APPROXIMATE SOLUTIONS FOR

REENTRY VEHICLE I N THE UPPER ATMOSPHERE

FLIGHT-PATH ANGLE OF A

by Jack A. White and Katherine G. Johnson

Langley Research Center Langley Station, Hampton, Va.

NATIONAL AERONAUTICS A N D SPACE A D M I N I S T R A T I O N WASHINGTON, D. C. 0 JULY 1964

https://ntrs.nasa.gov/search.jsp?R=19640016000 2020-03-26T19:15:59+00:00Z

APPROXIMATE SOLUTIONS FOR FLIGHT-PATH ANGLE O F A

REENTRY VEHICLE IN THE UPPER ATMOSPHERE

By Jack A. White and Katherine G. Johnson

Langley R e s e a r c h Center Langley Station, Hampton, Va.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sale by the Office of Technical Services, Department of Commerce, Washington, D.C. 20230 -- Price $1.25

APPROXIMATE SOIUTIONS FOR FLIGHT-PATH ANGIX OF A

REENTRY VEHICLF: I N THE UPPER ATMOSPHERE

By Jack A . White and Katherine G. Johnson Langley Research Center

SUMMARY

Two basic equations f o r approximating the fl ight-path angle of a reentry vehicle i n the upper atmosphere a re derived from the equations of motion. These solutions fo r fl ight-path angle a re obtained (1) by assuming constant velocity over the portion of the t ra jectory under consideration, and (2 ) approx- imating the velocity value fo r use i n the development of the equation f o r f l i gh t -path angle.

Solutions obtained from numerical integration of the nonlinear reentry equations a re used t o evaluate how well the derived equations approximate the t rue value of fl ight-path angle. The t ra jec tor ies selected were such tha t the space vehicle stayed within a safe f l i gh t corridor. Reentry t ra jec tor ies fo r a high-drag low-lift vehicle included i n the investigation covered a range of veloci t ies from suborbital t o hyperbolic, a l t i tudes from 200,000 fee t t o 3OO,OOO fee t , and desired fl ight-path angles from 00 t o 2 O . In the case of a h igh- l i f t reentry vehicle, t ra jec tor ies t ha t obtained level-fl ight conditions a t veloci t ies ranging from suborbital t o near-orbital veloci t ies fo r an a l t i t ude of 200,000 fee t were selected t o evaluate the closed form equations.

INTROIXJCT'ION

For an atmospheric reentry a t supercircular velocity and landing a t a pre- scribed point on the ear th 's surface, precise guidance of a space vehicle i s required. T h i s guidance must be effected i n the upper atmosphere t o bleed off excessive velocity before e i ther (1) descending d i rec t ly t o the landing point, or (2) performing a skipout maneuver t o extend the range.

Numerous prac t ica l reentry guidance gystems have been formulated i n recent years by experimental and analyt ical procedures. ance systems, such as those based on rapid time of predictions and those ut i - l i z ing reference t ra jec tor ies and t h e i r adjoint solutions require a complex computer. A guidance system based on a closed form solution of the equations of motion would be more a t t r ac t ive because of the reduction i n computer storage and computer component requirements. Because of the nonlinear nature of the equations of motion, closed form solutions f o r complete en t r ies cannot be

M a n y of these workable guid-

obtained; however, approximate relat ions f o r the t ra jec tory variables can be obtained f o r selected portions of an entry t ra jectory.

Two procedures where closed form solutions are ut i l ized f o r specif ic maneu- vers a re discussed i n references 1 and 2. I n reference 1, a space vehicle i s controlled t o prescribed e x i t conditions from the atmosphere t o achieve a desired range. I n reference 2, t he vehicle i s controlled t o level-f l ight con- dit ions a t a desired a l t i tude . In other studies i n which approximate solutions t o reentry equations are derived ( fo r example, refs. 3, 4, and 5 ) cer tain assump- t ions a re made such t h a t the closed form equations obtained a re valid f o r approx- imating quantit ies such as deceleration, t o t a l heat absorbed, heating ra te , and so forth, over a f a i r l y large in te rva l of time. The present study was made i n order t o develop a closed form solution fo r fl ight-path angle as a function of a l t i t ude and desired end conditions tha t could sa t i s f ac to r i ly approximate flight-path angle over a short period of time - 1 minute or less .

