Stability Analysis of Plane Vibrations of a Satellite in a Circular Orbit

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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

127

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.

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2. Ma

2.1 D

Denot satellit frame to the orbital inertia

࢞૚ǡ ࢟૚

compo relativ

ࢠ_૚െax betwe

ࢠ െaxi

Fig.1

In tho

ࢻ ࢼ ࢽ ࣓

I suppo uniform Earth

ࡸ ࢝

N 2281-4612 2281-3993

athematical For

Description of mot

e byࡻ the cente te describing a c

of reference, ࡻᇱ_࢞

e orbit in the dire l plane and let ࡻᇱ

a A,B and C, res

ǡ ࢠ૚ and ࣓ the

onents being refe ve to the orbital xis,ࣂthe angle b

en ࢠ andࢠ_૚െax is.

se variables we c

ࢻ ൌ ሺࢉ࢕࢙࣒ࢉ࢕࢙࣐ െ ࢼ ൌ ሺ࢙࢏࢔࣒ࢉ࢕࢙࣐ ൅ ࢽ ൌ ሺࢉ࢕࢙ࣂ࢙࢏࢔࣐ǡ ࢉ ࣓ ൌ ሺ࣒ሶࢉ࢕࢙ࣂ࢙࢏࢔࣐

In what follows w sed that the rot m angular speed [1]. The Lagrang

ࡸ ൌ૚

૛࢝۷Ǥ ࢝ െ ࢂ ࢝ ൌ ࣓ ൅ πࢽ

Academic J Published by MCS

rmulation

tion

er of the Earth a ircular orbit of ra

࢞_૚࢟_૚ࢠ_૚ be the or ection of motion

ᇱ_࢞࢟ࢠ_{be the syste}

spectively. Let a angular velocity erred to the body frame we shall between the axis o xis is࣊

૛െ ࣂ) and

can write

െ ࢙࢏࢔ࣂ࢙࢏࢔࣐࢙࢏࢔࣒ ൅ ࢙࢏࢔ࣂ࢙࢏࢔࣐ࢉ࢕࢙࣒

࢕࢙ࣂࢉ࢕࢙࣐ǡ ࢙࢏࢔ࣂሻ ࣐ െ ࣂሶࢉ࢕࢙࣐ǡ ࣒ሶࢉ࢕࢙ࣂ

we study the rota tational motion i

π (say), given b gian of the rotatio

Journal of Interdisciplin SER-CEMAS-Sapienza U

nd by ࡻᇱ_{the cur}

adius R with cent rbital system with of the satellite a em of central prin lsoࢻǡ ࢼǡ ࢽ be th vector of the s y systemࡻᇱ_࢞࢟ࢠ_.

use Euler’s ang of the body and t

࣐ the angle of

࣒ǡ െࢉ࢕࢙࣒࢙࢏࢔࣐ െ ࣒ǡ െ࢙࢏࢔࣒ࢉ࢕࢙࣐ ൅

ࣂࢉ࢕࢙࣐ ൅ ࣂሶ࢙࢏࢔࣐ǡ ࣒

ational motion of is independent o by the expression onal motion can b

nary Studies University of Rome

rrent position of tre atࡻ, see fig. h ࢞_૚െ axis along and ࢠ_૚െ in the d ncipal axes of the he three unit ve satellite relative To describe the gles ࣒the angle the orbital plane

proper rotation o

െ ࢙࢏࢔ࣂࢉ࢕࢙࣐࢙࢏࢔࣒ ൅ ࢙࢏࢔ࣂ࢙࢏࢔࣐ࢉ࢕࢙࣒

࣒ሶ࢙࢏࢔࣐ ൅ ࣐ሶ) the satellite abo of the orbital mo n π૛_ൌ ஜ

ࡾ૜ where μ be written as

A

the center of ma 1. Let ࡻࢄࢅࢆ be

ࡻࡻᇱ_,_࢟

૚ along th

direction orthogo e satellite with m ectors in the dire

to the orbital sy orientation of th of precession a

(so that the nuta of the satellite a

࣒ǡ ࢉ࢕࢙ࣂ࢙࢏࢔࣒ሻ

࣒ǡ െࢉ࢕࢙ࣂࢉ࢕࢙࣒ሻ

out its centre of m otion of the sat

μ is Gauss’ const

( (

Vol 2 No 6 ugust 2013

ass of the an inertial he tangent nal to the oments of ections of ystem, all he satellite round the ation angle around its

mass. It is ellite with ant of the

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129

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:

ࢂ ൌ_૛ࡾ૜ஜ૜ࢻ۷Ǥ ࢻ ൌ

૜

૛π૛ࢻ۷Ǥ ࢻ (3)

Substituting (2) and (3) in (1) we get

ࡸ ൌ૚_૛൫࡭࢖૛_{൅ ࡮ࢗ}૛_{൅ ࡯࢘}૛_{൯ ൅ ሺ࡭࢖ࢽ}

૚൅ ࡮ࢗࢽ૛൅ ࡯࢘ࢽ૜ሻ ൅π

૛

૛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 െ π}ଶ_{ࢽ ൈ ࢽࡵ}

ࢻሶ ൅ ࣓ ൈ ࢻ ൌ Ͳǡ ࢼሶ ൅ ࣓ ൈ ࢼ ൌ Ͳǡࢽሶ ൅ ࣓ ൈ ࢽ ൌ Ͳ (5) where۷ ൌଵ_ଶሺܶݎ۷ሻߜ െ ۷. This system admits Jacobi's integral

ቀଵ

ଶቁ ࣓۷Ǥ ࣓ ൅ πమ

ଶሺ͵ࢻ۷Ǥ ࢻ െ ࢽ۷Ǥ ࢽሻ ൌ ݄ (6)

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 ൏ ܣ ൌ ܤ

ͳ ൏ ߙ ൏ ʹ to the caseܥ ൐ ܣ ൌ 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:

ܮ ൌͳ

ʹܣሺ݌ଶ൅ ݍଶሻ ൅ ͳ

ʹܥݎଶ൅ πሾܣሺ݌ߛଵ൅ ݍߛଶሻ ൅ ܥݎߛଷሿ ൅ πଶ

ʹ ሾܣሺߛଵଶ൅ ߛଶଶሻ ൅ܥߛ_ଷଶ_{൅ ሺܥ െ ܣሻሺͳ െ ͵ߙ}

ଷଶሻሿ (7)

