Digital Computers 00 Jay n

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A DIGITAL COMPUTER STUDY OF THEFIRST-STAGE TRAJECTORIES OF H\BHINITIAL ACCELERATION ROCKETSGORDON ROWLAND JAYNEandJOSEPH BARBOUR WILKINSON, JR.


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LIBRARYU.S. NAVAL POSTGRADUATE SCHOOL

MONTEREY, CALIFORNIA


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A DIGITAL COMPUTER STUDY OF THE FIRST-STAGETRAJECTORIES OF HIGH INITIAL ACCELERATION ROCKETS

by

Lieutenant Gordon Howland Jayne, U.S. NavyB.S., U.S. Naval Academy, 1952

B.S., Aero. Eng., U.S. Naval Postgraduate School, i960and

Lieutenant Joseph Barbour Wilkinson, Jr., U.S. NavyB.S., U.S. Naval Academy, 1952B.S., Aero. Eng., U.S. Naval Postgraduate School, i960

Submitted in Partial Fulfillmentof the Requirements for theDegree of Master of Science

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGYJune 1961


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MV*-"

*


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T.ihrnryU. S. Naval Postgraduate SchoolMonterey, Ciilitocnia ii

A DIGITAL COMPUTER STUDY OF THE FIRST- STAGETRAJECTORIES OF HIGH INITIAL ACCELERATION ROCKETS

by

Gordon H. Jayneand

Joseph B. Wilkinson, Jr.

Submitted to the Department of Aeronautics and Astronautics onMay 20, 1961 in partial fulfillment of the requirements for the degree ofMaster of Science.

ABSTRACTThe first-stage, powered- flight trajectory of a large rocket pow-

ered vehicle is studied by varying the initial acceleration, the verticalflight time, and the initial tilt angle. Trajectories were computed on anIBM 65O digital computer. Specific areas of interest with respect to highinitial acceleration rockets are the feasibility of using the "gravity turn"maneuver to obtain low burnout flight path angles, and the determination ofmaximum energy trajectories for various values of initial acceleration.

Results indicate that a relatively low initial tilt angle followedby a "gravity turn" maneuver is not adequate to achieve low burnout flightpath angles for high initial acceleration vehicles. For values of initialacceleration of about 2.5 to 3'0 a large percentage of burning time is spentin the programmed tilting phase, which results in lift load factors of theorder of .8 to 1.2.

Maximum energy trajectories occur at specific values of burnoutflight path angle for the initial accelerations considered. These burnoutangles start at about fifty-five degrees for an initial acceleration of 3.0and decrease to approximately zero degrees for an initial acceleration of 1.5

Burnout conditions of velocity, altitude, energy, and flight pathangle are plotted for the trajectories computed. The trajectories most closelyapproximating the maximum energy cases are included in tabular form.

Thesis Supervisor: Paul E. SandorffTitle: Professor of Aeronautics and

Astronautics


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iii

ACKNOWLEDGEMENTS

The authors wish to express their appreciation to the followingpersons: Professor Paul E. Sandorff for his advice and guidance duringthis project, Mr. Lawrence J. Berman who supplied much background materialand encouragement, and Miss Mary Carlo who typed the manuscript.

Acknowledgement is also made to Mr. Richard Russell, Mr. HughBlair-Smith, and Mrs. R. H. Walker of the Mathematics Group of the MITInstrumentation Laboratory for their instruction in problem programmingfor the IBM-65O digital computer.

The graduate work for which this thesis is a partial requirementwas performed while the authors were assigned from the U. S. NavalPostgraduate School for graduate training at the Massachusetts Instituteof Technology.


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IV

TABLE OF CONTENTS

Page NoObject 1Introduction 2Vehicle Description and Aerodynamics 5Trajectory Analysis 12Equations of Motion IkComputer Program 18Results 21Discussion of Results 61Conclusions 69

Chapter No ,

123

k5

6

78

AppendixA Atmospheric Data 71

Figures1 Missile configuration 62 Zero-lift drag coefficient for two

cone-cylinder bodies 83 Cross-flow drag coefficient versuscross-flow Mach number 11k Simplified diagram of vector quantities

associated with the missile l65,6 Variation of V-u with U for variousvalues of n. 317,8,9>10 Variation of Y-u with U for variousvalues of ni 33


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TABLE OF CONTENTS (Continued)

Figures

11,12,13,1^

15,16,17,18

19,20,21,22

23,2^,25,26

27

28,29,30

313233

Variation of E, with U for variousvalues of n^Variation ofvalues of n^

., with U for various

Page No.

