THE ASCENDING TRAJECTORIES PERFORMANCE AND CONTROL TO MINIMIZE I H E HEAT LOAD FOR THE TRANSATMOSPHERIC AERO-SPACE PLANES Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2828 J. Barlow* and A. Al-Garni** University of Maryland, College Park, MD 20742 Abstract meters for the total numerical aerodynamic control law. An=[C1,/C;,]=bl+b2~2+b3q where The paper discusses the performance and control of ascending trajectories for Transatmospheric Vehicles (TAV) using air-breathing propulsion. The work looks to minimize the heat load per unit area near the stagnation point. The vehicle is modelled as a point variabie mass with drag polar and variable thrust. The earth is assumed spherical with exponential atmosphere. The research is structured in two parts. The first part is an analytical and seeks the two controls (aerodynamic and thrust) which are necessary to transfer this TAV from one specified state(e.g. h, = 20 km, MI=5, Q, = 5 kj/cm ,... etc.) to a second specific stale (e.g. h2= 63 km, M, = 25, Q2 = 300 kj/cm 2... etc.) while satisfying given equality constraints such as constant dynamic prcssurc, and conslant rate of climb. Certain algebraic relations among the state variables u, q, y, p, and Q and a fccdback for aerodynamic and thrust controls have been dcrivcd a closed form solutions. The heat load can be Q I 350kj/cm2 for rc 2 35 m/sec, q 5 0.2 atrn., 0 5 r, 5 1, and 0 I h, 5 1. This analytical approach was hclpful in the numcrical approach in the second pan. The second part was carried out by using extensive numcrical optimization algorithms. The conuol laws which minimize the total heat load were found as combinations of some state variables in a parametric way which give the heat load Q =300 kjl cm2 for the ~ w ocontrols; h , = b , + b2u2+ b3q. . s , = a , + a 2 u 2 + a 3 q and , A numerical example was worked out for illusuation, and in general all the constraints were satisfied. p = 0.000147 (l/m) - reciprocal of b e scale height. C = 1/I = (112250. sec.) - specific fuel consumption. - C, = 5.73 x 1 0 8- ((wlcm2)4 (m3/kg)(sec3/ m3)) -a constant in the heat equation, for r~ = 0.1 m (body nose radius), hw = 0.9 (wall enthalpy to total enthalpy) C, = 1.83 x lo-' (1 - h,) where C2 = 1000- constant for numerical purposes. C3 = 58-(m)-total length of the vehicle. C, = (p, V,2 /2 C2C*,, )-(N/mZ)-a constant in the unit of pressure. C, = D, /q S - aerodynamic drag coefficient. C*, =2 C, - aerodynamic drag cocflicient at maximum aerodynamic lift to aerodynamic drag. C,, = D, /qS - ram drag coefficient. C,, - zero lift aerodynamic drag coefficient, where C, = C,, + Ka C2,, C, = C, + C, = (Dl / q S) - total drag coefficient. C,, = L, / q S - aerodynamic lift coefficient. aerodynamic lift coefficient at C',, =GI- maximum aerodynamic lift to aerodynamic drag ratio. C,, = L, / q S - lift coefficient due to ram drag. C,, = C,, + C,, = L, / q S - total lift coefficient. dl, d, - coefficient as an optimization design parameters for ram drag, where Dr / W, = [dl + d2u2] d., - coefficient as an optimi~ationdesign parameters ' for lift due to ram drag, where (L, / La) = d3 0, = (P v2 S Ch/2) = q S C,- (Newton)- Aerodynamic drag. D, = f, flh = W o[dl + d2u2] - (Newton) - ram drag. Dl = D,+ D, - (Newton) - total drag. E = La / D,=(C,,/C,)-aerodynamic lift to aerodynamic drag ratio. EL=Emax =((La / DJmnX=(Calr/Cads)- maximum aerodynamic lift to aerodynamic drag ratio. a - (mIscc2) - acceleration component in direction of V. a,, a,, a, - cocfficicn~sas an optimization design paramctcrs for the total numcrical thrust control law. where r, = T g / w o = a l+ + u 2 + a l q b , , b2, b,- coefficients as an optimization design para- * ** Associate Professor of Aerospace Engineering. Ph. D. student of Aerospace Engineering. Member AIAA. Copyright @ 1990 by the American Institute o f Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner. 1 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2828 