E-ISSN 2281-4612
ISSN 2281-3993
Academic Journal of Interdisciplinary Studies
Published by MCSER-CEMAS-Sapienza University of Rome
Vol 2 No 6
August 2013
Stability Analysis of Plane Vibrations of a Satellite in a Circular Orbit
H.Yehia
H. Nabih
Department of Mathematics, Faculty of Science,
Mansoura University, Mansoura 35516, Egypt
Doi:10.5901/ajis.2013.v2n6p127
Abstract
In this paper, we analyze for stability the problem of planar vibrational motion of a satellite about
its centre of mass. The satellite is dynamically symmetric whose center of mass is moving in a
circular orbit. The in-plane motion is a simple pendulum-like motion in which the axis of symmetry
of the satellite remains in the orbital plane. It is expressed in terms of elliptic functions of time.
Using Routh’s equations we study the orbital stability of planar vibrations of the satellite, in the
sense that the stable in-plane motions remain under perturbation very near to the orbital plane.
The linearized equation for the out-of-plane motion takes the form of a Hill’s equation. Detailed
analysis of stability using Floquet theory is performed analytically and numerically. Zones of
stability and instability are illustrated graphically in the plane of the two parameters of the
problem: the ratio of moments of inertia and the amplitude of the unperturbed motion.
1. Introduction
Suppose that the satellite is a dynamically symmetric rigid body and its center of mass moves in a
circular orbit. A possible solution of the equations of motion represents what is called in-plane
motion. That is a pendulum-like motion in which the axis of symmetry of the satellite remains all
the time in the orbital plane (see e.g. [1]). The aim of this paper is to study the stability of the
planar periodic motion, in particular, the stability of vibration motion. Markeev [2] was the first to
study the problem in the above formulation. He studied the orbital stability of the planar periodic
motion for dynamically symmetric satellite whose polar axis of the ellipsoid of inertia is shorter than
the equatorial ones. But, digrams of stability contained some imperfections because of low
accuracy of numerical calculations. Akulenko, Nesterov and Shmatkov [3] pointed out these
imperfections. Markeev and Bardin [4] studied the problem of stability of the planer periodic
motions in the nonlinear setting using the normal forms of Birkhoff. Such analysis can be
performed only numerically.
In this paper, we study the stability of plane vibrational motions of a satellite in a circular
orbit. The problem is reduced to a form of Hill’s equation, i.e. a single second-order linear
differential equation with a periodic coefficient. It turned out that this equation is a generalization
of Lame’s equation of the first rank. To deal with this equation we combine Floquet theory [6] with
the method devised by Ince for Lame’s equation, which consists in finding primitive periodic
solutions in the form of Fourier series expansions. Point sets in the space of parameters,
corresponding to those solutions separate zones of stability and instability. We obtain a detailed
picture of those zones, analytically and in a purely numerical treatment. Both analyses are in
complete agreement.
127
E-ISSN
N 2281-4612
ISSN 2281-3993
2
Academic Journal
J
of Interdisciplinnary Studies
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U
of Rome
Vol 2 No 6
August 2013
2. Ma
athematical Forrmulation
2.1 Description
D
of mottion
er of the Earth and by ࡻᇱ the currrent position of the center of ma
ass of the
Denote by ࡻ the cente
adius R with centtre atࡻ, see fig.1. Let ࡻࢄࢅࢆ be an inertial
satellitte describing a circular orbit of ra
frame of reference, ࡻᇱ ࢞૚ ࢟૚ ࢠ૚ be the orrbital system with
h ࢞૚ െ axis alongࡻࡻᇱ , ࢟૚ along th
he tangent
e orbit in the dire
ection of motion of the satellite and
a ࢠ૚ െ in the d
direction orthogonal to the
to the
orbitall plane and let ࡻᇱ ࢞࢟ࢠ be the syste
em of central prin
ncipal axes of the
e satellite with moments of
inertia
a A,B and C, resspectively. Let also ࢻǡ ࢼǡ ࢽ be th
he three unit ve
ectors in the dire
ections of
࢞૚ ǡ ࢟૚ ǡ ࢠ૚ and ࣓ the angular velocity vector of the satellite
s
relative to the orbital syystem, all
compo
onents being refe
erred to the bodyy systemࡻᇱ ࢞࢟ࢠ. To describe the orientation of th
he satellite
relativve to the orbital frame we shall use Euler’s ang
gles ࣒the angle of precession around the
ࢠ૚ െaxxis, ࣂthe angle between
b
the axis of
o the body and the
t orbital plane (so that the nuta
ation angle
࣊
between ࢠ andࢠ૚ െaxxis is െ ࣂ) and ࣐ the angle of proper rotation o
of the satellite around
a
its
૛
ࢠ െaxiis.
Fig.1
c write
In those variables we can
ࢻ ൌ ሺࢉ࢕࢙࣒ࢉ࢕࢙࣐ െ ࢙࢏࢔ࣂ࢙࢏࢔࣐࢙࢏࢔࣒
࣒ǡ െࢉ࢕࢙࣒࢙࢏࢔࣐ െ ࢙࢏࢔ࣂࢉ࢕࢙࣐࢙࢏࢔࣒
࣒ǡ ࢉ࢕࢙ࣂ࢙࢏࢔࣒ሻ
ࢼ ൌ ሺ࢙࢏࢔࣒ࢉ࢕࢙࣐ ൅ ࢙࢏࢔ࣂ࢙࢏࢔࣐ࢉ࢕࢙࣒
࣒ǡ െ࢙࢏࢔࣒ࢉ࢕࢙࣐ ൅ ࢙࢏࢔ࣂ࢙࢏࢔࣐ࢉ࢕࢙࣒
࣒ǡ െࢉ࢕࢙ࣂࢉ࢕࢙࣒ሻ
ࢽ ൌ ሺࢉ࢕࢙ࣂ࢙࢏࢔࣐ǡ ࢉ࢕࢙ࣂࢉ࢕࢙࣐ǡ ࢙࢏࢔ࣂሻ
࣓ ൌ ሺ࣒ሶࢉ࢕࢙ࣂ࢙࢏࢔࣐
࣐ െ ࣂሶࢉ࢕࢙࣐ǡ ࣒ሶࢉ࢕࢙ࣂ
ࣂࢉ࢕࢙࣐ ൅ ࣂሶ࢙࢏࢔࣐ǡ ࣒ሶ࢙࢏࢔࣐ ൅ ࣐ሶ)
I what follows we
In
w study the rota
ational motion of the satellite abo
out its centre of mass.
