On stability of the Hill's equation with damping
We consider the Hill’s equation with damping describing the parametric oscillations of the torsional pendulum excited by means of varying the moment of inertia of the rotating body. Using the method of a small parameter we have calculated analytically a fundamental system of solutions of this equat...
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| Zitieren: | On stability of the Hill's equation with damping / C. Cattani, E.A. Grebenikov, A.N. Prokopenya // Нелінійні коливання. — 2004. — Т. 7, № 2. — С. 169-179. — Бібліогр.: 12 назв. — англ. |
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| author | Cattani, C. Grebenikov, E.A. Prokopenya, A.N. |
| author_facet | Cattani, C. Grebenikov, E.A. Prokopenya, A.N. |
| citation_txt | On stability of the Hill's equation with damping / C. Cattani, E.A. Grebenikov, A.N. Prokopenya // Нелінійні коливання. — 2004. — Т. 7, № 2. — С. 169-179. — Бібліогр.: 12 назв. — англ. |
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| container_title | Нелінійні коливання |
| description | We consider the Hill’s equation with damping describing the parametric oscillations of the torsional pendulum excited by means of varying the moment of inertia of the rotating body. Using the method of a small
parameter we have calculated analytically a fundamental system of solutions of this equation in the form
of power series in the excitation amplitude ε with accuracy O(ε²) and verified the conditions of its stability.
In the first-order approximation in ε, we have proved that the resonance domain exists only if the excitation
frequency Ω is sufficiently close to the double natural frequency of the pendulum, and the corresponding
equation of the stability boundary has been obtained.
Розглядається рiвняння Хiлла зi згасанням, що описує параметричнi коливання крутильного
маятника, якi збуджуються змiною моменту iнерцiї тiла, що обертається. За допомогою методу малого параметра аналiтичним шляхом отримано фундаментальну систему розв’язкiв
цього рiвняння у виглядi степеневих рядiв вiдносно амплiтуди збудження ε з точнiстю до O(ε²)
та перевiрено виконання умов його стiйкостi. У першому наближеннi по ε доведено, що область
резонансу iснує лише в областi частот збудження Ω, близьких до подвiйної власної частоти маятника, i отримано рiвняння межi областi стiйкостi.
|
| first_indexed | 2025-12-07T16:17:24Z |
| format | Article |
| fulltext |
UDC 517.9
ON STABILITY OF THE HILL’S EQUATION WITH DAMPING
ПРО СТIЙКIСТЬ РIВНЯННЯ ХIЛЛА ЗI ЗГАСАННЯМ
C. Cattani
DiFarma, Univ. Salerno
Via Ponte Don Melillo, I-84084 Fisciano (SA), Italy
e-mail: ccattani@unisa.it
E. A. Grebenikov
Computing Center Rus. Acad. Sci.
Vavilova Str., 37, 117312 Moscow, Russia
e-mail: greben@ccas.ru
A. N. Prokopenya
Brest State Techn. Univ.
Moskowskaya Str., 267, 224017 Brest, Belarus
e-mail: prokopenya@brest.by
We consider the Hill’s equation with damping describing the parametric oscillations of the torsional pen-
dulum excited by means of varying the moment of inertia of the rotating body. Using the method of a small
parameter we have calculated analytically a fundamental system of solutions of this equation in the form
of power series in the excitation amplitude ε with accuracy O(ε2) and verified the conditions of its stability.
In the first-order approximation in ε, we have proved that the resonance domain exists only if the excitation
frequency Ω is sufficiently close to the double natural frequency of the pendulum, and the corresponding
equation of the stability boundary has been obtained.