Case i (Ͳ ൏ ߙ ൏ ͳ ):

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Fig.2

ܮ

൅

N

݌

߮

N

ܴ ܣ

T A

ܴ ܣ

T

ܣ ܣ

D

τ

ߖ ߠ ൅

C I the po

߰the

N 2281-4612 2281-3993

ܮ ൌ ൤ܣ

ʹଶߠ ൅ ܥ ʹ ൅π߮ሶܥ ߠ ൅஼

ଶ߮ሶ

Now, we note tha

݌_ఝൌ߲ܮ

߲߮ሶൌ ܥൣ൫ߖሶ ൅ ߮ሶ ൌ ܥ_ଵെ ൫ߖሶ ൅ π൯

Next, we ignore ɔ ܴ ൌܣ

ʹൣߠሶଶ൅ ߖሶଶ ܣ ଶ_{ߠ ൅ ͵ሺܣ െ ܥ}

The satellite can p According to this

ܴ ൌܣ

ʹൣߠሶଶ൅ ߖሶଶ ܣ ଶ_{ߠ ൅ ͵ሺܣ െ}

The equations of

ܣߖሷ ଶ_{ߠ െ ʹܣߠሶ൫} ܣߠሷ ൅ ܣߖሶଶ_ߠ

Defining the

non-τ

by ሺሻᇱ_{the non-d} ߖᇱᇱଶ_{ߠ െ ʹߠ}ᇱ_ሺߖ ߠᇱᇱ_{൅ ߖ}ᇱଶ_ߠ ൅ ߠ ൌ Ͳ Case ii (ͳ ൏ ߙ ൏

In this case the e olar axis perform

angle of precessi

ଶ_{ߠ൨ ߖሶ}ଶ_{൅ ሾπሺܣ}

ଶ_൅πమ

ଶሺܣ െ ܥሻሾ

at ߮ is a cyclic coo

൅ π൯ ߠ ൅ ߮ሶ൧ ൌ

ݏ݅݊ ߠ

ɔ and construct th

ଶ_{ߠ൧ ൅ ሾπܣ}ଶ_ߠ ܥሻ ଶ_ߖଶ_{ߠሿ ൅}

perform the in-pla condition the Rou

ଶ_{ߠ൧ ൅ ሾπܣ}ଶ_ߠ ܥሻ ଶ_ߖଶ_ߠሿ

motion of the sat

ߖሶ ൅ π൯ ߠ ߠ ߠ ൅ ʹܣπߖሶ ߠ

dimensional time dimensional equat

ߖᇱ_{൅ ͳሻ ߠ ߠ} ߠ ൅ ʹߖᇱ_ߠ

ʹ ):

quatorial axis rem ms small angular

on around the ݖଵ

ܣ ଶ_{ߠ ൅ ܥ}ଶ_ߠ

ଶ_{ߠ ൅ ͵}ଶ_ߖ

ordinate where :

ܥ_ଵ

he Routhian:

ߠ ൅ ܥܥଵ ߠሿߖሶ ൅ ൅஼஼భ

ଶ ሺʹπ ߠ െ ܥ

ane motions only uthian takes the f

ߠ ൅ ܥ ߠሿߖሶ ൅π ʹ

tellite in this case

ߠ ൅ ͵πଶ_{ሺܣ െ ܥሻ} ߠ ൅ ͵πଶ_{ሺܣ െ ܥ}

e ߬ ൌ πݐ and deno tions of motion b

൅ ͵ሺͳ െ ߙሻ ଶ_ߠ ߠ ൅ ሺͳ െ ߙሻ ଶ_ߖ

mains approximat oscillations abou

ଵെaxis

ߠሻ ൅ ܥ߮ሶ ߠሿߖሶ ൅

ଶ_ߠሿ

൅πଶ

ʹ ሾܥ െ ܣ ൅ ܥଵሻ

y if the real consta form:

πଶ

ʹ ሾܥ െ ܣ ൅

e become

ଶ_{ߠ ߖ ߖ ൌ} ܥሻ ଶ_{ߖ ߠ}

oting derivatives w ecome:

ߠ ߖ ߖ ൌ Ͳ ߖ ߠ ߠ

tely orthogonal to ut this plane so w

A

ܣ ʹߠሶଶ

(

( ant ܥ_ଵ=0.

(

ൌ Ͳ (

ߠ ൅ πଶ_{ܣ ߠ ൌ}

with respect to

(

o the plane of the we shall use Eul

(8)

(9)

(10)

(11)

(12)

Ͳ (13)

(14)

(15)

e orbit and ler’s angle

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Fig.3

Now, t

ܮ π ଶ ݌

a

ܴ ሺ

T

ܣ ܣ െ

A

߰ ߠ ൅

T orbital

3. So

Assum

ߖ

I

ߖ

W classif

N 2281-4612 2281-3993

the Lagrangian fu

ܮ ൌ ሺ஺

ଶଶߠ ൅ ஼ ଶ πమ

ଶሾͶሺܥ െ ܣሻ ଶߠ ݌_ఝൌడ௅

డఝሶൌ ܥൣ൫߰ሶ ൅

and we get the Ro

ܴ ൌܣ

ʹൣߠሶଶ൅ ߰ሶଶ ሺͶܣ െ ͵ܥሻ ଶ_{ߠ ൅}

Then the equation

ܣ߰ሷ ଶ_{ߠ ൅ ൣܥܥ} ଶ ܣߠሷ ൅ ܣ߰ሶଶ_ߠ െܥܥଶπ ߠ ൅ πଶ

And in terms of th

߰ᇱᇱଶ_{ߠ െ ሾߙܥ} ଶ ߠᇱᇱ_{൅ ߰}ᇱଶ_ߠ ൅ሺͶ െ ͵ߙሻ ߠ

The equations (1 l frame.

