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 Published by MCSSER-CEMAS-Sapienza University 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 ൌ ሺࢉ࢙࣒ࢉ࢙࣐ െ ࢙ࣂ࢙࣐࢙࣒ ࣒ǡ െࢉ࢙࣒࢙࣐ െ ࢙ࣂࢉ࢙࣐࢙࣒ ࣒ǡ ࢉ࢙ࣂ࢙࣒ሻ ࢼ ൌ ሺ࢙࣒ࢉ࢙࣐ ࢙ࣂ࢙࣐ࢉ࢙࣒ ࣒ǡ െ࢙࣒ࢉ࢙࣐ ࢙ࣂ࢙࣐ࢉ࢙࣒ ࣒ǡ െࢉ࢙ࣂࢉ࢙࣒ሻ ࢽ ൌ ሺࢉ࢙ࣂ࢙࣐ǡ ࢉ࢙ࣂࢉ࢙࣐ǡ ࢙ࣂሻ ࣓ ൌ ሺ࣒ሶࢉ࢙ࣂ࢙࣐ ࣐ െ ࣂሶࢉ࢙࣐ǡ ࣒ሶࢉ࢙ࣂ ࣂࢉ࢙࣐ ࣂሶ࢙࣐ǡ ࣒ሶ࢙࣐ ࣐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 ሺࢽ 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 ൌ ሺǡ ࢉ࢙࣒ǡ ࢙࣒ሻǡࢼ ൌ ሺǡ െ࢙࣒ǡ ࢉ࢙࣒ሻǡࢽ ൌ ሺǡ 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 ܣሺଶ ݍଶ ሻ ݎܥଶ πሾܣሺߛଵ 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 129 (7) Academic Journal J of Interdisciplinnary Studies E-ISSN N 2281-4612 ISSN 2281-3993 2 Published by MCSSER-CEMAS-Sapienza University U of Rome Vol 2 No 6 August 2013 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 130 E-ISSN N 2281-4612 ISSN 2281-3993 2 Academic Journal J of Interdisciplinnary Studies Published by MCSSER-CEMAS-Sapienza University U 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 ݏൌ 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 E-ISSN N 2281-4612 ISSN 2281-3993 2 Academic Journal J of Interdisciplinnary Studies Published by MCSSER-CEMAS-Sapienza University U of Rome Vol 2 No 6 August 2013 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 ቂଵሺଵିఔ మ ୱ୧୬మ ఉሻమ భ మశయቀమశయഌమ భ ౙ౩ మഁቁሺభషഀሻశరഌభ ඥయሺభషഀሻ ౙ౩ ഁ 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 ଵሺଵିఔమమ ୱ୧୬మ ఉሻమ మ ౙ౩ మഁశరഌమ ඥయሺഀషభሻ ౙ౩ ഁ మశవഌమሺഀషభሻ ቃ݃ ൌ Ͳ లሺഀషభሻ൫భషഌమ ౩మ ഁ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 