Розглядається рiвняння Хiлла зi згасанням, що описує параметричнi коливання крутильного
маятника, якi збуджуються змiною моменту iнерцiї тiла, що обертається. За допомогою ме-
тоду малого параметра аналiтичним шляхом отримано фундаментальну систему розв’язкiв
цього рiвняння у виглядi степеневих рядiв вiдносно амплiтуди збудження ε з точнiстю до O(ε2)
та перевiрено виконання умов його стiйкостi. У першому наближеннi по ε доведено, що область
резонансу iснує лише в областi частот збудження Ω, близьких до подвiйної власної частоти ма-
ятника, i отримано рiвняння межi областi стiйкостi.
Introduction. We consider the second order linear differential equation of the form
d2θ
dt2
+ β(t, ε)
dθ
dt
+ κ(t, ε)θ(t) = 0, (1)
where β(t, ε) and κ(t, ε) are continuous periodic functions of time t with a period T , i. e., β(t+
+T, ε) = β(t, ε), κ(t+T, ε) = κ(t, ε) for all t and ε is a small parameter. Equations of this type
describe dynamical systems with intrinsic periodicity and appear in many branches of science
and engineering. A physical example which is considered here is the torsional oscillations of
the body mounted on an elastic shaft and excited by means of alternating its moment of inertia.
c© C. Cattani, E. A. Grebenikov, and A. N. Prokopenya, 2004
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2 169
170 C. CATTANI, E. A. GREBENIKOV, AND A. N. PROKOPENYA
In the case where β(t, ε) = 0, equation (1) reduces to the Hill’s equation which has been
the subject of many papers (see, for example, [1 – 3]). The case β(t, ε) = const was analyzed
by P. Pedersen [4]. It was shown that in both cases, depending on the parameters of the system,
there are values of the excitation frequency Ω = 2π/T and amplitude ε such that the solution
θ(t) increases unboundedly as t → ∞ and the motion of the system becomes unstable. This
phenomenon is known as a parametric resonance.
The parametric resonance in linear oscillating systems has been studied quite well and di-
fferent methods were developed [5]. The most general method is the classic Floquet method [6]
which is based on a calculation of the monodromy matrix and an analysis of the behaviour of its
eigenvalues. It was used for studying equation (1) and some more general systems of differential
equations in [7, 8]. But this method requires a large number of numerical integrations and this li-
mits its possibilities, especially, if coefficients of the equations depend on some parameters. The
main aim of the present paper is to study the stability of equation (1) in the case of parametric
oscillations of the torsional pendulum with damping and to determine analytically boundaries
of the domains of instability in the space of the parameters. It should be noted that the stability
analysis of differential equations with periodic coefficients is rather cumbersome but it can be
successfully done with a modern computer software such as, for example, the computer algebra
system Mathematica [9].
Criteria of the system stability. According to the general theory of linear differential equati-
ons with periodic coefficients (see, for example, [1]), behaviour of the solutions of equati-
on (1) is determined by its characteristic multipliers ρ which are just the eigenvalues of the
monodromy matrix X(T ) and, hence, are given by the characteristic equation
det(X(T )− ρI2) = 0, (2)
where I2 is an 2 × 2 identity matrix. Here X(t) is the principal fundamental matrix for the
equation (1) which is defined as
X(t) =
(
θ1(t) θ2(t)
θ′1(t) θ′2(t)
)
,
where θ1(t) and θ2(t) are two linearly independent solutions of equation (1) satisfying the
following initial conditions
θ1(0) = 1, θ′1(0) = 0,
(3)
θ2(0) = 0, θ′2(0) = 1.
Hence, the characteristic equation (2) can be written in the form
ρ2 − 2Aρ+B = 0, (4)
where
A =
1
2
(θ1(T ) + θ′2(T )),
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
ON STABILITY OF THE HILL’S EQUATION WITH DAMPING 171
B = θ1(T )θ′2(T )− θ′1(T )θ2(T ).