m
It is
supposed that the rottational motion is
i independent of
o the orbital mo
otion of the satellite with
ஜ
m angular speed π (say), given by
b the expression
n π૛ ൌ ૜ where μ is Gauss’ constant of the
uniform
ࡾ
Earth [1]. The Lagrang
gian of the rotatio
onal motion can be
b written as
૚
ࡸ ൌ ࢝۷Ǥ ࢝ െ ࢂ
૛
(
(1)
(
(2)
࢝ ൌ ࣓ ൅ πࢽ
128
Academic Journal of Interdisciplinary Studies
E-ISSN 2281-4612
ISSN 2281-3993
Published by MCSER-CEMAS-Sapienza University of Rome
Vol 2 No 6
August 2013
where ࢝ the absolute angular velocity of the satellite and V its potential function in the
gravitational field of Earth. A normally acceptable approximate form (see e.g. Beletskii [1]) is:
૜ஜ
૜
(3)
ࢂ ൌ ૜ ࢻ۷Ǥ ࢻ ൌ π૛ ࢻ۷Ǥ ࢻ
૛ࡾ
૛
Substituting (2) and (3) in (1) we get
૚
π૛
૛
૛
ࡸ ൌ ൫࡭࢖૛ ൅ ࡮ࢗ૛ ൅ ࡯࢘૛ ൯ ൅ ሺ࡭࢖ࢽ૚ ൅ ࡮ࢗࢽ૛ ൅ ࡯࢘ࢽ૜ ሻ ൅
ሺࢽ۷Ǥ ࢽ ൅ ૜ࢻ۷Ǥ ࢻሻ
(4)
The equation of motion of the satellite relative to the orbital frame, corresponding to the
Lagrangian can be written in the Euler-Poisson form [5]:
ࡳሶ ൅ ࣓ ൈ ൫ࡳ െ ૛πࢽ۷൯ ൌ ૜πଶ ࢻ ൈ ࢻ۷ െ πଶ ࢽ ൈ ࢽࡵ
ࢻሶ ൅ ࣓ ൈ ࢻ ൌ Ͳǡ ࢼሶ ൅ ࣓ ൈ ࢼ ൌ Ͳǡࢽሶ ൅ ࣓ ൈ ࢽ ൌ Ͳ
(5)
ଵ
where۷ ൌ ሺܶ‫ݎ‬۷ሻߜ െ ۷. This system admits Jacobi's integral
ଵ
ଶ
ቀ ቁ ࣓۷Ǥ ࣓ ൅
ଶ
πమ
ଶ
(6)
ሺ͵ࢻ۷Ǥ ࢻ െ ࢽ۷Ǥ ࢽሻ ൌ ݄
It was noted in [5] that this system of equations (6) of motion of the satellite on a circular
orbit is form-equivalent to the equations of motion of an electrically charged rigid body about a
fixed point of it, while acted upon by gravitational, electric and magnetic fields. The system (5)
admits a simple solution, which represents motion of the satellite with two of its principal axes
always in the orbital plane and the third orthogonal to it. Let the last axis be the ࢞ െaxis. It is not
hard to see that this solution is
࣓ ൌ ࣓ࢽǡ ࣓ ൌ ࣒ሶǡ
ࢻ ൌ ሺ૙ǡ ࢉ࢕࢙࣒ǡ ࢙࢏࢔࣒ሻǡࢼ ൌ ሺ૙ǡ െ࢙࢏࢔࣒ǡ ࢉ࢕࢙࣒ሻǡࢽ ൌ ሺ૚ǡ ૙ǡ ૙ሻ
When ߰ሶ ൌ Ͳ the satellite takes the relative equilibrium position with one of its axes directed to
the Earth’s centre and another directed along the tangent to the orbit. In general we have two
types of motion: vibration and rotation. Each is centered about one of the equilibrium positions.
஼
This occurs according to the ratio ߙ ൌ of moment of inertia of the satellite, restricted by the
஺
triangle inequality to the interval [0, 2]. The subinterval
Ͳ ൏ ߙ ൏ ͳ corresponds to the case ‫ ܥ‬൏ ‫ ܣ‬ൌ ‫ܤ‬
ͳ ൏ ߙ ൏ ʹ to the case‫ ܥ‬൐ ‫ ܣ‬ൌ ‫ ܤ‬.
For the first case the satellite is moving around the tangent of the orbital plane and the
second cases the satellite is moving aroundࡻࡻᇱ .
2.2 Equation of motion
We may express the Lagrangian of the problem in the form:
ͳ
πଶ
ͳ
ሾ‫ܣ‬ሺߛଵଶ ൅ ߛଶଶ ሻ
‫ ܮ‬ൌ ‫ܣ‬ሺ‫݌‬ଶ ൅ ‫ ݍ‬ଶ ሻ ൅ ‫ ݎܥ‬ଶ ൅ πሾ‫ܣ‬ሺ‫ߛ݌‬ଵ ൅ ‫ߛݍ‬ଶ ሻ ൅ ‫ߛݎܥ‬ଷ ሿ ൅
ʹ
ʹ
ʹ
൅‫ߛܥ‬ଷଶ ൅ ሺ‫ ܥ‬െ ‫ܣ‬ሻሺͳ െ ͵ߙଷଶ ሻሿ
Case i (Ͳ ൏ ߙ ൏ ͳ ):
గ
Under the condition i the Lagrangian is (ߖ ൌ ൅ ߰ )
ଶ
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Fig.2
‫ܣ‬
‫ܣ‬
‫ܥ‬
‘•ଶ ߠ ൅ •‹ଶ ߠ൨ ߖሶ ଶ ൅ ሾπሺ‫ܣ‬
‫•‘ ܣ‬ଶ ߠ ൅ ‫‹• ܥ‬ଶ ߠሻ
ߠ ൅ ‫߮ܥ‬ሶ •‹ ߠሿߖሶ ൅ ߠሶ ଶ
ʹ
ʹ
ʹ
஼
πమ
൅
൅π߮ሶ ‫ ߠ ‹• ܥ‬൅ ߮ሶ ଶ ൅ ଶ ሺ‫ ܣ‬െ ‫ܥ‬ሻሾ ‘•• ଶ ߠ ൅ ͵ ‘•ଶ ߖ ‘•
‘ ଶ ߠሿ
‫ܮ‬ൌ൤
ଶ
N
Now,
we note tha
at ߮ is a cyclic coo
ordinate where :
߲‫ܮ‬
ሶ
‫݌‬ఝ ൌ
ൌ ‫ܥ‬ൣ൫ߖ ൅ π൯ •‹ ߠ ൅ ߮ሶ ൧ ൌ ‫ܥ‬ଵ
߲߮ሶ
߮ ൌ ‫ܥ‬ଵ െ ൫ߖሶ ൅ π൯ ‫ߠ ݊݅ݏ‬
߮ሶ
N
Next,
we ignore ɔ and construct th
he Routhian:
πଶ
‫ ܣ‬ଶ
ܴ ൌ ൣߠሶ ൅ ߖሶ ଶ ‘•
‘ ଶ ߠ൧ ൅ ሾπ‫ •‘ ܣ‬ଶ ߠ ൅ ‫ܥܥ‬ଵ •‹ ߠሿߖሶ ൅
ሾ‫ ܥ‬െ ‫ ܣ‬൅
ʹ
ʹ
஼஼భ
ଶ
ଶ
ଶ
‫ ߠ •‘ ܣ‬൅ ͵ሺ‫ ܣ‬െ ‫ܥ‬ሻ
‫ߠ •‘ ߖ •‘ ܥ‬ሿ ൅
ሺʹπ •‹ ߠ െ ‫ܥ‬ଵ ሻ
ଶ
(
(8)
(
(9)
(
(10)
The satellite can perform
T
p
the in-pla
ane motions only
y if the real consta
ant ‫ܥ‬ଵ =0.