I n the development of the guidance system reported i n reference 2, an equa- t ion f o r approximating fl ight-path angle was derived and used t o predict when a control maneuver should be made t o bring a reentry vehicle t o leve l f l i g h t a t a specific a l t i tude . The present investigation evolved from t h i s phase of ref- erence 2. f i e d and other equations were derived.

The or iginal equation f o r approximating fl ight-path angle was modi-

In the present paper, t ra jec tor ies calculated from the equations of motion are used t o determine the re la t ive accuracy of the closed form equations. Tra- jector ies covering a wide range of conditions i n the upper atmosphere, t ha t is, velocity, a l t i tude , and fl ight-path angle, were selected i n order t o determine where these approximations a re valid. It should be noted tha t the range of parameters used i n t h i s study i s chosen within a previously established flight corridor.

A constant used t o simplify the second term of equation (16)

B constant used i n exponential approximation of atmospheric density, f t

C constant used i n the development of equation (17)

l i f t coefficient

CD drag coefficient

CL a

Q acceleration due t o gravity

h a l t i t ude above ear th ' s swface, f t

Ah = hd - h, f t

2

K132 constants defined i n equations (2 ) and (3)

constant used i n the development of equation (10) K3

L/D

m

n

r

S

t

V

V1

l i f t -drag r a t i o

mass of vehicle, slugs.

integer

radius of earth, f t

surface area of vehicle, sq f t

time, sec

velocity of vehicle, f t /sec

constant mean value of velocity, f t /sec

Wl ,w2 ,x , W weight of vehicle , l b

y = e-h/B

variables used i n development of equation (17)

Y fl ight-path angle, deg or radians

first approximation of fl ight-path angle, deg or radians 71

constant used i n exponential approximation of atmospheric density, P C slugs / f t 3

Subscript:

d values a t desired condltions

Derivatives with respect t o time a re denoted wdth dots over the variables. d

ANALYTICAL DEVELOPMENT

For the analysis made i n the present study, it was assumed that the ear th A constant gravitational f i e l d i s spherical with a radius of 20,908,800 fee t .

3

of 31.2 ft/sec2 was assumed for all the altitudes covered. It was also assumed that the earth is stationary in all respects and there is no relative motion of the atmosphere.

From the equations of motion, two basic closed form solutions for flight- path angle are obtained. In the first, constant velocity is assumed. In the second, an approximate solution for velocity is obtained and used to develop the closed form equation for flight-path angle. modified as is subsequently shown.

These two methods are further

Equations of Motion

With these assumptions, the two-degree-of-freedom equations of motion of an entry vehicle as derived in reference 6 are as follows:

1; = v sin 7

V = -K1V 2 e -h/* - g sin

where

A standard substitution used to make the equations more concise is:

-h/B y = e

Also, in the present investigation, the small-angle approximation of

sin 7 = 7

cos 7 = 1

4

i s used. Hence, equations (1) t o (3) become:

I n general, i n other studies velocity has been considered the independent variable. t ion i s of secondary importance t o the variation of flight-path angle and a l t i - tude. Accordingly, i n t h i s study, a l t i t ude is made the independent variable.

(See, f o r example, re f . 3 . ) However, f o r path control velocity varia-

Dividing equations ( 5 ) and (6) by equation (4 ) gives

dV - Bg W Y YV

It i s noted tha t these equations can not be evaluated when y = 0.

F i r s t derivation.- I n order t o obtain a closed form solution fo r f l i gh t - path angle from equation ( 8 ) , the following analysis i s made. be written i n the form

Equation (8) may

From the mean-value theorems f o r integrals , the second term on the right can be writ ten as

5

where K3 l ies between the greatest and the l ea s t values of -

may possibly equal one of them. Since V1 i s actual ly a variable i n this expression, the problem here i s tha t of velocity selection.