Thus, the characteristic multipliers ρ1,2 are functions of two parameters A and B and are given
by
ρ1,2 = A±
√
A2 −B. (5)
In order to determine ρ1,2 we should find two linearly independent solutions θ1(t) and θ2(t)
of the equation (1) satisfying the initial conditions (3). Although these solutions are not found
yet, we can characterize the properties of ρ1,2 in terms of the parameters A and B.
a) If 0 ≤ A2 < B then, according to (5), ρ1,2 are a complex-conjugate pair of characteristic
multipliers with absolute value |ρ1,2| =
√
B and can be represented as
ρ1,2 =
√
B exp (±i 2πσ
Ω
),
where i is the imaginary unit and σ is a real number. The corresponding characteristic exponents
µ1,2 are defined then as
µ1,2 =
1
T
ln ρ1,2 =
Ω
4π
lnB ± i σ
and the general solution of equation (1) may be written in the form
θ(t) = (C1Re(e(iσt)f(t)) + C2Im(eiσtf(t))) exp
(
t
Ω
4π
lnB
)
, (6)
where f(t) is a complex-valued periodic function with the period T =
2π
Ω
and C1, C2 are
arbitrary constants. It is obvious now from (6) that in the case where 0 ≤ A2 < B < 1 the
function θ(t) → 0 as t → ∞ and the motion of the system is asymptotically stable. For B = 1
solution (6) is bounded and oscillatory, and the motion of the system is stable. Thus, the system
becomes unstable only if B > 1.
b) In the case where A2 = B there is a single real characteristic multiplier ρ1 = A. It
can be regarded as the limit σ → 0 in case a). Again, the system is unstable for |A| > 1 and
asymptotically stable for |A| < 1. In the case where A = 1 and A = −1 there exists a periodic
solution with periods T and 2T , respectively. Besides, there may exist an additional solution
growing linearly with t → ∞ and the system will be unstable.
c) If A2 > B then, according to (5), the characteristic multipliers ρ1,2 are different real
numbers. They are both positive or negative if B > 0 and A > 0 or A < 0, respectively. In the
case where B < 0, the characteristic multipliers ρ1,2 have opposite signs. The general solution
of equation (1) can be written in the form
θ(t) = C1f1(t) exp
(
t
Ω
2π
ln |ρ1|
)
+ C2f2(t) exp
(
t
Ω
2π
ln |ρ2|
)
, (7)
where f1(t), f2(t) are real-valued periodic functions with the periods T =
2π
Ω
or T =
4π
Ω
depending on the sign of the corresponding characteristic multiplier. Hence, the system will be
unstable if at least one characteristic multiplier has the absolute value greater than 1.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
172 C. CATTANI, E. A. GREBENIKOV, AND A. N. PROKOPENYA
As a result, we can conclude that the domain of asymptotic stability of the equation (1) is
inside a triangle bounded by the lines B = 1, B = −1 ± 2A in the A − B plane. The points
that lie on the boundary of the triangle determine stable behaviour of its solutions, while the
domain being outside the triangle is just the domain of instability.
The parameter B can be found without solving equation (1). Indeed, since the functions
θ1(t) and θ2(t) are the solutions of equation (1), we can write
θ′′1(t) + β(t, ε)θ′1(t) + κ(t, ε)θ1(t) = 0, (8)
θ′′2(t) + β(t, ε)θ′2(t) + κ(t, ε)θ2(t) = 0. (9)
Multiplying equations (8), (9) by (−θ2(t)) and θ1(t), respectively, and adding them we obtain
the following relationship:
θ1(t)θ′′2(t)− θ2(t)θ′′1(t) =
d
dt
(θ1(t)θ′2(t)− θ2(t)θ′1(t)) =
= − β(t, ε)(θ1(t)θ′2(t)− θ2(t)θ′1(t)).
Hence, the function y(t) = θ1(t)θ′2(t)− θ2(t)θ′1(t) satisfies the following differential equation:
y′(t) = −β(t, ε) y(t). (10)
Using the initial conditions (3) we obtain the solution of equation (10) in the form
y(t) = exp
− t∫
0
β(τ, ε)dτ
.
