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Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-2828
J. Barlow* and A. Al-Garni**
University of Maryland, College Park, MD 20742
meters for the total numerical aerodynamic control law.
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,)
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,
C, = C,, + Ka C2,,
C, = C, + C, = (Dl / q S) - total drag coefficient.
C,, = L, / q S - aerodynamic lift coefficient.
aerodynamic lift coefficient at
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
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.
=((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.
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-2828
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | 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
dV/dt=a=(T,- D,)/m-gsiny
V d y /d t = L, /m - g cos y + (v2/r) cos y
d Old t = (V/r) cos y
dm/dt=-CTg/g,,=-Tg/(I go)
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,
Il =(C2= 1 0 ) C * l , ( ~ @ o )
u = v/JTs,r,-)
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),
d rld t = V, u sin y
-(I -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)
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)
Y 1 = Po V, S h,17 u
d Old t = Vc u ( cos y )/r
dp/dt=-C T
V .lq o 5u3
d Q/d t = C,
q = C , qt12
If also we use the variable'; and7 (of equations (9-a) and
( 9 4 ) in (1 -b)-(7-b), yield;
d r/df= u sin y
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-2828
dq/df= -
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
For given nonlinear equations, describing a
dynamical syst5m and of the form;
, j = 1,2,....,11
d xj/d t = fj( x f , t )
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
C (2 m31a XJ>
-( C)- = o
f ~X,
A more general result is obtained by using ?in (a-6) and
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;
sin y (duld t ) + u c o s y (dyld t) = O
and u2( dqld t)+ 2 q u (d uld t)=O
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;
r = r i + rc t
q = q . e-(P " ' )
by (2-b) and (a-2)
by (8-a-b
u = J(q/C,)
-(a- 12)
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;
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;
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;
( ClIP) @JC2C 1.1 V,3 (Q/Cq)'.'( 1 1 ~aq
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 )
results show that for rc 2 f5 mlsec and q I .15
total aerodynamic control -(a-6)
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 | | DOI: 10.2514/6.1990-2828
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
,, = 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
(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)
't,,(t) 5 (7,,,,,,(t)
= 1)
... 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.
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 +.
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 | | 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.
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.
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.
[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
[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.
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
[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 | | 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,
[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.
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[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.
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Mach 8 Blowdown and Shock Tunnels",NASA
TP 2159, July 1983.
Fig. 1 A conceptual aerospace plane, or transatmospheric
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.
local honwnlal
Fig.3 The analytical approach graphic result,Plot-AN]Plot-AN6.
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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
0 1 k , 11 for the constant rate of climb and
constant dynamic pressure case.
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.
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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.
- (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
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 | | DOI: 10.2514/6.1990-2828
Fig.4 The numerical approach graphic rcsullJ'lot-N1Plol-N35.
u-dlml.. speed
u-dlmlr meed
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1990-2828
u-dlmlr speed
udlmlr speed
udlmlr. speed
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