37

kl

53

Drag velocity loss as a function of-^ for various values of n^

Variation of maximum lift load factorwith U_ for various values of n.m 1Variation of tilting time with n^ forvarious values of Tv to reach a -u of 30Values of U and Tv required to obtain aspecified , under maximum energy conditions 5^Values of Y-u and V, at maximum energy 57Variation of density ratio with altitude 72Variation of speed of sound with altitude 7^-

Tables

II

III

IV

VI

Straight Line Approximations of the Zero-Lift Drag Coefficient Curve for VariousValues of Mach Number 9Straight Line Approximations of the CrossFlow Drag Coefficient Curve for VariousValues of Mach Number 9Numerical Values of Constants andParameters 19Trajectories Investigated Showing BurnoutValues for Various Values of Mass Ratios 23Computer Results for RepresentativeTrajectories 58Atmospheric Data Approximation Formulas 73

References 75


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VI

LIST OF SYMBOLS

oA Cross-sectional area, ftc Exhaust velocity, ft/secC-p, Drag coefficientCD Zero-lift drag coefficientCL. Cross-flov drag coefficientCL Lift coefficientD Drag force, lbDT Time interval, secE Total energy, ft-lb/slugE^ Total energy at burnout, ft-lb/slugF Thrust, lbgave Average acceleration of gravityg Gravitational conversion factor, 32.17^05 ft/sec'I Specific thrust, secL Lift, lbM Mach numberMc Cross-flow Mach numberMR Mass ratiom Mass, slugn^ Initial thrust to weight rationL Lift load factorR Density ratio, (?/,S Planform area, ft

o2


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VI

LIST OF SYMBOLS (Continued)

T Time, secT-^ Burnout time, secTu Fictitious burnup time, secT Vertical flight time, secU Angle of missile axis from vertical, deg or radUm Maximum programmed U, deg or radV Velocity, ft/secV^ Velocity at burnout, ft/secV_^ Velocity loss due to drag forceVq. Velocity loss due to gravityV Speed of sound, ft/secw Weight flow rate lb/secW. Initial weight, lbX Horizontal range, ftV Altitude, ftY^ Altitude of burnout, ftboc Angle of attack, deg or rad

Flight path angle, deg or rad0-5 Flight path angle at burnout, deg or rad\ Atmospheric density, slug/ftCq Atmospheric density at sea level, slug/ft


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OBJECT

The object of this thesis is to study the early powered-flighttrajectory of a large rocket powered vehicle. The effects on the first-stage trajectory of varying vertical flight time, initial tilt angle,and initial acceleration, are of primary interest, especially as theyaffect the maximum burnout energy conditions.

Of interest also is the feasibility of using a relatively smallinitial tilt angle followed by a "gravity turn" to reach practical burnoutconditions of velocity, altitude, and flight path angles for vehicles with

high initial acceleration.


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CHAPTER 1

INTRODUCTION

The study reported herein is concerned primarily with the initialportions of the powered-flight trajectory of a large, single stage, rocketpowered vehicle. Conceptually, this vehicle could he the "booster stage ofan ICBM or a satellite launcher.

Usually there are three phases to the initial flight trajectoryof a large ballistic missile or satellite launching vehicle. These phasesinclude a vertical flight phase, a tilt phase, and a gravity turn phase.The gravity turn phase is customarily followed by a period of "constant-

attitude thrust", during which the major portion of the flight velocity isachieved; this latter regime is not considered in this study.

A vertical launch for a large, rocketpowered vehicle of currentdesign is necessary due to the inability of the vehicle structurally to with-stand the transverse loads which would be present during an inclined launch.Vertical or near vertical flight is also necessary in order to achievealtitude. Usually the vertical flight path is followed for a short time, butthe time of vertical flight must be carefully selected in order to achieve atrajectory which minimizes propellant expenditure.

