Appendix A Gravity Gradient Torque

Transcription

1 Appendix A Gravity Gradient Torque We will now consider the gravity gradient torque on an earth satellite in a circular orbit. We assue that the earth is a perfect rigid sphere with a radially syetric ass distribution and denote R = vector fro Earth center to the center of ass of the satellite. r = vector fro Earth center to the ass eleent d of the satellite. ρ = position vector of ass eleent d with respect to the satellite center of ass. Then, if i, j, k are the unit vectors along the body axes X, Y, Z respectively, we have ρ = x i + y j + z k r = R + ρ (A.1) (A.2) The attractive force on the ass eleent d is d F = μ d r 3 r (A.3) where: μ = GM = Earth s gravitational paraeter, with G = universal gravitational constant, and M = total ass of the earth. The torque about the center of ass of the satellite caused by the force on d is d L g = ρ μ d r 3 r = μ d r 3 ρ ( R + ρ) = μ d r 3 ρ R (A.4) Since r 2 = ( R + ρ) ( R + ρ) = R (ρ/r) R ρ R 2 (A.5) R. V. Ranath, Coputation and Asyptotics, SpringerBriefs in Coputational Mechanics, 113 DOI: / , The Author(s) 2012

2 114 Appendix A: Gravity Gradient Torque r 3 can be written as r 3 = ρ 2 R ρ + 2 R R 2 1 R 1 3 ρ R 2 3/2 (A.6) Note that in the above expression ρ R and neglecting second and higher order ters in the binoial expansion. The total torque exerted on the satellite becoes L g = μ = μ R 1 3 cot ρ R 2 ( ρ R)d ρd R + 3μ 1 R 2 ( R ρ)( ρ R)d (A.7) Since the origin of the body axes coincides with the center of ass of the satellite, ρd = 0 (A.8) so that L g = 3μ 1 R 2 ( R ρ)( ρ R)d (A.9) Now, choose the orbit reference axes with the origin at the center of ass of the satellite. Note that Z 0 is directed fro the center of ass of the satellite to the center of the Earth, and X 0 is in the orbit plane along the forward direction noral to Z 0, and Y 0 is noral to the orbit plane. The attitude of the satellite can be identified by three successive rotations fro the orbit reference axes in the following sequence: ψ about Z 0,θabout the newly displaced Y 0 axis and φ about the final position of axis X 0 X. Therefore, R can be expressed as R = R(c 1 i + c 2 j + c 3 k) (A.10) where c 1, c 2 c 3 denote: c 1 sinθ c 2 cosθsinφ c 3 cosθcosφ Substituting into the torque equation and integrating, the coponents of the gravity gradient torque along the body axes can be written as Note: 2 = μ/

3 Appendix A: Gravity Gradient Torque 115 L g x = 3μ = 3μ (c 1 x + c 2 y + c 3 z)c 3 y c 2 z)d (A.11) (x 2 + y 2 )d (x 2 + z 2 )d c 2 c 3 (A.12) = (I z I y )sin2φcos 2 θ (A.13) L g y = 3μ = 3μ (c 1 x + c 2 y + c 3 z)c 1 z c 3 x)d (A.14) (y 2 + z 2 )d (x 2 + y 2 )d c 1 c 3 (A.15) = (I z I x )cosφsin2θ (A.16) L gz = 3μ = 3μ (c 1 x + c 2 y + c 3 z)c 3 x c 1 y)d (A.17) (x 2 + z 2 )d (y 2 + z 2 )d c 2 c 3 (A.18) = (I x I y )sinφsin 2 θ (A.19) In suary, we can write (a) Gravity Gradient Torque (GGT) Fro orbital dynaics 1, the agnitude of the position vector R can be expressed in ters of the eccentricity e, w3i-ajor axis a and true anoaly f as The orbital period is R = a(1 e2 ) (1 + ecosf ) p = 2π/ω orbit = 2π a 3 /μ (A.20) (A.21) It can be shown that the GGT can be written as 2, ( ) ( ) L G b = 3ωorbit 2 (1 + ecosf )3 /(1 e 2 ) 3 R b R R b I R (A.22)

4 116 Appendix A: Gravity Gradient Torque = O(ω 2 orbit ) (A.23) if e 1 and the inertia atrix I is not approxiately an identity atrix. It can be shown that the GGT can be expressed as a product of attitude-dependent ters with a higher frequency and the orbit-dependent ters with the relatively lower frequency 2. This is useful later in the developent of the solution.

