INTERNATIONAL CONFERENCE of SCIENTIFIC PAPER AFASES 2012 Brasov, May 2012

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1 HENRI OANDA AIR FORE AADEMY ROMANIA INTERNATIONAL ONFERENE of SIENTIFI AER AFASES Brasov, 4-6 Ma ENERAL M.R. STEFANIK ARMED FORES AADEMY SLOAK REUBLI A MATHEMATIAL MODEL FOR OMUTIN THE TRAJETORIES OF ROKETS IN A RESISTANT MEDIUM TAKIN INTO AOUNT THE EARTH'S ROTATION. THE SYSTEM OF DIFFERENTIAL EQUATIONS OF THE ROKET'S TRAJETORY Bogdan-Aleandru BELEA* *Militar Technical Academ, Bucharest, Romania Abstract: Within this aer, there is established the sstem of ordinar equations of the trajector of a rocket with resect to the Earth, while taking into account, the aerodnamic drag, the weight, and the Earth's rotation and curvature. The local latitude and the longitude during the, flight, which imlicitl aear in the eressions of the forces acting uon the rocket, are also calculated. The mathematical model thus obtained is rendered in forms utilied in ballistics. This sstem of equations can also he used for the calculation of the trajector of an active-reactive rojectile, and also, with some adjustments, for the calculation of the orbit of an aerosace vehicle. Kewords: rocket, Earth's rotation, trajector, rocket latitude and longitude.. THE EXRESSIONS OF THE FORES ATIN ON THE ROKET WITHIN THE HYOTHESIS OF THE FUNDAMENTAL ROBLEM OF THE EXTERNAL BALLISTIS The vector equation of the rocket s, d equation m T + R + + Fc of [9], contains the forces which effectivel act on the rocket: - engine s thrust, T ; - the aerodnamic resultant force, F a, which, within the main hothesis of the fundamental roblem of ballistics, reduces to the aerodnamic drag, R ; - the weight, ; - the oriolis force, F c... THE THRUST It is assumed that the thrust, T, is oriented along the ais of the rocket, which in turn (whithin the main hothesis of the fundamental roblem of the eternal ballistics is tangent to the trajector (Figure. Figure. Forces effectivel alied. FORES EFFETIELY ALIED Within the Earth-linked frame O (the

2 trajector is eressed with resect to Figure, the velocit vector can be written as i + j k ( + So the thrust vector is T T ( In general, the absolute value of the thrust is function of the flight s altitude, h, and time [6, ]. The absolute value of the thrust created b the rocket engine can be eressed as [5, 6] T T ( t + Se [ B( h ], ( where T ( t is the ground-level thrust (which deends uon the time t, S e - the area of the nole s eit section, - the ground-level air ressure, and B( h - the function which indicates the relative variation of the air ressure with resect to the altitude,, B h (4 h ( where is ressure at the fling altitude, h. For the constant thrust motor (which is often encountered the value of T is given b e ω [6] T Qeef ef I s (5 where Q e g τ and e are the mass flow rate and the weight flow rate of the ehaust gasses in the eit of the engine s nole, resectivel, ef is the effective seed of the gasses in the nole s eit, I s - the secific imulse of the engine, ω - the total amount of fuel, τ - the ballistic duration of the active eriod (of e ω engine running, while Qe, e, g τ I ef s g (6.. THE AERODYNAMI DRA Within the hotheses mentioned above, the aerodnamic drag, R, is oriented along the velocit vector, but it acts in oosite i + j + k direction, R R R (7, and the absolute value is R ρ S (8 The densit, ρ, and the secific weight, γ, at the altitude of the flight, h, H( h ρ ρ, γ g ρ, (9 where ρ is the densit of the air at the ground-level, while ( H h is the function which indicates the relative variation of air s densit (or secific weight with resect to the altitude, h [7, 6]. The smbol S of formula (8 is the reference aria, usuall the aria of the d transverse section of the bod, S π, 4 ( where d is the maimum diameter of the transverse section of the bod (fuselage, and - the dimension-less aerodnamic drag coefficient. Within the adoted hotheses, at a given altitude, h, the coefficient is function of the Mach number during the flight, M, a ( where the seed of sound, a, at the flight altitude, h, is given b a TRK, ( in which K is the ratio of the secific air s heats, J K.45, R 87. is the v kg K air s constant, and T T ( h is the air temerature at the fling altitude, h [7]. Substituting the numerical values, formula ( becomes a. 8 T ( h (. Within the same hothesis, the value of is eressed as a function (. It should be noticed that act, which is the function which corresond to the active hase of the flight, is different than as, which is the corresonding function during the assive hase. This is due to the fact that, during the active hase, the art of the drag roduced b the vortices which aear at the osterior art of the rocket (at the bottom while fling with engine off disaears due to

