AE 43 Space Technology I Chapte Satellite Dynamic.1 Intoduction In thi chapte we eview ome dynamic elevant to atellite dynamic and we etablih ome of the baic popetie of atellite dynamic.. Dynamic of a Rigid Body Conide the igid body B with ma cente B*. Refeence fame B i fixed in the body. Given the element of ma dm whoe location elative to B* i defined by = b 11+ b + 33 b (.1) B B dm B* B1 B3 J Inetial Ref Fame Figue.1 Refeence Fame and Ma Element The moment of inetia matix i defined by I I I I = I I I I I I 11 1 13 1 3 31 3 33 (.) AE 43 Fall 001-1
whee 11 ( 3 ) = ( 1 + 3 ) 33 = ( 1 + ) B B B I = + dm, I dm, I dm I = dm, i j ij B i j Note that I i ymmetic. (.3) We will alway ue the cente of ma a the efeence point. The angula momentum i H = Iω B/ J B/ J B/ J 1 11 1 1 13 3 H = I ω + I ω + I ω B/ J 1 1 3 3 H = I ω + I ω + I ω B/ J 1 31 1 3 33 3 H = I ω + I ω + I ω (.4) The otational kinetic enegy i E k 1 B J B J = ω / H / (.5) The equation of motion ae: J dh dt B/ J = T (.6) whee T i the um of the toque on the body. momentum o we will dop the J. H B/ J will alway be the inetial angula AE 43 Fall 001 -
dh dt dh dt J B B B B J B = + H = ω / T (.7) In cala fom thee ae H 1 + ωω 3( H3 H)= T1 H + ωω 1 3( H1 H3)= T H + ωω ( H H )= T 3 1 1 3 (.8) We deived thee equation fo a igid body but they apply to any ytem of paticle and/o igid bodie a long a the toque ae computed about the efeence point. In thi coue ou efeence point fo the angula momentum will alway be the cente of ma of the ytem. If the axe ae pincipal axe then H B = I ω, H B = I ω, H B = I ω (.9) 1 11 1 3 33 3 and the equation of motion become I ω + ω ω ( I I )= T I ω + ω ω ( I I )= T I ω + ω ω ( I I )= T 11 1 3 33 1 3 1 11 33 33 3 1 11 3 (.10) Thee ae called Eule' equation of motion fo a igid body. Toque Fee Motion of a Body of Revolution We want to conide pin about the axi of ymmety of an axiymmetic body when thee ae no toque. Let B3 be the axi of ymmety, then T = T = T = 0 1 3 I = I = I, I = I 11 T 33 (.11) AE 43 Fall 001-3
Define σ = I / I (.1) T Subtituting the above in Eq. (.10) give ω 1 + ωω3( σ 1)= 0 ω ω1ω3( σ 1)= 0 (.13) ω = 0, ω = ω 3 3 Thu, the pin ate i a contant. Let λ = ( σ ) ω 1 λ i called the nutation fequency. (.14) ω ω = λω 1 = λω 1 ω + λ ω = 1 1 0 (.15) The olution i ω = ω coλt ω in λt 1 10 0 ω = ω in λt + ω coλt 10 0 ω = ω co λt + α 1 ω = ω in λt + α T t ( ) 1 / = ω ω ω α ω T = 10 + 0,tan ω 0 10 (.16) α i a phae angle and ω T i called the tanvee angula velocity... Now detemine the angula velocity a a function of the Eule angle defined in Fig. AE 43 Fall 001-4
J 3 B 3 B φ θ θ ψ φ θ J ψ φ J 1 B 1 Figue. Eule Angle Since thee ae no extenal toque the total angula momentum H i contant. We will let the diection of the J 3 axi be along the angula momentum vecto. Then B ω = ( θ co φ + ψ inθinφ) b1 + θ in φ + ψ inθcoφ b φ ψ coθ b + ( + ) 3 (.17) ( 1 ) ψ = ω inφ + ω co φ / inθ θ = ω coφ ω inφ φ = ω ψ coθ 3 1 (.18) Alo, ince the pin ate, ω, i a contant, the component of H along the B 3 axi, H 3, i contant. With J 3 a the total angula momentum AE 43 Fall 001-5
co θ = H3 / H = Iω / H (.19) Thu, θ i contant and i called the nutation angle. ω + ω = ω = θ + ψ in θ 1 T φ + ψ coθ = ω (.0) Since θ i contant, ψ and φ mut be contant. ψ inθ = ω inφ + ω coφ 1 ψ inθ = ω in λt + α + φ T (.1) Theefoe, λt + α + φ = contant φ =- λ =-( σ -1) ω ψ = σω / coθ (.) Conide the plane fomed by the J 3 (angula momentum vecto) and the B 3 (pin) axi. Thi plane i called the nutation plane. The nutation plane otate at the ate ψ = σω / coθ. Fom the following figue AE 43 Fall 001-6