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2828 designed to spend the smallest possible time in the atmosphere which leads to small amount of heat during the ascending time. The transatmospheric vehicles(TAV) which use airbreathing propulsion are now being conceived (e.g. National Aerospace Plane (NASP) , X -30) as in Fig. 1. These vehicles are expected to accelerate for longer period of time of about 20 - 30 minutes in the denser part of the atmosphere (h I 70 km) to reach orbital speed. This would produce severe hating problems, particularly in the ascending phase. In [4] it was found that the heating load during entry of the TAVs is about three times the shuttle entry heating load, and during ascending phase of the TAVs the heating load is about three times the entry load (i.e. the heating lo& of a TAV during ascent is about nine times greater than the heating load of the shuttle entry).This represents a serious issue for any TAV. This paper deals with the performance and control of the ascending phase trajectories for TAVs, looking to minimize the heat load pcr unit area near the stagnation point. The research is structured into two parts. The first part is an analytical approach to scck two controls (acrodynamic and thrust) which arc ncccss,ary to transfer this TAV from one specified slate (e.g. hl, = 5, h, = 20 km, Q, = 5 kj/ cm2,...etc.), to a second specific state (e.g. M2 = 2 5, h2 = 63 km, Q2 = 320 kj/ crn2, etc), while satisfying given equality constrain~rsuch as constant dynamic pressure, and constant rate of climb. This analytical study is framed to produce the maximum generality for general TAV including high aerodynamic lift to aerodynamic drag ratio designs such as wave riders. The approach is similar to the study in [5] which was for reentry problems with only acrodynarnic lift as the trajectory controller. Here we have additional equations and two controls, to include variablc mass and thrust. The second part of this research is a trajectory optimization by numerical approach to find the suboptimal controllers (aerodynamic and thrust) to minimize the total heat load. It is similar to [ 6 ]where trajectories are optimized for aerodynamic and thrust controller while satisfying many selected equality and inequality physical and mission constraints such as limitation in spced, power, and altitude. This approach is necessarily carried out using extensive numerical computation. The results of this work are part of [7] which provides insight and specific oonstraints on thc design and operation of TAV class vehicles, and defines additional work needed in the area of trajectory sclcction including particularly high aerodynamic lift to aerodynarn~cdrag ratio designs such as wave riders. The vehicle is modelled as a point variable mass with drag polar and variable thrust. The earth planet is assumed spherical and non-rotating with an exponential atmosphere. The uajectory was taken in the equatorial plane. These last assumptions could be relaxed for the detailed studies of specific missions. Fig. 2 shows the sketch of the system where the equations which describe the system are given by (1-a)-(7-a): rate of climb-(1-a) d r/d t = V sin y dp/dt=-pj3Vsin y -(2-a) dV/dt=a=(T,- D,)/m-gsiny -(3-a) V d y /d t = L, /m - g cos y + (v2/r) cos y -(4-a) -@a) d Old t = (V/r) cos y dm/dt=-CTg/g,,=-Tg/(I go) -(6-a) heat rate -(7-a) d Q/d t = C, p0.5V3 dynamic pressure-(8-a) 4=(1/2) p v 2 Introducing the dimensionless variables; r = r/r, -(9-a) (r Il =(C2= 1 0 ) C * l , ( ~ @ o ) u = v/JTs,r,-) p = = v/vc -(9-b) -(9-c) m/mo C t = L/t* The dimensionless controls; ha = C,, /C*,, '5, =Tg/Wo 7,= ( Tg - Dr )/Wo h, = d3ha h, = ha + A,= h,( 1 + dJ A ppling the dimensionless variables equations 9's (without (9-a) &d (9-e) ) to equations (1-a)- (8-a), yield; d rld t = V, u sin y -(I -b) -(2-b) d q / d t = - P V , q usin y d u/d t = ( l/Vc ) ( ul/ p - u2 / u3 -g sin y ) -(3-b) where u l = g , n ~ ~ ( 1 - ( d , + d 2 u 2 )u)2,= p o S f V c 2 q u2, and u3 = 2 C2 mo q . d y/d t = y llu, - g cos y /(V, u) + V, u cos y /r -(4-b) where Y 1 = Po V, S h,17 u -(5-b) d Old t = Vc u ( cos y )/r /I -(6-b) dp/dt=-C T -(7-b) V .lq o 5u3 d Q/d t = C, q = C , qt12 -(8--b) If also we use the variable'; and7 (of equations (9-a) and ( 9 4 ) in (1 -b)-(7-b), yield; -(I-c) d r/df= u sin y e) =-+K Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2828 dq/df= - P roq u sin y d yld'i = Pa7(r012 C2 C,) At ( q u/ p) (cosy/ F u ) + u c o s y l ; ) d0/d 1 = u cos ylf d p / d e i = - n t * C r = - n t * r8 d ~/d/dC = C, m l l ) V: ro q0.5 u3 For the numerical approach we also have; A,= bl + b, u2 + b q -(24 -(34 -(4-c) -(5-c) -(6-c) -m) -(lo-a) The For given nonlinear equations, describing a dynamical syst5m and of the form; , j = 1,2,....,11 d xj/d t = fj( x f , t ) w+re X = ( X I , x2 , ......,xi]) E X" (state vector) and to found the controls (control vector) C = ( C I , c2 ....., Ci2 ) E C12 for thc_constraints Oi3( X ) = constant , i3 = 1, 2, .....,p , p c i l and p s i 2 to be satisfied if the system is controllable , [hen il C (2 m31a XJ> jz -( C)- = o f ~X, A more general result is obtained by using ?in (a-6) and (a-7). -(a-I) i This in general generate a trajectory which is not unique and the selection of the controls will be based on mission desirability and physical and mission constraints For this problem we consider the equations (1)'s(7 )'s and two constraints 0,= rc = V, u sin y = constant rate of climb -(a-2) and @, = q = C, q u2 = constant dynamic pressure-(a-3) Applying (a- 1) to(a-2) and(a-3) give; -(a4 sin y (duld t ) + u c o s y (dyld t) = O and u2( dqld t)+ 2 q u (d uld t)=O -(a-5) substituting for b, f ,and fi and solving for hanr and r, . After long algebraic manipulation , and using MACSYMA [8] and Mathmatica 191 to check the result, yield as closed form fecdback controls: Thus from (a-6 or a-8)-(a-7 or a-9) we have for given S, AM =fZIU, r, y,rc,Q)andr,,=f,@, r,q ,f(h) ,rc,q) Since we need A, = gx(rc,q) and r,, = gr (rc,q), then we must specify p, r,q, and A in term of rc and q .This can be done if we integrate the following equations; -(a-10) by(1-b)and(a-2) r = r i + rc t q = q . e-(P " ' ) -(a-11) by (2-b) and (a-2) by (8-a-b u = J(q/C,) -(a- 12) -(a-13) y = sin-'(rc I Vc u ) by (a-2) notice we also can get 0 = f, (t ). For p and t we can apply a physical and mission constraint as in [7],and [15]-1231, p, = .3 , and for 4 = 1280. sec with p decrease almost linearly with slope = - .0005 per second . Thus for first order approximation to CI; p pi - .0005t -(a- 14) Since the vehicle must not descend and is initially at about M I = 5, h,= 20 km , and q, = 1.01 atm.Wherc the final state will be about M,= 25, h2= 63 km, and q2 =.05 am., this give (rc), = 38 mlsec, which leads to the first order approximation for the time; -(a-15) t = (1.0982 - q )/(.00084) sec. where q in atm. For the heat equation we change the independent variable from t to q , and integrating from q i to qr for rc and q to be constants this ive; &t=Qi+ ( ClIP) @JC2C 1.1 V,3 (Q/Cq)'.'( 1 1 ~aq ) + -(a-16) where aq = (11 qf - 11 qi 1 Thus equations (a-6)-(a-15)give; A,, = gX(rc,q)and ~ U I I= & (rc,q), also (a-16) gives Q,, = & (rc,q) for given qi and q, . With the use of Mathmatica [91 the results for 3D and contours plots for Lnr, rat, and Q, A,, = ( mo CL C2/poS)[ ( 2 g C4 to-3 Y /Vc2q as functions of rc and q, are shown in Plot-Al-Plot-A6 ( 2 cos ylq r ) - ( C4 P c2/Vc2q cos Y ) These results show that for rc 2 f5 mlsec and q I .15 total aerodynamic control -(a-6) am. we can have Q I300 kj/cm2.We notice that if we ad ' ~ , = ( P ~ C C I V ~ / ~ ~ ~ ) ~ ) + use different average (rc), as (rc), > 38 mlsec or (rc), ( ~ , r ~ S f q / 2 CC,, m o ) + ( g r c P I g o vC)c 38 m/sec this generally make the Plot-A2 and Plottotal thrust control -(a-7) A4 for hat, and 7mt shifted to the right or to the left