Integrating equation ( 9 ) gives ( fo r constant VI)

Substi tuting Ah = hd - h and recal l ing tha t VI i s regarded as a constant value of velocity gives

L J

where, for the conditions chosen, 7 will take the sign of &h.

Second derivation.- Another approach t o a closed form solution of equa- t i on w e r i v e a successive approximation procedure. not constant and a closed form solution of equation (7) does not exi.st, an approximate solution of t h i s equation i s t o be obtained. A f i r s t approximation of flight-path angle obtained from equation (8) i s used i n equation (7) t o yield an approximate solution f o r velocity. "he velocity approximation thus obtained i s substi tuted i n equation (8) t o yield a second approximation f o r fl ight-path angle.

Since velocity i s

Equation (8) may be writ ten i n the form

6

where

It i s found from numerical calculations tha t the second term on the r ight of equation (7) i s usually s m a l l i n comparison with the f i r s t term. lec t ing t h i s term, equation (7) becomes

By neg-

The procedure t o be followed now is:

(1) Solve equation (11) fo r a f i r s t approximation of

(2) Use t h i s value of

7

7 i n equation (12) t o obtain a velocity approximation

( 3 ) Substitute t h i s value of V i n equation (11) t o obtain a second approximation fo r 7

The f i r s t approximation of f l ight-path angle i s obtained by integrating equa- t i on (11) under the assumption tha t dc/dy = 0. Hence,

A r e a l value of yl i s desired under a l l conditions; hence, i n determining yl the absolute value of the expression i n equation (13) i s taken. takes the sign of Ah.

The angle 71

The subst.itution of yl a s defined by equation (13) for y i n equa- t i on (12) yields:

This equation may be integrated with the a i d of the following substitutions:

7

z = 2K2By

Thus

Integrating yields

Since, by definit ion,

it follows tha t

2 1 c - z = y

it should be noted tha t yl = f i n . ) Hence,

or D

where vd and yd, the desired end conditions, a r e known.

This solution fo r V i s then used i n equation (11) t o yield a second By using equation (14), the second term of equation (11) approximation f o r y.

may be writ ten i n the form

8

Integrating gives

Since - * - - - dz the second term of t h i s equation becomes Y z '

where

T h i s in tegral may be evaluated as follows: Let x = ( c - z ) ' / ~ where c i s a constant. Then

z = c - x 2

dz = -2x dx

Hence the in tegra l becomes s.." (. -2x - x2) dx

or

By p a r t i a l fractions,

2Ax A A

( x + p ) ( x - fF) x + fl . x - p

Thus the integral may now be written

2CD -X

A ecL

I n order t o integrate t h i s expression, l e t

Then

The expression t o be integrated becomes

10

Integrating yields

n - -2CDfi

Ae cL log, wl + 2 n=l n n! L. J

wl,d

5 6 + AeCL

wl

Truncating the ser ies a t n = 1 gives

W2,d

w2

1

w1 yd w2 d Since w1,d - wl = w 2 , d - w2 and -- = -L, t h i s expression may be Wl,d y w2

writ ten i n the form:

o r

P

c Hence, equation (16) becomes:

where takes the sign of Ah.

11

It should be noted tha t equation (17) can be used t o evaluate E under a l l conditions except zero l i f t by assuming

Now tha t the in tegra l of

thus :

has been obtained, equation (11) can be integrated aY

'(7 2 d - Y2) = -K2(yd - y) k E

Solving for y

or

7 = ( Yl f 2 y 2

The posit ive sign i s chosen when E (obtained by eq. (17)) and Ah have l i k e signs, and the negative sign when E and Ah have unlike signs. The angle 7 takes the sign of Ah.

Procedure

A s a basis f o r evaluating the equations derived i n the preceding sections, the equations of motion (eqs. (1) t o ( 3 ) ) were solved i n negative time, by using a d i g i t a l computer, t o provide t ra jec tor ies with end conditions (Va,

specified f o r . t h i s study. exact solutions.