olution for in-pl

me thatͲ ൏ ߙ ൏ ͳ ߖᇱᇱ_{൅ ͵ሺͳ െ ߙሻ ݏ݅݊}

Integrating this eq

ߖᇱଶ_{൅ ͵ሺͳ െ ߙሻ ݏ݅݊}

Where h is a mea fied as follow:

unction takes the

ଶ_{ߠሻ ߰ሶ}ଶ_{൅ ሾπሺܣ} ߠ െ ʹܥ ൅ ܣ െ ͵ሺܥ ൅ π൯ ߠ ൅ ߮ሶ൧ ൌ ܥ

outhian

ଶ_{ߠ൧ ൅ ሾπܣ}ଶ_ߠ ൅ ͵ሺܥ െ ܣሻ ଶ_߰

ns of motion are:

ߠ െ ʹܣ൫߰ሶ ൅ π ߠ െ ሾܥܥଶെ ʹܣπ ሺͶܣ െ ͵ܥሻ ߠ

he non-dimension

ߠ െ ʹሺ߰ᇱ_{൅ ͳሻ} ߠ െ ሺߙܥ_ଶെ ʹ ߠ ߠ െఈ஼మ

π ߠ ൌ

4), (15) , (21) a

ane motion

, by backing to e

݊ ߖ ܿ݋ݏ ߖ ൌ Ͳ

quation we obtain

݊ଶ_{ߖ ൌ Ͳ}

asure of the ener

131

form:

ଶ_{ߠ ൅ ܥ}ଶ_ߠሻ െ ܣሻ ଶ_߰ଶ ܥ_ଶ

ߠ ൅ ܥܥ_ଶ ߠሿ߰ሶ ൅ ߰ ଶ_{ߠሿ ൅}஼஼మ

ଶ ሺʹπ

π൯ ߠ ߠ൧ߠሶ ൅ ͵ ߠሿ ߰ሶ ߠ ൅ ͵ ߠ ൌ Ͳ nal time

ߠ ߠሿߠᇱ_{൅ ͵} ߠሻ߰ᇱ_{ߠ ൅ ͵ሺߙ െ}

Ͳ

and (22) describe

quation (14) the n

rgy of the in-plan nary Studies University of Rome

ሻ ൅ ܥ߮ሶ ߠሿ߰ሶ ൅஺ ଶ ߠሿ

πଶ

ʹ ሾܥ െ ܣ ൅ π ߠ െ ܥ_ଶሻ

͵πଶ_{ሺܥ െ ܣሻ}ଶ_ߠ πଶ_{ሺܥ െ ܣሻ}ଶ_߰

͵ሺߙ െ ͳሻ ଶ_ߠ െ ͳሻ ଶ_{߰ ߠ}

e the motion of t

in-plane motion ߠ

ne motion. We w

A

஺

ଶߠሶଶ൅ π߮ሶܥ ߠ ൅

( (

(

ߠ ߰ ߰ ൌ Ͳ

ߠ ߠ

(

߰ ߰ ൌ0 (

ߠ

( the satellite relat

ߠ ൌ Ͳ is described (

( will show that the

൅஼ ଶ߮ሶଶെ

(16) (17)

(18)

(19)

(20)

(21)

(22) tive to the

d by (23)

(24) motion is

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Fig.4

From e

ߖ

a

න

݇

I

ߖ ݅ ߖ

F

߰ ߰ ߰ ݇

න

݅ ߰

݅

߰

4. St

4.1 St

The lin

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equation (24) und

ߖᇱ_{ൌ േඥ݄ െ ͵ሺͳ െ}

and on separation

න ݀ߖ ඥͳ െ ݇_ଵଶଶ_ߖ ݇_ଵଶ_ൌ͵

݄ሺͳ െ ߙሻ

If ȁ݇_ଵȁ ൐ ͳǡ ݄ ൏ ͵ሺ

ߖሺ߬ሻ ൌ ିଵ_൬ߥ ଵݏ ݂ȁ݇_ଵȁ ൏ ͳǡ ݄ ߖሺ߬ሻ ൌ ܽ݉൫േξ݄

For second case w

߰ᇱᇱ_{൅ ͵ሺߙ െ ͳሻ} ߰ᇱଶ_{൅ ͵ሺߙ െ ͳሻ} ߰ᇱ_{ൌ േඥ݄ െ ͵ሺߙ െ} ݇_ଶଶ_ൌ͵

݄ሺߙ െ ͳሻ න ݀ߠ

ඥͳ െ ݇ଶଶଶ߰ ൌ

݂ȁ݇ଶȁ ൏ ͳǡ ݄ ൐ ͵ ߰ሺ߬ሻ ൌ ܽ݉൫േξ݄߬

݂ȁ݇ଶȁ ൐ ͳǡ ݄ ൏ ͵ሺ

߰ሺ߬ሻ ൌ ିଵ_൬ߥ ଶݏ݊

udying of Vibra

Stability of in-plane

near stability equ

der the condition

െ ߙሻ ଶ_ߖ

n of variables

ൌ േξ݄ න ݀߬

ሺͳ െ ߙሻ we define

݊ ቀඥ͵ሺͳ െ ߙሻ߬ǡ ߥ_ଵ ൐ ͵ሺͳ െ ߙሻ ߬ǡ ݇ଵ൯ǡ

when we study th

߰ ߰ ൌ Ͳ ଶ_{߰ ൌ ݄}

െ ͳሻ ݏ݅݊ଶ_߰

ൌ േξ݄ න ݀߬

ሺߙ െ ͳሻ

߬ǡ ݇_ଶ൯ǡ߰ᇱ_{ሺ߬ሻ ൌ} ሺߙ െ ͳሻǡ ߥଶൌ

ͳ ݇_ଶ ݊ ቀඥ͵ሺͳ െ ߙሻ߬ǡ ߥଶ

ation motion

ne motion.

ation w. r. toߠ is

(

0

<

α

<

1

)

e ߥ_ଵൌ ଵ ௞భ

ଵቁ൰ ǡ ߖᇱሺ߬ሻ ൌ ξ݄ܿ

ߖᇱ_{ሺ߬ሻ ൌ േξ݄݀݊}

he in-plane motio

േξ݄݀݊ሺξ݄߬ǡ ݇ଶሻ

ଶቁ൰ ǡ ߰ᇱሺ߬ሻ ൌ ξ݄ܿ݊

s

ܿ݊ሺඥ͵ሺͳ െ ߙሻ߬ǡ ߥ_ଵ

݊ሺξ݄߬ǡ ݇ଵሻ

n ߠ ൌ Ͳ andܥ_ଶൌ

݊ ቀඥ͵ሺͳ െ ߙሻ߬ǡ ߥଶ

A

(

ଵሻ (

(

ൌ Ͳ then

(

ቁ (

(25)