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 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 ܣଶିଶ ܣଶିଵ ሺܾ െ Ͷ݊ଶ ሺʹ݊ଶ െ ͳሻߥଵଶ ሻܣଶ ܣଶାଵ െ ʹ ʹ ʹ ݊ሺʹ݊ ͵ሻ ଶ ߥଵ ܣଶାଶ ൌ Ͳ െ ʹ ݀ ሺʹ݊ ͳሻଶ െ ʹሻ ଶ ݊ሺʹ݊ െ ͳሻ െ ͳ ଶ ߥଵ ܣଶିଵ ܣଶ ቆܾ െ ሺʹ݊ ͳሻଶ ߥଵ ቇ ܣଶାଵ െ ʹ ʹ ʹ ௗ ܣଶାଶ െ ଶ ሺଶାହሻାଶ ଶ ߥଵ ܣଶାଷ 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 ܣଶିଶ ܣଶିଵ ሺܾଵ െ Ͷ݊ଶ ሺʹ݊ଶ െ ͳሻߥଶଶ ሻܣଶ ܣଶାଵ െ ʹ ʹ ʹ ݊ሺʹ݊ ͵ሻ ଶ െ ߥଶ ܣଶାଶ ൌ Ͳ ʹ ሺଶିଵሻିଵ ௗ ሺଶାଵሻమ ିଶሻ ଶ ௗ െ ሺʹ݊ଶ െ ͳሻߥଶଶ ܣଶିଵ భ ܣଶ ቀܾଵ െ ሺʹ݊ ͳሻଶ ߥଶ ቁ ܣଶାଵ భ ܣଶାଶ െ ଶ ሺଶାହሻାଶ ଶ ߥଶ ܣଶାଷ 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 ܤଶିଶ ܤଶିଵ ሺܾ െ Ͷ݊ଶ ሺʹ݊ଶ െ ͳሻߥଵଶ ሻܤଶ ܤଶାଵ െ ʹ ʹ ʹ 134 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 ݊ሺʹ݊ ͵ሻ ଶ ߥଵ ܤଶାଶ ൌ Ͳ ʹ ݊ሺʹ݊ െ ͳሻ െ ͳ ଶ ݀ ሺʹ݊ ͳሻଶ െ ʹሻ ଶ െ ߥଵ ܤଶିଵ ܤଶ ቆܾ െ ሺʹ݊ ͳሻଶ ߥଵ ቇ ܤଶାଵ ʹ ʹ ʹ െ ௗ ሺଶାହሻାଶ ଶ ଶ ܤଶାଶ െ ߥଵଶ ܤଶାଷ ൌ 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 ܤଶିଶ ܤଶିଵ ሺܾଵ െ Ͷ݊ଶ ሺʹ݊ଶ െ ͳሻߥଶଶ ሻܤଶ ܤଶାଵ ʹ ʹ ʹ ݊ሺʹ݊ ͵ሻ ଶ ߥଶ ܤଶାଶ ൌ Ͳ െ ʹ ݀ଵ ሺʹ݊ ͳሻଶ െ ʹሻ ଶ ݊ሺʹ݊ െ ͳሻ െ ͳ ଶ ߥଶ ܤଶିଵ ܤଶ ቆܾଵ െ ሺʹ݊ ͳሻଶ ߥଶ ቇ ܤଶାଵ െ ʹ ʹ ʹ ௗభ ଶ ܤଶାଶ െ ሺଶାହሻାଶ ଶ ߥଶଶ ܤଶାଷ ൌ Ͳ (47) 4.2.3 Even Ͷߨ -periodic solution The solution in this case has the form భ ஶ (48) ߠ ൌ σஶ ୀ ܥ ݊ߚ σୀ ܹ 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 ܥଶିଶ ܥଶିଵ ሺܾ െ Ͷ݊ଶ ሺʹ݊ଶ െ ͳሻߥଵଶ ሻܥଶ ܥଶାଵ െ ʹ ʹ ʹ ݊ሺʹ݊ ͵ሻ ଶ ߥଵ ܥଶାଶ ൌ Ͳ െ ʹ ݀ ሺʹ݊ ͳሻଶ െ ʹ ଶ ݊ሺʹ݊ െ ͳሻ െ ͳ ଶ ߥଵ ܥଶିଵ ܥଶ ቆܾ െ ሺʹ݊ ͳሻଶ ߥଵ ቇ ܥଶାଵ െ ʹ ʹ ʹ ݊ሺʹ݊ ͷሻ ʹ ଶ ݀ ߥଵ ܥଶାଷ ൌ Ͳ ܥଶାଶ െ ʹ ʹ 135 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 ሺʹ݊ െ ͷሻሺʹ݊ ͳሻ ଶ ݀ ሺʹ݊ ͳሻଶ ሺʹ݊ ͳሻଶ െ ͺ ଶ ߥଵ ܹିଶ ܹିଵ ቆܾ െ ߥଵ ቇ ܹ ͺ Ͷ ͳ ʹ ௗ ଶ ܹାଵ െ ሺଶାଵሻሺଶାሻ 