3 HENRI OANDA AIR FORE AADEMY ROMANIA INTERNATIONAL ONFERENE of SIENTIFI AER AFASES Brasov, 4-6 Ma ENERAL M.R. STEFANIK ARMED FORES AADEMY SLOAK REUBLI jet of gasses eiting the engine. onsequentl, the drag coefficient of the rocket can be eressed as act, t < t < t (4 where as, t > t t and t are the value of the time at the beginning and the end of the active hase, resectivel, while act as ost, (5 in which ost (also noted is the osterior drag coefficient (bottom drag which disaears during the active hase, when the engine is running. The function, which is usuall given numericall, is established using eerimental data and/or calculations. alculation in the ballistic ractice often emlo drag functions like π et F(, a γ (6 or 8 π et (, a γ, 8 π et K 8 γ (7, where γ is the secific weight of the air at ground level in normal conditions (for the ballistic standard atmoshere, according to [6], kgf et et γ. 6, and (M is m the standard drag coefficient or the drag law which corresonds to certain class of aerodnamic shaes [5, 6, 8,,, 5]. In the case, the coefficient is eressed et as ( M i (M, (8 where i is the shae inde (or coefficient of the rocket and it corresonds to the drag law which was chosen (usuall using eerimental data. From formulae (6 and (7, we can obtain F(, a (, a K (9 et 8 ( M F(, a π γ and ( 8 8 (, a K π γ π γ where function K K(M is a dimensional drag coefficient. onsequentl, the drag, defined b formula (8, can be eressed as 8 R ρ S i F(, a π γ ( i S γ 4F(, a γ where γ ρ g is the secific weight of the air at flight altitude. γ ρ If H ( h ( is the function γ ρ n that indicates the variation of the relative secific weight of the air with resect to the height, then the eression of R, eression (, becomes 4 i S R H ( h F(, a 4 i S H ( h (, a ( 4 i S H ( h K Taking as reference surface the area of the maimum cross-section of rocket s bod, formula (, the eression of the drag, formula (, becomes

4 i d R i d H H ( h F(, a ( h (, a (4 i d H ( h K The function F (, a, (, a and K are given numericall or in closed form (for various laws [5,6,8,,4,], while the shae inde, i, must corresond to the drag law which was adoted. If the function of rocket s drag coefficient is available, then, in equation ( and (4, the shae inde must be chosen i, while K is relaced b π K, h γ (5 8.. THE WEIHT As resented in [9], it is considered that the weight,, is oriented toward the center of the Earth (oint, figure, so m g m g i + g j + g k, (6 ( where m is the mass of the rocket; the weight s acceleration vector, g, at the current location O of the rocket in flight [9], is ( + R j + + ( + R + O i + k g g g (7 O If the distance O between the rocket and the Earth s center is written as k ( R + O D, O D r + + (8 then the weight s acceleration vector becomes R + g g i + j + k. (9 D D D In oint O, located at the distance r and at the latitude λ from the center of the Earth, the value of g can be obtained from Eression (4 [9] (taking into account Equation (7 as well. Neglecting the small terms and combining Equations (4 and (7, we have r g a r λ a λ a ( f M r λ r f M Now, r R h, where, usuall, R ( λ, μ + R is the distance between the center of the Earth and the surface of the Earth, along the geocentric vertical which asses through O, while h is the altitude of oint O, so Formula ( becomes f M ( R + h g λ ( + ( R h f M For altitudes h between ero and u to the order of magnitude of tenths of kilometers, and even higher, Equation ( becomes [6] f M cosλ ( g ( R + h 7 At the ground level, where h, from Equation ( we have f M R g cos λ ( R f M so ( R + h λ g R f M (4 g + R h R λ f M The fraction which aears in the brackets in the above equation is racticall equal to for fling altitudes h of u to several hundred kilometers, so in such cases Formula (4 is R h g g ( + g, (5 R + h R which allows for the calculation of the variation of g with resect to the height. In such situations, the series eansion of the brackets in Equation (5 (using the binomial formula is