(26)

(27)

(28)

(29)

(30)

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133

ߠᇱᇱ_{൅ ሾߖ}ᇱଶ_{൅ ͳ ൅ ʹߖ}ᇱ_{൅ ͵ሺͳ െ ߙሻ െ ͵ሺͳ െ ߙሻ}ଶ_{ߖሿߠ ൌ ͲሺͲ ൏ ߙ ൏ ͳሻ} ₍₃₂₎ ߠᇱᇱ_{൅ ሾ߰}ᇱଶ_{൅ ͳ ൅ ʹ߰}ᇱ_{െ ͵ሺߙ െ ͳሻ}ଶ_{߰ሿߠ ൌ Ͳሺͳ ൏ ߙ ൏ ʹሻ} ₍₃₃₎

Let ݓ ൌ ඥ͵ሺͳ െ ߙሻ߬ andߖ ൌ ܽ݉ሺݓሻ. Using (26) and (32) we get

ௗమ_ఏ

ௗ௪మ൅ ቂͳ ൅ ߥଵଶ൅

ଵ

ଷሺଵିఈሻെ ʹߥଵଶݏ݊ଶሺݓǡ ߥଵሻ ൅ඥయሺభషഀሻమഌభ ܿ݊ሺݓǡ ߥଵሻቃ ߠ ൌ Ͳ (34) In trigonometric form with ߖ ൌ ߥଵ ߚ

ሺͳ െ ߥ_ଵଶଶ_ߚሻ݀ଶߠ

݀ߚଶെ ߥଵଶ ߚ ߚ ݀ߠ

݀ߚ൅ ሺͳ ൅ ߥଵଶ൅ ͳ

͵ሺͳ െ ߙሻെ ʹߥଵଶଶߚ ൅ మഌభ

ඥయሺభషഀሻ ߚሻߠ ൌ Ͳ (35) This equation can be transformed to hill's equation as follow:

Let ߠ ൌ ݂݃ ݃ᇱᇱ_{൅ ቂ} ଷఔభరୱ୧୬మଶఉ

ଵ଺ሺଵିఔభమୱ୧୬మఉሻమ൅

మశయቀమశయഌభమ ౙ౥౩ మഁቁሺభషഀሻశరഌభඥయሺభషഀሻౙ౥౩ഁ

లሺభషഀሻ൫భషഌభమ ౩౟౤మ ഁ൯ ቃ ݃ ൌ Ͳ (36) When ݓ ൌ ඥ͵ሺߙ െ ͳሻ߬ and ߰ ൌ ܽ݉ሺݓሻ then use equations (30) and (33) to get

ௗమ_ఏ

ௗ௪మ൅ ቂߥଶଶ൅

ଵ

ଷሺఈିଵሻെ ʹߥଶଶݏ݊ଶሺݓǡ ߥଶሻ ൅ඥయሺഀషభሻమഌమ ܿ݊ሺݓǡ ߥଶሻቃ ߠ ൌ Ͳ (37) We will use the same previous method to get the trigonometric form

߰ ൌ ߥ_ଶ ߚ Ƭ߰ᇱ_{ൌ ξ݄ܿ݋ݏߚ}

ሺͳ െ ߥଶଶଶߚሻ ݀ଶ_ߠ

݀ߚଶെ ߥଶଶ ߚ ߚ ݀ߠ ݀ߚ൅ ሺߥଶଶ൅

ͳ

͵ሺߙ െ ͳሻെ ʹߥଶଶଶߚ ൅ మഌమ

ඥయሺഀషభሻ ߚሻߠ ൌ Ͳ (38) Transform it to Hill's equation

݃ᇱᇱ൅ ቂ ଷఔమరୱ୧୬మଶఉ

ଵ଺ሺଵିఔమమୱ୧୬మఉሻమ൅

మశవഌమమሺഀషభሻ ౙ౥౩ మഁశరഌమඥయሺഀషభሻౙ౥౩ഁ

లሺഀషభሻ൫భషഌమమ ౩౟౤మ ഁ൯ ቃ ݃ ൌ Ͳ (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

ሺͳ െ ߥ_ଵଶଶ_ߚሻௗమఏ

ௗఉమെ ߥଵଶ ߚ ߚ

ௗఏ

ௗఉ൅ ሺܾ െ ʹߥଵ

ଶଶ_{ߚ ൅ ݀ ߚሻߠ ൌ Ͳ} ₍₄₀₎

ܾ ൌ ͳ ൅ ߥ_ଵଶ_൅ ͳ ͵ሺͳ െ ߙሻ ݀ ൌ ʹߥଵ

ඥ͵ሺͳ െ ߙሻ

and equation (38) in the form

ሺͳ െ ߥ_ଶଶଶ_ߚሻௗమఏ

ௗఉమെ ߥଶଶ ߚ ߚ

ௗఏ

ௗఉ൅ ሺܾଵെ ʹߥଶ

ଶଶ_{ߚ ൅ ݀}

ଵ ߚሻߠ ൌ Ͳ (41) ܾ_ଵൌ ߥ_ଶଶ_൅ ͳ

͵ሺߙ െ ͳሻ ݀_ଵൌ ଶఔమ

ඥଷሺఈିଵሻ

and the non-trivial periodic solutions of equation (40) are:

4.2.1 Odd ૛࣊ periodic solution:

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ߠ ൌ σஶ௡ୀ଴ܣ௡ ݊ߚ (42)

Substitution from (42) in (40), yields the following recurrence relations for coefficientsܣ௡ ሺܾ െ ͳ െ ߥଵଶሻܣଵ൅

