Periodic Solutions for Rotational Motion of. an Axially Symmetric Charged Satellite

Transcription

1 Applied Mathematical Science Vol no HIKARI Ltd Periodic Solution for Rotational Motion of an Axially Symmetric Charged Satellite Yehia A. Abdel-Aziz National Reearch Intitute of Atronomy and Geophyic (NRIAG) Cairo Egypt Univerity of Hail Department of Mathematic P.O. Box 44 Kingdom of Saudi Arabia Muhammad Shoaib Univerity of Ha il Department of Mathematic P.O. Box 44 Kingdom of Saudi Arabia Copyright 14 Yehia A. Abdel-Aziz and Muhammad Shoaib. Thi i an open acce article ditributed under the Creative Common Attribution Licene which permit unretricted ue ditribution and reproduction in any medium provided the original work i properly cited. Abtract The preent paper i devoted to the invetigation of ufficient condition for the exitence of periodic olution in the vicinity of tationary motion of a charged atellite in elliptic orbit. Lorentz force which reult from the motion of a charged atellite relative to the magnetic field of the Earth are conidered. An axial ymmetry i aumed about the center of ma of the atellite. In addition to the Lorentz force the perturbation caued by gravitational and magnetic field of the Earth are conidered. The tationary olution and periodic orbit cloe to them are obtained uing the Lyapunov theorem of holomorphic integral. Numerical reult are ued to explain the periodic motion of a certain atellite. It i hown that the charge-to-ma ratio ha a ignificant influence on the periodic motion of atellite. Keyword: Periodic olution Lorentz force Rotational motion charged pacecraft gravitational force 1. Introduction When a pacecraft i moving in geomagnetic field in Low Earth Orbit ( LEO ) then due to the accumulation of charging on the urface of the pacecraft the Lorentz

2 155 Yehia A. Abdel-Aziz and Muhammad Shoaib force i generated. A detailed analyi of Lorentz Augmented Orbit (LAO) i available in [1].The rotational motion of a rigid artificial atellite ubject to gravitational and magnetic torque wa treated in detail in [-6]. Chen and Liu [7] ue the Poincare map technique to invetigate the behavior of periodic and chaotic motion of the pinning gyrotat atellite. The attitude dynamic of a atellite i decribed by the ixth order Euler- Poion equation. To undertand the periodic behavior of a rigid body Kolooff [8] tranformed it equation of motion to a planar ytem of equation a wa done by Lyapunov in [9]. Abdel-Aziz [1] found the periodic olution of a pacecraft uing an approximated model of the Earth magnetic field model. Abdel-Aziz and Shoaib [15] tudied the effect of Lorentz force on the attitude dynamic of an electyrotatic pacecraft moving in a circular orbit and ued charge to ma ratio a emi-paive control. In thi paper we ue a imilar model of Lorentz force but the pacecraft i conidered to be in an elliptic orbit. In [16] Abdel-Aziz and Shoaib devolped the torque due to Lorentz force a a function of orbital element and invetigated the effect of lorentz force on the exitance and tability of equilibrium poition in Pitch-Roll-Yaw direction. In the preent work we conider a atellite moving in an elliptic orbit under the influence geomagnetic field Lorentz force and gravitational field. Uing the iothermal coordinate we reduce the equation of motion of the atellite to the planar ytem of equation. The exitence of periodic orbit in the neighborhood of tationary olution i obtained with the help of the Lyapunov method. The numerical reult are ued to how the ignificance of Lorentz force on the poition and tability of periodic orbit.. Equation of motion and their reduction of order Conider an axially ymmetric charged atellite under the action of Lorentz force gravitational and geomagnetic field. Two Carteian coordinate ytem are introduced. The origin of the coordinate ytem i taken to be at the center of ma ' ' ' O of the atellite. The orbital ytem of coordinate i named a OX Y Z and the coordinate ytem for the body of atellite i named a OX Y Z. In the orbital ' ' ' ytem OX OY and OZ are in the orbital direction normal to the orbit and along the radiu vector of the atellite. The attitude motion of atellite i defined by three Euler angle namely and. Here i the angle of preceion and i the angle of elf-rotation. The angle between the Z -axi of both the coordinate ytem i taken to be. The Z -axi i taken to be the axi of ymmetry. The principal moment of inertia of the atellite are taken to be A B C. Due to the ymmetry about Z -axi A = B. The unit vector of the orbital coordinate ytem are named α β γ. where α = (α 1 α α ) β = (β 1 β β ) and γ = (γ 1 γ γ ) (1) α 1 = coψ coφ inψ inφ coθ