5 HENRI OANDA AIR FORE AADEMY ROMANIA INTERNATIONAL ONFERENE of SIENTIFI AER AFASES Brasov, 4-6 Ma ENERAL M.R. STEFANIK ARMED FORES AADEMY SLOAK REUBLI The velocit is eressed using the h h h g g (6 comonents in the Earth-linked frame, R R R O For heights of the order of magnitude of with resect to which the orbit is eressed as tenths of kilometers (u to km it can be well. admitted that Let [ ] be the column matri containing h the comonents of the vector on the aes g g. (7 R of the geocentric Earth linked frame For oints located on the Earth surface or (Figure, close to it (at altitudes of the orders of tenths of kilometers, u to aroimatel km [ ] calculations can also emlo the formula used (4 in geodesics [4] g cos λ Since the aes of frame O (8,7h arallel to the aes of the frame [ ] which, for the latitude λ 45 and h, (figure, the column matri of the rovides g m s. The standard comonents of the vector with resect to value g m s ma be used the former frame is identical to [ ]. instead, as well. If [ ] is the column matri containing the orresondingl, g ma also be estimated using [] comonents of the vector with resect to g the frame O (figure, then ( sin λ (9.59 sin λ [ ] [ ] which, for, ields Γβ Γλ λ 45 (44 g m s Γβ Γλ [ ] Γλ β [ ]. THE INERTIAL ORIOLIS FORE Taking into account Equation ( [9], Eression (44 becomes The oriolis force acting uon the rocket cosβ cosλ cosβ cosλ sinβ which has the velocit with resect to the sinλ cosλ (45 Earth, which, in turn, rotates with the angular transort velocit sinβ sinλ sinβsinλ cosβ about the [9] has the eression F which gives the comonents: m a, (4 where m is the mass of the rocket and a cos β cosλ is the oriolis acceleration, a sin λ (46,(4 so sin β cosλ F The eression of the orollas force, (4, m m. (4 can be further eended as

6 i j k where F m m [( + +. (5 (47 Substituting the drag R with the general eression, Equation (8, and taking into i] + ( j+ ( k account Equations (7 and (5, the sstem of Net, taking into account (46, the oriolis force can be eressed as differential equations of the rocket's trajector becomes: F m ( sinβ λ sinλ i + d T ρs g + ( + m m m T ( β λ + sinβ λ j+ (48 c + m d T ρs R + ( sinλ β λ k + g ( which, defining the smbol a m m Tc Fc a can be d T ρs further written as g + ( F m m Tc m afc m ( afc i + afc j + afc k, (49 (5 d where a Fc ( d ( sin β cos λ sin λ a d Fc ( (5 ( cosβ cosλ sin cos + β λ afc ( where the traction, T, is determined as shown in., the acceleration of the weight, g, can ( sin λ cos β cosλ be obtained using Equation (4 [9] or (, ( or (6, the distance to the center of the. THE SYSTEM OF DIFFERENTIAL Earth, D is given b Equation (8, while EQUATIONS OF THE ROKET S,, and, are calculated using TRAJETORY Equation (46. Eressing the drag from one of Equations (4, which are usuall utilied in ballistics, and taking into account Equation (46, trajector's differential equations sstem becomes The differential equations sstem of rocket's trajector fling within a resistant medium and taking into account Earth's rotation and curvature is obtained from vector Equation (9 [9] and using Formulae (, (, (7, (6 and (49. ollecting the results, the trajector with resect to the Earth-linked O d d d frame is described b: T m T m T m R + g m R + g m R + g m d d d + a + a + a Fc Fc Fc (5 d T id H ( h K m mg g ( sin β cos λ + sin λ Tc d T id H ( h K m mg R + g + ( sin λ + sin β cos λ Tc d T id H ( h K m mg g + ( sin λ cos β cos λ Tc d d d (54