H T θ H H 3 Figue.3 Nutation Plane H = Hj3 = H3b3 + HT H = I ( ω1b1+ ωb)= I ω T T T T Thu, ω T mut be in the plane fomed by the angula momentum vecto and the pin axi. Thi i why it i called the tanvee angula velocity. Ou contol ytem will have to pefom two function. The fit will be to damp out the nutation, that i, educe the angle θ to zeo. Then, if the angula momentum vecto i not in the deied diection, e.g., along the obit nomal, it will have to move the angula momentum vecto to it deied diection. We will adde thee iue in Chapte 4. Now conide the following figue AE 43 Fall 001-7
H T H H 3 θ H H 3 α ω HT ITωT ωt tanθ = = = H I ω σω tanα ω T = = σtanθ ω (.3) The angula velocity vecto i in the nutation plane, the plane fomed by the pin axi and the angula momentum vecto. Thi plane otate at the ateψ = σω / coθ. Since φ = λ = ( σ 1) ω an obeve fixed to the body would ee himelf otating in the negative diection elative to the nutation plane when σ > 1, and in the poitive diection when σ < 1..3 Stability of Spin We want to invetigate the tability of the pin of a igid body about a pincipal axi. The body i no longe axiymmetic a in the peviou ection. Let the igid body be otating about the B 3 axi and let the thee axe be pincipal axe. We want to detemine the tability of thi pin. In tate pace fomat ω f ω = f = ωω I I / I 1 3 33 11 f = ωω I I / I 1 3 33 11 f = ωω I I / I 3 1 11 33 (.4) AE 43 Fall 001-8
An equilibium point of thi ytem i any one of the ω i unequal to zeo, and the othe two ω i equal to zeo, i.e., pin about any of the axe. We ae conideing pin about the B 3 axi. Expand Eq. (3.3) in a Taylo eie about the equilibium point and etain only the linea tem in the ω i. ω ω = 0 + δω 1 1 = 0 + δω ω = ω + δω 3 3 The equation of motion become δω = ω I I / I δω δω = ω I I / I δω (.5) δω 1 33 11 33 11 1 3 = 0 [ ] [ ] Diffeentiating the fit equation and ubtituting fo δω give [ ( ) ] = δω 1 + ω I33 I11 I33 I / I11I δω1 0 (.6) If the coefficient multiplying δω 1 i poitive the motion i decibed by a hamonic ocillato, thu it i table if I33 > I11 and I33 > I o I33 < I11 and I33 < I. Othewie it i untable. Theefoe, we get the eult that pin about the axi of minimum o maximum moment of inetia i table, and pin about the axi of middle moment of inetia i untable. We will ee in Chapte 4 that if thee i any enegy diipation in the body, which alway i the cae, then pin about the axi of minimum moment of inetia i alo untable. A heuitic agument howing thi to be the cae i now peented. Fo pin about a pincipal axi the kinetic enegy i E = 1 k Iω = 1 H I AE 43 Fall 001-9
Since the enegy diipation i intenal (no extenal foce o toque) the angula momentum i contant. Spin about a pincipal axi i an equilibium configuation and enegy cannot diipate. Howeve, let thee be a mall petubation fom thi configuation. Then enegy will diipate and the ytem will eek out the equilibium configuation with the minimum kinetic enegy. Thi i pin about the axi of maximum moment of inetia. Thi fact, although well known to atonome, wa not known by ome of the ealy pacecaft deigne and became known by a atellite becoming untable. The time contant fo the untable motion may be minute o hou depending upon the ouce of diipation..4 Dynamic of a Satellite with Reaction o Momentum Wheel Conide a atellite with a eaction o momentum wheel with it pin axi defined in the B-fame by the unit vecto n. The angula momentum of the ytem i H = HB + h (.7) whee H B i the angula momentum of the igid body and h i the angula momentum of the wheel elative to the body and i given by h = hn (.8) Equation (.8) ae applicable ince they ae fo a ytem, not jut a igid body. Howeve, ince thee ae fou unknown, the wheel peed elative to the body and the thee component of the angula velocity o angula momentum, we till need anothe equation. If the wheel i axiymmetic then we can wite ḣ = T M (.9) AE 43 Fall 001-10