݀

ʹܣଶെ ߥଵଶܣଷൌ Ͳ ݀

ʹܣଵ൅ ሺܾ െ Ͷ ൅ ߥଵଶሻܣଶ൅ ݀ ʹܣଷെ

ͷ

ʹߥଵଶܣସൌ Ͳ ݀

ʹܣଶ൅ ൬ܾ െ ͻ ൅ ͹

ʹߥଵଶ൰ ܣଷ൅ ݀ ʹܣସെ

ͻ

ʹߥଵଶܣହൌ Ͳ

For݊ ൒ ʹ

െ݊ሺʹ݊ െ ͵ሻ

ʹ ߥଵଶܣଶ௡ିଶ൅ ݀

ʹܣଶ௡ିଵ൅ ሺܾ െ Ͷ݊ଶ൅ ሺʹ݊ଶെ ͳሻߥଵଶሻܣଶ௡൅ ݀ ʹܣଶ௡ାଵ െ݊ሺʹ݊ ൅ ͵ሻ

ʹ ߥଵଶܣଶ௡ାଶൌ Ͳ െ݊ሺʹ݊ െ ͳሻ െ ͳ

ʹ ߥଵଶܣଶ௡ିଵ൅ ݀

ʹܣଶ௡൅ ቆܾ െ ሺʹ݊ ൅ ͳሻଶ൅

ሺʹ݊ ൅ ͳሻଶ_{െ ʹሻ}

ʹ ߥଵଶቇ ܣଶ௡ାଵ ൅ௗ

ଶܣଶ௡ାଶെ

௡ሺଶ௡ାହሻାଶ ଶ ߥଵ

ଶ_ܣ

ଶ௡ାଷൌ Ͳ (43)

For the second case the solution in a Fourier series gives the recurrence relations for coefficients ܣ_௡ in the form

ሺܾଵെ ͳ െ ߥଶଶሻܣଵ൅ ݀ଵ

ʹܣଶെ ߥଶଶܣଷൌ Ͳ ݀_ଵ

ʹ ܣଵ൅ ሺܾଵെ Ͷ ൅ ߥଶଶሻܣଶ൅ ݀_ଵ

ʹ ܣଷെ ͷ

ʹߥଶଶܣସൌ Ͳ ݀_ଵ

ʹ ܣଶ൅ ൬ܾଵെ ͻ ൅ ͹

ʹߥଶଶ൰ ܣଷ൅ ݀_ଵ

ʹ ܣସെ ͻ

ʹߥଶଶܣହൌ Ͳ

For݊ ൒ ʹ

ʹ ߥଶଶܣଶ௡ିଶ൅ ݀ଵ

ʹܣଶ௡ିଵ൅ ሺܾଵെ Ͷ݊ଶ൅ ሺʹ݊ଶെ ͳሻߥଶଶሻܣଶ௡൅ ݀ଵ

ʹܣଶ௡ାଵ െ݊ሺʹ݊ ൅ ͵ሻ

ʹ ߥଶଶܣଶ௡ାଶൌ Ͳ െ௡ሺଶ௡ିଵሻିଵ

ଶ ሺʹ݊ଶെ ͳሻߥଶ ଶ_ܣ

ଶ௡ିଵ൅ௗ_ଶభܣଶ௡൅ ቀܾଵെ ሺʹ݊ ൅ ͳሻଶ൅ሺଶ௡ାଵሻ

మ_ିଶሻ

ଶ ߥଶ ଶ_{ቁ ܣ}

ଶ௡ାଵ൅ௗ_ଶభܣଶ௡ାଶെ ௡ሺଶ௡ାହሻାଶ

ଶ ߥଶଶܣଶ௡ାଷൌ Ͳ (44)

4.2.2 Even ૛࣊ -periodic solution

Seeking a solution of the form

ߠ ൌ σஶ ܤ_௡

௡ୀ଴ ݊ߚ (45)

the substitution in equation (40) yields the recurrence relations for coefficients ܤ_௡:

ሺܾ െ ߥଵଶሻܤ଴൅ ݀ ʹܤଵൌ Ͳ ݀ܤ_଴൅ ሺܾ െ ͳሻܤ_ଵ൅݀

ʹܤଶെ ߥଵଶܤଷൌ Ͳ ߥ_ଵଶ_ܤ

଴൅ ݀

ʹܤଵ൅ ሺܾ െ Ͷ ൅ ߥଵଶሻܤଶ൅ ݀ ʹܤଷെ

ͷ

ʹߥଵଶܤସൌ Ͳ ݀

ʹܤଶ൅ ൬ܾ െ ͻ ൅ ͹

ʹߥଵଶ൰ ܤଷ൅ ݀ ʹܤସെ

ͻ

ʹߥଵଶܤହൌ Ͳ

For݊ ൒ ʹ

ʹ ߥଵଶܤଶ௡ିଶ൅ ݀

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135

െ݊ሺʹ݊ ൅ ͵ሻ

ʹ ߥଵଶܤଶ௡ାଶൌ Ͳ െ݊ሺʹ݊ െ ͳሻ െ ͳ

ʹ ߥଵଶܤଶ௡ିଵ൅ ݀

ʹܤଶ௡൅ ቆܾ െ ሺʹ݊ ൅ ͳሻଶ൅

ʹ ߥଵଶቇ ܤଶ௡ାଵ ൅ௗ

ଶܤଶ௡ାଶെ

௡ሺଶ௡ାହሻାଶ

ଶ ߥଵଶܤଶ௡ାଷൌ Ͳ (46)

The recurrence relations for coefficientsܤ_௡for the second case

ሺܾଵെ ߥଶଶሻܤ଴൅ ݀_ଵ

ʹܤଵൌ Ͳ ݀_ଵܤ_଴൅ ሺܾ_ଵെ ͳሻܤ_ଵ൅݀ଵ

ʹܤଶെ ߥଶଶܤଷൌ Ͳ ߥ_ଶଶ_ܤ

଴൅ ݀_ଵ

ʹܤଵ൅ ሺܾଵെ Ͷ ൅ ߥଶଶሻܤଶ൅ ݀_ଵ

ʹܤଷെ ͷ

ʹߥଶଶܤସൌ Ͳ ݀_ଵ

ʹ ܤଶ൅ ൬ܾଵെ ͻ ൅ ͹

ʹߥଶଶ൰ ܤଷ൅ ݀_ଵ

ʹܤସെ ͻ

ʹߥଶଶܤହൌ Ͳ

For݊ ൒ ʹ

ʹ ߥଶଶܤଶ௡ିଶ൅ ݀ଵ

ʹ ܤଶ௡ିଵ൅ ሺܾଵെ Ͷ݊ଶ൅ ሺʹ݊ଶെ ͳሻߥଶଶሻܤଶ௡൅ ݀ଵ

ʹܤଶ௡ାଵ െ݊ሺʹ݊ ൅ ͵ሻ

ʹ ߥଶଶܤଶ௡ାଶൌ Ͳ െ݊ሺʹ݊ െ ͳሻ െ ͳ

ʹ ߥଶଶܤଶ௡ିଵ൅ ݀_ଵ

ʹܤଶ௡൅ ቆܾଵെ ሺʹ݊ ൅ ͳሻଶ൅

ʹ ߥଶଶቇ ܤଶ௡ାଵ ൅ௗభ

ଶܤଶ௡ାଶെ

௡ሺଶ௡ାହሻାଶ

ଶ ߥଶଶܤଶ௡ାଷൌ Ͳ (47)