A
According
to this condition the Rou
uthian takes the form:
f
πଶ
‫ܣ‬
ܴ ൌ ൣߠሶ ଶ ൅ ߖሶ ଶ ‘•
‘ ଶ ߠ൧ ൅ ሾπ‫ •‘ ܣ‬ଶ ߠ ൅ ‫ߠ ‹• ܥ‬ሿߖሶ ൅
ሾ‫ ܥ‬െ ‫ ܣ‬൅
ʹ
ʹ
ଶ
ଶ
ଶ
‫ܣ‬
‫ ߠ •‘ ܣ‬൅ ͵ሺ‫ ܣ‬െ ‫ܥ‬ሻ ‘• ߖ ‘• ߠሿ
(
(11)
T equations of motion of the sattellite in this case
The
e become
‫ ܣ‬ሷ ‘•ଶ ߠ െ ʹ‫ߠܣ‬ሶ൫ߖሶ ൅ π൯ •‹ ߠ ‘• ߠ ൅ ͵πଶ ሺ‫ ܣ‬െ ‫ܥ‬ሻ ‘•
‫ߖܣ‬
‘ ଶ ߠ •‹ ߖ ‘• ߖ ൌ Ͳ
(
(12)
ଶ
ଶ ሺ‫ܣ‬
ሷ
ሶ
ሶ
െ ‫ܥ‬ሻ
‫ ܣ‬൅ ‫ ߠ •‘ ߠ ‹• ߖܣ‬൅ ʹ‫ܣ‬πߖ •‹ ߠ ‘• ߠ ൅ ͵π
‫ߠܣ‬
‫•‘ ܥ‬ଶ ߖ •‹ ߠ ‘•• ߠ ൅ πଶ ‫ ߠ ‹• ܣ‬ൌ Ͳ (13)
D
Defining
the non-dimensional time
e ߬ ൌ π‫ ݐ‬and deno
oting derivatives w
with respect to
τ by ሺሻᇱthe non-ddimensional equattions of motion become:
ߖ ᇱᇱ ‘•ଶ ߠ െ ʹߠ ᇱ ሺߖ
ߖ ᇱ ൅ ͳሻ •‹ ߠ ‘• ߠ ൅ ͵ሺͳ െ ߙሻ ‘•ଶ ߠ •‹ ߖ ‘• ߖ ൌ Ͳ
(
(14)
ᇱᇱ
ᇱଶ
ߠ ൅ ߖ •‹ ߠ ‘• ߠ ൅ ʹߖ ᇱ •‹ ߠ ‘• ߠ ൅ ሺͳ െ ߙሻ ‘•ଶ ߖ •‹ ߠ ‘• ߠ
൅ ߠൌͲ
൅•‹
(
(15)
C
Case
ii (ͳ ൏ ߙ ൏ ʹ ):
I this case the equatorial axis rem
In
mains approximattely orthogonal to
o the plane of the
e orbit and
the po
olar axis perform
ms small angular oscillations abou
ut this plane so w
we shall use Euller’s angle
߰the angle of precession around the ‫ݖ‬ଵ െaxis
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of Rome
Vol 2 No 6
August 2013
Fig.3
unction takes the form:
Now, tthe Lagrangian fu
஺
஼
஺
஼
‫ ܮ‬ൌ ሺ ‘•ଶ ߠ ൅ •‹
• ଶ ߠሻ ߰ሶ ଶ ൅ ሾπሺ‫•‘ ܣ‬ଶ ߠ ൅ ‫‹• ܥ‬ଶ ߠሻሻ ൅ ‫߮ܥ‬ሶ •‹ ߠሿ߰ሶ ൅ ߠሶ ଶ ൅ π߮ሶ ‫ ߠ ‹• ܥ‬൅ ߮ሶ ଶ െ
ଶ
ଶ
ଶ
ଶ
πమ
ଶ
ሾͶሺ‫ ܥ‬െ ‫ܣ‬ሻ ‘•ଶ ߠ െ ʹ‫ ܥ‬൅ ‫ ܣ‬െ ͵ሺ‫ ܥ‬െ ‫ܣ‬ሻ ‘•ଶ ߰ ‘•ଶ ߠሿ
‫݌‬ఝ ൌ
డ௅
డఝሶ
ൌ ‫ܥ‬ൣ൫߰ሶ ൅ π൯ •‹ ߠ ൅ ߮ሶ ൧ ൌ ‫ܥ‬ଶ
(
(16)
(
(17)
a we get the Ro
and
outhian
πଶ
‫ ܣ‬ଶ
ܴ ൌ ൣߠሶ ൅ ߰ሶ ଶ ‘• ଶ ߠ൧ ൅ ሾπ‫ •‘ ܣ‬ଶ ߠ ൅ ‫ܥܥ‬ଶ •‹ ߠሿ߰ሶ ൅
ሾ‫ ܥ‬െ ‫ ܣ‬൅
ʹ
ʹ
஼஼మ
ଶ
ଶ
ଶ
ሺ െ ͵‫ܥ‬ሻ ‘• ߠ ൅ ͵ሺ‫ ܥ‬െ ‫ܣ‬ሻ ‘• ߰ ‘• ߠሿ ൅
ሺͶ‫ܣ‬
ሺʹπ
π •‹ ߠ െ ‫ܥ‬ଶ ሻ
ଶ
(
(18)
Then the equation
T
ns of motion are:
‫ ܣ‬ሷ ‘•ଶ ߠ ൅ ൣ‫ܥܥ‬ଶ ‘• ߠ െ ʹ‫ܣ‬൫߰ሶ ൅ π൯
‫߰ܣ‬
π •‹ ߠ ‘• ߠ൧ߠሶ ൅ ͵π
͵ ଶ ሺ‫ ܥ‬െ ‫ܣ‬ሻ ‘•ଶ ߠ •‹ ߰ ‘• ߰ ൌ Ͳ (19)
ଶ
ሷ
ሶ
ሶ
‫ ܣ‬൅ ‫ ߠ ••‘ ߠ ‹• ߰ܣ‬െ ሾ‫ܥܥ‬ଶ െ ʹ‫ܣ‬π •‹ ߠሿ ߰ ‘• ߠ ൅ ͵πଶ ሺ‫ ܥ‬െ ‫ܣ‬ሻ ‘•ଶ ߰ •‹ ߠ ‘• ߠ
‫ߠܣ‬
െ ଶ π ‘• ߠ ൅ πଶ ሺͶ‫ ܣ‬െ ͵‫ܥ‬ሻ •‹ ߠ ‘•
െ‫ܥܥ‬
‘ ߠൌͲ
(
(20)
A in terms of th
And
he non-dimension
nal time
߰ ᇱᇱ ‘•ଶ ߠ െ ሾߙ‫ܥ‬ଶ ‘• ߠ െ ʹሺ߰ ᇱ ൅ ͳሻ •‹ ߠ ‘• ߠሿߠ ᇱ ൅ ͵
͵ሺߙ െ ͳሻ ‘•ଶ ߠ •‹
 ߰ ‘• ߰ ൌ0
(
(21)
ᇱᇱ
ᇱଶ
ᇱ
ߠ ൅ ߰ •‹ ߠ ‘• ߠ െ ሺߙ‫ܥ‬ଶ െ ʹ •‹ ߠሻ߰
ߠ
‘• ߠ ൅ ͵ሺߙ െ ͳሻ ‘•ଶ ߰ •‹ ߠ ‘• ߠ
ఈ஼
൅ െ ͵ߙሻ •‹ ߠ ‘•
൅ሺͶ
‘ ߠ െ మ ‘• ߠ ൌ Ͳ
(
(22)
π
The equations (14), (15) , (21) and
T
a
(22) describe
e the motion of tthe satellite relattive to the
orbitall frame.
olution for in-plane motion
3. So
me thatͲ ൏ ߙ ൏ ͳ, by backing to equation (14) the in-plane motion ߠ ൌ Ͳ is described
d by
Assum
ߖ ᇱᇱ ൅ ͵ሺͳ െ ߙሻ ‫݊݅ݏ‬
݊ ߖ ܿ‫ ߖ ݏ݋‬ൌ Ͳ
(
(23)
I
Integrating
this eq
quation we obtain
n
ߖ ᇱଶ ൅ ͵ሺͳ െ ߙሻ ‫݊݅ݏ‬
݊ଶ ߖ ൌ Ͳ
(
(24)
W
Where
h is a mea
asure of the enerrgy of the in-plan
ne motion. We w
will show that the motion is
classiffied as follow:
131
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Fig.4
0 < α < 1)
From e
equation (24) und
der the condition (
ߖ ᇱ ൌ േඥ݄ െ ͵ሺͳ െ ߙሻ •‹ଶ ߖ
a on separation
and
n of variables
݀ߖ
ൌ േξ݄ න ݀߬
න
ඥͳ െ ݇ଵଶ •‹ଶ ߖ
͵
݇ଵଶ ൌ ሺͳ െ ߙሻ
݄
I ȁ݇ଵ ȁ ൐ ͳǡ ݄ ൏ ͵ሺሺͳ െ ߙሻ we define
If
e ߥଵ ൌ
(
(25)
ଵ
௞భ
ൌ •‹ିଵ ൬ߥଵ ‫ ݊ݏ‬ቀඥ͵ሺͳ െ ߙሻ߬ǡ ߥଵ ቁ൰ ǡ ߖ ᇱ ሺ߬ሻ ൌ ξ݄ܿܿ݊ሺඥ͵ሺͳ െ ߙሻ߬ǡ ߥଵ ሻ
ߖ
ߖሺ߬ሻ
݄ ൐ ͵ሺͳ െ ߙሻ
݂݅ȁ݇ଵ ȁ ൏ ͳǡ
ൌ ܽ݉൫േξ݄߬ǡ ݇ଵ ൯ǡ ߖ ᇱ ሺ߬ሻ ൌ േξ݄݀݊
ߖ
ߖሺ߬ሻ
݊ሺξ݄߬ǡ ݇ଵ ሻ
F second case when
For
w
we study th
he in-plane motion ߠ ൌ Ͳ and‫ܥ‬ଶ ൌ Ͳ then
߰ ᇱᇱ ൅ ͵ሺߙ െ ͳሻ •‹ ߰ ‘• ߰ ൌ Ͳ
߰ ᇱଶ ൅ ͵ሺߙ െ ͳሻ •‹
ଶ ߰ ൌ ݄
߰ ᇱ ൌ േඥ݄ െ ͵ሺߙ െ ͳሻ ‫݊݅ݏ‬ଶ ߰
͵
݇ଶଶ ൌ ሺߙ െ ͳሻ
݄
݀ߠ
න
ൌ േξ݄ න ݀߬
ඥͳ െ ݇ଶଶ •‹ଶ ߰
݂݅ȁ݇ଶ ȁ ൏ ͳǡ ݄ ൐ ͵ሺߙ െ ͳሻ
ൌ ܽ݉൫േξ݄߬߬ǡ ݇ଶ ൯ǡ߰ ᇱ ሺ߬ሻ ൌ േξ݄݀݊ሺξ݄߬ǡ ݇ଶ ሻ
߰
߰ሺ߬ሻ
ͳ
݂݅ȁ݇ଶ ȁ ൐ ͳǡ ݄ ൏ ͵ሺሺߙ െ ͳሻǡ ߥଶ ൌ
݇ଶ
ൌ •‹ିଵ ൬ߥଶ ‫݊ݏ‬
߰
߰ሺ߬ሻ
݊ ቀඥ͵ሺͳ െ ߙሻ߬ǡ ߥଶ ቁ൰ ǡ ߰ ᇱ ሺ߬ሻ ൌ ξ݄ܿ݊
݊ ቀඥ͵ሺͳ െ ߙሻ߬ǡ ߥଶ ቁ
ation motion
4. Studying of Vibra
4.1 Stability
St
of in-plane
ne motion.