Upon completion of the vertical flight phase, a tilting phase iscommenced. Tilting is normally accomplished by deflecting the thrust vectorof the vehicle to produce a tilting moment according to some selected program;


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this changes the attitude of the vehicle, and subsequent thrusting changesthe velocity vector. This maneuver is non-optimum and is best completedquickly; however, the tilt rate must not be so rapid as to exceed practical

limitations of the vehicle control and structure. The tilting phase is com-pleted when the vehicle body axis and the thrust vector are both alignedwith the vehicle velocity vector.

The third phase of the conventional trajectory concerns the flightregime where this alignment exists and a relatively slow turning path follows,brought about by the component of gravity transverse to the flight path.This phase hopefully terminates at an altitude above the sensible atmosphere,and with an attitude that matches the subsequent constantattitude thrust re-gime in such a manner that the best overall trajectory performance is obtained,

The important problem of proceeding from the earth's surface,through the three phases of the trajectory outlined, to arrive at a desirablealtitude, velocity and attitude, is complicated because of the external forcesacting on the vehicle. The major forces affecting the vehicle during thesephases are thrust, the earth's gravitational force, and the aerodynamic forcesof lift and drag, which act in a direction perpendicular and parallel to theinstantaneous direction of flight, respectively. Gravitational and dragforces acting on the vehicle result in velocity losses during the flight andthus detract from the efficiency of the launch. The lift force may in certaincases be beneficial in that it may aid in turning the vehicle.

For a specific vehicle, the important trajectory design parametersare velocity, altitude, and flight path angle at burnout. It is only possiblehowever, to compute trajectory characteristics by numerical integration of theequations of motion from specified initial conditions. In this paper numerous


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trajectories are developed, using vehicle characteristics which are approx-imately representative of large chemical rockets of contemporary design, andvarying the time of vertical flight and the maximum tilt angle. The effectsof variations in two important vehicle design parameters are also included:namely, the initial thrust-to-weight ratio, n^, and the mass ratio; the valueof n^ is introduced as an additional initial variable, while with the assump-tion of constant mass flow every point in each computed trajectory correspondsto burnout for some specific mass ratio. The burnout conditions are thenexamined as functions of the initial variables by using burnout angle as agoverning parameter and cross-plotting. The nature of trajectory optimizationto maximize burnout velocity or burnout energy is of particular concern tothis study.


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CHAPTER 2

VEHICLE DESCRIPTION AND AERODYNAMICS

This study is intended to derive conclusions applicable to rocketvehicles similar to contemporary long range "ballistic missiles, satellitelaunchers, and space vehicle boosters. Development of trajectory data re-quires the use of certain vehicle design parameters which identify aerody-namic and engine performance. To simplify preliminary work the "high-drag"configuration missile of Ref . 1 is selected as a model. It is believedthat this design has aerodynamic characteristics representative of the classof vehicles described above. Rocket engine specific impulse is taken to be

300 lb-sec/lb, and, again for simplification, this value is considered con-stant throughout the flight regime. A value of 10 is selected for theratio of initial weight to burnout weight, defined as the mass ratio. Thismakes the propellant factor, the ratio of initial fuel weight to initialvehicle weight, equal to .9. Figure 1 shows the physical dimensions of theselected vehicle.

Since the vehicle of this study is similar to the high drag missileof Ref. 1, the aerodynamic coefficients utilized are extracted from thissource. A detailed explanation of the methods and procedures used to arriveat these values are set forth in Appendix G of that report.

The missile, being axially-symmetric, has only a drag force imposedon it during the vertical flight phase and the gravity turn phase, since


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Uo (

*-

SO/ \

15'

120

II

501+.9"

1,270.5

Wi/A = 3,010 lb/ft'

S/A 11.0

W. = 236,700 lbs

I = 300 secs

Fig. 1 Missile configuration. (Reference l)


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during these phases the angle-of- attack is zero. The zero-lift drag forceis made up of three parts: base drag, skin friction drag, and form drag.The zero-lift drag coefficient, based upon both theoretical and empiricaldata, and representing the sum of these forces, is plotted versus Mach num-ber in Fig. 2.

In order to simplify computer programming, the curve of C-p, versusMach number is divided into five segments. Each segment of the curve isthen represented by a straight line function. The breakdown of the Cp)ocurve and the approximating straight line functions are shewn in Table I.This straight line approximation of the curve representing Cn , while beingan approximation, is considered to be sufficiently accurate for the problemat hand.

