Stability of flight paths of lifting vehicles during entry into planetary atmospheres

ENTRY INTO PLANETARY ATMOSPHERES

JULY, 1961

by J. H. Fine

UTIA TECHNICAL NOTE NO. 48 AFOSR 1488

ENTRY INTO PLANETARY ATMOSPHERES

JULY, 1961

by

J. H. Fine

UTIA TECHNICAL NOTE NO. 48 AFOSR 1488

•

ACKNOWLEDGMENT

The author wishes to express his thanks to Dr. G. N. Patterson for the opportunity to pursue this investigation .

A great deal of appreciation is-also extended to Professor B. Etkin for suggesting the problem and for the generous amount of time which he spent in giving capable direction and providing constructive criticism of the paper.

Mention should also be made of the com putational facilities which were made available by Dr. C. -Co Gotlieb who is in charge of the Computation Centre at the University of Toronto.

Small pitching oscillations in a manned lifting vehicle about its trim angle of attack were analyzed for their ability to deflect the vehicle from its trim trajectory. Several planetary atmospheres along with both circular and hyperbolic velocities were .considered.

The results of thisinvestigation were conclusive. Except for those cases when the static_gnargin of the vehiclé was so small as to be impractical, i. e. less than 10 , the pitching oscillations were unable to cause an appreciable deviation from the reference path.

,

I. II.

lIl.

IV.

V.

VI. VII. (i) TABLE OF CONTENTS NOTATION INTRODUCTION BACKGROUND THEORY 2. 1 Analysis 2. 2 Equations of Motion 2.3 Statie Trajeetory 2.4 Oseillatory Motion

2. 5 Entry at Hyperbolic Veloei ty

PERTURBATION TECHNIQUE 3. 1 Analys is

3.2 Assumptions

3.3 Perturbation Equations

DEPENDENCE OF LIFT UPON ANGLE OF ATTACK 4.1 The Choice of the Lift Funetion

4. 2 The Models Chosen for CL (0<)

4.3 Correlation Between the Two Lift Functions DET AILS OF THE CALCULATION

5. 1 Overall Method 5. 2 Starting Point

5.3 The Bessel Function and th,e Three Integrals NUMERICAL DATA USED

6. 1 The Atmosphere 6.2 The Vehicle 6.3 Initial Conditions EVALUATION CRITERIA

Hi

1 2 2 3 4 5 6 9 9 9 9 12 12 13 13 13 13 14 14 15 15 16

17

18

7. 1 Criterion for Deviation 18

7.2 Criterion Used in Determining the Effeetiveness

VIII. RESULTS 19

IX.

X.

8. 1 Deviations for a Reasonable Statie Margin 19 8. 2 The Influenee of a Deereasing Statie Margin 19 8.3 The Influenee of Other Parameters at Very

Small Statie Margins 19

8.4 The Influenee of the Wing Loading 20 8.5 Effect of the Pitehing Oscillations Upon the Velocity 20 DISCUSSION OF RESULTS

9. 1 Major Charaeteristies 9.2 Other Characteristies

9.3 A Comparison of the Two Lift Models 9.4 Dependenee Upon the Statie Margin

9.5 Influenee of Other Parameters at Very Small Sta tic Margins

CONCL USIONS 10. 1 Major Charaeteristies 10. 2 Academie Results REFERENCES APPENDIX A APPENDIX B APPENDIX C TABLE I TABLE II 20 . 20 22 23 23 24 24 24 25 27 28 29 31 34 35

CLO( C _m~ C mS; D e f(o(o) fl (t), f2(t) g g(O(o) NOTATION

constants of integration used to satisfy the two arbi-trary conditions of the initial error angle of attack and .

the initial angular pitching velocity of the vehicle in Eq. 2.4, la

constants of integration used to match the oscillatory angle of attack and the angular pitching velo city of the vehicle at the beginning of the ascent with the same quantiti~s at the end of the descent in Eq. 2.4,· lb the lift coefficient

(L/!

fy2S)

(

~h~:)ate of change of lift coefficient with angle of

,,0(

0<

-+-0

o

the drag coefficient (D

I! fV

2S) the moment coefficient

(MI!

fy2SJ)

attack,

the rate of chan~ of pitching moment coefficient with angle of attack, (" C:; )

0 ( 0 - .0

the rate of change of pitching moment coefficient with time rate of change of angle of attack parameter Ó(

i

{o

Cm)

v'

\oó<J/V

Ö<o

~O

the rate of change of pitching moment coefficient with the pitching velocity

parameter

~.

9i"

("cm)

y

oé..e/V

9-0

the rate of change of pitching moment coefficient with control deflection

(0

Cm)

o

b

,

b~O

the drag on the vehicle, Ibs.

Naperian base

function defined hy Eq. D, 3

functions defined by Eq. 2. 2, 3

the acceleration due to gravity,

ftl

sec2 function defined by Eq. D. 8

1 1_{1, 12, 13} J_{o (} ) j{s) Kn Kl, K2 k(s) L M m

P,Q

R

s

s T

u

v

x y Greek Letters

the pitching moment of inertia about the centre of gravity, slugs ft2

integrals defined in Eqs. 3.3, 16

Bessel Function of the first kind of zero order function defined by Eq. 2.5, 15, slugs/ft the static margin

constants defined in Eqs. 2.4, 1

function defined by Eq. 2. 5, 15, slugs /ft

the aerodynamic lift acting on the vehicle, Ibs.

the aerodynamic pitching moment acting on the vehicle, lb. ft.

the wing Ioading parameter defined by Eq. 2.3, 16a the mass of the vehicle, slugs

parameters defined by Eq. C, 1 the radius of the earth, ft.

the reference area of the vehicIe, ft2.

the distance measured along the flight path, ft. a parameter defined by Eq, B,4 and.Eq. B, 5 a parameter defined by Eq. B,4 and Eq. B, 5 the flight velocity, ft/sec

the distance measured along the earth, ft.

Bessel Function of the second kind of zero order the altitude above the earth's surface, ft.

the angie of attack of the vehicie defined in Figure 1 the density parameter defined in Eq. 6. 1, 1, ft-1

'Y

rE

Ö ~ 9

'f

?

Sb

C

Superscripts (

.

) Subscripts ( _)i (

)

. m ( _)max ( _)min ( ₎₀ ( _)s ( _)T

the flight path angle defined in Figure 1

the flight path angle on entrance to the atmosphere the control angle deflection

a dummy variabie

the ç,ngle öf pitch defined in Figure 1 a dummy variabie

the air density, slugs/ft3

the air density at sea level as required by Figure 2, slugs/ft 3

the radius of gyration of the vehicle in pitch, ft.

derivative at a pointwith respect to time

the initial value

the mean value as defined by Eq. D, 12 the maximum vallie

the minimum value an o'scillatory variabie a non-oscillatory variabie

the sum of the oscillatory plus the non-oscillatory parts of the variabie

the ratio of the ~alue of this param eter to the value for the earth .of the same parameter

1. INTRODUCTION

The desire of man to achieve interplanetary flight brings with it the problems which are encountered in achieving orbits about other planets under the conditions of hyperbolie approach velocities along with descent into the surrounding atmosphere. A decelerat~on system which em-ploys any form of propulsion in use or under development, whether chemical or even nuclear in nature, is extremely uneconomieal to employ. It see:ms likely, therefore, that in the foreseeable future, the benefits to be gained from atmospheric braking will make this a technique of major importance in "any of man's attempts to achieve interplanetary contact (References 4, 5, 6) or to return from orbital flights about the earth.