3 Periodic olution for rotational motion 155 where α = coψ inφ α = coθ inψ coφ inθ inψ) β 1 = inψ coφ + coθ coψ inφ β = inψ inφ + coθ coψ coφ β = inθ coψ γ 1 = inθ inφ γ = inθ coφ γ = coθ. The Lagrangian of the ytem can be written a in Yehia [11] L = 1 ω T Iω V () V = VG VL VM i the total potential of the force acting on the pacecraft. V G V L and V M are the potential due to the gravitation of the Earth Lorentz force and magnetic field repectively I i the interia matrix of the pacecraft. Let ω = (p q r) and ω = (p q r ) are the angular velocitie of the atellite in the inertial and orbital reference frame repectively. A the orbital ytem rotate in pace with an orbital angular velocity about the axi which i perpendicular to the orbital plane. The relation between the angular velocitie in the two ytem i given by ω = ω + Ω β. According to Abdel-Aziz [1] we can write. (p q r) = (ψ inθinφ + θ coφ + Ωβ 1 ψ inθcoφ θ inφ + Ωβ ψ coθ + φ + Ωβ ) () The gravitation potential of the Earth i [] T V = I (4) G.1 Potential due to Lorentz force The magnetic field i expreed a [1] B = B r [coφ r + inφφ + θ ] (5) where B i the trength of the magnetic field in Wpm. The acceleration in inertial coordinate i given by a = F m = μ r r + q m (V rel B ). (6) The Lorentz force (per unit ma) can be written a

4 1554 Yehia A. Abdel-Aziz and Muhammad Shoaib F L = q m (V rel B ) (7) V rel = V ω e r (8) where V i the inertial velocity of the pacecraft ω e i the angular velocity vector of the Earth and V rel i velcoity of the pacecraft relative to the geomagnetic field. In agreement with [1] we have ued V = r r + rφ φ + rθ inφ θ (9) r = r r (1) ω e = ω e z (11) z ˆ = co rˆ in ˆ. (1) Therefore the acceleration due to Lorentz force in inertial coordinate i given by F L = qb m r [ (θ ω e )(in φ r + inφ φ ) + ( r r inφ φ coφ) θ ]. (1) A tated in [16] the Lorentz acceleration experienced by the geomagnetic field i decompoed to the three component R T N (radial trnveral and normal) repectively a function of orbital element a follow: R = qb T = qb r r (1 i f ) / p co i 1 e co f ) (14) in in ( mr e m μp ( / p co i 1 e co f / p in e iin f co f 1 e co f (15) N = qb m μp [1 ] / co 1 co / p co i 1 e co f r e in iin f p i e f / p co i r i f 1 in in + in in ico f f 1 e co f 1 e co f p 1in ico f 4 ) (16)

5 Periodic olution for rotational motion 1555 where i f a and e are the inclination of the orbit on the equator argument of the perigee true anomaly the emi-major axi and the eccentricity of the atellite orbit repectively. The final form of the potential due to Lorentz force can be written a below. V L = ρ A T (R T N) T (17) where ρ = (x y z )i the radiu vector of the pacecraft relative to the center T of ma of the pacecraft A 1.. Potential of the magnetic field of the Earth Let a dipole magnetic field B = (B 1 B B ) and the magnetic moment M = ( m ) of the atellite. Therefore the potential of the geomagnetic field i T V M = M B (18) A in Wertz [1] we can write geomagnetic field and the total magnetic moment of the orbital ytem directed to the tangent of the orbital plane normal to the orbit and in the direction of the radiu repectively a the following: a M ' B1 = in [co( ) co ] m f m m r (19) = a M ' B co m r () a M ' B = in [in( ) in ]. m f m m r (1) m = ' co () where m g m 15 a M = ' m =168.6 i co-elevation of the dipole α m = 19. and f i the true anomaly meaured from acending node and m g i the magnitude of the total magnetic moment. Hence we can write the potential due to the Earth magnetic field a follow V = m B. () M Therefore the final form of the potential of the problem i V = ( C A) co 1 Ain co ( mb f )in co z co [ R T co N in ] (4)

6 1556 Yehia A. Abdel-Aziz and Muhammad Shoaib It i clear that in addition to the firt term of gravitational effect the main parameter of the potential of the atellite motion i the charge-to-ma ratio and z.the Lagrangian can be written a below. 1 1 L = A( p q ) Cr V. (5) It i clear that i a cyclic variable therefore we can ue = Cr = f (arbitrary contant).uing equation (4) and the cyclic integral we can contruct the Routhian a follow R = L f = R R1 R t where = A [ R ] in R = A in [ A in co co f co ] 1 R = ( C A) co 1 Ain co ( mb f )in co z co [ R T co N in ] (7). Reduction of the equation of motion Uing Lyapunov method of the holomorphic integral [9] the ytem of equation of motion are reduced to another impler form. Let in x = u (8) L (6) y = A C v dv (1 v ) 1 mv (9) dt = d () = arcin u (1) v A/ C = arcco( ) () 1 m v m A C 1 v = A[ ]. () C 1 m v = In the new coordinate [14] the Routhian function can now be written a below. 1 x y y x R = (( ) ( ) ) 1 U (4)