Thus, the parameter B is determined only by the function β(τ, ε) and is given by
B = exp
− T∫
0
β(τ, ε)dτ
. (11)
Now we can conclude thatB > 0 for any β(t, ε). Hence, we can formulate the following criteria
for stability and instability of the system.
1. If the average value of the function β(t, ε) is negative, i.e.
T∫
0
β(t, ε)dt < 0, then B > 1
and the system is unstable.
2. If the average value of the function β(t, ε) is equal to zero then B = 1 and the system is
stable for |A| ≤ 1 and unstable for |A| > 1.
3. If the average value of the function β(t, ε) is positive then 0 < B < 1 and the system is
asymptotically stable for |A| < 1
2
(B + 1), stable for |A| =
1
2
(B + 1), and unstable for |A| >
>
1
2
(B + 1).
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
ON STABILITY OF THE HILL’S EQUATION WITH DAMPING 173
Equation of motion of the system. Let us imagine that a disk having a moment of inertia J0
is mounted on an elastic shaft and two point bodies of equal masses m are placed on its surface
symmetrically with respect to the axis of the shaft and can move without friction along radius
of the disk. The distance of each body from the axis of the shaft oscillates near the equilibrium
value r0 according to the law
r(t) = r0 + ε ϕ(Ωt),
where Ω and ε are the excitation frequency and amplitude respectively and ϕ(t) is a continuous
periodic function of time t with the period 2π. Hence, the moment of inertia of the system J
varies as
J(t) = J0 + 2m(r0 + ε ϕ(Ωt))2. (12)
Denoting the twisting angle of the disk by θ we can write the equation of motion of the system
in the form
d
dt
(
(t)
dθ
dt
)
= −γ dθ
dt
− cθ, (13)
where γ and c are the coefficient of viscous friction and the stiffness of the shaft respectively.
Substituting (12) into equation (13) we see that it is just the equation (1) with the coefficients
β(t, ε) =
γ + 4εmΩ(r0 + ε ϕ(Ωt))ϕ′(Ωt)
J0 + 2m(r0 + ε ϕ(Ωt))2
, κ(t, ε) =
c
J0 + 2m(r0 + ε ϕ(Ωt))2
, (14)
that are periodic functions of the period T =
2π
Ω
. Substituting β(t, ε) in (11) we can represent
the parameter B in the form
B = exp
− T∫
0
γ + 4εmΩ(r0 + ε ϕ(Ωt))ϕ′(Ωt)
J0 + 2m(r0 + ε ϕ(Ωt))2
dt
. (15)
Since the function ϕ(t) is supposed to be periodic with the period 2π, we have
ϕ(Ω T ) = ϕ
(
Ω
2π
Ω
)
= ϕ(2π) = ϕ(0)
and
T∫
0
4εmΩ(r0 + ε ϕ(Ωt))ϕ′(Ωt)
J0 + 2m(r0 + ε ϕ(Ωt))2
dt =
T∫
0
d(ln(J0 + 2m(r0 + ε ϕ(Ωt))2)) = 0.
Thus, the relationship (15) can be rewritten as
B = exp
− T∫
0
γ
J0 + 2m(r0 + ε ϕ(Ωt))2
dt
. (16)
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
174 C. CATTANI, E. A. GREBENIKOV, AND A. N. PROKOPENYA
The denominator in the expression under the integral sign in (16) is a positive function of t.
Hence, if γ < 0 then B > 1 and the system will be unstable for any values of other parameters.
So, we’ll suppose further that γ ≥ 0 and, hence, 0 < B ≤ 1. This means that the system can be
unstable only if |A| > 1
2
(B + 1). And the lines
B = −1− 2A, −1 ≤ A < 0, (17)
and
B = −1 + 2A, 0 ≤ A ≤ 1, (18)
are stability boundaries in the A−B plane.