During the tilting phase of the trajectory, the missile is sub-jected to lift forces, as well as drag forces, since the missile has an angleof attack during transition from the vertical flight phase to the zero-liftphase. Reference 1 outlines a cross-flow method of predicting lift and dragon bodies of revolution at an angle of attack. In this method the flow overthe missile is separated into two components: one along the axial directionof the body, and one component normal to the axis. The axial flow exerts aforce on the body in the axial direction while the cross flow exerts a force

in the normal direction. Reference 1 derives equations for C-r and C^ usingthis theory of cross and axial flow.

oCL (2 - CD )


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8

oo (DH O


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TABLE I

STRAIGHT LINE APPROXIMATIONS OF THE ZERO-LIFT DRAG COEFFICIENT CURVEFOR VARIOUS VALUES OF MACH NUMBER

M %o

0.72 .130

1.25.85OM - A82

1.90 .983- -323M

l.k .522- .080M

7.3.328 - .023M

.155

TABLE II

STRAIGHT LINE APPROXIMATIONS OF THE CROSS-FLOW DRAG COEFFICIENT CURVEFOR VARIOUS VALUES OF MACH NUMBER

M \.80

753.23Mc - .2*92.36 - .573Mc


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10

The terra C-p, is a drag coefficient due to the cross flow component. A plotucof C-r, versus Mach number, assumed to apply for this study, is shown inFig. 3- This plot is a series of straight line approximations of the cross-flow drag characteristics derived in Ref . 1 for the missile configuration ofFig. 1. These straight line approximations are described by functions asset forth in Table II. The straight line approximations are consideredsufficiently accurate since, as it can be seen from the above equation, theeffect of CD is minor at small angles of attack.


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11

dH

o

CDVa


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12

CHAPTER 3

TRAJECTORY ANALYSIS

The portion of the powered flight trajectory of interest in thisstudy is considered in three phases, as discussed in Chapter 1. The firstphase or vertical flight regime is followed by the tilt phase during whichthe vehicle is tilted from the vertical at a rate of two degrees per second.

In Ref . 1 the tilt phase is approximated by impulsive tilting to5.5 degrees from the vertical during a one second time interval at the endof vertical flight time, followed by a gravity turn which continues untilthe desired conditions of attitude, altitude, and velocity are reached. The

vehicle is assumed to be in the gravity turn as soon as the impulsive tilt-ing is accomplished. The impulsive tilting during a one second time intervalis justified by determining that the required vehicle response time is lessthan one second for the 5-5 degree tilt angle. This computation is made onthe basis of the time required to tilt through the specified angle with themaximum tilting moment available acting on the moment of inertia of thevehicle. For the present study, which is concerned with higher values of n.and consequent higher dynamic pressures during tilting, a tilt rate of 2degrees per second is selected as a reasonable maximum value. This is per-haps lower than necessary for tilting at sea level, but to have a basis forcomparison, this rate is used for all trajectories computed. In the computerthis tilt rate is approximated by increasing the tilt angle, U, two degrees


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13

per second until the tilt angle reaches the specified maximum programmedtilt angle, U . This value of tilt angle is then held constant until theangle-of-attack of the missile becomes zero. At this time the tilt phaseends and the zero angle-of-attack or gravity turn phase begins. Duringthe tilt phase thrust is considered to act parallel to the vehicle axis.The component of thrust required for tilt is considered a negligible losscompared to the total thrust vector.

In the zero angle-of-attack phase thrust acts in the direction ofthe instantaneous velocity vector, which is also parallel to the missileaxis. Turning is accomplished by the action of the earth's gravitationalfield. This part of the trajectory would logically be followed by a con-stant attitude or a "linear with time" thrust program, depending on themission of the vehicle. In this study the gravity turn is continued untilninety percent of the missile mass is consumed. Since this paper deals onlywith single stage characteristics, staging is not considered and all resultspertain to first-stage values.