Td, hd) These t ra jec tor ies a re hereinafter referred t o as

Exact a l t i t ude and velocity values (h, V) a t various

positions along these t ra jec tor ies were used t o calculate the approximate value of f l i gh t -path angle.

In order t o f a c i l i t a t e the discussion, the approximations obtained by the In method I, equa- various equations sha l l be referred t o as methods I t o IV.

t i o n (13) i s evaluated t o provide a rapid f i r s t approximation fo r equation becomes a par t of the approximations used i n succeeding methods. Method I1 evaluates equation (10) i n which the present velocity i s chosen fo r the constant-velocity value.' I n method 111, which a l so evaluates equation (lo), the velocity as obtained from equation (14) i s used fo r the constant-velocity value. An approximate solution fo r velocity i s used i n the development of the closed form equation fo r fl ight-path angle. All four methods were programed on a d i g i t a l computer. the program logic fo r method IV w a s considerably more complicated than tha t fo r the other methods.

7. This

Method IV evaluates equation (18).

It i s t o be noted tha t

RESULTS AND DISCUSSION

Results of the study are presented as time h is tor ies of the exact and the calculated values of fl ight-path angle. Time h is tor ies of the exact velocity and a l t i t ude values used i n the approximate equations a re included for reference.

For the most par t , the r e su l t s of the present study a re given fo r a high- drag low-lift vehicle having a of 20.5. Limited resu l t s are shown fo r t h i s vehicle where the l i f t -drag r a t i o w a s trimmed a t 0.2. which has a W/S

W/S r a t i o of 60 lb / f t2 and a l i f t -drag r a t io

Results a re a lso presented fo r a h igh- l i f t reentry vehicle r a t i o of 27.88 lb / f t2 and i s trimmed at a l i f t -drag r a t i o

of 1.5.

Desired conditions were selected a t three al t i tudes: 200,000 f ee t , 250,000 fee t , and 3OO,OOO f ee t . For tlie high-drag low-l i f t vehicle, these a l t i - tudes a re a l t i tudes where precise path control i s of primary concern. An a l t i - tude of 200,000 f ee t i s the (approximate) minimum a l t i t ude allowable during the in i t ia l dive in to the atmosphere where overdeceleration (greater than 8g) of the vehicle 's passenger w i l l not occur. An a l t i t ude of 250,000 fee t i s con- sidered the appropriate a l t i t ude a t which t o leve l off ( a f t e r bleeding off excessive velocity a t a lower a l t i t ude ) i n order t o obtain certain desired ranges. A t 3OO,OOO fee t , the density i s very low; i n many studies, this a l t i - tude i s regarded as the upper boundary of the usable atmosphere.

Hence, fo r desired end conditions for the high-drag low-lift vehicle a t hd = 200,000 fee t , velocity values ranging from 20,000 ft/SeC t o 40,000 f t / sec with Td = 0 were chosen. For hd = 250,000 fee t , veloci t ies were varied from 26,000 f t /sec t o 30,000 f t / sec while Td varied from 00 t o 1'. For

hd = 3OO,OOO feet, f l ight-path angles of lo and 2O were selected for veloci t ies of 25,000 f t /sec and 26,000 f t /sec. were calculated fo r veloci t ies of 26,000 f t /sec, 20,000 f t /sec, and 14,000 f t /sec f o r hd = 200,000 f ee t and Td = 00.

Results fo r the h igh- l i f t reentry vehicle

Other cases were investigated wherein sat isfactory correlation between the exact and approximate values w e r e derivedj however, cases where tolerable g values were exceeded have been omitted.

High-Drag Low-Lift Vehicle

L/D = kO.5.- In figure 1, results a re shown fo r cases where the desired end conditions a re l e v e l f l i gh t a t an a l t i t ude of 200,000 f ee t for four speci- f i ed veloci t ies . A t Vd = 40,000 f t /sec, 30,000 f t /sec, and 26,000 f t /sec f o r CL = S . 5 , excellent agreement with the exact values i s obtained by methods I1 and IV. A s noted i n the figure, a low value of y i s approximated by method I f o r these veloci t ies . approximations a re obtained by methods I1 and 111. and l ( e ) , values obtained by methods I and IV are considerably higher than the exact values f o r CL = -0.3, these two methods give low values.