The lin
near stability equation w. r. toߠ iss
132
(
(26)
(
(27)
(
(28)
(
(29)
(
(30)
(
(31)
Academic Journal of Interdisciplinary Studies
E-ISSN 2281-4612
ISSN 2281-3993
Published by MCSER-CEMAS-Sapienza University of Rome
ߠ ᇱᇱ ൅ ሾߖ ᇱଶ ൅ ͳ ൅ ʹߖ ᇱ ൅ ͵ሺͳ െ ߙሻ െ ͵ሺͳ െ ߙሻ •‹ଶ ߖሿߠ ൌ ͲሺͲ ൏ ߙ ൏ ͳሻ
ߠ ᇱᇱ ൅ ሾ߰ ᇱଶ ൅ ͳ ൅ ʹ߰ ᇱ െ ͵ሺߙ െ ͳሻ •‹ଶ ߰ሿߠ ൌ Ͳሺͳ ൏ ߙ ൏ ʹሻ
Let ‫ ݓ‬ൌ ඥ͵ሺͳ െ ߙሻ߬ andߖ ൌ ܽ݉ሺ‫ݓ‬ሻ. Using (26) and (32) we get
ௗమ ఏ
ௗ௪ మ
൅ ቂͳ ൅ ߥଵଶ ൅
ଵ
ଷሺଵିఈሻ
మഌభ
െ ʹߥଵଶ ‫݊ݏ‬ଶ ሺ‫ݓ‬ǡ ߥଵ ሻ ൅ ඥయሺభషഀሻ
ܿ݊ሺ‫ݓ‬ǡ ߥଵ ሻቃ ߠ ൌ Ͳ
In trigonometric form with •‹ ߖ ൌ ߥଵ •‹ ߚ
݀ଶߠ
݀ߠ
ͳ
ሺͳ െ ߥଵଶ •‹ଶ ߚሻ ଶ െ ߥଵଶ •‹ ߚ ‘• ߚ
൅ ሺͳ ൅ ߥଵଶ ൅
െ ʹߥଵଶ •‹ଶ ߚ
݀ߚ
݀ߚ
͵ሺͳ െ ߙሻ
మഌభ
൅ඥయሺభషഀሻ
‘• ߚሻߠ ൌ Ͳ
Vol 2 No 6
August 2013
(32)
(33)
(34)
(35)
This equation can be transformed to hill's equation as follow:
Let ߠ ൌ ݂݃
ଷఔర ୱ୧୬మ ଶఉ
భ
݃ᇱᇱ ൅ ቂଵ଺ሺଵିఔ
మ ୱ୧୬మ ఉሻమ ൅
భ
మశయቀమశయഌమ
భ ౙ౥౩ మഁቁሺభషഀሻశరഌభ ඥయሺభషഀሻ ౙ౥౩ ഁ
మ
లሺభషഀሻ൫భషഌమ
భ ౩౟౤ ഁ൯
ቃ݃ ൌ Ͳ
(36)
When ‫ ݓ‬ൌ ඥ͵ሺߙ െ ͳሻ߬ and ߰ ൌ ܽ݉ሺ‫ݓ‬ሻ then use equations (30) and (33) to get
ௗమ ఏ
ௗ௪ మ
൅ ቂߥଶଶ ൅
ଵ
ଷሺఈିଵሻ
మഌమ
െ ʹߥଶଶ ‫݊ݏ‬ଶ ሺ‫ݓ‬ǡ ߥଶ ሻ ൅ ඥయሺഀషభሻ
ܿ݊ሺ‫ݓ‬ǡ ߥଶ ሻቃ ߠ ൌ Ͳ
We will use the same previous method to get the trigonometric form
•‹ ߰ ൌ ߥଶ •‹ ߚ Ƭ߰ ᇱ ൌ ξ݄ܿ‫ߚݏ݋‬
݀ଶߠ
݀ߠ
ͳ
ሺͳ െ ߥଶଶ •‹ଶ ߚሻ ଶ െ ߥଶଶ •‹ ߚ ‘• ߚ
൅ ሺߥଶଶ ൅
െ ʹߥଶଶ •‹ଶ ߚ
݀ߚ
͵ሺߙ െ ͳሻ
݀ߚ
మഌమ
‘• ߚሻߠ ൌ Ͳ
൅ඥయሺഀషభሻ
(37)
(38)
Transform it to Hill's equation
݃ᇱᇱ ൅ ቂ
ଷఔమర ୱ୧୬మ ଶఉ
ଵ଺ሺଵିఔమమ ୱ୧୬మ ఉሻమ
మ
ౙ౥౩ మഁశరഌమ ඥయሺഀషభሻ ౙ౥౩ ഁ
൅ మశవഌమሺഀషభሻ
ቃ݃ ൌ Ͳ
లሺഀషభሻ൫భషഌమ ౩౟౤మ ഁ൯
మ
(39)
The equations (36 & 39) are periodic with periods ʹߨ and Ͷߨ and depend on two
parametersሺߙǡ ߰଴ ሻ. In the following subsection we deduce the equations of the boundaries between
zones of stability and instability in the plane of those parameters.