A great wealth of papers pertaining, in particular, to this latter subject now exists in the current literature and many sophisticated refinements have been made to the few "originals". The scope of the

analysis covered by these reports is extremely wide and varied. It includes the prediction of the flight paths which various types of reentry vehicles will travel under many reentry conditions . Extensive dynam ie and thermo - " dynamic studies have also been made of the pitching and heating of tnese vehicles during their trips through the atmosphere. In particular, the effort spent upon the dynam ie aspect of reentry flight is necessary due to the fact thai a vehic1e may be quite arbitrarily oriented, and in addition be tumbling, upon its "entrand! into the atmosphere. Consequently, it is necessary to inc1ude the moment equation in any analysis for which the aerodynamic forces are a function of the vehicle's angle of attack.

However I a division exists among the papers which has not

heretofore been justified to any degree of rigor. This division is the

separation between the static trajectory which a vehicle will follow and the dynamic pitching motions to which the vehic1e is subject during its reentry into the atmosphere. The former is specified by only the force equations while the latter also inc1udes the moment equation. This separation was questioned for several reasons. One was the fact that the fundamental nature of the pitching oscillations is described by a Bessel function of zero order rather than the ordinary trigonometrie functions. Since the charac-teristic property of th is curve is a decreasing"amplitude which is coupled with an increasing frequency, the area under each loop of the function exceeds that under the following loop. Thus. the possibility existed that the net lift acting on the vehiele might be substantially altered if oscillatory behaviour were present.

A further cor'nplication arises, however, because of the change in the density of the air which takes place as the vehic1e descends into the atmosphere. It might be expected that the perturbing effect of the pitching oscillations which are due to the properties of the Bessel function would be offset, at least in part, by the increasingly greater density of the air through whieh the vehiele is travelling.

A second reason was the nature of the perturbation Eq. 3.3.5 itself. The explanation of this factor will bedëaltwith in Section 9.1.

Consequently, this study was initiated to determine for a typical winged vehic1e the conditions under which its oscillatory behaviour would significantly affect its reentry flight path. Although the vehicle and the type of trajectory chosen may seen restrictive, nevertheless, the simplicity of the governing equations far outweighed the advantage of including additional terms since any additional complexity could affect the final answers only to a very limited extent. In addition, extremely small values of the static margin were found to be necessary before any signifi-cant deviations from the reference path were observed. Consequently, it was felt that restrictive though these choices were. the conclusions will apply to almost any other vehic1e. as weU as to other flight paths.

Initia1ly, it was thought that the static trajectory would be sensitive to the type of perturbation which was studied. Only later did it become evident. as indicated above, that this would not be the case. Thus, a fairly comprehensive study of the significant parameters involved was made. It provided sufficient evidence to prove that the osc.i1lations did not affect the flight path to any significant extent as long as the vehic1e possessed a reasonable value for the statie margin. An attempt was then made to

determ ine if other parameters were also able to produce deviations from the equilibrium path. The criteria employed in this regardmay be found in Section 7. 2. Since this investigation was quite time consurriing and only of academic interest, only a few of the major parameters were assessed for this effect.

One rather interesting result was the complete independence of the deviations with respect to the wing loading of the vehicle. Although this fact can easily'be deduced from the perturbation equations. for the flight path, nevertheless it is significant that this parameter, which is so impor-tant to the static trajectory, has absolutely no effect in producing any change in this static trajectory as a result of the oscillations of the vehicle.

Il. BACKGROUND THEORY 2. 1 Analysis

The standard three equations of motion, the two force equa-tions perpendicular and parallel to the flight path and the moment equation. are first separated into two sections. One is the set of equations describ-ing the static trajectory; the other is the set of equations describing the dynam ic pitching osciUations. Both sets of equations are then sim plified in order to produce an analytical result for both types of motion.

A typical entry vehicle is pictured in Figure 1. With refer-ence. to the notation which is defined therein, where angles are positive in the direction indicated by the arrow, the following are the governing equa-tions of m otion:

mY.y

₊

L ₊

m(

y2 _g

)cos

₁

₌

₀ (2. 2, la)

R+y

.

mV + D _mgsinr

₌

₀ (2. 2, lb)

..

I fI)

₌

_{0 (2. 2, Ic)} where: L D M

+

Cm .--=~-

«),

. Q( Y

Following closely the analysis of Tobak and Allen (Ref. 1), the following assumptions are made in order to solve Eqs. 2, 2, 1 which in their present form are nonlinear:

(i) none of the aerodynamic coefficients in Eqs. 2.2, 1 vary

with Mach Number due to the high velocities being considered, (U) ·the drag coefficient, CD, does not vary with the pitching

velocity nor the oscillatory angle of attack, which is valid if these variables remain small, and

(Ui) the division between the static trajectory and the dynamic pitching motions, the very question which this paper seeks to answer, is valid, thus allowing a separation in the variables between the parts that are changing uniformly and the parts that are oscillatory in nature; as a consequence, the result-ing oscillatory section of Eqs. 2. 2, 1 will be independent of the section which changes uniformly, providing only that the oscillatory part of the flight path angle is small compared to the uniformly varying part.

Under these assumptions, the resulting equations for the statie trajeetory are:

+

+

-1

_{cos 75} =

o

(2. 2, 2a) mV mgsin 'Ys =

o

(2. 2, 2b) =

o

(2.2,2e) where:

=

:: ::

In addition, the equation for the oseillatory angle of attaek is: where: 0(0

+

=

~(lpV~CT.

_ ) d t J m "'"-1)("5 2.3" Statie Trajeetory

+

=

o

(2.2,3)

Sinee only the existanee of deviations from the statie tra-jeetory was in question in this analysis, the aetual seareh was for the

smallest value of the statie margin whieh would be suffieient, when eoupled with an error in the entry angle of attaek (i. e. a value different from that required for trim), to eause a departure from the non-oseillatory path. Thus, although the deviations might beeome quite large, it is unlikely that

the results obtained will be sensitive to the type of static trajeçtory that is chosen. Consequently, the simplest possible path, a skip trajectory, is used. In order to further sim plify the analytical solution, four additional assumptions are made:

(iv) the uniformly varying part of the angle of attack, O(S' is

assumed to be constant, thereby eliminating the moment Eq. 2. 2, 2c,

(v) the air density can be accurately expressed as an exponential

functionof the altitude which is:

~

=

~ 0 e-PY

where ~o and (3 are chosen to obtain a best fit for the altitude range under consideration,

(vi) the difference between the centrifugal force and the

gravita-tional force is negligible when compared with the lift force;

this assumption is exact when the vehicle is travelling at satellite velocity and is still quite accurate as long as the flight velocity is reasonably high and the aerodynam ic forces are large,

(vii) with reference to Eq. 2. 2, lb, the effect of gravity in reduc-ing the 'velocity is negligible .when compared with the effect of the aerodynamic drag produced by the vehicle at the high

velocities being considered.