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Ik

CHAPTER k

EQUATIONS OF MOTION

The equations of motion are developed using an inertial X, Ycoordinate frame. This assumes a "flat", non-rotating earth, which is agood approximation during the early powered-flight phase of the type ofrocket vehicle considered. The gravitational acceleration due to the earthis assumed to be constant during the portion of the trajectory of interestin this paper. This also is a reasonable assumption when the altitudereached is small compared with the radius of the earth, as it is in thisstudy.

Rocket engine characteristics are simplified by assuming constantthrust and constant mass flow rate. Both of these quantities usually varywith atmospheric pressure, thrust increasing and mass flow rate decreasingas altitude is increased. This means that the specific impulse actually in-creases with altitude and that the initial thrust-to-weight ratio is basedon the lower level of thrust found at sea level. The simplifications madein this study specify a constant specific impulse of 300 seconds, which maybe thought of as representing an average value. The initial thrust-to-weightratio in this study is therefore somewhat larger than it would be for anactual vehicle of comparable performance.


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15

Considering the vehicle as a point mass the equations of motion are

D2Y/DT2 ( F/m ) cos u _ gave _(D/m)sin^ -(L/rrOcostf (l)

D2X/DT2 * (F/m) sin U -(D/m)costf + (L/ra)sintf (2)wherein

F/m = thrust per unit massSave " graviNational acceleration due to the earthD/ra = drag per unit mass

L/m = lift per unit massThe lift terms are considered positive in sign for the negative angle-of-attack condition which occurs during the tilt phase of the trajectory. Figureh shows the vector relationships involved.

Thrust, lift, and drag forces per unit mass are computed using thenomenclature of Ref . 2, in which Tu is defined as a fictitious time when thetotal mass of the vehicle would be consumed.

From the conventional relationships between rocket parameters

I s = F/w (h)

m = W./go (l - T/Tu ) (5)

c I sgQ (6)

Applying (k) , (5), and (6) to the various accelerations due to thrust, lift,and drag in (l) and (2) gives

F/m = c/(Tu - T) (7)


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Fig. h Simplified diagram of vector quantities associated withthe missile.


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17

l 2D/m = (iev%Ag Tu )/W.(Tu-T)i o2L/ra = (lev^LAg^J/WiCVT)

(8)

(9)


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18

CHAPTER 5

COMPUTER PROGRAM

The computer used in this study is an IBM 65O digital computerlocated in the computation center of the Instrumentation Laboratory of theMassachusetts Institute of Technology. Although not comparable in speed tothe larger digital computers such as the IBM 70^, it is adequate for thisstudy, computing an average trajectory in about ten minutes. The computerprogram is prepared using the MAC programming system developed by theInstrumentation Laboratory computation center.

Time intervals for integration are varied according to the phaseof the trajectory. During the vertical flight phase the time interval isset at four seconds for vertical flight times of four seconds and above, andone second for vertical flight times less than four seconds. The time in-terval is reduced to one second during the tilt phase to maintain comparableaccuracy in computing the rapidly changing trajectory quantities. At thecompletion of the tilt phase the time interval is increased again to fourseconds and is held constant until burnout.

The initial conditions for the equations of motion are set equalto zero for each run. Parameters held constant for all runs are: S/A, W^/A,DU/DT, I , and w. Variable parameters for each run are: T , Tu , and Um .The fictitious burn-up time, Tu , equals I s/n^. Since I s is held constant,T is directly proportional to n . . Table III lists the numerical values ofthe constants and parameters used.


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19

TABLE III

NUMERICAL VALUES OF CONSTANTS AND PARAMETERS

Symbol Value Description

e .0023769 slug/ft 3 Atmospheric density at sealevelSo 32.17^05 ft/sec2 Gravitational conversionfactor

gave 32.0 ft/sec2 Gravitational accelerationacting on vehicle (assumedconstant)

S/A 11.0 Ratio of planform area to crosssection area

W./A 3010 lb/ft2

Ratio of initial weight tocross section areadu/dt 2 deg/sec Tilt rateh 300 sec Average specific impulse

n.1 Tusec sec3.02.52.01.5

100120150200

901C8135180


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20

Tilt rate is held constant at two degrees per second until U isreached. This is mechanized on the computer by increasing U instantaneouslyat the beginning of each one second time interval until U equals U . Atthis point Um is held constant until the angle of attack becomes zero. Liftand drag are computed during the tilt phase in the manner shown in Chapters2 and k. These calculations are made at the beginning of each time intervaland are integrated as constants within the differential equation loop of theprogram. The error introduced by this approximation was small, as a resultof the selection of integration intervals; in general the change in aerody-namic force from one interval to the next did not exceed three percent.When the angle-of-attack becomes zero, U is set equal to (90 - ~& ) , andthereafter varies directly with tf . This point marks the beginning of thegravity turn phase.