A t suborbital velocity (vd = 20,000 f t /sec) , the best A s shown i n figures l ( d )

CL = 0.5, whereas for

Cases were calculated for desired a l t i t ude of 250,000 fee t for Vd = 30,000 f t /sec and 26,000 f t /sec fo r yd = 10 as w e l l as yd = 00 f o r

both posit ive and negative values of For

t i v e and negative l i f t ; excellent resu l t s a r e obtained by method N for posit ive l i f t , as noted i n figures 2(a) and 2(c) . the most accurate r e su l t s a re obtained by using methods I1 and IV. gave very poor approximations for the cases where Vd = 30,000 ft /sec; however, a l l methods were sat isfactory fo r

CL. These resu l t s a re shown i n figure 2. yd = Oo, methods I1 and I11 give consistently good results f o r both posi-

For Td = 10 ( f igs . 2(e) t o 2 (h ) ) , Method I

Vd = 26,000 f t /sec.

Figure 3 shows resu l t s f o r t h e cases where hd = 3OO,OOO feet f o r vd = 26,000 f t /sec and 25,000 f t /sec. For vd = 26,000 f t /sec and yd = lo, excellent predictions of fl ight-path angle a re obtained by methods I11 and N fo r both posit ive and negative l i f t . (See f ig s . 3(a) and 3(b) . )

For yd = 20, a l l methods except method I yie ld satisfactory resu l t s a s noted i n figures 3(c) and 3(d). For the case where V d = 25,000 f t /sec and

= 20, excellent agreement with the exact values i s obtained by method I11 and good agreement by method 11. values calculated by methods I and IV are unsatisfactory f o r both posit ive and negative l i f t .

7d It i s noted i n figures 3(e) and 3(f) tha t

L/D = 50.2.- The four approximate equations were evaluated fo r selected cases where the l i f t -d rag r a t i o of the reentry vehicle was trimmed t o 50.2. A s i l l u s t r a t ed i n figures 4( a ) and 4(b), a l l methods provide sat isfactory resu l t s fo r e i ther yd = 00 or yd = 2O, fo r ha = 200,000 ft, vd = 26,000 f t /sec, and CL = 0.2. Shown i n figures 4(c) and 4(d) a re cases fo r hd = 3OO,OOO ft,

14

Vd = 25,000 f t /sec, and method I1 good results f o r both posit ive and negative lift. hd = 200,000 f t , gave excellent resu l t s fo r CL = 0.2. (See f ig . 4 (e ) . )

7d = 2O, CL = 50.2; method I11 provided excellent resu l t s

For V d = 20,000 f t /sec, and yd = Oo, both methods I1 and I11

In a comparison of figures 3(e) and 3 ( f ) and figures 4(c) and 4(d), it can be seen tha t methods I and IV are unsatisfactory i n predicting flight-path angle f o r suborbital veloci t ies a t l ( d ) , and l ( e ) , this i s a l so t rue f o r suborbital veloci t ies a t h = 200,000 fee t .

h = 3OO,OOO feet . A s noted i n figures &(e) ,

High -Li f t Vehicle

Shown i n figure 5 a re resu l t s calculated fo r a high-l i f t reentry vehicle (L/D = 1.5) where desired end conditions were leve l f l i gh t a t an a l t i tude of 200,000 fee t . It may be seen i n figure 5(a) t ha t excellent carrelation with the exact values i s provided by methods I and IV f o r Method I1 provided excellent resu l t s a l so fo r suborbital veloci t ies , vd = 20,000 f t /sec, and

vd = 26,000 f t /sec.

Vd = 14,000 f t /sec, as shown i n figures 5(b) and 5(c) .