A paper by Eggers, Allen, and Niece (Ref. 2) gives the

following results which are easily obtained af ter introducing the above

simplifications, by straightforward integration:

y

₌

-l-ln CL(o(~)9Q S

(.!> 2~m ( cos 'Ys - cos 'rE) (2. 3, la)

-

~

[rs

+

'Ys

t

tany

~

+

tanf?tl

J]

x

=

in

/'Ys

I

tan 'YE tan

'1

_tan/-}/.

(2.3, lb)

(2.3, Ic)

'Ys - 'YE V

₌

_VEe CL(o<g)

7

CD(a<s)

2.4 Oscillatory Motion

The results for the skip trajectory were then utilized by Tobakand Allen to make a change of independent variable from time, t, to

the flight path angle, rs. A transformation suggested by Friedrich and Dore (Ref. 3) followed by the elimination of negligible terms (see Appendix C) gives the characteristic Bessel Function solution which is:

(a) for the descent phase when . rE

>

rs

>

0:

(b) for the ascen.t phase when 0

>

rs

> -

rE:

OCo(~ = (2.4, 1b)

where:

Thus the angle of attack of the vehicle can now be expressed as a continuous function of the flight pa th angle,

r. ,

which is:

s

(2. 4, 2)

2.5 Entry at Hyperbolic Velocity

In order for this investigation to be applicable to.a vehicle entering a planetary atmosphere at all velocities, it must be shown th at entry at a hyperbolic velocity, or any other speed, presents no difficulties with re gard to the size of the deviations which are produced. Consequently, it is necessary to establish that the velocity of the vehicle has a negligible effect upon d ro / d s , the oscillatory change in the flight path angle per unit distance travelled along the flight path. However, it must first be shown that V 0 is negligible at all reference velocities. This can be done

in the following manner. Equation 2.2, 1b gives:

mgsin(rs

+

"0) = 0 (2.5,1) Subtracting the non-oscillatory part and assuming that

"0«

1 gives:

The gravity term, mg '10 is small cornpared to Do; therefore, Eq. 2. 5, 2 rn ay be approxima ted by:

_ Ib

rn Therefore: where: k(s)

=

= V dVo s ds

Do

rn V_s = ~ (Vs

+

Vo ) rn _ k(s) _V s rn = 1

=2f

C D(s)S o V_s (2.5,3) (2. 5, 4) (2.5, 5) (2. 5, 6) 2 (2.5,7) (2. 5,8) (2. 5, 9)

Thus, at a given altitude, the oscillatory velocity, V 0' is approximately proportional to the reference path velo city, V s, if it is a valid assumption that the former is small compared to the latter. Section 8.5 gives the oscillatory velocity as approximately one foot per second for a circular satellite speed; thus, it will be no more than about two feet per second for a speed of 50, 000 fps. Therefore, it is avalid assumption that the oscillatory velocity is small compared to the reference path velocity for all values of the entry veiocity, VE'

It can 'now be shown that by neglecting the oscillatory velocity, V 0' relative to the reference path velocity, V s' the latter will have no effect up on d '10/ ds. Eq. 2. 2, la gives:

m(Vs

+ Vo

l (is

+

7

₀)

+

(Ls

+ Lo

)

+

m

t

(i :

:o)~

gJ

cos(rs

+

rol •

0(2. 5, 10)

Subtracting the non-oscillatory parts and neglecting V 0 gives:

+

Therefore:

dro

=

Lo ds mv:_{, . s}

2

_-, (2.5, 12)

=

_ jes) (Vs

+

Vo)2 m _{V s} 2 (2. 5, 13) ~ _ jes) m (2. 5, 14) where: j(s)

₌

_'2 1 _fC (s) S Lo (2. 5, 15)

The function j(s) is, proportional to CLoand

f>;

thus, at a given altitude j(s) is proportional to CLo which is a function of

«0.

Conse-quently, for any given height, it is necessary to show that to a good approxi-mation, 0( o(s) is independent of the velocity. .

Neglecting :;, the oscillatory angle of attack is given by:

6<0

=

Thus, it follows that:

v...Lfv dO<o_\

= dS\ ds} M 1

Cm(~) ~

f(Y) V 2 S J, I (2.5, 16) (2. 5, 17)

Further differentation of the left hand side andrearranging terms gives:

where:

But it can also be shown that:

1 d(V2) 2 ds

=

=

=

=

=

Crn.(O<o) S.J. 21 V dV ds

fT

D m

~CD(O()~

f(Y) V2 S

o

(2.5, 18) (2. 5, 19) (2. 5, 20) (2.5,21) (2.5, 22) (2.5, 23)

which gives:

= ~ (y) g(OCo) (2.5, 24)

where:

g(~) = (2.5, 25).

Inserting Eq. 2. 5, 24 into Eq. 2. 5, 18 gives:

+

g (y)

g(o(o)

d~

+

~

(y) f(O(o)olo

ds

=

o

(2.5,26)

Therefore, at a given altitude, the variation of 0(0 with

dis-tance, s, is independent of speed; hence, the variation of CLo with the dis-tance, s, is also independent of the velocity of the vehicle.

As a consequence, dro/ds will be independent of the speed. Therefore, the results of this paper will apply to vehicles entering a plane-tary atmosphere at all velocities.

IIl. PERTURBATION TECHNIQUE 3. 1 Analysis

The solution by Tabak and Allen for the pitching oscillations which were intially assumed to have a negligible effect uPQn the static tra-jectory is now utilized to pr~vide the perturbations upon the static trajectory. A test as to whether or not this effect is indeed negligible can be made by inserting a new aerodynamic lift coefficient which is now a function of the oscillatory plus the static angles of attack. If replacing the,old constant lift coefficient with its new variable counterpart does not alter the flight path when a solution is effected tothe new perturbation equations, then it can safely be assumed that the or.iginal separation in Eqs. 2,. 2, 1 between the section which changed uniformly ·and the oscillatory section is valid. 3. 2 Assumptions .

In accordance with the previous analysis, a new group of assumptions is made which is quite similar to the former set since it includes (i). (v), (vi), and. (vii). A single addition is sufficient to complete the list. It is assumed that the angle of attack of the vehicle is given as a continuous function of the flight pa~h angle, rs ' by Eq. 2.4, 2.

3.3 Perturbation Equations

The new set of equations is now formulated below. Starting with Eq. 2. 4, la and applying assumptions (v) and (vi), we have:

~

fo e-flY y2 SCL~(ru

Rearranging the terms gives: .PoS e-/fYdy 2m Integration starting at r where _Y gives: .PoS _e-py 2~m or Y

=

=

=

=

=

=

=

=

- mYr _ m y2 d

r

d s m y2 sinr

~r

.