Calculation of the parameters A and B with the method of a small parameter. The coeffi-
cients β(t, ε) and k(t, ε) defined in (14) can be represented as series expansions in powers
of ε
β(t, ε) = 2β0 +
∞∑
j=1
βj(t)εj , κ(t, ε) = ω2
0 +
∞∑
j=1
κj(t)εj , (19)
where
β0 =
γ
2p
, ω2
0 =
c
p
, p = J0 + 2mr2
0, β1 =
4mr0
p
(−2β0ϕ(Ωt) + Ωϕ′(Ωt)),
β2 =
4mϕ(Ωt)
p2
(−β0ϕ(Ωt)(p− 8mr2
0) + Ω(p− 4mr2
0)ϕ′(Ωt)),
κ1 = −4
p
mr0ω
2
0ϕ(Ωt), κ2 =
2m
p2
(−p+ 8mr2
0)ω2
0ϕ(Ωt)2, . . . .
The series (19) converge for any t and sufficiently small ε and βj(t), κj(t) are continuous functi-
ons. So, according to Poincare – Liapunov theorem [10 – 12], a general solution of equation (1)
can be also represented as a power series
θ(t) =
∞∑
j=0
θj(t)εj (20)
that converges for any t and sufficiently small ε with θj(t) being continuous functions.
In order to obtain differential equations determining functions θj(t) let us substitute expansi-
ons (19), (20) into equation (1). Then, equating coefficients of εj , j = 0, 1, 2, ... , in the left- and
the right-hand sides of the equation we obtain the following system of differential equations:
θ′′0(t) + 2β0θ
′
0(t) + ω2
0θ0(t) = 0,
(21)
θ′′j (t) + 2β0θ
′
j(t) + ω2
0θj(t) = fj(t), j = 1, 2, . . . ,
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
ON STABILITY OF THE HILL’S EQUATION WITH DAMPING 175
where
fj(t) = −
j∑
n=1
(
βn(t)θ′j−n(t) + κn(t)θj−n(t)
)
. (22)
Two linearly independent solutions θ0(t) of the first equation in (21) must satisfy the initial
conditions (3). The corresponding functions are easily found and are given by
θ0(t) = e−β0t
(
cos(ωt) +
β0
ω
sin(ωt)
)
, (23)
θ0(t) =
1
ω
e−β0t sin(ωt), (24)
where ω =
√
ω2
0 − β2
0 . Initial conditions for the functions θj , j = 1, 2, . . . , can be written
then as
θj(0) = θ′j(0) = 0. (25)
Solving the second equation in (21) with initial conditions (25) we obtain the following expressi-
on for the functions θj j = 1, 2, . . . ,
θj(t) = − 1
2i ω
e−(β0+i ω)t
t∫
0
j(τ) e(β0+i ω)τ dτ +
+
1
2i ω
e−(β0−i ω)t
t∫
0
fj(τ) e(β0−i ω)τ dτ, (26)
where the functions fj(t) are defined in (22). Using the recurrence relation (26) we can successi-
vely calculate the coefficients θj in the expansion (20). But, as j is growing the calculations
become more and more cumbersome. So this method can be reasonably realized only with a
computer software.
Using the system Mathematica we have done the above calculations in the case of ϕ(t) =
= cos t for the initial functions θ0 given in (23), (24) with accuracy of ε2. As a result we have
found the parameter A as a power series in ε,
A = exp
(
−2πβ0
Ω
)(
cos
(2πω
Ω
)
+ ε2 πm
p2ωΩ(Ω2 − 4ω2)
(
−p(4ω2 − Ω2)×
×
(
(ω2 − β2
0) sin
(2πω
Ω
)
+ 2ωβ0 cos
(2πω
Ω
))
+ 8mr2
0
(
(ω2(3ω2 − Ω2)−
− β2
0(6ω2 − Ω2)− β4
0) sin
(2πω
Ω
)
+ 2ωβ0(4ω2 − Ω2) cos
(2πω
Ω
))))
, (27)
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
176 C. CATTANI, E. A. GREBENIKOV, AND A. N. PROKOPENYA
where the error term isO(ε3). Substituting the expansion (19) into (16) we obtain the parameter
B in the form
B = exp
(
−4πβ0
Ω
)(
1 + ε2 4πmβ0
p2Ω
(p− 8mr2
0)
)
. (28)
It should be noticed that the series (27), (28) converge for any Ω and sufficiently small ε. And
calculating the parameters A and B we can easily find the characteristic multipliers ρ1,2 accor-
ding to (5).