7 HENRI OANDA AIR FORE AADEMY ROMANIA INTERNATIONAL ONFERENE of SIENTIFI AER AFASES Brasov, 4-6 Ma ENERAL M.R. STEFANIK ARMED FORES AADEMY SLOAK REUBLI The latitude λ and the longitude μ of the rocket at a certain instant during the flight are obtained using Formulae (6 or (7 and (4 [9], while taking into account Eressions (4, (9 and (4 [9]. The altitude h, of the rocket during the flight is h T c R (55 where T c is the distance between the rocket and the center of the Earth [9], while R which is generall a function R ( λ, μ, is the distance from the center of the Earth,, u to a oint, located on the surface of the Earth, at the same latitude λ and longitude μ as the rocket. Air ressure,, and air densit, ρ (or the secific weight, γ, at the altitude h (which occur in form (5 of the equations sstem are obtained from Equations (4 and (9. The functions B ( h and H( h, which reresent the relative variation with altitude of the ressure and densit, resectivel, can be eressed as on B ( h B( h ρ ρ ρ ρ H ( h H ( h (56 ρ ρ ρ ρ where and ρ are the ground-level air ressure and densit n, and ρ, are the ground-level air ressure and densit in normal conditions, while the functions ρ ρ γ B( h and H ( h are the ρ n ρ γ n functions that describe the relative variations of the air ressure and densit (or secific weight, with resect to the altitude within the standard atmoshere model which was adoted [7,, ]. For flight distances (ranges of 4-5 km or even larger it can be admitted that is a R constant equal to the sherical homogeneous Earth ( R 67 m, without affecting meaningfull the accurac of the results. In such a case the variation of g with resect to the latitude λ can also be neglected, however, its variation with resect to the altitude h must be taken into account. The sstem of differential equations that was obtained reresents the mathematical model of rocket's trajector (in a resistant medium, which takes into account the dail Earth's rotation and Earth's curvature. The calculations are further erformed via the numerical integration of this sstem. The numerical integration of the differential equations sstem begins at the moment t t, when the rocket looses contact with the launching device (e.g., the ram, and the elements of the motions at the end of the motion of the launching device are used as the initial conditions, i.e., t t, ( cosθ, cosθ,,,, ( (57 where and θ are the seed while leaving the launcher and the launching angle, resectivel. During the assive hase of the flight, in other words after the end running time of the rocket engine, for t > t (where t is the etent of the active hase, the traction term in the equations of motion have to be voided ( T. The elements of the motion (state variables as well as other arameters at the moment t t have to be recorded and the integration further roceeds using these ( t t data as initial conditions. The sloe θ of the tangent to the trajector with resect to the horiontal lane at the launching osition, O is given b the eression

8 tgθ (58 + The equations of motion are usuall integrated until a redetermined altitude, h, is reached on the descending side of the trajector (when θ <. alculation of the range requires integrating until h on the descending art of the trajector. Adequate choice of the initial conditions and of the end conditions allows the ordinar differential equations sstem obtained herein to be used for calculation of the active-reactive rojectile's trajector as well. So, if t is the moment when the rojectile eits the barrel, t - the start of the rocket engine and t corresonds to the end of the engine run, the ordinar differential equations sstem (with the aroriatel selected boundar conditions can be integrated choosing T for the time-frame [ t,t], net setting, T, for the time-frame [ t,t ] and, finall T, again for t > t, when the engine is no longer running. The mathematical model resented in this aer can also be used (with the adequate adjustments for calculating the trajector of an aerosace vehicle within the dense atmoshere. III. ONLUSION The mathematical model obtained is rendered in forms utilied in ballistics. This sstem of equations can also be used for the calculation of the trajector of different rockets, active-reactive rojectile, and also, with some adjustments, for the calculation of the trajectories of an aerosace vehicle. For distances of -5 km, the inertial oriolis force introduced b the Earth's rotation can be neglected. For bigger distances, the oriolis force must be also considered. The variation of g with resect to the altitude h must be taken into account. For flight distances (ranges of 4-5 km or even larger, it can be admitted that R is a constant equal to the sherical homogeneous Earth ( R 67 m, without affecting, meaningfull, the accurac of the results. For distances over 5-6 km, the variation of g with resect to the latitude λ and altitude h must be taken into account. REFERENES [] B.A. BELEA - Eterior Ballistics of rockets, MTA, Bucharest, (in Romanian [] F. MORARU, L. MORARU, B. SOBENIU (BELEA - A Mathematical Model for omuting the Trajectories of Rockets in a Resistant Medium Taking into Account the Earth's Rotation. art I: The ector Equation of Rocket's Motion with Resect to the Earth. The osition of the Rocket with Resect to an Earth-Linked eocentric Frame, MTA Review, ol. XX, No., , Jun.. [] F. MORARU - Eterior Ballistics and Rocket Flight Dnamics, Militar Academ ublishing House, Bucharest, Romania, 97 (in Romanian [4] F. MORARU - Handbook of Eterior Ballistics, Militar ublishing House, Bucharest, Romania, 976 (in Romanian [5] F. MORARU -Aerodnamics: art I, Militar Academ ublishing House, Bucharest, Romania, 98 (in Romanian [6] F. MORARU - haters of Numerical Analsis: Alications in Ballistics and Flight Dnamics, ed Edition, Militar Technical Academ ublishing House, Bucharest, Romania, (in Romanian [7] M.M. NI-FA, D. St. ANDREESU - The uided and Unguided Rocket '.s Flight, Militar ublishing House, Bucharest, Romania, 964 (in Romanian [8] M.M. NITĂ F. MORARU, R.N. ATRAULEA - Airlanes and Rockets: Design oncets, Militar ublishing House, Bucharest, Romania, 985 (in Romanian [ 9] I.M. SAIRO - Eterior Ballistics, I.D.T., Bucharest, Romania, 954 (in Romanian