CONCLUDING REMARKS

An analysis of the equations of motion fo r a space vehicle reentering the ear th 's atmosphere has been made. a closed form solution f o r fl ight-path angle t h a t could be used t o predict the path parameters over a short time interval . were investigated.

The objective of the analysis was t o develop

Only acceptable reentry conditions

The resu l t s of t h i s analysis have shown tha t the fl ight-path angle of a reentry vehicle can be approximated with some degree of accuracy by several methods. The approximations a re va l id i n the a l t i t ude range between 200,000 fee t and 3OO,OOO fee t , f o r veloci t ies ranging from suborbital t o hyper- bolic, f o r a high-drag low-lift vehicle. For a high l i f t i n g vehicle, the approximations are val id for suborbital t o o rb i t a l veloci t ies a t an a l t i t ude of 200,000 fee t .

Langley Research Center, National Aeronautics and Space Administration,

Langley Station, Hampton, Va., Apr i l14 , 1964.

1. Dunning, Robert S.: Study of a Guidance Scheme Using Approximate Solutions of Trajectory Equations To Control the Aerodynamic Skip Flight of a Reentry Vehicle. NASA TN D-1923, 1963.

2. White, Jack A.: Feasibility Study of a Bang-Bang Path Control for a Reentry Vehicle. NASA TN D-2049, 1963.

3 . Chapman, Dean R.: An Approximate Anaaytical Method for Studying Entry Into Planetary Atmospheres. NASA TR R-11, 1959. (Supersedes NACA TN 42'16.)

4. Loh, W. H. T.: A Second Order Theory of Ehtry Mechanics Into a Planetary Atmosphere. June 1961.

Paper No. 61-116-1810, S.M.F. Pub. Fund, Inst. Aerospac-e Sci.,

5. Wang, Kenneth,-and Ting, Lu: Approximate Solutions for Reentry Trajectories With Aerodynamic Forces. Polytechnic Inst. Brooklyn, May 1961.

PIBAL Rep. No. 647 (Contract No. AF 49(638)-445),

6. Eggleston, John M., and Young, John W. : Trajectory Control for Vehicles Entering the Earth's Atmosphere at Small Flight-Path Angles. 1961.

NASA TR R-89, (Supersedes NASA MEMO 1-19-591;. )

16

I

Exact Method

I I1 I11 IV

~ -__ _ _ _ _ -

Approximate - - __ i

0

Figure 1.- Time his tory of exact t ra jec tory and calculated values of f l ight-path angle for desired a l t i t ude of 20O,OOO f ee t . yd = 00.

I 1 - 1 I I I

Exact Method

(--- I

a" bo - 2 -;r -I-

I

IV

bo a" - 2 -

-1 -

0 10 20 Time, t, sec

I

(b) Vd = 30,000 ftlsec; CL = 0.5.

Figure 1.- Continued.

18

1 I- - _ t 1

Exact

Method

I I1 Apnrnximate I--- - - - - -

r - 2.4

d M-2-o t h"

-1.6 -

-1.2 -

-. 8 .-

-.4 -

0 i o 20 Time, t, sec

( c ) vd = 26,000 ft/sec; CL = 0.5.

Figure 1.- Continued.

I I I I I 1

-2.1

-2 . I

- 2. (

-1.6 bD 3 2 -1.2

-.8

-.4

0

Method

I I1 I11 Iv

<by\, -

-

I 10 20

Time, t, sec

(a) V d = 20,000 f t /sec; cL = 0.5.

Figure 1.- Continued.

20

,

I I I ~... . 1 .~ - .. I I

M

a 2=

Method

I1 Approximate I - - - " 11-

Time, t, sec

vd = 20,000 ft/sec; cL = -0.5.

Figure 1.- Concluded.

21

3

n F: I I I .~ . I - -

- 1.

- 1.

Method

I

0 10 20 30 Time, t, sec

( a ) V d = 30,000 f t / sec ; 7d = Oo; CL = 0.5.

40 50

Figure 2.- Time history of exact t ra jec tory and calculated values of fl ight-path angle for desired a l t i t ude of 250,000 f ee t .