whee T M i the wheel moto toque applied to the wheel. An equal and oppoite toque i applied to the atellite. Thu, the equation of motion of a ytem with N wheel ae H + ω H ω H = T H + ω H ω H = T H + ω H ω H = T h = T, = 1,,..., N i 1 3 3 1 3 1 1 3 3 1 1 3 Mi (.30) The atellite o body angula momentum i Ḣ + ωω( H H )= T 1 3 3 1 N H = H + hn e 1 B1 i i 1 i= 1 N H = H + hn e B i i i= 1 N H = H + hn e 3 B3 i i 3 i= 1 Fom the atellite angula momentum we obtain the angula velocity. (.31) ω = I 1 H B In thi coue we will ue a (1,,3)=(yaw, oll, pitch) equence of otation. The angula velocity i ψ ω = R ( θ) R ( φ) R( ψ) 0 n 0 0 R θ R φ φ θ + R 0 0 θ + 3 1 3 3 ψ coθcoφ in φ θ inθinψ coθinφcoψ ω = ψ inθcoφ + φ co θ + n coθinψ inθinφcoψ θ + ψ inφ coφcoψ (.3) AE 43 Fall 001-11
whee n i the obital angula velocity o mean motion. Thi tem i needed ince we ae efeencing the attitude with epect to a otating (Eath pointing) efeence fame. The angula ate ae ( 1 ) φ = ω inθ + ω coθ ninψ ψ = ω coθ + ω inθ + ninφco ψ / coφ 1 θ = ω ncoφcoψ ψ inφ 3 θ = ω ω coθ + ω inθ tanφ nco ψ / coφ 3 1 (.33) Note the ingulaity when ψ =±90deg. Thee i alway a ingulaity with any et of Eule angle. The poce i given the extenal toque and the moto toque: 1. Integate Eq. (.30) to obtain the ytem angula momentum and eaction wheel angula momenta.. Solve fo the body o atellite angula momentum with Eq. (.31). 3. Solve fo the angula velocity uing Eq. (.3). 4. Solve fo the angula ate uing Eq. (.33) and integate to obtain the attitude. In the cla poject we will neglect the mean motion tem. Thi i jutified becaue the tem i o mall fo geoynchonou atellite..4 Ditubance Toque.4.1 Sola Peue Toque (Agawal, pg 133) The toque eulting fom ola peue i the majo long-tem ditubance toque fo geoynchonou pacecaft. The dominant toque on LEO pacecaft i the gavity gadient o aeodynamic. The ola adiation foce eult fom the impingement of photon on the pacecaft. A faction, ρ S, ae peculaly eflected, a faction, ρ d, ae diffuely eflected and a faction, ρ a, ae abobed by the uface. AE 43 Fall 001-1
In the figue below thee i a uface of aea A on which the photon ae impinging. The nomal to the uface i n and the unit vecto fom the un to the pacecaft i S. The angle between the unit vecto n and S i ψ. The ola flux i P= PS The foce ceated by the abobed photon eult fom the tanfe of momentum of the photon to the pacecaft and i given by F a = ρ PA n S S a (.34) S A n Speculaly eflected photon ψ ψ Incoming photon whee P i the ola adiation peue. Note that the foce i in the diection along the un line. When the photon ae peculaly eflected they tanfe twice the momentum and the diection i nomal to the uface. Thee i no momentum tanfe tangent to the uface. F = ρ PA n S n (.35) Fo that potion that i diffuely eflected the photon' momentum may be conideed topped at the uface and eadiated unifomly into the hemiphee. Thu the foce ha a AE 43 Fall 001-13