Y

00

sb

S· 2f3m

f;~ig(f)J

rE

(3.3, 1) (3.3,2) (3.3,3) (3.3,4) (3.3,5) (3.3,6)

The corresponding equation for the

x

coordinate is derived in a similar fashion. Following the same procedure and, in addition, sub-stituting Eq. 3.3, 5 in the relation for the density gives:

=

Rearranging the terms gives: dx

₌

. 2 dr - m Y cos

r - -

_dx cos r d r (3.3, 7) (3.3,8) (3.3,9)

Assuming C L

~T(r)J

constant between rE and ri and starting the integration at

where: x

=

=

the va lue of the

x x· ₁

=

r..

₁ cos r dr -t-an-\-'E

l~

:::

t

+

tant)

t

- tan _1_ 2

+

(3.3,10)

The final equation which gives the velocity is found by applying assumption (vii) to Eq. 2.2, 16 giving:

=

-mV

Dividing by Eq. 3.3, 1 and inverting gives: cL[O<T(r)] = V

-{f

C D[ o(T(r)]

dd~

= V...QL dt Re'arranging the terms gives:

dV ₌ CD [O(T(r)]

-V

_cL[~(r~

Integra tion starting at

r = _rE where V = V E gives: V = V E

ex{[

CD

lOT

Cr>]

d

r)

CL

~T(t)]

'rE (3. 3, 11) (3. 3, 12) (3.3, 13) dr (3.3,14) (3. 3, 15)

"

(12)

Equation 3.3. 16. Eq. 3.3. 10 and Eq. 3. 3. 15 give the path of the vehicle and its velocity as a function of the flight path ängle. ?'. as it executes a skip trajectory into and out of the atmosphere. The notation of these equations can be considerably compressed to:

y

₌

1 1 MI (3. 3, 16a)

7fn~

1 _{+ (3.3. l6b)} x

₌

7

12 x. 1 V

₌

V e 13 (3.3. l6c)· E where:

IV. DEPENDENCE OF LIFT UPON ANGLE OF ATTACK 4. 1 The Choice of the Lift Function

The functional relationship between the lift coefficient. CL. and the angle of attack, 0(, has purposely been left unspecified up to this point in order to allow for complete freedom of choice in the selection of a mathematical model to relate the lift coefficient to the angle of attack of the vehicle. In order to be completely assured that the dynamic oscillations are truly unable to significantly affect the flight path. two mathematical models are selectedwhich are completely different. However. due to the limited range of values allowed for the oscillatory angle of attack. the resultant values for the lift coefficient are almost identical for both m odeis.

4. 2 The Models Chosen for CL (0( )

The two mathematical models chosen are the linearized theory in which the lift coefficient is given by:

(4. 2. 1) and the Newtonian Flow Theory in which the lift coefficient is given by:

(4.2. 2)'

In addition, it should be stated at this point that for the former case the drag coefficient, CD, is assumed a constant equal to the static or equilibrium drag coefficient, CDs' while ,for the latter case the drag coefficient is given by:

(4. 3, 3)

4. 3 Correlation Between the Two Lift Functions

Since it is desirable to produce two sets of trajectory calcu-lations which could be compared directly, it is necessary for each calculation in one set that is to be compared to a corresponding calculation in the other set that the CLs be the sam e. It is also necessary that the initial values of CL produced by the sum of the static and the oscillatory angles of attack be equal. Two parameters are available, Ci.( and 0(. which can be varied at will to produce the two above mentioned conditions. However, since yet another parameter, Cma( , is directly proportional to the value of CLO( , the best course is to use equal values of CL., in the corresponding calcu-lation in each set while the angle of attack in the first model is varied so as to produce the desired value of the lift coefficient as outlined above. Consequently, for a given value of o(s in the second mathematical model,

the Newtonian Flow Theory, a value of CLoc scan be found from Eq. 4.2, 2 which is then used in the first model, whereupon the corresponding values of cts and o(o(

r

)i are found from Eq. 4.2, 1 which produce the necessary.

values for CLs and CL [o(T(r~ i. V . DETAILS OF THE CALCULATION 5. 1 Overall Method

The actual calculations were performed on an IBM 650 digital computer. A step by step method of evaluation was used to determine a sufficient number of points on the trajectory so as to retain the desired

accuracy. However, an actual record was taken only for a very smal! number of positions on the flight path since such a large volume of output was not

only unnecessary to determine the path taken by the vehicle, but was, in addition; far too burdensome to handle.

Major alterations were also made to the form of Eq. 2.3, la and Eq. 2. 3, 1b since they were quite unsuitable for machine ca1culation. Using standard trigonometric identities, these twoequations were trans-formed int 0:

1 . CL(<<S)

fo

S

7

In

(1,

+

11. I)

(1,

-

11. I

4

P

m sin E 2 s sin E 2

S)

-

~

[" l;

+

~

I

(Ijl

tan

r~

> X

In [sm

'E;

Irs

l )

cos~

rE -2

I

si)

+

~

sin

(,.~

+

\rsl

>J1

[Sint

lj;: ;

11'51) COS

~

lj;: ;

l1'slj

+!

sin (1'E - j1'sP

J

S

y = (5. 1, la)

x

₌

(5.1, 1b)

These equations were in a form which could be evaluated directly on the computer.

5.2 Starting Point

Since all the equations are dependent up on one variabie. the flight path angle. r. the most convenient point to start the calculation was found to be at a flight path angle which was an extremely small and fixed amount less than the theoretical entrance value, l' E , upon entrance into the atmosphere. This system was adopted due to the fact that the vehicle's initial angle of attack. and not its initial flight path angle, produced the largest variations in the initial altitude of the vehicle. As a consequence, the vehicle could be confined initially to a region between about 400. 000 feet and 600, 000 feet. Due to the negligibly small values of dynamic pressure at this extreme height, the initia 1 part of the trajectory was essentially a straight Hne; consequently, it was assured that all of the significant lift perturbation due to the oscillatory angle of attack would be present to disturb the path of the vehicle.

5. 3 The Bessel Function and the Three Integrals

The evaluation of the integrals involved the numerical evaluation of the Bessel functions. contained in the expres sion for the

oscillatory angle of attack. A standard program available at the University of Toronto Computing Centre was used to ca1culate values of Jo and Y o for arguments up ta' a value of seven. For values greater than seven, an asymptotic expansion was employed that was developed from. Watson

(Ref. 8). The details of the calculation and the accuracy obtained are con-tained in Appendix B.

The evaluation of the Bessel Functions was the major com-putational problem. Once this step was carried out, the int~grals were obtained by employing a simple trapezoidal rule. Other methods of num-erical integratiQn w:ere found impractical due to the fact th at the interval of integration was constantly changing at indeterminate values of ,.. Con-sequently, it was felt that any additional complication which would have been caused by the use of other numerical integration formulae would not have been worthwhile.

VI. NUMERICAL DATA USED 6. 1 The Atmosphere

As stated in previous sections, the density relation for the atmosphere is assumed to be:

=

(6.1, 1) where: fo = 0.003684 slugs/cu. ft. 1

=

-22,000 ft.

This assum ption is excellent over the altitude range under consideration as can be seen in Fig. 2 which compares the density of the ARDe 1959 Model Atmosphere (Ref. 7) with the density given by the above relation.