22

& a c t - - 7- t- \ Method

-.4 M a, a 2 -. 2

0 10 20 30 Time, t, s e c

(b) V d = 30,000 f t /sec; yd = 00; cL = -0.5.

Figure 2. - Continued.

23

1 I I __ 1 I I I

.. s -- I I

--1. 0 Exact

Method

- .8 I1 I11 \----- IV

-.6

2 -. 4

-. 2

0 10 20 40 Time, t, s e e

24

( c ) vd = 26,000 f t l s e c ; yd = oO; CL = 0.5.

Figure 2.- Continued.

,11111 11.1111 I

.8

.6

. 2

0

- &act

.\ Method

10 20 30 40 Time, t, sec

(d) Vd = 26,000 f t /sec; 7d = 00; cL = -0.5.

50

Figure 2.- Continued.

25

...

1. c

.e

.6 M a, a h" .4

.2

0

Method

I11 Approximate

(----.--- IV I

5 10 15 20

Time, t, sec

26

2.2

2.0

1.8

1.6 ho 3 2 1.4

1.2

1. c

\ Exact - \

Method I I1 I11 IV

_ _ _ _

-_--

\ '\ \

-

- \ '\

\

.a 0

_ _ I I I I 23 - 3-0 40 50

1 __ - 10

Time, t, see

(f) Vd = 30,000 ft/sec; yd = 1'; CL = -0.5.

Figure 2.- Continued.

27

I

4-1

I I I 1

1. c

. a

.6 M

2 2 .4

. 2

0

Exact

Approximate

--

{=I

Method

I I1 I11 IV

(g) vd = 26,000 ft/sec; 7d = io; cL = 0.5.

Figure 2. - Continued.

28

2.

1.

1.

-.w_ Y

Method

:\ \ \ \ I - - - 1

0 10 20 30 Time, t, see

(h) Va = 26,000 Ft/sec; yd = 10; cL = -0.5.

Figure 2.- Concluded.

29

u e, m --. iJ k

I I

Exact Method

(- - - I I1 I11 IV

--- Approximate

I I I 1 I 1 .4 0 10 20 30 40 50 60

Time, t, see

( a ) Vd = 26,000 ft /sec; 7d = lo; CL = 0.5.

Figure 3 . - Time history of exact t ra jec tory and calculated values of fl ight-path angle fo r desired a l t i t ude of 300,000 f ee t .

2.

Exact

Method 2.

I1 I11

I V

_ - - -

1.

1.

a - \

. I . . . \ I - u 60 90 120 150

8 . . 1 0 30

Time, t, sec

. I . . . \ I - u 60 90 120 150

1 30

8 I . . 0

Time, t, sec

(b) Vd = 26,000 ft /sec; T~ = 10; cL = -0.5.

Figure 3 . - Continued.

7

I

I I I I - - ~- 1

1.8

2.0

1.9'

- -

/ I I I 1.5, 10 20 30 40 50

Time , t, sec

I I I . . . 1

( e ) Vd = 26,000 f t l s ec ; 7d = 2 O ; CL = 0.5.

CJ

I I I ~ . I

( d ) vd = 26,000 ftlsec; 7d = 2 O ; cL = -0.5.

Figure 3 . - Continued.

32

I I I 1 111 I 1 I l l I 1 1 1 1 1 1 1 1 1 1 1

Exact

-P k

d

Method

/ 0

1 - I I I I I

I1 I11

I- ! 1 1

( e ) Vd = 25,000 ft /sec; 7d = 20; CL = 0.5.

Figure 3.- Continued.

33

V

I I I I 4

34

I

-2 .0

-1.6

-1.2 M

2 ??

-.8

-. 4

0

Exact

Method

I1 Approximat e { - - - - I11 \---- rv

Time, t, sec

(a) & = 200,000 ft; vd = 26,000 ft/sec; yd = 00; CL = 0.2.

Figure 4.- Time history of exact trajectory and calculated values of flight-path angle.