4.2 The stability of vibration motion analytically and numerically
Using Floquet's theorem we will study the stability of equation (35) both analytically and
numerically. For simplicity we write the equation in the form
ௗమ ఏ
ௗఏ
ሺͳ െ ߥଵଶ •‹ଶ ߚሻ మ െ ߥଵଶ •‹ ߚ ‘• ߚ ൅ ሺܾ െ ʹߥଵଶ •‹ଶ ߚ ൅ ݀ ‘• ߚሻߠ ൌ Ͳ
(40)
ௗఉ
ௗఉ
ͳ
ܾ ൌ ͳ ൅ ߥଵଶ ൅
͵ሺͳ െ ߙሻ
ʹߥଵ
݀ൌ
ඥ͵ሺͳ െ ߙሻ
and equation (38) in the form
ௗమ ఏ
ௗఏ
ሺͳ െ ߥଶଶ •‹ଶ ߚሻ మ െ ߥଶଶ •‹ ߚ ‘• ߚ ൅ ሺܾଵ െ ʹߥଶଶ •‹ଶ ߚ ൅ ݀ଵ ‘• ߚሻߠ ൌ Ͳ
(41)
ௗఉ
ܾଵ ൌ
݀ଵ ൌ
ߥଶଶ
ͳ
൅
͵ሺߙ െ ͳሻ
ௗఉ
ଶఔమ
ඥଷሺఈିଵሻ
and the non-trivial periodic solutions of equation (40) are:
4.2.1 Odd ૛࣊ periodic solution:
Such solution can be expressed as a Fourier series:
133
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E-ISSN 2281-4612
ISSN 2281-3993
Vol 2 No 6
August 2013
Published by MCSER-CEMAS-Sapienza University of Rome
ߠ ൌ σஶ
(42)
௡ୀ଴ ‫ܣ‬௡ •‹ ݊ߚ
Substitution from (42) in (40), yields the following recurrence relations for coefficients‫ܣ‬௡
݀
ሺܾ െ ͳ െ ߥଵଶ ሻ‫ܣ‬ଵ ൅ ‫ܣ‬ଶ െ ߥଵଶ ‫ܣ‬ଷ ൌ Ͳ
ʹ
݀
݀
ͷ
ଶ
‫ܣ‬ଵ ൅ ሺܾ െ Ͷ ൅ ߥଵ ሻ‫ܣ‬ଶ ൅ ‫ܣ‬ଷ െ ߥଵଶ ‫ܣ‬ସ ൌ Ͳ
ʹ
ʹ
ʹ
͹
݀
ͻ
݀
‫ܣ‬ଶ ൅ ൬ܾ െ ͻ ൅ ߥଵଶ ൰ ‫ܣ‬ଷ ൅ ‫ܣ‬ସ െ ߥଵଶ ‫ܣ‬ହ ൌ Ͳ
ʹ
ʹ
ʹ
ʹ
For݊ ൒ ʹ
݀
݀
݊ሺʹ݊ െ ͵ሻ ଶ
ߥଵ ‫ܣ‬ଶ௡ିଶ ൅ ‫ܣ‬ଶ௡ିଵ ൅ ሺܾ െ Ͷ݊ଶ ൅ ሺʹ݊ଶ െ ͳሻߥଵଶ ሻ‫ܣ‬ଶ௡ ൅ ‫ܣ‬ଶ௡ାଵ
െ
ʹ
ʹ
ʹ
݊ሺʹ݊ ൅ ͵ሻ ଶ
ߥଵ ‫ܣ‬ଶ௡ାଶ ൌ Ͳ
െ
ʹ
݀
ሺʹ݊ ൅ ͳሻଶ െ ʹሻ ଶ
݊ሺʹ݊ െ ͳሻ െ ͳ ଶ
ߥଵ ‫ܣ‬ଶ௡ିଵ ൅ ‫ܣ‬ଶ௡ ൅ ቆܾ െ ሺʹ݊ ൅ ͳሻଶ ൅
ߥଵ ቇ ‫ܣ‬ଶ௡ାଵ
െ
ʹ
ʹ
ʹ
ௗ
൅ ‫ܣ‬ଶ௡ାଶ െ
ଶ
௡ሺଶ௡ାହሻାଶ ଶ
ߥଵ ‫ܣ‬ଶ௡ାଷ
ଶ
(43)
ൌͲ
For the second case the solution in a Fourier series gives the recurrence relations for
coefficients ‫ܣ‬௡ in the form
݀ଵ
ሺܾଵ െ ͳ െ ߥଶଶ ሻ‫ܣ‬ଵ ൅ ‫ܣ‬ଶ െ ߥଶଶ ‫ܣ‬ଷ ൌ Ͳ
ʹ
݀ଵ
ͷ
݀ଵ
ሺܾ
‫ ܣ‬൅ ଵ െ Ͷ ൅ ߥଶଶ ሻ‫ܣ‬ଶ ൅ ‫ܣ‬ଷ െ ߥଶଶ ‫ܣ‬ସ ൌ Ͳ
ʹ ଵ
ʹ
ʹ
͹ ଶ
݀ଵ
ͻ
݀ଵ
‫ ܣ‬൅ ൬ܾଵ െ ͻ ൅ ߥଶ ൰ ‫ܣ‬ଷ ൅ ‫ܣ‬ସ െ ߥଶଶ ‫ܣ‬ହ ൌ Ͳ
ʹ ଶ
ʹ
ʹ
ʹ
For݊ ൒ ʹ
݀ଵ
݀ଵ
݊ሺʹ݊ െ ͵ሻ ଶ
ߥଶ ‫ܣ‬ଶ௡ିଶ ൅ ‫ܣ‬ଶ௡ିଵ ൅ ሺܾଵ െ Ͷ݊ଶ ൅ ሺʹ݊ଶ െ ͳሻߥଶଶ ሻ‫ܣ‬ଶ௡ ൅ ‫ܣ‬ଶ௡ାଵ
െ
ʹ
ʹ
ʹ
݊ሺʹ݊ ൅ ͵ሻ ଶ
െ
ߥଶ ‫ܣ‬ଶ௡ାଶ ൌ Ͳ
ʹ
௡ሺଶ௡ିଵሻିଵ
ௗ
ሺଶ௡ାଵሻమ ିଶሻ ଶ
ௗ
െ
ሺʹ݊ଶ െ ͳሻߥଶଶ ‫ܣ‬ଶ௡ିଵ ൅ భ ‫ܣ‬ଶ௡ ൅ ቀܾଵ െ ሺʹ݊ ൅ ͳሻଶ ൅
ߥଶ ቁ ‫ܣ‬ଶ௡ାଵ ൅ భ ‫ܣ‬ଶ௡ାଶ െ
ଶ
௡ሺଶ௡ାହሻାଶ ଶ
ߥଶ ‫ܣ‬ଶ௡ାଷ
ଶ
ଶ
ଶ
ଶ
(44)
ൌͲ
4.2.2 Even ૛࣊ -periodic solution
Seeking a solution of the form
ߠ ൌ σஶ
(45)
௡ୀ଴ ‫ܤ‬௡ ‘• ݊ߚ
the substitution in equation (40) yields the recurrence relations for coefficients ‫ܤ‬௡ :
݀
ሺܾ െ ߥଵଶ ሻ‫ܤ‬଴ ൅ ‫ܤ‬ଵ ൌ Ͳ
ʹ
݀
݀‫ܤ‬଴ ൅ ሺܾ െ ͳሻ‫ܤ‬ଵ ൅ ‫ܤ‬ଶ െ ߥଵଶ ‫ܤ‬ଷ ൌ Ͳ
ʹ
݀
݀
ͷ
ଶ
ߥଵ ‫ܤ‬଴ ൅ ‫ܤ‬ଵ ൅ ሺܾ െ Ͷ ൅ ߥଵଶ ሻ‫ܤ‬ଶ ൅ ‫ܤ‬ଷ െ ߥଵଶ ‫ܤ‬ସ ൌ Ͳ
ʹ
ʹ
ʹ
͹ ଶ
݀
ͻ ଶ
݀
‫ ܤ‬൅ ൬ܾ െ ͻ ൅ ߥଵ ൰ ‫ܤ‬ଷ ൅ ‫ܤ‬ସ െ ߥଵ ‫ܤ‬ହ ൌ Ͳ
ʹ
ʹ
ʹ
ʹ ଶ
For݊ ൒ ʹ
݀
݀
݊ሺʹ݊ െ ͵ሻ ଶ
ߥଵ ‫ܤ‬ଶ௡ିଶ ൅ ‫ܤ‬ଶ௡ିଵ ൅ ሺܾ െ Ͷ݊ଶ ൅ ሺʹ݊ଶ െ ͳሻߥଵଶ ሻ‫ܤ‬ଶ௡ ൅ ‫ܤ‬ଶ௡ାଵ
െ
ʹ
ʹ
ʹ
134
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E-ISSN 2281-4612
ISSN 2281-3993
Published by MCSER-CEMAS-Sapienza University of Rome
Vol 2 No 6
August 2013
݊ሺʹ݊ ൅ ͵ሻ ଶ
ߥଵ ‫ܤ‬ଶ௡ାଶ ൌ Ͳ
ʹ