4.2.3 Even Ͷߨ -periodic solution

The solution in this case has the form

ߠ ൌ σஶ ܥ_௡

௡ୀ଴ ݊ߚ ൅ σஶ௡ୀ଴ܹ௡൫݊ ൅భమ൯ߚ (48) Substitution in equation (40) yields the recurrence relations for coefficientsܥ_௡and ܹ_௡ ሺܾ െ ߥ_ଵଶ_ሻܥ

଴൅ ݀ ʹܥଵൌ Ͳ ݀ܥ଴൅ ሺܾ െ ͳሻܥଵ൅

݀

ʹܥଶെ ߥଵଶܥଷൌ Ͳ ߥ_ଵଶ_ܥ

଴൅ ݀

ʹܥଵ൅ ሺܾ െ Ͷ ൅ ߥଵଶሻܥଶ൅ ݀ ʹܥଷെ

ͷ

ʹߥଵଶܥସൌ Ͳ ݀

ʹܥଶ൅ ൬ܾ െ ͻ ൅ ͹

ʹߥଵଶ൰ ܥଷ൅ ݀ ʹܥସെ

ͻ

ʹߥଵଶܥହൌ Ͳ ൬ܾ ൅݀

ʹെ ͳ Ͷെ

͹

ͺߥଵଶ൰ ܹ଴൅ ሺ ͷ ͳ͸ߥଵଶ൅

݀ ʹሻܹଵെ

͹

ͳ͸ߥଵଶܹଶൌ Ͳ ൬݀

ʹ൅ ͻ

ͳ͸ߥଵଶ൰ ܹ଴൅ ൬ܾ െ ͻ Ͷ൅

ͳ

ͺߥଵଶ൰ ܹଵ൅ ݀ ʹܹଶെ

ʹ͹

ͳ͸ߥଵଶܹଷൌ Ͳ

For݊ ൒ ʹ

ʹ ߥଵଶܥଶ௡ିଶ൅ ݀

ʹܥଶ௡ିଵ൅ ሺܾ െ Ͷ݊ଶ൅ ሺʹ݊ଶെ ͳሻߥଵଶሻܥଶ௡൅ ݀ ʹܥଶ௡ାଵ െ݊ሺʹ݊ ൅ ͵ሻ

ʹ ߥଵଶܥଶ௡ାଶൌ Ͳ െ݊ሺʹ݊ െ ͳሻ െ ͳ

ʹ ߥଵଶܥଶ௡ିଵ൅ ݀

ʹܥଶ௡൅ ቆܾ െ ሺʹ݊ ൅ ͳሻଶ൅

ሺʹ݊ ൅ ͳሻଶ_{െ ʹ}

ʹ ߥଵଶቇ ܥଶ௡ାଵ

൅݀ ʹܥଶ௡ାଶെ

݊ሺʹ݊ ൅ ͷሻ ൅ ʹ

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െሺʹ݊ െ ͷሻሺʹ݊ ൅ ͳሻ

ͳ͸ ߥଵଶܹ௡ିଶ൅ ݀

ʹܹ௡ିଵ൅ ቆܾ െ

ሺʹ݊ ൅ ͳሻଶ Ͷ ൅

ሺʹ݊ ൅ ͳሻଶ_{െ ͺ} ͺ ߥଵଶቇ ܹ௡ ൅ௗ

ଶܹ௡ାଵെ

ሺଶ௡ାଵሻሺଶ௡ା଻ሻ ଵ଺ ߥଵ

ଶ_ܹ

௡ାଶൌ Ͳ (49)

And for second case the recurrence relations for coefficients ܥ_௡and ܹ_௡ ሺܾଵെ ߥଶଶሻܥ଴൅