respectively. This because the result is constraint dependent. This analytical approach will be a good help in the numerical approach next Functional Ob-iective Minimiiration; Qm,,(t) 5 0.5 kj/cm2 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2828 ' Minimization: Final Ob'lechve Qhlx < 310 kj/cm2 The numerical approach was done on the general equations (1-b or c)- (7-b or -c) as a trajectory optimization problem to minimize the total hcat load. This was done with the use of Console [lo] (where the gradient is often computed by finite differences), Simnon [l 11, IMSL [12], and a personal interface and code program as in [7]. The computation for these ODE'S has the phenomenon of stiffness( see [13] and [14] ), which made this problem very time consuming. The change of the independent variable from t tofwas good help in this matter. The gradient in [lo] is often computed by finite differences. The parameters a,'s and bi's for the control equations (10-a) and (10-b) had bcen found after the physical and mission constraints have been satisfied. These consuaints can be found in [7], [15]-(231,as following; Final Constraints: (ufmin= 1) I Uf I (I+,,, = 1.1) = 60 km) I h, I (h,,,, = 65 km (h,,, ,, = 0.9 rad) (Of,, = 0.6 rad) I 8, I (Of,, @f. min = 0.28) 5 Pf I (Pf,,,, = 0.38) (tfm, = 1000 sec) I 4 I (4, ,,, =I500 sec) ... etc. Functional Constraints (%in(t) = 0.2) I u(t) I (u,,,(t) = 1.I) (&,,(I) = 20 km) I h(t) I (h,,,(t) = 65 km) (ymin(t)= 0 rad) 5 y (t) I (ymax(t)= .05 rad) (O,,(t) = 0 rad) I O(t) I OmaX(t) = 0.9 rad) @,in(t> 0.28) I P(t) I (~,,,(t) = 0.94) = 150 m/sec) (rc,,(t) = 0 m/sec) I rc(t) I (rcmax(t) (q,,(t) = .02 atm) I q(t) I (q,,,,,(t) = 1.1 atm) [ (a(t)/gJ,, = 0 I 5 a(t)/g, (a(t)/g,),,, = 11 P a , , ,(O = 5) I Pa, (0 I (Pa,, ,,,(t) = 300) (PG, ,,(t) = 1500 kg,m2) I Pa4(t) I Pg,*,,(t) (Pa,, ,,,(t) = 12000 kg/m2) (Pa,, ,,,(t) = .4) = .01> I Pa,(t> 5 Pa,, ,,(t) = 0) I Pa,(t) I (Pa,, ,,,(t) = 0.1) (ha,, ,(O = 0) 5 ,A 0) I (k,,,,(t) = 1) =0 't,,(t) 5 (7,,,,,,(t) = 1) ('t,,,-(t) ... etc. The numerical inp- W, = 1,000,000 lbf = 4,450,000 N, m, = 454,000 kg, S = 860 m2, C,= 58 m, C = .OW444 Isec, E' = 6, p = 0.000147/m, V = 7900 mlsec. C The results for the controls found to be: = 0.9345 + 0.35 u2 + O.O973033q, h, = 0.6 + 0.33808u2- 0.0106353q also, dl=.6, d2=.2, and d3=.0545. The constraints are generally satisfied and Q,, I 300 kj/cm2 which is about 25 % less than [4]'s result. Also we can gel Q 5 100 kj/cm2 as in [7] if increase r,,by 20 %, and decrease tf by 20 %, which lead the vchiclc to climb from h = 20 km to h 2 48 km with u I .4, then accelerate to u = .8 for h = 60 krn, and gct to orbital speed u = 1 for h 2 75 krn. The heat can even reduce to = 50 kj/cm2 for if we increase T, by Qf = Qshul,le.reenrry 50 % and reduce 4 by 40 C/c as in [7] keeping u I 0.4, then accelerate to u = 0.8 until h = 60 km, and then hit orbital speed u = 1 at h 2 75 km.Thus to minimize the heat we need in general to incrcase (T, - Dl) specialy by decreasing D, and +. 7, 3- The Result The result for the analytical approach shown in Fig.3 (Plot AN-1 to Plot AN-6), and for the numerical approach shown in Fig. 4 (Plot N-1 to Plot N-35). The analytical approach gives a closed form solution for the controls and the heat ,(seeequations (a-6 or -8),(a-7 or 9), and (a-16) respectively ). It shows also that we can have Q c300 kj/cm2 for rc > 35 m/sec and q 5 0.15 atm, and for Q= conslant q varies almost linearly with rc. The analytical approach help in the numerical approach which give good result in the reachable domain of h, and 7,. The heat load found to be Q,, I 300 kj/cm2 which is less by 25 % than [4], and Q,, can be reduced to Q,, ,. = 50 kj/cm2 if increase t, by 50 % and reduce + by 40 % as in [7].Generally To minimize the heat we need to decrease 4 and/or increase T, by increasing T, or decreasing D, . The Plot N-2 shows that the vehicle trajectory which sometime ,,, Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2828 