Determination of the domains of instability. It follows from (27), (28) that in the case of
ε = 0, the parameters A and B take the form
A = exp
(
−2πβ0
Ω
)
cos
(2πω
Ω
)
, B = exp
(
−4πβ0
Ω
)
.
Hence,
B + 1± 2A = 1 + exp
(
−4πβ0
Ω
)
± 2 exp
(
−2πβ0
Ω
)
cos
(2πω
Ω
)
≥
(
1− exp
(
−2πβ0
Ω
))2
> 0
for any Ω and β0 > 0. This means that the corresponding point (A,B) in the A − B plane
belongs to the domain of asymptotic stability of the system. This has been expected because
for ε = 0 equation (1) reduces to the damped oscillator equation whose general solution is
well-known. Since the parameters A and B are continuous functions of ε, the point (A,B) will
belong to the asymptotic stability domain for sufficiently small ε > 0 and β0 = const > 0 as
well. Only in the case of β0 = 0, ε = 0, the point (A,B) belongs to the stability boundaries
(17), (18) if
Ω = Ω0 =
2ω0
n
, n = 1, 2, 3, . . . . (29)
Hence, the domains of instability in the space of the parameters (Ω, β0, ε) can exist only in a vi-
cinity of the points (29). The boundary of every such domain is some surface which degenerates
into a point as ε → 0. Thus, considering these domains we can represent Ω = Ω(ε), β0 = β0(ε)
for sufficiently small ε as the power series
Ω = Ω0 + Ω1ε+ Ω2ε
2 + . . . ,
(30)
β0 = β01ε+ β02ε
2 + . . . .
Substituting (30) into (27), (28) and expanding their right-hand sides in powers of ε we
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
ON STABILITY OF THE HILL’S EQUATION WITH DAMPING 177
obtain the parameters A and B in the form
A = cos
(2πω0
Ω0
)
− 2πε
Ω2
0
(
β01Ω0 cos
(2πω0
Ω0
)
− ω0Ω1 sin
(2πω0
Ω0
))
+
+
πε2
ω0Ω4
0
(
β2
01Ω2
0
(
2πω0 cos
(2πω0
Ω0
)
+ Ω0 sin
(2πω0
Ω0
))
−
− 2β01ω0Ω0Ω1
(
2πω0 sin
(2πω0
Ω0
)
− Ω0 cos
(2πω0
Ω0
))
−
− 2ω0(β02Ω3
0 + πω2
0Ω2
1) cos
(2πω0
Ω0
))
+
+
πω0ε
2
p2Ω3
0(Ω2
0 − 4ω2
0)
(
4ω2
0(mΩ2
0(−p+ 6mr2
0) + 2p2Ω2
1 − 2p2Ω0Ω2) +
+ Ω2
0(mΩ2
0(p− 8mr2
0)− 2p2Ω2
1 + 2p2Ω0Ω2)
)
sin
(2πω0
Ω0
)
,
(31)
B = 1− 4πβ01ε
Ω0
+
4πε2
Ω2
0
(2πβ2
01 − β02Ω0 + β01Ω1).