7 Periodic olution for rotational motion 1557 where 1/ = u(1 m v ) /[1 1/ ] (5) 1 v δ = (1 u )[Ω A/ C + U = μ ( Ω (C A) f A C (1+m v ) 1/ (1 v ) 1/ (1 u ) 1/]v/[1 v ] 1/ (6) v C(1+m v ) 1 Ω (1 u + 1 A [m B Ω f ] [1 u ] 1 [ 1 v 1+m v ]1 ) + z [1 u ] 1 [ R T v A C 1 v 1+m v + N [ 1+m v ]1 ]). (7) The equation derived from the Routhian can be written a below. d ( R R dτ x ) = d ( R R x dτ y ) =. (8) y The above equation are tranformed to the following planar ytem of equation. x y τ = Γ + U τ x Where Γ = (δ x 1) (δ y ) = co x (1 v 1 ) y = Γ x + U (9) τ τ y ( Ω v 1 co 1 x) [Ω m ( f A ) co x] (v v 4 )v 1 [Ω (1 v ) 1 v 1 + ( f A ) co x v 1 ] (1 + v v ) v 1 = (1 + m v ) 1. The Jacobi integral of the new ytem i x y h = ( ) ( ) U. (41) The region of poible motion can be determined by the inequality U(x y h f ).. The periodic olution and numerical reult (4) The potential function in iothermal coordinate [9 14] can be written a below. U(x y) = μ (h + w) (4) Ω w = Ω (C A) v (xy) Cv + 1/A(m 1 (1 u (xy)) B Ωf )(1 u (x y)) v 1 (1 v (x y)) + z (1 u (x y)) ( R Tv 1 v (x y) A + Nv C (1 v (x y)) ). (4)

8 1558 Yehia A. Abdel-Aziz and Muhammad Shoaib The tationary olutuon of the new ytem of a given in Yehia [14] are U U U = = =. x y z (44) Here Γ u v are function of x and y; U i the force function. The ytem (44) can be written a w w = = h w =. x y (45) We aume that P x y P f i are the tationary olution and i( i i) = i( ) =1... w w hi = hi ( f ) i the correponding energy contant. Equation = and = x y will give the value of x i and y i and h w = will determine the energy contant. To find periodic olution around the tationary point for ome value of f let x x x y = y. (46) = y The ytem of equation given by (4) become x y U ( x x y y) ( x x y x y y U ( x x y y) ( x x y y x y ( ) ( ) U ( x x y y; h)=. y; h)= y; h) = (47) With the aid of Lyapunov theorem [9] we find periodic olution in the following way. x = =1 c x ( ) y = =1 c y ( ) h = h c h. (48) =1 Here c are parameter h are contant and ( ) ( ) ( ) x ( ) = x ( T) y ( ) = ( T). The period T can alo be written a T = = (1 Tc =1 ) (49) To implify the ytem tranform the variable u. u =. (5) In the ame way a in [14] we approximate () x and () y a follow.

9 Periodic olution for rotational motion 1559 x y ( ) ( ) = a 1 = a r=1 r=1 ( a ( a ( r) 1 ( r) coru b ( r) 1 co ru b ( r) in ru) in ru). (51) For = 1 equation (46) take the following form (1) (1) x = x cx y = y c (5) y The time t and the coordinate atify the following relationhip. = t t ( x( ) y( )) d. (5) Therefore equation (41) reduce to: (1) (1) d x d y = du du (1) (1) d y d x = du du (54) The zero ubcript in the above equation correpond to the value x x y = = y w = x = x y = y x w = x = x y = y (55) y w = x = x y = y. xy Equation (54) ha the following characteritic equation. 1 = ( ) 4( ). (56) Let conider the two poible cae. 1) > ( w ha extremum at P ). In thi cae two different frequencie are poible if the following condition i atified. >. (57) The above inequality hold at all point where w ha a maximum. ) < ( w ha addle point at P ). The only poible frequency in thi cae i when the poitive ign i taken in equation (56).