݀ଵ ʹܥଵൌ Ͳ ݀_ଵܥ_଴൅ ሺܾ_ଵെ ͳሻܥ_ଵ൅݀ଵ

ʹܥଶെ ߥଶଶܥଷൌ Ͳ ߥ_ଶଶ_ܥ

଴൅ ݀_ଵ

ʹܥଵ൅ ሺܾଵെ Ͷ ൅ ߥଶଶሻܥଶ൅ ݀_ଵ

ʹܥଷെ ͷ

ʹߥଶଶܥସൌ Ͳ ݀_ଵ

ʹ ܥଶ൅ ൬ܾଵെ ͻ ൅ ͹

ʹߥଶଶ൰ ܥଷ൅ ݀_ଵ

ʹܥସെ ͻ

ʹߥଶଶܥହൌ Ͳ ൬ܾଵ൅

݀ ʹെ

ͳ Ͷെ

͹

ͺߥଶଶ൰ ܹ଴൅ ሺ ͷ ͳ͸ߥଶଶ൅

݀_ଵ ʹሻܹଵെ

͹

ͳ͸ߥଶଶܹଶൌ Ͳ ൬݀ଵ

ʹ ൅ ͻ

ͳ͸ߥଶଶ൰ ܹ଴൅ ൬ܾଵെ ͻ Ͷ൅

ͳ

ͺߥଶଶ൰ ܹଵ൅ ݀_ଵ

ʹܹଶെ ʹ͹

ͳ͸ߥଶଶܹଷൌ Ͳ

For݊ ൒ ʹ

ʹ ߥଶଶܥଶ௡ିଶ൅ ݀ଵ

ʹ ܥଶ௡ିଵ൅ ሺܾଵെ Ͷ݊ଶ൅ ሺʹ݊ଶെ ͳሻߥଶଶሻܥଶ௡൅ ݀ଵ

ʹܥଶ௡ାଵ െ݊ሺʹ݊ ൅ ͵ሻ

ʹ ߥଶଶܥଶ௡ାଶൌ Ͳ െ݊ሺʹ݊ െ ͳሻ െ ͳ

ʹ ߥଶଶܥଶ௡ିଵ൅ ݀_ଵ

ʹܥଶ௡൅ ቆܾଵെ ሺʹ݊ ൅ ͳሻଶ൅

ʹ ߥଶଶቇ ܥଶ௡ାଵ

൅݀ଵ ʹܥଶ௡ାଶെ

݊ሺʹ݊ ൅ ͷሻ ൅ ʹ

ʹ ߥଶଶܥଶ௡ାଷൌ Ͳ െሺʹ݊ െ ͷሻሺʹ݊ ൅ ͳሻ

ͳ͸ ߥଶଶܹ௡ିଶ൅ ݀_ଵ

ʹܹ௡ିଵ൅ ቆܾଵെ

ሺʹ݊ ൅ ͳሻଶ_{െ ͺ} ͺ ߥଶଶቇ ܹ௡ ൅ௗభ

ଶܹ௡ାଵെ

ሺଶ௡ାଵሻሺଶ௡ା଻ሻ

ଵ଺ ߥଶଶܹ௡ାଶൌ Ͳ (50)

4.2.4 Odd ૝࣊ -periodic solution

Assuming such solution in the form

ߠ ൌ σஶ ܩ_௡

௡ୀ଴ ݊ߚ ൅ σஶ௡ୀ଴ܪ௡ሺ݊ ൅భమሻߚ (51)

Substitution in (40) yields the following recurrence relations for coefficients ܩ_௡and ܪ_௡ǣ ሺܾ െ ͳ െ ߥଵଶሻܩଵ൅

݀

ʹܩଶെ ߥଵଶܩଷൌ Ͳ ݀

ʹܩଵ൅ ሺܾ െ Ͷ ൅ ߥଵଶሻܩଶ൅ ݀ ʹܩଷെ

ͷ

ʹߥଵଶܩସൌ Ͳ ݀

ʹܩଶ൅ ൬ܾ െ ͻ ൅ ͹

ʹߥଵଶ൰ ܩଷ൅ ݀ ʹܩସെ

ͻ

ʹߥଵଶܩହൌ Ͳ ൬ܾ െ݀

ʹെ ͳ Ͷെ

͹

ͺߥଵଶ൰ ܪ଴൅ ሺ ݀ ʹെ

ͷ

ͳ͸ߥଵଶሻܪଵെ ͹

ͳ͸ߥଵଶܪଶൌ Ͳ ൬݀

ʹെ ͻ

ͳ͸ߥଵଶ൰ ܪ଴൅ ൬ܾ െ ͻ Ͷ൅

ͳ

ͺߥଵଶ൰ ܪଵ൅ ݀ ʹܪଶെ

ʹ͹

ͳ͸ߥଵଶܪଷൌ Ͳ

For݊ ൒ ʹ :

െ݊ሺʹ݊ െ ͳሻ െ ͳ

ʹ ߥଵଶܩଶ௡ିଵ൅ ݀

ʹܩଶ௡൅ ቀܾ െ ሺʹ݊ ൅ ͳሻଶ൅

ሺଶ௡ାଵሻమ_ିଶ

ଶ ߥଵଶቁ ܩଶ௡ାଵ൅ ݀ ʹܩଶ௡ାଶ െ݊ሺʹ݊ ൅ ͷሻ ൅ ʹ

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137

ʹ ߥଵଶܩଶ௡ିଶ൅ ݀

ʹܩଶ௡ିଵ൅ ሺܾ െ Ͷ݊ଶ൅ሺଶ௡మିଵሻߥଵଶሻܩଶ௡൅ ݀ ʹܩଶ௡ାଵ െ݊ሺʹ݊ ൅ ͵ሻ

ʹ ߥଵଶܩଶ௡ାଶൌ Ͳ െሺʹ݊ െ ͷሻሺʹ݊ ൅ ͳሻ

ͳ͸ ߥଵଶܪ௡ିଶ൅ ݀

ʹܪ௡ିଵ൅ ቆܾ െ

ሺʹ݊ ൅ ͳሻଶ_{െ ͺ} ͺ ߥଵଶቇ ܪ௡

൅ௗ_ଶܪ௡ାଵെሺଶ௡ାଵሻሺଶ௡ା଻ሻ_ଵ଺ ߥଵଶܪ௡ାଶൌ Ͳ (52)

For second case the recurrence relations for coefficients ܩ_௡and ܪ_௡ are :

ሺܾଵെ ͳ െ ߥଶଶሻܩଵ൅ ݀ଵ

ʹ ܩଶെ ߥଶଶܩଷൌ Ͳ ݀_ଵ

ʹ ܩଵ൅ ሺܾଵെ Ͷ ൅ ߥଶଶሻܩଶ൅ ݀_ଵ

ʹ ܩଷെ ͷ

ʹߥଶଶܩସൌ Ͳ ݀_ଵ

ʹ ܩଶ൅ ൬ܾଵെ ͻ ൅ ͹

ʹߥଶଶ൰ ܩଷ൅ ݀_ଵ

ʹ ܩସെ ͻ

ʹߥଶଶܩହൌ Ͳ ൬ܾ_ଵെ݀ଵ

ʹ െ ͳ Ͷെ

͹

ͺߥଶଶ൰ ܪ଴൅ ሺ ݀_ଵ

ʹ െ ͷ

ͳ͸ߥଶଶሻܪଵെ ͹

ͳ͸ߥଶଶܪଶൌ Ͳ ൬݀ଵ

ʹ െ ͻ

ͳ͸ߥଶଶ൰ ܪ଴൅ ൬ܾଵെ ͻ Ͷ൅

ͳ

ͺߥଶଶ൰ ܪଵ൅ ݀ଵ

ʹܪଶെ ʹ͹

ͳ͸ߥଶଶܪଷൌ Ͳ

For݊ ൒ ʹ :