Now one can easily see that, in fact, the relations (17), (18) can be fulfilled only in a vicinity
of the points (29). Moreover, only one of them can be fulfilled in every point (29). Substituting
(31) into (17), (18) we obtain, successively for n = 1, 2, 3, 4, . . . , the following equations:
B + 1 + 2A =
1
4
π2ε2
(4β2
01 + Ω2
1
ω2
0
− 4m2r2
0
p2
)
= 0, (32)
B + 1− 2A =
4π2ε2
ω2
0
(β2
01 + Ω2
1) = 0, (33)
B + 1 + 2A =
9π2ε2
4ω2
0
(4β2
01 + 9Ω2
1) = 0, (34)
B + 1− 2A =
16π2ε2
ω2
0
(β2
01 + 4Ω2
1) = 0, (35)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equations (33) – (35) have the only solution β01 = Ω1 = 0. This means that the domains of
instability in a vicinity of the points (29) for n = 2, 3, 4, . . . can be found only if we take into
account the third and higher order terms in the expansion (20). Equation (32) shows that the
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
178 C. CATTANI, E. A. GREBENIKOV, AND A. N. PROKOPENYA
Fig. 1. The cross-section of the boundary surface
by the plane β0 = const.
Fig. 2. The cross section of the boundary surface
by the plane β0 = const.
domain of instability, where the inequality B + 1 + 2A < 0 is fulfilled, exists in a vicinity on the
point Ω0 = 2ω0. The boundary of this domain in the space of the parameters (Ω, β0, ε) is given
by the equation
4β2
0 + (Ω− 2ω0)2 =
4m2r2
0ω
2
0
p2
ε2, (36)
where we have taken into account the relations β0 = εβ01, Ω = 2ω0 +εΩ1. Thus, we have found
the stability boundary (36) in linear approximation in ε.
Equation (36) determines a cone in three-dimensional space (Ω, β0, ε) and the system will
be unstable if the point determined by these parameters lies inside the cone. The cross sections
of the cone by the planes ε = const and β0 = const are shown in Fig. 1, 2 respectively. The first
graph shows that for any value of the excitation frequency Ω from the interval
|Ω− 2ω0| ≤
2mr0ω0ε
p
and amplitude ε there exists a maximal value of the damping coefficient β0 for which the
parametric resonance can still occur. On the other hand, if the coefficient β0 is small enough
and fixed then the parametric resonance can occur only if the excitation amplitude ε is greater
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
ON STABILITY OF THE HILL’S EQUATION WITH DAMPING 179
then some threshold value. Recall that the equation for the instability boundary (36) has been
obtained in the first approximation in the excitation amplitude ε. Taking into account the hi-
gher approximations we can notice that it is deformed with ε growth. Besides, the domains of
instability can arise in a vicinity of other points (29).
Conclusion. In the present paper we have studied the parametric oscillations of the torsi-
onal pendulum with damping which are described by the second order differential equati-
on with periodic coefficients. The excitation of the pendulum is realized by means of vary-
ing the moment of inertia of the rotating body. We have calculated analytically two linearly
independent solutions of the equation of motion in the form of power series in the excitati-
on amplitude ε with accuracy O(ε2) and verified the conditions of its stability. It has been
shown that the domains of a parametric resonance can exist only in a vicinity of the points
Ω =
2ω0
n
n = 1, 2, 3, . . . , where Ω and ω0 are the excitation frequency and natural frequency
of the pendulum, respectively. In the first approximation in the excitation amplitude ε it has
been proved that the resonance domain exists only in a vicinity of the point Ω = 2ω0 and the
corresponding equation of the stability boundary was obtained.
1. Yakubovich V. A., Starzhinskii V. M. Linear differential equations with periodic coefficients. — Moscow:
Nauka, 1972. — 720 p. (in Russian).
2. Grebenikov E. A., Prokopenya A. N. Determination of the boundaries between the domains of stability and
instability for the Hill’s equation // Nonlinear Oscillations. — 2003. — 6, № 1. — P. 42 – 51.