݊ሺʹ݊ െ ͳሻ െ ͳ ଶ
݀
ሺʹ݊ ൅ ͳሻଶ െ ʹሻ ଶ
െ
ߥଵ ‫ܤ‬ଶ௡ିଵ ൅ ‫ܤ‬ଶ௡ ൅ ቆܾ െ ሺʹ݊ ൅ ͳሻଶ ൅
ߥଵ ቇ ‫ܤ‬ଶ௡ାଵ
ʹ
ʹ
ʹ
െ
ௗ
௡ሺଶ௡ାହሻାଶ
ଶ
ଶ
൅ ‫ܤ‬ଶ௡ାଶ െ
ߥଵଶ ‫ܤ‬ଶ௡ାଷ ൌ Ͳ
(46)
The recurrence relations for coefficients‫ܤ‬௡ for the second case
݀ଵ
ሺܾଵ െ ߥଶଶ ሻ‫ܤ‬଴ ൅ ‫ܤ‬ଵ ൌ Ͳ
ʹ
݀ଵ
݀ଵ ‫ܤ‬଴ ൅ ሺܾଵ െ ͳሻ‫ܤ‬ଵ ൅ ‫ܤ‬ଶ െ ߥଶଶ ‫ܤ‬ଷ ൌ Ͳ
ʹ
݀ଵ
݀ଵ
ͷ
ଶ
ߥଶ ‫ܤ‬଴ ൅ ‫ܤ‬ଵ ൅ ሺܾଵ െ Ͷ ൅ ߥଶଶ ሻ‫ܤ‬ଶ ൅ ‫ܤ‬ଷ െ ߥଶଶ ‫ܤ‬ସ ൌ Ͳ
ʹ
ʹ
ʹ
͹ ଶ
݀ଵ
ͻ ଶ
݀ଵ
‫ ܤ‬൅ ൬ܾଵ െ ͻ ൅ ߥଶ ൰ ‫ܤ‬ଷ ൅ ‫ܤ‬ସ െ ߥଶ ‫ܤ‬ହ ൌ Ͳ
ʹ
ʹ
ʹ ଶ
ʹ
For݊ ൒ ʹ
݀ଵ
݀ଵ
݊ሺʹ݊ െ ͵ሻ ଶ
െ
ߥଶ ‫ܤ‬ଶ௡ିଶ ൅ ‫ܤ‬ଶ௡ିଵ ൅ ሺܾଵ െ Ͷ݊ଶ ൅ ሺʹ݊ଶ െ ͳሻߥଶଶ ሻ‫ܤ‬ଶ௡ ൅ ‫ܤ‬ଶ௡ାଵ
ʹ
ʹ
ʹ
݊ሺʹ݊ ൅ ͵ሻ ଶ
ߥଶ ‫ܤ‬ଶ௡ାଶ ൌ Ͳ
െ
ʹ
݀ଵ
ሺʹ݊ ൅ ͳሻଶ െ ʹሻ ଶ
݊ሺʹ݊ െ ͳሻ െ ͳ ଶ
ߥଶ ‫ܤ‬ଶ௡ିଵ ൅ ‫ܤ‬ଶ௡ ൅ ቆܾଵ െ ሺʹ݊ ൅ ͳሻଶ ൅
ߥଶ ቇ ‫ܤ‬ଶ௡ାଵ
െ
ʹ
ʹ
ʹ
൅
ௗభ
ଶ
‫ܤ‬ଶ௡ାଶ െ
௡ሺଶ௡ାହሻାଶ
ଶ
ߥଶଶ ‫ܤ‬ଶ௡ାଷ ൌ Ͳ
(47)
4.2.3 Even Ͷߨ -periodic solution
The solution in this case has the form
భ
ஶ
(48)
ߠ ൌ σஶ
௡ୀ଴ ‫ܥ‬௡ ‘• ݊ߚ ൅ σ௡ୀ଴ ܹ௡ ‘•൫݊ ൅ మ൯ߚ
Substitution in equation (40) yields the recurrence relations for coefficients‫ܥ‬௡ and ܹ௡
݀
ሺܾ െ ߥଵଶ ሻ‫ܥ‬଴ ൅ ‫ܥ‬ଵ ൌ Ͳ
ʹ
݀
݀‫ܥ‬଴ ൅ ሺܾ െ ͳሻ‫ܥ‬ଵ ൅ ‫ܥ‬ଶ െ ߥଵଶ ‫ܥ‬ଷ ൌ Ͳ
ʹ
݀
݀
ͷ
ଶ
ߥଵ ‫ܥ‬଴ ൅ ‫ܥ‬ଵ ൅ ሺܾ െ Ͷ ൅ ߥଵଶ ሻ‫ܥ‬ଶ ൅ ‫ܥ‬ଷ െ ߥଵଶ ‫ܥ‬ସ ൌ Ͳ
ʹ
ʹ
ʹ
݀
͹ ଶ
݀
ͻ ଶ
‫ ܥ‬൅ ൬ܾ െ ͻ ൅ ߥଵ ൰ ‫ܥ‬ଷ ൅ ‫ܥ‬ସ െ ߥଵ ‫ܥ‬ହ ൌ Ͳ
ʹ ଶ
ʹ
ʹ
ʹ
ͷ
݀
͹
݀ ͳ ͹
൬ܾ ൅ െ െ ߥଵଶ ൰ ܹ଴ ൅ ሺ ߥଵଶ ൅ ሻܹଵ െ ߥଵଶ ܹଶ ൌ Ͳ
ʹ Ͷ ͺ
ͳ͸
ʹ
ͳ͸
݀
ͻ
ͻ ͳ
݀
ʹ͹
൬ ൅ ߥଵଶ ൰ ܹ଴ ൅ ൬ܾ െ ൅ ߥଵଶ ൰ ܹଵ ൅ ܹଶ െ ߥଵଶ ܹଷ ൌ Ͳ
ʹ ͳ͸
Ͷ ͺ
ʹ
ͳ͸
For݊ ൒ ʹ
݀
݀
݊ሺʹ݊ െ ͵ሻ ଶ
ߥଵ ‫ܥ‬ଶ௡ିଶ ൅ ‫ܥ‬ଶ௡ିଵ ൅ ሺܾ െ Ͷ݊ଶ ൅ ሺʹ݊ଶ െ ͳሻߥଵଶ ሻ‫ܥ‬ଶ௡ ൅ ‫ܥ‬ଶ௡ାଵ
െ
ʹ
ʹ
ʹ
݊ሺʹ݊ ൅ ͵ሻ ଶ
ߥଵ ‫ܥ‬ଶ௡ାଶ ൌ Ͳ
െ
ʹ
݀
ሺʹ݊ ൅ ͳሻଶ െ ʹ ଶ
݊ሺʹ݊ െ ͳሻ െ ͳ ଶ
ߥଵ ‫ܥ‬ଶ௡ିଵ ൅ ‫ܥ‬ଶ௡ ൅ ቆܾ െ ሺʹ݊ ൅ ͳሻଶ ൅
ߥଵ ቇ ‫ܥ‬ଶ௡ାଵ
െ
ʹ
ʹ
ʹ
݊ሺʹ݊ ൅ ͷሻ ൅ ʹ ଶ
݀
ߥଵ ‫ܥ‬ଶ௡ାଷ ൌ Ͳ
൅ ‫ܥ‬ଶ௡ାଶ െ
ʹ
ʹ
135
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െ
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Vol 2 No 6
August 2013
ሺʹ݊ െ ͷሻሺʹ݊ ൅ ͳሻ ଶ
݀
ሺʹ݊ ൅ ͳሻଶ ሺʹ݊ ൅ ͳሻଶ െ ͺ ଶ
ߥଵ ܹ௡ିଶ ൅ ܹ௡ିଵ ൅ ቆܾ െ
൅
ߥଵ ቇ ܹ௡
ͺ
Ͷ
ͳ͸
ʹ
ௗ
൅ ଶ ܹ௡ାଵ െ
ሺଶ௡ାଵሻሺଶ௡ା଻ሻ ଶ
ߥଵ ܹ௡ାଶ
ଵ଺
(49)
ൌͲ
And for second case the recurrence relations for coefficients ‫ܥ‬௡ and ܹ௡
݀ଵ
ሺܾଵ െ ߥଶଶ ሻ‫ܥ‬଴ ൅ ‫ܥ‬ଵ ൌ Ͳ
ʹ
݀ଵ
ሺܾ
݀ଵ ‫ܥ‬଴ ൅ ଵ െ ͳሻ‫ܥ‬ଵ ൅ ‫ܥ‬ଶ െ ߥଶଶ ‫ܥ‬ଷ ൌ Ͳ
ʹ
݀ଵ
݀ଵ
ͷ
ଶ
ߥଶ ‫ܥ‬଴ ൅ ‫ܥ‬ଵ ൅ ሺܾଵ െ Ͷ ൅ ߥଶଶ ሻ‫ܥ‬ଶ ൅ ‫ܥ‬ଷ െ ߥଶଶ ‫ܥ‬ସ ൌ Ͳ
ʹ
ʹ
ʹ
݀ଵ
͹ ଶ
݀ଵ
ͻ ଶ
‫ ܥ‬൅ ൬ܾଵ െ ͻ ൅ ߥଶ ൰ ‫ܥ‬ଷ ൅ ‫ܥ‬ସ െ ߥଶ ‫ܥ‬ହ ൌ Ͳ
ʹ
ʹ
ʹ ଶ
ʹ
݀ ͳ ͹ ଶ
ͷ ଶ ݀ଵ
͹
൬ܾଵ ൅ െ െ ߥଶ ൰ ܹ଴ ൅ ሺ ߥଶ ൅ ሻܹଵ െ ߥଶଶ ܹଶ ൌ Ͳ
ʹ Ͷ ͺ
ͳ͸
ͳ͸
ʹ
ͻ
ͻ ͳ
݀ଵ
ʹ͹
݀ଵ
൬ ൅ ߥଶଶ ൰ ܹ଴ ൅ ൬ܾଵ െ ൅ ߥଶଶ ൰ ܹଵ ൅ ܹଶ െ ߥଶଶ ܹଷ ൌ Ͳ
Ͷ ͺ
ͳ͸
ʹ ͳ͸
ʹ
For݊ ൒ ʹ
݀ଵ
݀ଵ
݊ሺʹ݊ െ ͵ሻ ଶ
ߥଶ ‫ܥ‬ଶ௡ିଶ ൅ ‫ܥ‬ଶ௡ିଵ ൅ ሺܾଵ െ Ͷ݊ଶ ൅ ሺʹ݊ଶ െ ͳሻߥଶଶ ሻ‫ܥ‬ଶ௡ ൅ ‫ܥ‬ଶ௡ାଵ
െ
ʹ
ʹ
ʹ
݊ሺʹ݊ ൅ ͵ሻ ଶ
ߥଶ ‫ܥ‬ଶ௡ାଶ ൌ Ͳ
െ
ʹ
݀ଵ
ሺʹ݊ ൅ ͳሻଶ െ ʹ ଶ
݊ሺʹ݊ െ ͳሻ െ ͳ ଶ
ߥଶ ‫ܥ‬ଶ௡ିଵ ൅ ‫ܥ‬ଶ௡ ൅ ቆܾଵ െ ሺʹ݊ ൅ ͳሻଶ ൅
ߥଶ ቇ ‫ܥ‬ଶ௡ାଵ
െ
ʹ
ʹ
ʹ
݊ሺʹ݊ ൅ ͷሻ ൅ ʹ ଶ
݀ଵ
ߥଶ ‫ܥ‬ଶ௡ାଷ ൌ Ͳ
൅ ‫ܥ‬ଶ௡ାଶ െ
ʹ
ʹ
ሺʹ݊ െ ͷሻሺʹ݊ ൅ ͳሻ ଶ
݀ଵ
ሺʹ݊ ൅ ͳሻଶ ሺʹ݊ ൅ ͳሻଶ െ ͺ ଶ
െ
ߥଶ ܹ௡ିଶ ൅ ܹ௡ିଵ ൅ ቆܾଵ െ
൅
ߥଶ ቇ ܹ௡
ͺ
ʹ
Ͷ
ͳ͸
൅
ௗభ