5 Appendix B Geoagnetic Torque (GMT) Torque Siilarly, the Geoagnetic Torque Torque (GMT) can be written as follows: An earth satellite interacts with the geoagnetic field resulting in a torque L M = V M B (B.1) where B = geoagnetic field and V M = agnetic oent of the spacecraft. The latter could arise fro any current-carrying devices in the satellite payload or by eddy currents in the etal structure, which cause undesirable torques. On the other hand, the vehicle agnetic oent could also be purposely generated by passing an electric current through an onboard coil to create a torque for attitude control. This has been discussed in Chap. 8. The geoagnetic field odeled as a dipole has the for B = (μ B /R 5 ) R 2 e B 3( e B R) R (B.2) where e B is a unit vector in the direction of the geoagnetic dipole axis, which is inclined about 11.5 degrees fro the geophysical polar axis. R is the satellite position vector and μ B = gauss c 3. The GMT can be written in the for 2 L M b = V M b C ib (μ B /R 5 ) R 2 e i B 3( ei B R i ) R i (B.3) Although neither the geoagnetic field nor the body agnetic oent can be precisely deterined in general, odeling the as dipoles is sufficiently accurate for our purpose. Two points are worthy of note. First, both GGT and GMT are of order of ω 2 orbit, provided that the eccentricity is not too high and if the satellite ass distribution is not too nearly spherical. Second, both GGT and GMT can be expressed in a for separating the attitude and orbital frequencies. R. V. Ranath, Coputation and Asyptotics, SpringerBriefs in Coputational Mechanics, 117 DOI: / , The Author(s) 2012

6 Appendix C Floquet s Theory Floquet s theory 3 establishes the for and nature of the solutions of a linear differential equation of any order, with coefficients which are periodic functions of a fixed period T. The theory is sufficiently explained if we liit its application to a differential equation of second order. Let u 1 (t) and u 2 (t) be any linearly independent solutions of the differential equation We can write the general solution as U(t) = Au 1 (t) + Bu 2 (t) (C.1) where A and B are arbitrary constants. Note that the coefficients of the equation are periodic functions with period T. Therefore, both u 1 (t + T) and u 2 (t + T) are also solutions of the equation. Hence these functions can be expressed linearly in ters of the fundaental set as follows. u 1 (t + T) = a 1 u 1 (t) + a 2 u 2 (t); u 2 (t + T) = b 1 u 1 (t) + b 2 u 2 (t) (C.2) The general solution can be written as U(t + T) = (Aa 1 + Bb 1 )u 1 (t) + (Aa 2 + Bb 2 )u 2 (t) U(t + T) = ku(t) (C.3) (C.4) A and B ust satisfy the equations Ak = Aa 1 + Bb 1 Bk = Aa 2 + Bb 2 (C.5) (C.6) These are hoogenous equations in A and B. The necessary and sufficient conditions for the existence of nontrivial solutions is given by ( ) a1 k b det 1 = 0 (C.7) a 2 b 2 k R. V. Ranath, Coputation and Asyptotics, SpringerBriefs in Coputational Mechanics, 119 DOI: / , The Author(s) 2012

7 120 Appendix C: Floquet s Theory If k is one of the roots of this equation, then the general solution of the differential equation will satisfy Eq. C.4. Let us now write k = e λt and define the function W(t) = e λt U(t) (C.8) We then have, fro Eq. C.4, W(t + T) = e λ(t+t) U(t + T) = e λt U(t) = W(t) (C.9) Therefore, the differential equation has a solution of the for U(t) = e λt W(t) (C.10) where W(t) is a periodic function. Proceeding in a siilar anner, Floquet generalized the result to a linear differential equation of any order with coefficients that are periodic in one period. He showed that the solutions have the for of a product of an exponential function and a periodic function. The principal difficulty concerns the deterination of λ, which is called the Floquet exponent. Ifλ = 0, the solution is periodic. Otherwise, it is either stable, if λ < 0, and unstable if λ > 0. Usually, λ cannot be deterined analytically. One has to resort to asyptotic approxiations or nuerical approaches to deterine λ. References 1. R. H. Battin, An Introduction to the Matheatics and Methods of Astrodynaics (AIAA, New York, 1987) 2. Y. C. Tao, Satellite Attitude Prediction by Multiple Scales Sc.D. Dissertation, Massachusetts Institute of Technology, Cabridge, MA, G. Floquet, Annales de l Ecole norale superiere. Sup. 2, 12 (1883) 4. R. V. Ranath, A Multiple Scales Approach to the Analysis of Linear Systes, (USAFFDL-TR Wright-Patterson AFB, OH, 1960) also R. V. Ranath, G. Sandri, A generalized ultiple scales approach to a class of linear differential equations. J. Math. Anal. Appl. 28, (1969) 5. J. R. Wertz (eds), Spacecraft Attitude Deterination and Control : Appendix H (D. Reidel Publishing Copany, Dordrecht, 1978) 6. Ince, Ordinary Differential Equations, Dover, New York 7. P. F. Byrd, M. D. Friedan, Handbook of Elliptic Integrals for Engineers and Scientists. Die Grundlehren der atheatischen Wissenschaften in Einstelldarstellungen Band M. Abraowitz, I. A. Stegun, Handbook of Matheatical Functions. Nat. Bur. Standards, Appl. Math. Series