35

F-r

2.0

1.8

1.6 w Q) a h" 1.4

1.2

I I I I I

1. oo

/

@' /

Approximat e \=-- , /

2 4 6 8 10 T i m e , t, sec

I I1 111 N

- I 12

(b) ha = 200,000 ft; Vd = 26,000 f t / s e c ; yd = 2O; CL = 0.2.

Figure 4.- Continued.

.\ s 200 - - 1 I

2.

1.

M W 1.

h" I

1.76 t / /

/ /

/

/- Method

I1 I11 Approximate{ - - - -

1. 600 1 I ._ - -1- I I I 1 10 20 30 40 50 EO

Time, t, sec

( c ) hd = 3OO,OOO ft; Vd = 25,000 ft/sec; yd = 2O; CL = 0.2.

Figure 4.- Continued.

37

2.40

2.32

2.24

bo

2.16 h"

2.08

2.00

1.92[

Exact

Method

(--- I I1 I1 I IV

---- -- Approximate \

+/ ----- I 1 .. . I - .~

10 20 40 50 60 Time, t, see

(a) ha = 300,000 f t ; Vd = 25,000 ft /sec; yd = 2'; CL = -0.2.

Figure 4.- Continued.

. I

I

\ \

&act

Method

I11 Approximate r---

\ I . I - _ _ . . I - - 1 I

0 5 10 1 5 20 30 Time, t, sec

( e ) ha = 200,000 f t j Vd = 20,000 ft /sec; 7d = 00; cL = 0.2.

Figure 4.- Concluded.

39

I

-5

-4

-3 ho a, 6

s

Exact Method

Time, t, sec

( a ) vd = 26,000 ft /sec.

Figure 5.- Time history of exact t r a j ec to ry and calculated values of fl ight-path angle f o r high l i f t i n g vehicle. hd = 200,000 f ee t ; CJ, = 1.5; 7d = 0'.

..

-4

-3 M a, a

Exact Method

--

-3 - M a, a

-1 -

0 10 20 30 40 Time, t, sec

(b) Vd = 20,000 ft /sec.

Figure 5.- Continued.

41

-

--I

-3

M (u

a -2 h"

-1

0

\

Exact Method

I 1- - - I1 ----

Approximat e {-- 111

- I

10 20 30 Time, t, sec

( c ) Vd = 14,000 f%/sec.

40 -. 50

Figure 5.- Concluded.

42 NASA-Langley, 1964 L-3795

I

“The aeronautical and space activities of the United States shall be conducted so as to contribute . . . t o the expansion of human Rnowl- edge of phenomena in the atmosphere and space. The Administration shall provide for the widest practicable and appropriate dissemination of information concerning its activities and the results thereof .”

-NATIONAL AERONAUTICS AND SPACE ACT OF 1958

NASA SCIENTIFIC AND TECHNICAL PUBLICATIONS

TECHNICAL REPORTS: important, complete, and a lasting contribution to existing knowledge.

TECHNICAL NOTES: of importance as a contribution to existing knowledge.

TECHNICAL MEMORANDUMS: Information receiving limited distri- bution because of preliminary data, security classification, or other reasons.

CONTRACTOR REPORTS: Technical information generated in con- nection with a NASA contract or grant and released under NASA auspices.

TECHNICAL TRANSLATIONS: Information published in a foreign language considered to merit NASA distribution in English.

TECHNICAL REPRINTS: Information derived from NASA activities and initially published in the form of journal articles.

SPECIAL PUBLICATIONS: Information derived from or of value to NASA activities but not necessarily reporting the results -of individual NASA-programmed scientific efforts. Publications include conference proceedings, monographs, data compilations, handbooks, sourcebooks, and special bibliographies.

Scientific and technical information considered

Information less broad in scope but nevertheless

Details on the availability of these publications may be obtained from:

SCIENTIFIC AND TECHNICAL INFORMATION DIVISION

N AT1 0 N A L A E RO N A UT I CS A N D SPACE A D M I N I STRATI 0 N

Washington, D.C. PO546