3. Nayfeh A. H., Mook D. T. Nonlinear oscillations. — New York: John Wiley, 1979. — 704 p.
4. Pedersen P. On stability diagrams for damped Hill equations // Quart. Appl. Math. — 1985. — 42. —
P. 177 – 195.
5. Yakubovich V. A., Starzhinskii V. M. Parametric resonance in linear systems. – Moscow: Nauka, 1987. —
274 p. (in Russian).
6. Cesari L. Asymptotic behaviour and stability problems in ordinary differential equations. — Second ed. —
New York: Springer, 1962. — 271 p.
7. Seyranian A. P., Solem F., and Pedersen P. Multi-parameter linear periodic systems: sensitivity, analysis and
applications // J. Sound and Vibr. — 2000. – 229. — P. 89 – 111.
8. Mailybaev A. A., Seyranian A. P. Parametric resonance in systems with small dissipation // J. Appl. Math. and
Mech. — 2001. — 65, No. 5. — P. 755 – 767.
9. Wolfram S. The Mathematica book. – Cambridge: Univ. Press, 1999. — 1470 p.
10. Erouguine N. P. Linear systems of differential equations. — Minsk: Byelorus. Acad. Sci., 1963. — 272 p. (in
Russian).
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12. Grimshaw R. Nonlinear ordinary differential equations. – CRC Press, 2000. — 328 p.
Received 24.02.2004
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
|
| id | nasplib_isofts_kiev_ua-123456789-177003 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-07T16:17:24Z |
| publishDate | 2004 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cattani, C. Grebenikov, E.A. Prokopenya, A.N. 2021-02-09T18:39:25Z 2021-02-09T18:39:25Z 2004 On stability of the Hill's equation with damping / C. Cattani, E.A. Grebenikov, A.N. Prokopenya // Нелінійні коливання. — 2004. — Т. 7, № 2. — С. 169-179. — Бібліогр.: 12 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/177003 517.9 We consider the Hill’s equation with damping describing the parametric oscillations of the torsional pendulum excited by means of varying the moment of inertia of the rotating body. Using the method of a small parameter we have calculated analytically a fundamental system of solutions of this equation in the form of power series in the excitation amplitude ε with accuracy O(ε²) and verified the conditions of its stability. In the first-order approximation in ε, we have proved that the resonance domain exists only if the excitation frequency Ω is sufficiently close to the double natural frequency of the pendulum, and the corresponding equation of the stability boundary has been obtained. Розглядається рiвняння Хiлла зi згасанням, що описує параметричнi коливання крутильного маятника, якi збуджуються змiною моменту iнерцiї тiла, що обертається. За допомогою методу малого параметра аналiтичним шляхом отримано фундаментальну систему розв’язкiв цього рiвняння у виглядi степеневих рядiв вiдносно амплiтуди збудження ε з точнiстю до O(ε²) та перевiрено виконання умов його стiйкостi. У першому наближеннi по ε доведено, що область резонансу iснує лише в областi частот збудження Ω, близьких до подвiйної власної частоти маятника, i отримано рiвняння межi областi стiйкостi. en Інститут математики НАН України Нелінійні коливання On stability of the Hill's equation with damping Про стійкість рівняння Хілла зі згасанням Об устойчивости уравнения Хилла з затуханием Article published earlier |
| spellingShingle | On stability of the Hill's equation with damping Cattani, C. Grebenikov, E.A. Prokopenya, A.N. |
| title | On stability of the Hill's equation with damping |
| title_alt | Про стійкість рівняння Хілла зі згасанням Об устойчивости уравнения Хилла з затуханием |
| title_full | On stability of the Hill's equation with damping |
| title_fullStr | On stability of the Hill's equation with damping |
| title_full_unstemmed | On stability of the Hill's equation with damping |
| title_short | On stability of the Hill's equation with damping |
| title_sort | on stability of the hill's equation with damping |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/177003 |
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