In the gravity turn phase lift is set equal to zero and drag iscalculated in the same manner as above using the zero angle-of-attack dragcoefficient. The program is terminated when T equals .9 T , which corres-ponds to the mass ratio of ten mentioned in Chapter 2. Values of velocity,altitude, range, flight path angle, and energy are punched for each computertime interval, and for burnout.


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21

CHAPTER 6

RESULTS

For the study and results as presented here, all vehicle andtrajectory parameters are held constant, except the initial thrust-to-weight ratio of the vehicle, the time of vertical flight, and the maxi-mum programmed tilt angle. The initial thrust-to-weight ratios used are1.5, 2.0, 2.5, and 3-0. The vertical flight times are varied from 1second to 2k seconds, and the maximum programmed tilt angle is variedfrom 2 to 90 degrees. The trajectory calculations are continued in allcases for a total time, T = 0.9TU , i.e., a mass ratio of 10.

Approximately one hundred trajectories were computed for thispaper. Table IV lists values of mass ratio, velocity, altitude, flightpath angle, energy, scalar velocity loss due to drag, and scalar velocityloss due to gravity for some of the more useful trajectories. The effectsof varying the parameters, ni , Ty , and Um , are shown in Figs. 5 through 18,which display burnout values of velocity, altitude, energy, and flightpath angle plotted against maximum programmed tilt angle for a mass ratioof 10. Separate plots are shown for each vertical flight time used.

Scalar velocity loss due to drag is shown in Fig.5.19 through 22.V-pj, divided by the ballistic coefficient, W./C^ A, is plotted versus burnoutflight path angle for each n- and each Tv . A more general presentation isobtained with the ballistic coefficient, which uses the value of Cn at Mach2.0.


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22

Since side loading is an important consideration in large rocketdesign, plots of the maximum lift load factor versus U for each n. and each ' miT'v are included as Figs. 23 through 26.

The length of time that the rocket is subjected to lift loads isalso of interest. Figure 27 shows the time required to tilt the missile sothat it will attain a burnout flight path angle of thirty degrees. This isplotted against n^ to show the large increase in time required for tiltingas n. is increased.i

An optimization study is made, based on finding the combination ofparameters which would give the highest specific energy at burnout for eachn. investigated. Energy is first maximized for fixed values of ft-u by plot-ting energy versus the value of U corresponding to particular verticalflight times. The maximum energy points are then cross-plotted against ~8 ^and the value of U corresponding to the maximum energy point for the fixedvalues of tf-u. As a cross-check, this procedure is reversed so that energyis maximized for fixed values of U , and cross-plotted in the same manner.Also included in these figures are the values of burnout velocity and burn-out altitude which occur at the maximum energy points. The optimizationresults are shown in Figs. 29 through 31.

Representative trajectories for various values of n. are identifiedin detail in Table V. These particular trajectories were chosen because theywere closest to the maximum energy trajectories for each case.