ଶ
ܹ௡ାଵ െ
ሺଶ௡ାଵሻሺଶ௡ା଻ሻ ଶ
ߥଶ ܹ௡ାଶ
ଵ଺
(50)
ൌͲ
4.2.4 Odd ૝࣊ -periodic solution
Assuming such solution in the form
భ
ஶ
(51)
ߠ ൌ σஶ
௡ୀ଴ ‫ܩ‬௡ •‹ ݊ߚ ൅ σ௡ୀ଴ ‫ܪ‬௡ •‹ሺ݊ ൅ మሻߚ
Substitution in (40) yields the following recurrence relations for coefficients ‫ܩ‬௡ and ‫ܪ‬௡ ǣ
݀
ሺܾ െ ͳ െ ߥଵଶ ሻ‫ܩ‬ଵ ൅ ‫ܩ‬ଶ െ ߥଵଶ ‫ܩ‬ଷ ൌ Ͳ
ʹ
݀
ͷ
݀
‫ܩ‬ଵ ൅ ሺܾ െ Ͷ ൅ ߥଵଶ ሻ‫ܩ‬ଶ ൅ ‫ܩ‬ଷ െ ߥଵଶ ‫ܩ‬ସ ൌ Ͳ
ʹ
ʹ
ʹ
݀
͹ ଶ
݀
ͻ
‫ ܩ‬൅ ൬ܾ െ ͻ ൅ ߥଵ ൰ ‫ܩ‬ଷ ൅ ‫ܩ‬ସ െ ߥଵଶ ‫ܩ‬ହ ൌ Ͳ
ʹ ଶ
ʹ
ʹ
ʹ
݀
ͷ ଶ
͹
݀ ͳ ͹ ଶ
൬ܾ െ െ െ ߥଵ ൰ ‫ܪ‬଴ ൅ ሺ െ ߥଵ ሻ‫ܪ‬ଵ െ ߥଵଶ ‫ܪ‬ଶ ൌ Ͳ
ʹ ͳ͸
ͳ͸
ʹ Ͷ ͺ
݀
ͻ
ͻ ͳ
݀
ʹ͹
൬ െ ߥଵଶ ൰ ‫ܪ‬଴ ൅ ൬ܾ െ ൅ ߥଵଶ ൰ ‫ܪ‬ଵ ൅ ‫ܪ‬ଶ െ ߥଵଶ ‫ܪ‬ଷ ൌ Ͳ
ʹ ͳ͸
Ͷ ͺ
ʹ
ͳ͸
For݊ ൒ ʹ :
݀
݀
݊ሺʹ݊ െ ͳሻ െ ͳ ଶ
ሺଶ௡ାଵሻమ ିଶ ଶ
ߥଵ ‫ܩ‬ଶ௡ିଵ ൅ ‫ܩ‬ଶ௡ ൅ ቀܾ െ ሺʹ݊ ൅ ͳሻଶ ൅
ߥଵ ቁ ‫ܩ‬ଶ௡ାଵ ൅ ‫ܩ‬ଶ௡ାଶ
െ
ଶ
ʹ
ʹ
ʹ
݊ሺʹ݊ ൅ ͷሻ ൅ ʹ ଶ
െ
ߥଵ ‫ܩ‬ଶ௡ାଷ ൌ Ͳ
ʹ
136
Academic Journal of Interdisciplinary Studies
E-ISSN 2281-4612
ISSN 2281-3993
Vol 2 No 6
August 2013
Published by MCSER-CEMAS-Sapienza University of Rome
݊ሺʹ݊ െ ͵ሻ ଶ
݀
݀
ߥଵ ‫ܩ‬ଶ௡ିଶ ൅ ‫ܩ‬ଶ௡ିଵ ൅ ሺܾ െ Ͷ݊ଶ ൅ ሺଶ௡మ ିଵሻߥଵଶ ሻ‫ܩ‬ଶ௡ ൅ ‫ܩ‬ଶ௡ାଵ
ʹ
ʹ
ʹ
݊ሺʹ݊ ൅ ͵ሻ ଶ
ߥଵ ‫ܩ‬ଶ௡ାଶ ൌ Ͳ
െ
ʹ
ሺʹ݊ െ ͷሻሺʹ݊ ൅ ͳሻ ଶ
݀
ሺʹ݊ ൅ ͳሻଶ ሺʹ݊ ൅ ͳሻଶ െ ͺ ଶ
ߥଵ ‫ܪ‬௡ିଶ ൅ ‫ܪ‬௡ିଵ ൅ ቆܾ െ
൅
ߥଵ ቇ ‫ܪ‬௡
െ
ͺ
ʹ
Ͷ
ͳ͸
െ
ௗ
൅ ‫ܪ‬௡ାଵ െ
ଶ
ሺଶ௡ାଵሻሺଶ௡ା଻ሻ ଶ
ߥଵ ‫ܪ‬௡ାଶ
ଵ଺
(52)
ൌͲ
For second case the recurrence relations for coefficients ‫ܩ‬௡ and ‫ܪ‬௡ are :
݀ଵ
ሺܾଵ െ ͳ െ ߥଶଶ ሻ‫ܩ‬ଵ ൅ ‫ܩ‬ଶ െ ߥଶଶ ‫ܩ‬ଷ ൌ Ͳ
ʹ
݀ଵ
݀ଵ
ͷ
‫ܩ‬ଵ ൅ ሺܾଵ െ Ͷ ൅ ߥଶଶ ሻ‫ܩ‬ଶ ൅ ‫ܩ‬ଷ െ ߥଶଶ ‫ܩ‬ସ ൌ Ͳ
ʹ
ʹ
ʹ
͹ ଶ
݀ଵ
ͻ
݀ଵ
‫ ܩ‬൅ ൬ܾଵ െ ͻ ൅ ߥଶ ൰ ‫ܩ‬ଷ ൅ ‫ܩ‬ସ െ ߥଶଶ ‫ܩ‬ହ ൌ Ͳ
ʹ
ʹ
ʹ ଶ
ʹ
݀ଵ ͳ ͹ ଶ
݀ଵ
ͷ ଶ
͹
൬ܾଵ െ െ െ ߥଶ ൰ ‫ܪ‬଴ ൅ ሺ െ ߥଶ ሻ‫ܪ‬ଵ െ ߥଶଶ ‫ܪ‬ଶ ൌ Ͳ
ͳ͸
ʹ Ͷ ͺ
ʹ ͳ͸
݀ଵ
ͻ
ͻ ͳ
݀ଵ
ʹ͹
൬ െ ߥଶଶ ൰ ‫ܪ‬଴ ൅ ൬ܾଵ െ ൅ ߥଶଶ ൰ ‫ܪ‬ଵ ൅ ‫ܪ‬ଶ െ ߥଶଶ ‫ܪ‬ଷ ൌ Ͳ
Ͷ ͺ
ͳ͸
ʹ ͳ͸
ʹ
For݊ ൒ ʹ :
݀ଵ
ሺʹ݊ ൅ ͳሻଶ െ ʹ ଶ
݊ሺʹ݊ െ ͳሻ െ ͳ ଶ
ߥଶ ‫ܩ‬ଶ௡ିଵ ൅ ‫ܩ‬ଶ௡ ൅ ቆܾଵ െ ሺʹ݊ ൅ ͳሻଶ ൅
ߥଶ ቇ ‫ܩ‬ଶ௡ାଵ
െ
ʹ
ʹ
ʹ
݊ሺʹ݊ ൅ ͷሻ ൅ ʹ ଶ
݀ଵ
ߥଶ ‫ܩ‬ଶ௡ାଷ ൌ Ͳ
൅ ‫ܩ‬ଶ௡ାଶ െ
ʹ
ʹ
݀ଵ
݀ଵ
݊ሺʹ݊ െ ͵ሻ ଶ
ߥଶ ‫ܩ‬ଶ௡ିଶ ൅ ‫ܩ‬ଶ௡ିଵ ൅ ሺܾଵ െ Ͷ݊ଶ ൅ ሺʹ݊ଶ െ ͳሻߥଶଶ ሻ‫ܩ‬ଶ௡ ൅ ‫ܩ‬ଶ௡ାଵ
െ
ʹ
ʹ
ʹ
݊ሺʹ݊ ൅ ͵ሻ ଶ
ߥଶ ‫ܩ‬ଶ௡ାଶ ൌ Ͳ
െ
ʹ
ሺʹ݊ െ ͷሻሺʹ݊ ൅ ͳሻ ଶ
݀ଵ
ሺʹ݊ ൅ ͳሻଶ ሺʹ݊ ൅ ͳሻଶ െ ͺ ଶ
െ
ߥଶ ‫ܪ‬௡ିଶ ൅ ‫ܪ‬௡ିଵ ൅ ቆܾଵ െ
൅
ߥଶ ቇ ‫ܪ‬௡
ͺ
ͳ͸
ʹ
Ͷ
൅
ௗభ
ଶ
‫ܪ‬௡ାଵ െ
ሺଶ௡ାଵሻሺଶ௡ା଻ሻ ଶ
ߥଶ ‫ܪ‬௡ାଶ
ଵ଺
(53)
ൌͲ
The systems of equations (43,46,49,52) and also (44 , 47, 50, 53) are homogeneous infinite
systems. The boundary curves can be drawn by solving the determinately equations. We shall
study the stability and instability zones in the plane of parametersߙ,߰଴ where߰଴ is the amplitude
of vibrations ሺߥ ൌ •‹ ߰଴ ሻ.
4.3 Numerical and analytical diagrams of stability
Let the symbolsܹ௦ ǡ ‫ܩ‬௦ denote the (s) zones of stability and instability respectively. The zones of
stability and instability are separated by the curves ±
∏
2s
,
±
∏
2 s +1
whose equations describe the
distribution of eigen-values for boundary problems of periods 2π ,