It should be noted tha t the actual value of ~o is quite im-material since it appears only in Eq. (3.3. 16a) incombination with the wing loading parameter,

mis,

which was found to have no effect upon the size of the deviations that were produced by the pitching oscillations of the

vehicle.

.-On the other hand, this same equation shows the deviations to be inversely proportional to the value of (i since the subtraction of Eq. 2. 3, la from Eq. 3 .. 3, 16a gives the deviations at

r

=

0

as:

Ay = Ymin with - Ymin without oscillations oscillations

(6. I, 2)

l ' 1 cos?E

=

(Fln

f,0

(6. 1,3)

CL{Ofs )

rECLs[~(r~

d r

However, the density versus altitude relation givenby Fig. 2 shows that in the region from about 380, 000 ft. down toabout 230, 000 ft .• the actual value.of {3 is greater than the one used in this paper. Above 380,000 ft. , the aerodynamic lift force on the vehicle is too small to affect its path. Thus, the only problem is the region down to about 140, 000 ft. since the actual value of f3 again becomes larger below this altitude. and.for a vehicle with a minimum altitude above 230, 000 ft., the deviation should always be less than that predicted by Eq. 6. 1,3. However. even for vehicles which deseend into the questionable region, the actual deviations

will never be greatly different from those which have been calculated since the actual value of (j in this vicinity is never less than seventy-five percent of the assumed one. In addition, the higher values of ($ above 230, 000 ft .

. will, to some extentt counteract the lower values below this level.

Conse-quently, it is quite iIllpossible for lar~e errors to exist in the calculated deviations as a result of the approximation made in this report to the true density versus altitude curve given in Fig. 2.

Entry into other planetary atmospheres requires onlya change in the value of (! since it has already been shown that the value of

~ 0 has no effect upon.the size of the deviations which are produced in this study.

Tabie llliis.tsthe values of

13

-1 for a number of planetary atmospheres (Réf. 4). It shows that the Iargest value of this parameter is the one for the atmos'phere of Titan; however, it is only 4.5 times as large as the value for the earth's atmosphere. All the other values of (J -1 are less than three times as large as the value for earth.

It has already been shown in.Eq. 6. 1,3 that the deviations are inversely proportional to the value of

f3.

It will also he shown that as long

as

the vehicle possesses a reasonahle static margin, there. will he a sufficient number of pitching oscillations to keep the deviations helow a

negligible value. Consequently, deviations from the reference path for entries into the planetarJ) atmospheres listed in Table II should he approximately in the ratio of

(3

e:

1, i. e. they will alBO be negligible.

6. 2 The Vehicle

The parameters representative of the type of vehicle chosen, a sort of delt~ winged glider, are not at all critical. .This fact can easlly be seen in the parameters Kl and K2 as given in Eqs. 2.4, 1. The primary parameter in Kl is essentially the reciprocal of the angle of attack, CLJ Cu

while the primary parameter in K2 is the static stability, Cm • . Thus, K 1 can v.ary only by small amounts while K2' due to changes in Cm _{. ~} of many orders of magnitude, ranges from values as high as around one thou-sand to as lowas ten. Therefore, the main parameter in this study will

be the static stability, Cm.c , o r the static margin, .K_{n . The following, then,}

are the values which were taken as representative

tor

the parameters of the.

vehicle which was em ployed.

(i)

(H)

The geometric parameters:

the length of the vehicle was taken as

.I

= 50 ft. ,

the ratio of the length of thf vehicle to the radius of gyration in pitch was taken as

{..I/cd

= 22, and

'

.

.

(Hi) _{the wing loading was varied from} S

W

= 5 psf. to T

W

= 100 psf. The lift and dragcoefficients have already been dealt with in

. Section IV.

The stability parameters:

(i)

the dam ping coefficients in pitch were taken as Cm _q

+

Cm

Ó(

-0.015, and

·(ii) the statie margin was varied from Kn = 10- 1 to Kn= 10- 5 .

6.3 Initial Conditions

Certain restrictions dictated the choke of the initial condi-tions . While many of these requirements appear self -evident, a brief resum'ee will be helpful.

=

(i) Conformity with human endurance limits dictated that the entry flight path angle for the vehicle be sufficiently small so as not to im pos e too great a load factor up on the occupants. As a consequence, the values of the entrance flight path angles which were chosen lay between rE = 0.573 degrees and rE = 5. 157 degrees.

(ii) The range of values chosen for the equilibrium or static angle of attack was chosen so as to provide a reasonably representative range of values for the lift coefficient. They lay between Q(s = 11. 5 degrees and Cis = 45.8 degrees. (iii) The accuracy with which the static angle of attack can be

set will play the major role in determining the initial oscillatory angle of attack. While present technology is accurate to at least one degree, nevertheless, it seemed desirable to extend the study to include values from

(iv)

C:(O ( r)i = -14. 3 degrees to

+

14. 3 degrees.

While a non-zero value for the angular veloclty of the

vehicle is a distinct possibility as it enters the atmosphere, an initial oscillatory angle of attack will provide exactly the same type of lift perturbation to the vehicle. Thus, in order to simplify the analysis, the angular v.elocity of the vehicle as it entered the atmosphere was assumed to be negligibly small and was set equal to zero. A few sample studies for vehicles spinnirig about their pitch axis at very low angular velocities confirmed that an initial oscillatory angle of attack and

turn bling were interchangeable in their effect upon the. vehicle.

'.

(v) Assumption (vi) in Sec. 2.3 requires that the vehicle's velocity be close to circular satellite speed at all times; accordingly, the velocity at entrance to the atmosphere was set at VE = 26, 000 fps. which .will also be, since the drag is negligible up to the· initial position, the veloc ity at that point. The principal reason for the choiee of this particular figure

is that 26, 000 fps is one percent greater than circular satellite speed at an altitude of 325, 000 ft.

VII EVALUATION CRITERIA 7. 1 Criterion for Deviation

The criterion which was used in evaluating the extent to which the lift perturbation produced by the oscillatory angle of attack affected the flight path was the difference between the lowest points to which the vehicle would deseend while first Under the influence of the total angle of attack,

oC.T(r), as given by Eq. 2.4, 2 and second while at the equilibrium or statie angle of attack, O{s.

The reason for this choice was twofold. In the first place, the only other choice available was one which was connected with the x rather than the y coordinate; however, as indicated in Appendix A, the accuracy of this coordinate was very P90r unless an exorbitant amount of time was spent for a calculation. Consequently, it would be almost

impossible to arrive at an accurate evaluation of the effect of ~he lift per-turbations and it would even be possible, due to the inaccuracies involved, that a difference could be found where none really existed. In the second place, a typieal reentry trajeetory is not the type of manoeuvre which was used in this paper for the purpose of studying the effect of the oscillatory angle of attack up on the flight path. In most cases, the vehicle will not skip out of the earth's atmosphere. This result might be accomplished by programming the angle of attack in a manner which prevents the vehicle from ascending and at the same timeproduces the proper reentry conditions for the vehicle. For example, one condition might be to keep the skin tempera-ture below a specified value. This condition, as weil as many others whieh could also be specified, will depend upon the minimum altitude to whieh the vehicle will deseend under a given angle of attack. Thus it will be extremely important to ascertain the effect which ;an;y initial misalignment from the specified trim angle of attack will have upon this minimum altitude.