10 156 Yehia A. Abdel-Aziz and Muhammad Shoaib In the cae of non-commenurable frequencie the value of h () c and can eaily be obtained. For a bounded c the erie will abolutely converge. The following value of x x (1) y (1) and h are obtained from equation (54). (1) = 1 () T (1) ( ) y = inu cou h =. (58) The ytem of equation given below i the firt approximation of the olution of equation (41). x = x y = y h = h. ( ) cinu c( in u cou) (59) The periodic olution in equation (59) could be tranformed again a a olution in and. To aid the undertanding of the exitence of periodic orbit ome numerical example are given in figure (1-) for variou charge to ma ratio. In figure (1a) the charge to ma ratio i taken to be.1. It can be een here that two type of orbit exit in thi pecial cae. One type i the uual elliptical orbit and the other i a figure 8 type of orbit. Figure 8 type of orbit are not very common. The numerical example in figure (1b 1c) how the dependence of the poition of the periodic orbit around the tationary poition on the charge-to-ma ratio. Thee figure explain that the Lorentz force have ignificant influence on the poition of the periodic olution. For different value of charge-to- ma ratio figure (1-) how the ocillation of the energy integral h and the maximum and addle point of the energy integral a in equation (69). Figure 1. The phae plane of at a) q /m =.1 b) q /m =. c) q /m =..

11 Periodic olution for rotational motion 1561 Figure. The maximum and addle point of the energy integral h at q /m =.1.1. Figure. The maximum and addle point of the energy integral h at q / m =. q/ m = Concluion Thi paper preented the ufficient condition for the exitence of periodic olution cloe the tationary motion of an axially ymmetric charged atellite in an elliptic orbit. We invetigated the rotational motion of the atellite under the action of the Lorentz force gravitational and magnetic field of the Earth. The Routhian function of the atellite motion i contructed. The equation of motion of the atellite are reduced to the planar equation of motion. The exitence of periodic olution are obtained uing Lyapunov method of the holomorphic integral. Numerical

12 156 Yehia A. Abdel-Aziz and Muhammad Shoaib reult have hown the effect of Lorentz force in particularly the charge to ma ratio on the poition of periodic olution and the maximum and addle point of the energy integral. In addition to the elliptical orbit ome very exotic figure 8 orbit have alo been identified in the vicinity of tationary motion. Reference [1] Schaffer L. and J. Burn Charged dut in planatery magnetophere: Hamiltionan dynamic and numerical imulation for highly charged grain. J. Geophy Re. 99 (A9): (1994). [] Beletky V. V. The motion of artificial atellite about it center of ma. Mocow Nauka (1965). [] Kovalenko A. P. Magnetic ytem of control of flying apparati. Mocow Machinit (1975). [4] Guran A. Claification of ingularitie of a torque-free gyrotat atellite. Mechanic Reearch Communication 19 : (199). [5] Sanguk Lee and Jeong-Sook Lee. Attitude control of geotationary atellite by uing liding mode control. J. Atron. Space Sci. 1: (1996). [6] Panagioti T. and Haijun S. Satellite attitude control and power tracking with energy/momentum wheel. Journal of Guidance Control And Dynamic 4: -4 (1). [7] Li-Qun Chen and Yan-Zhu Liu Chaotic attitude motion of a magnetic rigid pacecraft and it control Non-Linear Mechanic 7: (). [8] Kolooff G. V. On ome modification of hamilton principle applied to the problem of mechanic of a rigid body. Tr. Otd. Fizich.Nauk Ob-valiubit.Etetvozn 11:5-1 (19). [9] Lyapounov A. M. The general problem of tability of motion. Collected work. Vol. M.-L. AN.USSR (1956). [1] Abdel-Aziz Y. A. Perodic motion of pinning rigid pacecraft under influence of gravitational and magnetic field. Applied Mathematic and Mechanic. 7 : (6).

13 Periodic olution for rotational motion 156 [11] Yehia H. M. On the reduction of the order of equation of motion of a rigid body about a fixed point. Vetn. MGU er. Mat. Mech. 6: (1976). [1] Peck M. A. 5 in AIAA Guidance Navigation and Control Conference. San Francico CA. AIAA paper 5995 (5). [1] Wertz J. R. Spacecraft attitude determination and control. D. Reidel Publihing Company Dordecht Holland (1978). [14] Yehia H. M. On the periodic nearly tationary motion of rigid body a bout a fixed point Prikl.Math.Mech 41(): (1977). [15] Abdel-Aziz Y. A. and Shoaib M. Equilibria of a charged artificial atellite ubject to gravitational and Lorentz torque Reearch in Atronomy and Atrophyic 14 Vol. 14 (7) [16] Abdel-Aziz Y. A. and Shoaib M. Numerical analyi of the attitude tability of a charged pacecraft in Pitch-Roll-yaw direction International Journal of Aeronautical and Space Science Vol 15 (1) Received: June 15 15; Publihed: February 7 15