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23

TABLE IV

TRAJECTORIES INVESTIGATED SHOWING BURNOUTVALUES FOR VARIOUS VALUES OF MASS RATIO

]n. s 1-51

Ty 3l ..\ MR \ \ A -8E^xlO VD VG8 2 13*lAl

l6l180

1.073.395.1310.0

2327,19010,68416,700

1,463285,739423,565616,000

87.853.550.949.0

.001.350

.7071.590

59422422422

4204,1884,6945,078

8 6 14l42162180

1.083^55.2610.0

2528,16912,08618,130

1,703165,530207,113250,500

83.614.8ll.l8.9

.001

.387

.7971.670

78749779798

4833,0323,1553,272

8 12 17"1*1161

1.093.395.13

31^4,8432,970

2,532363,871140,962

78.2-11.3-20.3

.001

.129

.0481.6

5,08311,142

5161,8741,688

16 4 26*lk2162180

1.153.455.2610.0

5067,204

10,73116, 4oo

6,21k305,300458,176650,700

86.061.459.658.O

.003

.358

.7231.553

5.91407407407

838M394,8825,393

16 8 27*139159180

1.163.284.8810.0

5297,136

10,56517,100

6,715248,475358,888525,000

82.041.838.436.0

.oo4

.334

.6741.628

6.80465465465

8943,8994,220M35

16 12 29*l4l161180

1.173.395.1310.0

5777,75^11,484

17,700

7,782212,551290,231388,000

78.O27.023.321.0

.004

.369

.7531.680

8.79556559559

9493,4903,7573,9^1

16 18 31*139159180

1.183.28if. 8810.0

6277,72711,39218,181

8,900147,717180,348216,304

72.812.99.05.8

.005

.346

.7061.722

11.3821878937

9522,9522,9803,082

* Completion of tilt phase. (


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2k

TABLE IV (Conti:aued)n = 1.5

TV Um \ MR J^ Yb 4 v10 "8 i VG2k 6 39' :

139159180

1.2k3.28^. 8810.0

82U6,755

10,011*16,300

ll*,793291,072^39,719676,000

83.666.063.662.3

.008

.321

.61*31.51*6

21.51*01*1*01*1*01*

1,230k,3klM325,!+96

2k 12 1*1*l6l180

1.263.395.1310.0

8787,31010,85916,830

16,392270,316395,063566,000

78.01*6.5^3.51*1.5

.009

.35^ 717

1.597

26.51*1*31*1*31*1*3

1,3263,7^71*,1*981*,927

2k 2k ^5*lUl161180

1.293.395-1310.0

9887,81*9

11,63017,850

19,658190,631252,1*72326,500

66.021.717.815.5

.Oil

.369 757

1.695

1*2.9626631*636

1.1*293,0253,5363,711*

2k 32 1*9*ll*l161180

1.333-395-1310.0

1,1077,996

11,81518,095

23,08611*1,693168,1*29192, k8k

58.010.86.9k.l

.011*

.365 752

1.699

66.9891979

1,092

1,5762,6133,0063,013

* Completion of tilt phase (


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25

TABLE IV (Continued)n - 2.0

Tv um Tb MR Zl A *b I^xlO"8 3l VG1 4 4*

104120135

1.033-265.010.0

1337,785

11,52017,853

263235,616351,320510,742

86.051-249.147.5

.0002.379.777

1.758

.05658659659

1522,9573,3413,688

1 10 7"103119135

1.053. OPif. 8310. u

2387,86111,626l8,4o6

811175, 494247,918350,294

80.230.427.625.3

.0005.365.756

1.807

.34892902903

2352,0472,6522,891

1 20 12 -x-io4120135

1.093.265-010.0

4277,97911,871

18,300

2,392125,765162,216200,000

69.I16.012.911.0

.002 359 7571.748

2.01,4191,5321,623

4oi2,0022,1172,277

1 24 Ik*106118135

1.103.404.6810.0

5078,21611,04217,980

3,250114,092133,796165,825

66.912.09.66.9

.002 374.653

I.669

3.41,7201,9012,203

4io1,8441,9372,017

8 10 20*104120135

1.153.265.010.0

7327,67011,35717,620

6,989251,704380,350559,000

80.060.158.958.0

.007 375 767

1.729

16.5616616616

6023,1143,5473,964

8 20 25*105121135

1.203.335.1610.0

9458,08511,99818,040

10,914217,596318,203443,500

69.84l . 539.1

38.6

.008

.397

.8221 . 762

22.3738739739

7932,7973,1133,421

8 30 30*106118135

1.253. to4.6810.0

1,1648,42211,31718,350

15,467179,763231,661332,000

59.828.126.023.9

.012

.413

.7151.786

54.2938948950

9372,4202,6152,900

8 50 4o*104120135

1.363.265.010.0

1,6007,66611,28017,353

25,169101,365121,199140,363

40.910.37.04.5

.021

.326

.6751.551

2231,8322,2752,848

1,1421,9021,9651,999

* Completion of tilt phase. ( c< = 0)


