P. 100 – 106.
11. Pouch N., Abetsc P., and Lalia M. Direct method of Liapounov in the theory of stability. — Moscow, 1980. —
300 p.
12. Mitchenko E.F., Rosov N.K. Differential equations with a small parameter and relaxation oscillations. —
Moscow, 1975. — 148 p.
13. Lefschetz S. Geometrical theory of differential equations. — Moscow, 1961.
14. Andronov A.A., Vitt A.A., and Haikin S.E. Theory of vibrations. — Moscow: Fizmatgiz, 1959.
15. Filatov A.N. Averaging methods in differential and integro-differential equations. — Tashkient: Fan, 1971. —
279 p.
16. Belmann R., Cook K.L. Difference differential equations. — Moscow: Mir, 1967. — 548 p.
Received 05.07.2002
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
|