െ݊ሺʹ݊ െ ͳሻ െ ͳ

ʹ ߥଶଶܩଶ௡ିଵ൅ ݀_ଵ

ʹܩଶ௡൅ ቆܾଵെ ሺʹ݊ ൅ ͳሻଶ൅

ʹ ߥଶଶቇ ܩଶ௡ାଵ

൅݀ଵ ʹܩଶ௡ାଶെ

݊ሺʹ݊ ൅ ͷሻ ൅ ʹ

ʹ ߥଶଶܩଶ௡ାଷൌ Ͳ െ݊ሺʹ݊ െ ͵ሻ

ʹ ߥଶଶܩଶ௡ିଶ൅ ݀ଵ

ʹ ܩଶ௡ିଵ൅ ሺܾଵെ Ͷ݊ଶ൅ ሺʹ݊ଶെ ͳሻߥଶଶሻܩଶ௡൅ ݀ଵ

ʹ ܩଶ௡ାଵ െ݊ሺʹ݊ ൅ ͵ሻ

ʹ ߥଶଶܩଶ௡ାଶൌ Ͳ െሺʹ݊ െ ͷሻሺʹ݊ ൅ ͳሻ

ͳ͸ ߥଶଶܪ௡ିଶ൅ ݀ଵ

ʹܪ௡ିଵ൅ ቆܾଵെ

ሺʹ݊ ൅ ͳሻଶ_{െ ͺ} ͺ ߥଶଶቇ ܪ௡ ൅ௗభ

ଶܪ௡ାଵെ

ሺଶ௡ାଵሻሺଶ௡ା଻ሻ

ଵ଺ ߥଶଶܪ௡ାଶൌ Ͳ (53)

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 symbolsܹ_௦ǡ ܩ_௦ denote the (s) zones of stability and instability respectively. The zones of

stability and instability are separated by the curves

∏

± s

2 , ±

+

∏

1

2s whose equations describe the

distribution of eigen-values for boundary problems of periods

2

π

,

4

π

. We shall draw these curves by solving the last four determinants. We classify the curves as:

+

∏

s

2 for even

2

π

−

periodic solutions and −

∏

s

(12)

E-ISSN ISSN 2 W equati

g

le ߙ a throug

toɗ଴ൌ

π

4

a

Fig.5 In the 0

=

ψ

Equat

g

L N 2281-4612 2281-3993 + +

∏

1

2s for even

4

We now intend to

ion of vibration (3

1

(

3

1

[

−

+

′′

g

α

et

1

₃₍₁1 ₎

ω

α

=

+

−

ߙ_ଵൌͺ ͻǡ ߙଶൌ

ʹ͵ ʹͶǡ ߙ

and so on. There gh each of those

ൌ Ͳ. For example

at ߙ ൌ଼ ଽ and

+

∏

s

2

e same manner a

0

to determine th tion (39) become

)

1

(

3

1

−

+

′′

g

α

Let₃₍_α1₋₁₎

=

n

2wh

−

π

4

periodic solu

o find the intersec

36) takes the form

0

]

)

g

=

α

2

ω

where

ω

is

ߙ_ଷൌͶͶ Ͷͷǡ ߙସൌ

͹ͳ ͹ʹ

e exist two curve e values of Į. We

e, two curves, o

,

∏

−

s

2 with period

as in the case of

he zones of stabi es:

0

=

g

here

n

(+) ve int

utions and − +

∏

1 2s fo

ctions of

∏

±

s

2 , ±

∏

2s

m

s constant , then

ǡ ߙ_ହൌͳͲͶ ͳͲͷǡ ǥ ǥ

es with the sam e note that the

ne odd − +

∏

1 2s and

d

2

π

at ߙ ൌଶଷ ଶସ (

f (

0

<

α

<

1

) w

lity and instability

teger .then 3

=

α

or odd

4

π

−

pe

±

+

∏

1

s with

ψ

0

=

0

.

) 1 ( 3 1 2

1

−

₋

=

_ω

α

e period one eve stability zones oc

the other even

2

∏

(see Fig. 5).

we get the inters

y for the case

1

<

1

2

3

1

+

n for

n

=

A

eriodic solutions.

On the line

ψ

₀

(

, for

ω

= 2, 3,

en and other od ccupy the lower

.

1 2 + +

∏

s with the sam

sections of

∏

±

s

2 ,

2

<

α

(

= 1 , 2 , 3 , 4 , ...

0

=

, the

(54)

4, 5,6 ….

dd passing part near me period , ± +

∏

1 2s with

(55)

(13)

E-ISSN ISSN 2

ߙ

a

Fig.6

Refer

Beletsk R Markee K Akulen m M Markee c Yehia

M Arscott

N 2281-4612 2281-3993

ߙଵൌ Ͷ ͵ǡ ߙଶൌ

ͳ͵ ͳʹǡ ߙ

and so on. The zo

rences

kii V.V. 1965, Mo Russian).

ev, A. p.: 1975, Kosmich. Issled. 13 nko, L.D., Nestero mechanical system Mekh., 63, 705-713 ev A.P., and Bard circular orbit, Celes , H.M. :1986 ,On Mecan. Theor. App t F. M . 1964, Peri

ߙଷൌ ʹͺ ʹ͹ǡ ߙସൌ

Ͷͻ Ͷͺ

ones of stability a

otion Of An Artifi

Stability of plane 3, 322-336 (in Rus ov, S. V. and Sh ms, Prikl. Mat. Me

3 .

din 2003, On the st. Mech. Dyn. Ast the motion of a pl., 5, 747-754.

odic Differential E

139

ǡ ǥ ǥǤ

and instability are

icial Satellite Abo

e oscillations and ssian), English tran hmatkov, A.M. 19 ekh. 63, 746-756

stability of plana t. 85, 51-66.

rigid body acted

quations .

shown in Fig. 6.

ut Its Center Of

rotations of a s nsl. Cosmic Res., 999 , Generalized 6 (in Russian) ;En

ar oscillations and

upon by potentia

A

Mass. Nauka, M

satellite in a circu 285 – 298 . d parametric osci nglish transl J. A

rotations of a sa

al and gyroscopic

Moscow (In

ular orbit ,

illations of Appl. Math.

atellite in a

(14)