component due to the tanfe of momentum and a component due to the eadiation. Since it i eadiated unifomly the eadiation component will be nomal to the uface. The foce i Fd = dpa n S ρ S + n 3 (.36) The total ola adiation foce i FS = PA( n S) ( ) S + 1 ( n S) + 1 ρ ρ ρ d n 3 (.37) whee ρa + ρ + ρd =1 ha been ued. The ola peue i uually aumed to be contant and to have the value P=4.644*10-6 N/m. Let F = PA n S F S F n S1 + S FS = ( ) FS = 1 ( n S) + 1 1 ρ, ρ ρ d 3 (.38) Refeing to the figue below and with δ a the declination of the un, poitive fom the venal equinox to the autumnal equinox, and α a the obit angle meaued fom local noon, the unit vecto to the un i S = coδi0 inδk0 = coδcoαo1 + coδinαo inδo3 (.39) K 0 δ Obit Plane I 0 α AE 43 Fall 001-14
The (O 1, O, O 3 ) ae the axe of the obital fame and coincide with the (I 0, J 0, K 0 ) axe when α = 0. Fo momentum bia and thee axi tabilized pacecaft the majo potion of the ola peue toque i uually due to the ola panel that tack the un. Uually the ola panel only have a ingle gimbal, which i about the 3-axi o pitch axi. Thu, they ae alway within 3 degee of the Sun and the lo of powe fom not pointing diectly at the un i offet by not having the added complexity and weight of a double gimbal ytem to diectly tack the Sun. Thu, Then I0 = coαo1 inαo J0 = inαo1 + coαo K0 = o3 n = I = coαo + inα o (.40) 0 1 F = PA K K K I [ 0 1 0] = K = F coδ + F co δ, K F coδinδ 1 S1 S S1 (.41) In the obital fame the ola peue foce i F = PA K coαo K inα o + K o (.4) 1 1 1 3 Auming no attitude eo the vecto fom the cente of ma to the cente of peue of A i x o y o z o cp = cp 1 + cp + cp 3. Then the ola peue toque i AE 43 Fall 001-15
T = F T S cp S S ycpk z K cp 1inα = PA zcpk + xcpk 1coα x K inα + y K coα (.43) cp 1 cp 1 In the toque i I, J, K ( 0 0 0) ytem which otate at appoximately 1 deg/day the ola adiation T S ycpk x K coα cp inα = PA K ( xcp ycp ) zcpk coα inα 1 x K inα + y K coα (.44) cp 1 cp 1 Note that the pitch toque i peiodic, hence it effect ove one obit i zeo. Howeve, the oll and yaw toque have ecula component which mean that ove one obit thee will be a net change in the pacecaft angula momentum. To detemine the citical o deign condition one mut evaluate the toque at equinox and oltice to detemine the maximum condition. Thi will be addeed in Chapte 3..4. Gavity Gadient Toque The gavity gadient toque on a igid body in a cicula obit of adiu R 0 and obital angula velocity ω 0 can be expeed a _ T 3ω 0 o I o G = 1 1 (.45) whee I _ i the inetia dyadic I bb I bb I bb 11 1 1 1 1 11 1 3 _ I = I b b I b b I b b 1 1 3 3 I b b I b b I b b 31 3 1 3 3 33 3 3 (.46) AE 43 Fall 001-16
To pefom the dot and co poduct o 1 mut be tanfomed into the body fame. With BO R = ( Rij ) o = R b + R b + R b 1 11 1 1 31 3 (.47) Two cae of inteet ae the ditubance toque due to poduct of inetia and thoe eulting fom an attitude eo when the axe ae pincipal axe. Poduct of Inetia With R = R = 0, R = 1and ψ = φ = θ = 0the gavity gadient toque i 1 31 11 T ω I b I b G = 3 0 ( 1 3 13 ) (.48) Note that thee i no yaw toque. Alo note that thi i a contant toque which can only be counteed in the long tem with extenal toque, e.g. thute. Pincipal axe The gavity gadient toque in thi cae become T G = 3 0 R R I I ω R R I I R R I I 1 31 33 31 11 11 33 1 11 11 (.49) Fo the 1--3 otation R R R 11 1 31 = coθcoφ = inθcoφ = inφ (.50) AE 43 Fall 001-17
T G = 3 0 Fo mall angle ( I33 I) inθinφcoφ ω ( I11 I33) coθinφcoφ ( I I ) 11 inθcoθco φ (.51) 0 TG = 3ω0 I I I I ( 11 33) 11 φ θ (.5) Note that the yaw toque i zeo. Thi i why thee i vey little contol of yaw when uing gavity gadient. The only contol toque i fom the nonlinea coupling of oll and pitch. AE 43 Fall 001-18