7. 2 Criterion Used in Determining the Effeetiveness of Parameters Other Than the Statie Margin

The results shown in Figure 7a indicate that a value of the statie margin exists for whieh there will be no deviation from the statie trajectory even though an initial error angle of attack exists. Extremely large deviations will be produced if the statie margin is smaller while only negligible or at the very most small deviations will result if the statie

margin is larger. Consequently, this value of the static margin was employed to test the effectiveness of other parameters in controlling the size of the deviations which were produced. The decrease or increase in

the static margin which was necessary to maintain a zero deviation from the static trajectory also gives a direct measure of the respective destabilizing or stabilizing influence due to a change in these other parameters. Notice that this change in the static margin in response to a stabilizing effect is the opposite of that which is expected; normally, the statie margin would be allowed to decrease if the vehicle became more stabie. The reason behind this effect is explained in Section IX .

VIII. RESULTS

8. 1 Deviations for aReasonabie Static Margin

It was firmly established that for va lues of the static margin of the order of ten per cent, the deviations in the minimum altitude reached due to the oscillatory behaviour of the vehicle were of little or no conse-quence. Fig. 3 gives a detailed account of the deviations under conditions for which the initial error angle of attack was 8.594 degrees.

Evidence that an initia 1 angular pitching velocity is equivalent to an initial error angle of attack is given in Fig. 4. In order to provide a direct comparison between these two initial conditions, a value for the initial angular velocity was used which would produce a maximum oscilla-tory angle of attack equal to the corresponding initial error angle.

Having firmly established that the deviations were insignifi-cant for error angles even as large as ten degrees, the question still

re-mained as to how the deviations per degree of error angle varied with respect to the initial oscillatory angle of attack. Fig. 5 gives the details of an

investigation intp this question.

A further study of the effects of a variabie initial angular pitching velocity was made; these results are given in Fig. 6.

8.2 The Influence. of a Decreasing Static Margin

As indicated in the introduction, a study was made to deter-mine the effect of extremely small values of the static QlB,rkin upon the

deviations produced due to the oscillatory angle of attack. Fig. 7 shows the values which are found at astatic angle of attack, O{s' of 28.648 degrees. 8.3 The Influence of Other Parameters at Very Smalt Statie Margins

An investigation was also made into the stabilizing or destabi-lizing influence of three major parameters; those chosen were the flight path angle at entrance to the atmosphere, 'YE' the static angle of attack, O(S' and the initial error angle of attack, ~('Y)i' The results appear in Figs. 8, 9 and 10.

8. 4 The Influence of the Wing Loading

It was impossible to devise separate figures to present these results since the influence of the wing loading of the vehicle upon the value

of the deviations produced was absolutely nil. Accordingly, all figures are universal curves as far as wing loading is concerned.

8.5 Effect of the Pitching Oscillations Upon the Velocity

The velocity of a vehicle subj.ect to pitching oscillations was compared to the velocity of a vehicle in which the angle of attack was held equal to its equilibrium or trim value. The position at which the comparison wasmade.was at the minimum altitude of the perturbed and static trajec-tories, respectively.

Most cases studied showed differences of less than one foot per second although a very few instances wererecorded in which the

difference lay between 1. 0 and 1. 5 fps. Since the differences in velocit;y were so small, no further discussion is necessary.

IX. DISCUSSION OF RESULTS 9. 1 Major Characteristics

Two major characteristics were evident.

(i) Figures 3 and 5 show that the deviations are extremely small. Measurements using a unit of length based on the length of the vehicle would in most cases yield a value less than unity. while only a very few of the deviations would amount to values even as high as ten. It is obvious, therefore, that the size of the deviations which can be expected.is negligible for all practical purposes .

(ii) While the deviations are in themselves quite small, never-theless, it is extremely interesting to note that for reasonable values of the static margin, all of the deviations produced are negative; that is to say, the vehicle wil! descend into the

atmosphere to a lower height due to the presence of an initial error angle of attack which is either positive or negative. Furthermore, the size of the deviations produced is very nearly equal in both cases.

The reason that the deviations are always negative lies with the governing Eq. 3.3, 16a. Setting

o(T('Y) = O(S

in 11 without and giving o(,T(r) _{in 11 OSwcil,tlhlatl'ons by Eq,} 3.4,2 gives oscillations

the deviations at

o

as:

n.y

:::

where: CL(O(s)

2 sin2 [OI's

+

Ofocril cos

~S

+

c(O('r~

2 Sin2«S

eos~

Linearizing the above expression by setting

=

sine

O(o('r~

=

~('r)

(9.1, 1)

1

and negleeting the seeond order terms of o(o( r) gives the lift perturba-tion, when the initial angular pitehing velocity is zero, as:

C e K l('E - r) J (K

J'Y. -

r)

1 0 2 E (9. 1, 2)

where:

=

=

CLo<~s) _{C L (o(s) C(O(r)i} .

If.ACL were simply a trigonometrie function, then integration of Eq. (9.1, 1) would give:

L:::,.y (9. 1,3)

However, sinee this expression negleets entirely.the deeay factor present in the Bessel function as weU as the preeeeding exponential term: this estimate of the deviations wlll be far too large: But, Eq. 9.1, 3 does show that the deviations should be negative whenever the angle of attack is charaeterized by oscillations about some mean value.

There is yet another reason as to why the deviations are always negative as long as the vehiele possesses a reasonable value for its statie margin. This elue concerns the change in the minimum altitude which oceurs as a result of errors in the trim or statie angle of attaek. It is qui~e

easily shown that for positive and negative errors in the trim lift eoeffieiertt which are equal, the rise in the minimum altitude due to the positivè error will always be less than the drop in the minimumaltitude due to the negative error. Consequently, whenever any pitehing oscillations are present, the

(22)

negative part of the oscillations will tend to overpower the positive parts of the cycles and produce a fall in the minimum altitude to which the vehicle

. will descend.

9. 2 Other Characteristies

The purpose of this paper was to ascertain the effect of various parameters upon the deviations produced as a result of an initial oscillatory angle of attaek. Consequently, an evaluation of the seeondary parameters .. füllows:

(i) While the number of investigations into the dependence of the deviations upon the wing loading of the vehicle was small, it was found, and quite emphatically so (down to the last

hundredth of a foot), tha t there is no eonnection between the wing loading of the vehicle and the deviations which result. This result is, of'course, in agreem ent with Eq. 9. 1, 1. Any other result would have cast doubts upon the validity of the calculations .

(ii) All of Figures 3 to 10 confirm that while there is some depend-ence between the initial flight path angle and the deviations produced, this dependence is slight.

(iii) On the other hand, Figure 5 shows a marked dependenee between the value of the initial error angle of attaek and the deviations produced. While it is natural to expect a larger deviation as a result of a greater initial oscillatory angle of attack, these curves demonstrate a definite amount of non-linearity.

(iv) Perhaps the greatest change which occurs in the deviations, aside from a decreasing statie margin, is that due to a decreasing equilibrium or static angle of attack. Figure 3 elearly shows that for a given initial error angle of attack, there is a close relationship between the. statie angle of attack and the deviations whieh are produeed.