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 ܥଶିଶ ܥଶିଵ ሺܾଵ െ Ͷ݊ଶ ሺʹ݊ଶ െ ͳሻߥଶଶ ሻܥଶ ܥଶାଵ െ ʹ ʹ ʹ ݊ሺʹ݊ ͵ሻ ଶ ߥଶ ܥଶାଶ ൌ Ͳ െ ʹ ݀ଵ ሺʹ݊ ͳሻଶ െ ʹ ଶ ݊ሺʹ݊ െ ͳሻ െ ͳ ଶ ߥଶ ܥଶିଵ ܥଶ ቆܾଵ െ ሺʹ݊ ͳሻଶ ߥଶ ቇ ܥଶାଵ െ ʹ ʹ ʹ ݊ሺʹ݊ ͷሻ ʹ ଶ ݀ଵ ߥଶ ܥଶାଷ ൌ Ͳ ܥଶାଶ െ 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 െ ሺଶାଵሻሺଶାሻ 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 σୀ 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 ሺଶାଵሻమ ିଶ ଶ ߥଵ ܩଶିଵ ܩଶ ቀܾ െ ሺʹ݊ ͳሻଶ ߥଵ ቁ ܩଶାଵ ܩଶାଶ െ ଶ ʹ ʹ ʹ ݊ሺʹ݊ ͷሻ ʹ ଶ െ ߥଵ ܩଶାଷ ൌ 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 ܩଶିଶ ܩଶିଵ ሺܾ െ Ͷ݊ଶ ሺଶమ ିଵሻߥଵଶ ሻܩଶ ܩଶାଵ ʹ ʹ ʹ ݊ሺʹ݊ ͵ሻ ଶ ߥଵ ܩଶାଶ ൌ Ͳ െ ʹ ሺʹ݊ െ ͷሻሺʹ݊ ͳሻ ଶ ݀ ሺʹ݊ ͳሻଶ ሺʹ݊ ͳሻଶ െ ͺ ଶ ߥଵ ܪିଶ ܪିଵ ቆܾ െ ߥଵ ቇ ܪ െ ͺ ʹ Ͷ ͳ െ ௗ ܪାଵ െ ଶ ሺଶାଵሻሺଶାሻ ଶ ߥଵ ܪାଶ 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 ܩଶିଵ ܩଶ ቆܾଵ െ ሺʹ݊ ͳሻଶ ߥଶ ቇ ܩଶାଵ െ ʹ ʹ ʹ ݊ሺʹ݊ ͷሻ ʹ ଶ ݀ଵ ߥଶ ܩଶାଷ ൌ Ͳ ܩଶାଶ െ ʹ ʹ ݀ଵ ݀ଵ ݊ሺʹ݊ െ ͵ሻ ଶ ߥଶ ܩଶିଶ ܩଶିଵ ሺܾଵ െ Ͷ݊ଶ ሺʹ݊ଶ െ ͳሻߥଶଶ ሻܩଶ ܩଶାଵ െ ʹ ʹ ʹ ݊ሺʹ݊ ͵ሻ ଶ ߥଶ ܩଶାଶ ൌ Ͳ െ ʹ ሺʹ݊ െ ͷሻሺʹ݊ ͳሻ ଶ ݀ଵ ሺʹ݊ ͳሻଶ ሺʹ݊ ͳሻଶ െ ͺ ଶ െ ߥଶ ܪିଶ ܪିଵ ቆܾଵ െ ߥଶ ቇ ܪ ͺ ͳ ʹ Ͷ ௗభ ଶ ܪାଵ െ ሺଶାଵሻሺଶାሻ ଶ ߥଶ ܪାଶ 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. Academic Journal J of Interdisciplinnary Studies E-ISSN N 2281-4612 ISSN 2281-3993 2 + ∏ 2 s +1 Vol 2 No 6 August 2013 Published by MCSSER-CEMAS-Sapienza University U of Rome 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 E-ISSN N 2281-4612 ISSN 2281-3993 2 Academic Journal J of Interdisciplinnary Studies Published by MCSSER-CEMAS-Sapienza University U 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