climbs and another time keeps constant altitude which has been reflected on all the numerical result (e.g. Plot N- 12, and Plot N- l3), when the vehicle climbs q decreases and when there is no climb q increases, this is because at climb p decreases faster than the increase in u2 then q decreases, and at no climb u2 increase faster than the decrease in p then q increases). The numerical aerodynamic and thrust total controls are A, = 0.6 + 0.33808 u2- 0.0106353 Q, 7, = 0.9345 + 0.35 u2+ 0.0973033 q. and Where all the constraints are satisfied specially 0.01 5 Pd = (T,- DJ / D, 5 0.4. The analytical closed form solution for,T and A,, are used in the numerical example, where they show an' acceptable result in Plot N- 15 and Plot N- 17 where 0 <T,,, I 2, and 0 < A,, I 1 except when approaching orbital speed where ha,,< 0, this is due to the high rc and low q. This can be fixed (is. h 2 0) by decreasing rc or increasing q by flying at lower altitude. This shows how the analytical approach help in the numerical approach. 4- Conclusion The result of this work shows for givcn real and reachable controls h and 7 we can get Q 5300 kj/cm2, and even less Q = Q,,.,, = 50 kj/cm2. The main problem is hen the wade between propulsion and heat systems in the ascending trajectory as discussed before. The uajcctory in Plot-N2 is within a reasonable flight corridor with current technology (see[l5], and [16] ). Improvements are clearly needed in material and cooling system to provide better margin in handling high heat 1oad.Improvementsin propulsion will allow lower heat load by allowing higher flight altitude. Thus the researche in this area including this, have revealed no fundamental problems that would keep a practical hypersonic air-breathing vehicle from operating in the future taking a payload from regular runway airport to orbital speed and out to space. Then come back to the atmosphere and land again to earth's airports as today conventional airplanes do. References [l] Chapman, R. R., "An approximate analytical method for studying entry into planetary atmosphere", NACA TN 4276 1958. [21 Vinh, Buseman & Culp, "Hypersonic and planetary entry flight mechanics", University of Michigan press, 1980. [3] Vinh, N. X., "Optimal trajectories in atmospheric flight", Elseveir, 1981. [4] Tauber, M. E., & Menees, G.P.,"Acrotherrnodynamics of transatmospheric vehicles", AIAA paper 86-1257. [5] Blestos, N. A.,"Performance and control with lift modulation of hypervelocity entry vchicles",Ph.D. thesis, University of Michigan, 1976. [6] Miele, T., Wang, C. Y. Tzcng, and W. W. Melvin, " Optimal abort landing trajectories in the prcscnce of wind shear", journal of optimi~ationtheory and applications; vol. 55, No. 2, Novcmbsr 1987. [7] Al-Garni, Ahmed,"The Ascending Trajectories Performance And Control To Minimize The Heat Load For Thc Transatmosphcric Acro-Space Plancs", Ph.D. Disscrtation, Dcpartmcnt of Acr Space Engineering , University of Maryland, College Park, Maryland, to be complctcd in December 1990. 181 The Mathlab Group Laboratory for Computer Science, MIT, "MACSYMA:Refcrcnce Manual;" Version 10, 2nd printing, volume 1 and 2. December 1983. MACSYMA, Group,Symbolics, Inc., 257 Vassar street, Cambridge, Mass., 02139. [9] Stephen Wolfram, "Mathematica - A system for doing mathematics by computer", Addison-Wesley Publishing Company, Inc.,1988. [lo] M. K. H. Fan, L. S. Wang, J. Konicka, and A. L. Tits,"Console User's Manual", System Research Center, Technical Report University of Maryland, College Park, Maryland 20742, 1987. [ I 11 H. Elmqvist, "Simnon, An interactive simulation program for nonlinear systems: User's Manual", Department of Automatic Control, Lund Institute of Technology, Report 7502, April 1975. L problem-solving software [12] I ~ S Mathbibrary system, Fortran Subroutines for Mathematical Applications" , pp. 640-651, DIVPAG - Versio 1.0, April 1987. IMSL, Inc., Houston, Texas 77042-3020. [13] C. William Gear,"Numerical Initial Value Problems In Ordinary Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, New Jersey. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2828 [14] Richard L. Burden, and J. DouglasFaires,"Numerical Analysis", third edition, Prindle, Wcbcr & Schmidt, division of Wadsworth, Inc.. Boston, 1985. [15] Manuel Martinez-Sanchez,"Fundamentals of Hypersonic Airbreathing propulsion",Deparunent of Aeronautics and Astronautics, MIT. AIAA Professional Study Series, July 9-10,1988 precedcing the 24th joint propulsion conference in Boston. [16] Isaiah M. Blankson, Office of Aeronautics and Space Technology," Hypersonic Technology Needs For the National Aerospace Plane", presentation to Department of Aerospace Engineering, University of Maryland. By Dr. Isaiah M. BlanksonAssistant Director, National Aerospace Plane, office, NASA October 25, 1988. [17] John D. Anderson, Jr.,"Hypersonic and High Temperature Gas Dynamics, McGraw-Hill, New York, 1988. [18] Robert A. Jones, and Coleman dup. Donaldson, "From earth to orbit in a single stage", aerospace America, vol. 25, no. 8, August 1987, pp.32-34. 1191 J. Swithenbank,"Hypersonic air-breathing propulsion", Progress in the aeronautical sciences, 229294, 1966. [20] C. L. W. Edwards, W. J. Small, J. P. Weinder, and P.J. Johnston,"Studies of scrarnjet/Airframe Integration Techniques for Hypersonic Aircraft", NASA Langley Research Center, Hampton, %rginia 23665, presentcd at the AIAA 13th Aerospace Sciences Meeting, Pasadena, California, January 20- 22, 1975. [21] Patrick J. Johnston, Allen H.Whitehead, Jr., and Gary T. Chapman,"Fitting Aerodynamic and Propulsion into the Puzzle", Aerospace America, vo1.25 , no. 9, September 1987, pp.32-37. U. S. Standard Atmospheres, 1976. NOAAJASA, and USAF, Washington, D. C., October 1976. Jim A. Penland,Don C. Marcum, Jr., and SharonH. Stack.,"Wall-TemperatureEffects on the Aerodynaic far a Hydrogen-Fueled Transport Concept in Mach 8 Blowdown and Shock Tunnels",NASA TP 2159, July 1983. Figures Fig. 1 A conceptual aerospace plane, or transatmospheric vehicle. Fig2 The system is the non-rotating earth and atmosphere with the Aero-Space plane of mass m with the forces T, D(a1ong and oppsite to V-velocity vector rcspcctively), and L(pcrpendicu1ar to V).V making y (flight path angle) with local horizontal plan.All forces, V, y, r(position vector), and B(1ongitudinal angle) arc in he equatorial plane. - Y local honwnlal II Fig.3 The analytical approach graphic result,Plot-AN]Plot-AN6. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2828 Plot-AN1 3D-Plot for A,, , as fun. of rc and q, in the range 05 A, I 1 for the constant rate of climb and constant dynamic pressure case. Plot-AN:! Contour-Plot for Aant as fun. of rc and q, in ~ h range c 0 1 k , 11 for the constant rate of climb and constant dynamic pressure case. 0.6 6crl V & 0.4 Plot-AN3 3D-Plot for T,,, as fun. of rc and q, in the range 01 T, I1 for the constant rate of climb and constant dynamic pressure case. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2828 Plot-AN? Conlour-Plot for T , as fun. of rc and q, in the mngc 05 rut 51 for h e conslant rate of climb and constant dynamic pressure case. q-(atm.) Q,; - (kj/cm? Plot-AN5 3D-Plot for Q,, as fun ofrc and q, in h c range 01 Q,, Y O 0 I k j / ~ i ~Ii ?for L ~ I Cconhlanl rate 0s climb and constant dynamic pressurc case k Q,=~w-(k,/cml Plot-AN6 Contour-Plot for Qan,as fun. of rc and q, in the range 05 Qml 1300 (kj/cm?) l for the constanl ratc of climb 2nd constant dynamic pressure case Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2828 Fig.4 The numerical approach graphic rcsullJ'lot-N1Plol-N35. . .4 2 A , - .d a u-dlml.. speed 1 2 a A . .6 u-dlmlr meed 4 . B n a I 600- 1 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2828 a .4 u-dlmlr speed .4 s 1 udlmlr speed udlmlr. speed

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