(v) Figures 4 and 6 shows the deviations whieh are produeed by initial angular pitching veloeities in the vehicle. Table I

gives a complete resum~e of the values obtained along with the aetual angular velocities which were used.

The peculiar shape of Figure 4b at small statie angles of attack is due to the great reduction (by a factor of 3.2) in the initial angular velocity of the vehicle whieh was necessary to achieve the speeified m~nimum

Figure 6 is identical in all respects to Figure 5 except tha t the deviations are almost exactly twice as large in every case. : The qualitative aspect of this result is in agreement with the expectation that for equal oscillatory angles of attack at different altitude, the one which is operating within a higher density atmosphere will produce the greater deviation. Since both vehicles "start" at the same altitude, the one with an initial angular pitching velocity will reach its maximum oscillatory angle of attack which is equal to the initia I error angle of attack of the other vehicle, at a lower altitude at which the density of the air will, of course, be greater.

9. 3 'A Comparison of the Two Lift Models

The correlation between the deviations produced by the two lift models is difficult to ascertain. , While they agr.ee quite well at static angles of attack of about thirty degrees, there is a marked difference for other values. Moreover, due to the relationship between the two lift coefficients as outUned in Section IV, it is probably impossible for ratios greater than ten to exist between the two results. However, due to the

existence of factors of two or three between the deviations produced at static angles of attack of about tèn degrees, it can be concluded that the model which specifies the functional relationship between the lift coefficient and the vehicle' s angle of attack can greatly influence the actual value of the deviations produced.

9.4 Dependence Upon the Static Margin

It is rather noteworthy that the deviations are always negative for reasonable values of the static margin. The explanation for this result has already been given; it needs no further elaboration.

It is not until impractically small values of the order of 10-4 for the static margin are reached that this behaviour is reversed due to the fact that the motion is no longer oscillatory in nature. Indeed, for this range of values for the static margin, the angle of attack of the vehicle changes only extremely slowly; so slow in fact is the change that the oscillatory angle of attack has not the time to complete even half a cycle. This result is in agree,ment with the type of motion which,is expected of any vehicle which possesses such an inherently small degree of static stability.

Included in Fig. 7 are the asymptotes which the values of the deviations will reach when the static margin approaches zero. It can easily be seen that an approach to these lines does not begin until after the static margin falls below a value equal to about 10- 3 . . Since the theory of Tobak and Allen is still correct to at least ten percent for static margins of 10- 4 , and furthermore, since it is at the very least qualitatively correct for values of 10- 5, the curves shown in Figure 7 indicatethat the deviations produced by an initial error angle of attack are not at all serious

for

stat~2

margins greater than 10- 3, and the deviations for values greater than 10 are negligibly smalL

9.5 Influence of Other Parameters at Very SmaU Statie Margins

Figures 8, 9 and 10 may weU be misleading at first sight since, as mentioned in Section VIII, a stabilizing effect such as an increase in the entrance flight path angle,

'E

'

likewise requires an increase

in the statie margin, Kn' to maintain a zero deviation. This difficulty is easily mastered with the aid of Figure 7a. Reference to this curve together with Fig. 3 shows that the greater the value of rE' the smaUer the

absolute value of the deviation. Consequently, the way in which to interpret Fig. 8 is that it gives the vertical shift in the curves of Fig. 7a which are required in order to achieve a crossover point at the correct value of Kn after a change in the value of

'E

.

Compatibility of Fig. 8 with the informa-tion contained in Fig. 3 is thereby achieved. It should be noted, however, that a certain amount of distortion in the curves of Fig. 7a will take place during the vertical shift since the portions of the curves below the Kn axis must always remain there reguardless of the extent of the shift.

In addition, it can be seen that for reasonable values of the statie margin the normal correlation is found between the increase or decrease in Kn which is required to maintain an equal (but now small) deviation as a result of a change in rE' Thus, an increase in

'E

which is a stabilizing effect according to Fig. 3 will, if an equal deviation is to be maintained, call for a decrease in Kn' which is destabilizing. This result is in agreement with the positive slope which exists for values of Kn down to about 10-

3

in the curves of Fig. 7a.

Exactly the same interpretation should be applied to Figs. 9 and 10 as was given for Fig. 8. However, in Fig. 9 and especially in Fig. 10

~ much greater change in Kn is seen due to the changes.in ~(r) and ~,

tespectively. As a consequence, there wiU be even greater distortion during the vertical shift of the curves in Fig. 7a due to changes in these two parameters. But even here, the maximum value of the statie margin is significantly less than 10-3, indicating that a large safety façtor is present before the critical value of Kn is reached at which point the peviations are zero and below which the deviations start to increase rapidIy.

x

.

CONCLUSIONS

10. 1 Major Characteristics

As a result of the foregoing analysis, the following conc lu

-sions were reached concerning the values of the deviations which were pro-duced by an initial error angie of attack for the winged vehicie possessing the characteristics outlined in Section VI.

(i) Of prime importance is the fact that the deviations produced in the minimum altitude reached by the vehic Ie during entry

~nto a planetary atmosphere will always be negligible when the vehicle possesses areasonabie amount of static stability. This result confirms the conclusions reached by both Etkin (Ref. 9) and Rangi (Ref. 10) in which it was found that there is no coupling-;':between the static trajectory or the phugoid and the short period mode below an altitude of 300, 000 feet.

It should be noted, however, that in these two investigations, it was the phugoid which drove the short period mode, while in the present investigation, it is the short period mode which drives the phugoid.

(ii) Of secondary interest, due to the fact that the deviations are negligible, isthe result that these deviations will always be negative. Thus, the vehicle will always descend to a lower minimum altitude whenever an oscillatory angle of attack is introduced.

(iii) In addition, the functional relationship between the lift coefficient and the vehicle's angle of attack will influence the value of the deviations, although the order of magnitude of these deviations will remain unchanged.

(iv) Finally, the total effect of the oscillatory angle of attack upon the velocity of the vehicle during its descent is quite insignificant compared to its average reference path velo city . 10.2 Academic Results

A num ber of other characteristics were also found which are of only ac;ademic interest due to the fact that the deviations are, in any case, ne gligible .

(i) The most important is the result that the only parameter which aas a significant effect upon the magnitude of the deviations produced is the static margin of the vehicle; for values greater than 10- 3, the deviations will always be quite small and negative while for values less than 10- 5 the deviations will rapidly approach an asym ptotic value which is governed by the minimum altitude which is reached by the vehicle due to a static angle of attack equal to the original static angle of attack plus the initial error angle of attack ..

(ii) The only other variabie which is able to produce even a second order effect in the values of the deviations which are obtained is the static angle of attack.

(Hi) It was also found that a very small initial angular pitching velocity is equivalent to an initial error angle of attack in its ability to produce deviations from the statie trajectory. (iv) In addition, the dev:Î!ation per degree of the initial error

angle of attack increases moderately with inereases in the initial error angle of attack.

(v) The dependence of the V'alue of the deviations up on the initia 1 flight path angle is very limited.