curves by solving the last four determinants. We classify the curves as:
+
∏ for even 2π −
2s
periodic solutions and
−
∏ for odd 2π −
2s
137
4π
. We shall draw these
periodic solutions.
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+
∏
2 s +1
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for even
−
4π − periodic soluutions and ∏
2 s +1
±
∏
W now intend to
We
o find the intersecctions of
2s
,
fo
or odd
±
∏
2 s +1
4π −
withψ 0
pe
eriodic solutions.
= 0 . On the line ψ 0 = 0
, the
equatiion of vibration (3
36) takes the form
m
1
]g = 0
3(1 − α )
1 + 3(11−α ) = ω 2 where ω
g ′′ + [1 +
le
et
(
(54)
iss constant , then
α = 1 − 3(ω1 −1)
2
, for
ω
= 2, 3, 4, 5,6 ….
ͺ
ʹ͵
ͶͶ
͹ͳ
ͳͲͶ
ǡߙ ൌ
ߙଵ ൌ ǡ ߙଶ ൌ
ǡߙ ൌ
ǡߙ ൌ
ǡǥǥ
ͻ
ʹͶ ଷ Ͷͷ ସ ͹ʹ ହ ͳͲͷ
a
and
so on. There
e exist two curve
es with the same period one eve
en and other od
dd passing
throug
gh each of those
e values of Į. We
e note that the stability zones occcupy the lower part near
toɗ଴ ൌ Ͳ. For example
e, two curves, one odd
4π
a ߙൌ
at
଼
ଽ
and
+
∏
,
2s
−
∏
2s
with period
d
2π
−
∏
2 s +1
and the other even
at ߙ ൌ
ଶଷ
ଶସ
+
∏.
2 s +1
with the sam
me period
(see
(
Fig. 5).
Fig.5
e same manner as
a in the case off ( 0 < α
In the
<1
) we
w get the interssections of
±
∏
2s
,
±
∏
2 s +1
ψ 0 = 0 to determine thhe zones of stability and instabilityy for the case 1 < α < 2
Equattion (39) become
es:
1
g =0
3(α − 1)
2
L 3(α1−1) = n wh
Let
here n (+) ve intteger .then α = 1 2 + 1
3n
g ′′ +
138
(
(55)
for
n
= 1 , 2 , 3 , 4 , .......
with
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of Rome
Vol 2 No 6
August 2013
Ͷ
ͳ͵
ʹͺ
Ͷͻ
ߙଵ ൌ ǡ ߙଶ ൌ
ǡ ߙଷ ൌ
ǡ ߙସ ൌ
ǡ ǥ ǥǤ
͵
ͳʹ
ʹ͹
Ͷͺ
a so on. The zo
and
ones of stability and
a instability are shown in Fig. 6.
Fig.6
Referrences
Beletskkii V.V. 1965, Mo
otion Of An Artifiicial Satellite About Its Center Of Mass. Nauka, Moscow
M
(In
R
Russian).
Markee
ev, A. p.: 1975, Stability of plane
e oscillations and rotations of a ssatellite in a circu
ular orbit ,
K
Kosmich.
Issled. 13
3, 322-336 (in Russsian), English tran
nsl. Cosmic Res., 285 – 298 .
Akulen
nko, L.D., Nestero
ov, S. V. and Sh
hmatkov, A.M. 19
999 , Generalized
d parametric osciillations of
m
mechanical
system
ms, Prikl. Mat. Me
ekh. 63, 746-756
6 (in Russian) ;En
nglish transl J. Appl.
A
Math.
M
Mekh.,
63, 705-713
3.
Markee
ev A.P., and Bard
din 2003, On the stability of plana
ar oscillations and rotations of a sa
atellite in a
c
circular
orbit, Celesst. Mech. Dyn. Astt. 85, 51-66.
Yehia , H.M. :1986 ,On the motion of a rigid body acted upon by potentia
al and gyroscopic forces , J.
M
Mecan.
Theor. App
pl., 5, 747-754.
Arscottt F. M . 1964, Periodic Differential Equations .
139