(vi) Finally, the wing loading of the vehicle has no effect upon the values of the deviations produced by an initial oscillatory angle of attack.

1. Allen, H. J. Tobak, M. 2. Allen, H. J. Eggers. A. J. Neice, S. E. 3. Friedrich,

H.

R. 4. Chapman, D. R. 5. Wong, T. J. Slye, R. E. 6. Luidens, R. W. 7. Minzer, R. A. Champion, K. S. W. Pond, H. L. 8. Watson, G. N. 9. Etkin, B. 10. Rangi, R. S. REFERENCES

Dynamic Stability of Vehicles Traversing

. Ascending or Descending Paths Through the Atmosphere, NACA TN 4275, 1958.

A Comparative Analysis of the Performance of Long Range Hypervelocity Vehic1es,

NACA TN 4046, 1957.

The Dynamic Motion of a Missile Descending Through the Atmosphere, Journal Aero-Sci., vol. 22, no. 9, Sept., 1955, pp. 628-632,638. An Analysis of the Corridor and Guidanc e Requir.ements for Supercircular Entry into Planetary Atmospheres, NASA TR R-55, 1960. The Effect of Lift on Entry Corridor Depth and Guidance Requirements for the Return Lunar Flight, NASA TR R-80, 1960.

Approximate Analysis of Atmospheric Entry Corridors and Angles, NASA TN D-590, 1961. ARDC Model Atmosphere, Air Force Surveys in Geophysics, No. 115, AFCRC-TR-59-267,

1959.

A Treatise on the Theory of Bessel Functions, Second ed., Cambridge Univ. Press, 1944. Longitudinal Dynamics of a Lifting Vehicle in Orbital Flight, lAS Paper No. 60-82, lAS National Summer Meeting, Los Angeles. Calf., June, 1960.

Non-Linear Effects in the Longitudinal Dynamics of a Lifting Vehicle in Orbital Flight, University of Toronto, UTlA TN 40, Oct., 1960 . .

APPENDIX A Check on Accuracy

In order tornaintain as accurate a calculation as was rea-sonably possible without rather time consuming apalysis, a parallel calcu-lation was made for the static trajectory by exactly the same rnethod as that for the disturbed path. This parallel calculation was then compared . with the values obtained from Eq .. 2. 3, la and Eq. 2.3, lb.

In virtually all the cases considered, the y coordinate was exactly the same giving, with eight figures, a possible accuracy of one-tenth of an inch. Certainly therefore, the result could safety be assumed accurate to at least the order of a foot or less.

The accuracy of the x coordinate left much more to be desired during calculations employing a rather coarse interval which was, however, sufficient to obtain eight figure accuracy in the y coordinate. However, sufficient refinement of the interval in the integral 12 for the

x coordinate would produce as many figures of accuracy as was deern ed desirable. Since an inordinate length of time would have been required to produce even five figure accuracy, only an approximate value for the x coordinate, accurate to be only about one per cent, was found which could then be corrected to three figures on the basis of the error in the static trajectory which used the same rnethod of calculation.

The accuracy of the speed of the vehicle was also checked in this manner and here the result was again extremely favourable. Eight figure accuracy was achieved at all times for the parallel calculation giving, with much room to spare, an error in the speed of less than one foot per second.

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APPENDIX B

Bessel Function Calculation and Accuracy

For values of the argument which were less than seven, the following series expansions were used for J o( x ) and Y o ( x):

(B, 1)

rr

Yü(x) (B,2) where: +~. (-1)

(jU

J...

+

_1

+M'-1.

00 n+l 2nG

j

bo

(nl)2 \2) 1 2 n :: 0.57721567 is Euler' s Constant.

Whenever any term of an infinite series became less than 10-8 times the sumLOf the series, the subsequent terms were neglected.

Using the series expansions, the errors for large ar.guments were as follows:

Argum.ent

=

5 Error

=

+

O. 0000004 Argument = 6 Error =

!

O~ 0000008

Whenever the argument of Jo(x) and Yo(x ) exceeded seven, the following asymptotic expansions given by Watson (Ref. 8) were used:

1

Jo(X) =

(7T

2xt{!tCOS(X -

in)

+

Ssin(x

-in)]

(B,4) 1

Yo(X)

=

(7f

2x) 2[R sin (x -

i7T)

.:.

S cos(x

-i

7f)1

(B,5)

where: _{12.32 12.3}2_.52 _{. .(} R = 1

-2! (8xf

+

4! (8x)4 12 12.~.52 222",92 S =

+

1 .3 .5. . 3! (8xr I! (8x) 5! (8xf

The series expressions for Rand S were truncated after· five terms in. each since the error involved in their evaluation was always less than 10-8 at that point.

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The errors for arguments in this range were always less

+

APPENDIX C

Applicability of the Tabak and·Allen Approximations

The sim plification made by Tabak and Allen in order to obtain a solution for the oscillatory angle of attack was applied to the differential equation obtained for the motion af ter the tran,sformàtion suggested by Friedrich and Dore was utilized. The actual equation which resulted thereby was:

where:

M(r)

and: P Q M (r) 0(" (r)

+

M(r)OC(r)

=

o

=

~

_ _ _

l _ _

~fsin

r(CLv.

+

p)

+

i

cos

r

(cos

r -

cos

rE)

L-

\

CL

+

=

1 - cos r cos '>E 4 (cos

r -

cos

r

E)2

C~

+

+

This expression was then simplified to:

=

Q

+

1 - cos

r

cos

?E

(cos

r

-

cos

rE)

4 (cos

r -

cos

r

E

)2

+

(C, 1)

(C,2)

.due to the fact that for all reasonabie values of the static margin, the third term is always very much smaller than the first term. The second term

was kept due to the fact that for values of

r

near

rE'

it becomes extremely large. In addition, the last part, Q , of the first term is al ways very

much larger than the rest of the expressions within this bracket; conse-quently, only the value of Q is important in the numerator of the first

. term. It must now be shown that the above approximation to the original

expression is accurate for values of Kn down to 10- 3, that it is reasonabie for values of Kn down to 10-4 , and that it is at the very least in error by no more than about ten percent for values of Kn down to 10- 5 The last

(32)

stipulation applies to an assumption in Section IX in which the results for the oscillatory angle of attack were taken to be at Ieast.qualitatively if not quantatively correct.

The following represent ative values will, therefore, be used:

rE

= 1. 179 degrees O(S = 28. 648 degrees

Cm

+

_Cm

= - 0.015 q 4(

(11

= 22.

.i

= 50. feet

The following values can then be derived:

L

CL

D

= CD =1.8305

CL

= 0.4034 1

CL

= 1. 2565 ti(

Cm

₀₍ = -1..2565 Kn

Using the above values, it can then be shown that:

1 0.95534 X

M(r)

= _cos

_r

[1.4215 sin

r +

0.25 cos

r

+

30,150 Kn} 1 - 0.95534 cos

r

+

4 (cos

r -

0.95534)2 (C,3)

Then, in order for equation (C, 2) to be a valid approximation of equation (C, 1), the following criteria must be fullfilled:

(i) The percent variation within the numerator of the first term of equation (C, 1) from

r

=

rE

to

r

=

°

must be negligible.