70/176


71/176

26

TABLE IV (Continued)r^ 2.0

T um Tb MR Vb Yb 4 E^xlO"8 Zl VG16 16 37*

105121135

1.313-335.1610.0

1,1+227,8^9

11,65517,715

2^,960255,783386,0^5557,000

7I+.I59-*+57.856.5

.018

.390

.8031.700

132625625625

1,0513,l J+63,5703,860

16 36 k6*106118135

l.kk3. kok.6Q10.0

1,7828,32711,19518,180

36,270196,701+257,1+36376,000

5I+.I33.331.1+29.0

.028

.1+10

.7101.795

328833837838

2,1+052,6202,81+83,182

16 kQ 51*103119135

1.523.081+.8310.0

2,0307,80711,60918, kkk

1+1,96611+7,81+3200,871273,ll+l

1+2.622.719.717.3

.031+

.352 739

1.790

1+1+71,0311,0681,081

1,5831,9622,5032,675

16 60 58*106118135

1.633A0Ik 6810.0

2,k638,25311,10718,102

1+2,291118,317139,002173,138

30.112.1+10.07.1+

.01+6 379.661

1.690

601+1,1+361,5871,828

1,6632,0912,1862,270

2k 12 1+9*105121135

1.1+83.335.1610.0

1,9057,7^5

11,50)+17,1+20

1^,785271,1981+13,801+597,ooo

78.171.270.169.3

.033

.387

.7951.759

3*+7590590590

1,5333,2853,756M90

2k 30 56*10U120135

1.593.265.010.0

2,2817,757ll,5H

17,863

56,032226,562336,1+81+1+87,562

60.0k8.21+6.01+1+.3

.01+1+

.371+

.7711.750

1+68670671671

1,7262,9733,3383,666

2k 50 65*105121135

1.763.335.1610.0

2,9058,136

12,11+018,317

68,81+6176,97921+7,291+332,589

lK).228.1+25.623.6

.061+

.388

.8161.780

62681+285385I+

1,9292 f 6k22,8573,029

2k 70 77*105121135

2.053.335.1610.0

1+,0128,10512,09918,222

80,808121,801+ll+8,53l+175,681+

20.12.19.16.9

.106

.367

.7801.717

8I+51,1901,3501,531

2,0732,3252,1+012,1+1+7



72/176


73/176

27

TABLE IV (Continued)ni " 2 ->

T %_ Jk MR vb \ xb E^xlO"8 JSl VG1 10 8*

8496

108

1.073-335.010.0

4io8,34311,92718,292

3,634223,832326,512476,217

19.759.858.657-6

.003

.420

.8161.826

3.5794795795

2422,4832,7983,113

1 20 15"8395108

1.143.244.7810.0

8108,20011,71118,511

5,638189,882272,461402,888

69.945.944.242.7

.005

.397 7731.843

9.32926931931

4452,2242,4l82,758

1 4o -St278395108

1.293.244.7810.0

1,5278,12511,69618,595

17,249136,994187,321264,277

50.427.325.123.2

.017 374.744

1.814

138139914471465

7951,8261,9172,140

1 6o 39*8395108

1.483.244.7810.0

2,2187,450io,54716,722

30,09386,735105,551129,978

30.512.49.87.5

.034

.305

.5901.440

546241930383888

1,0161,4741,4851,590

8 20 29*8597108

1.323.425.1810.0

1,5848,55612,28618,200

21,891225,510328,567465,000

70.057.756.556.0

.020.438

.8601.670

167834835835

9292,4902,7793,165

8 30 34*8294

108

i.4o3.164.5810.0

1,8467,89811,28418,437

28,971182,871263,301402,241

60.246.544.843.2

.026

.371

.7211.829

31094l947947

1,0842,2612,4492,816

8 50 44*8496

108

1.583.335.010.0

2,4978,36712,07418,593

42,862143,370195,898269,376

40.527.625.523.7

.045

.396 792

1.815

623128413241338

1,3001,9692,1222,269

8 6o 50*8294

108

1.723.164.5810.0

3,0127,75011,15018,300

50,266112,553146,717201,300

30.119.817.516.0

.062

.336

.6691.740

804155417001823

1,4141,8061,8302,077



74/176


75/176

28

TABLE IV (Continued)n * 2.5

T Um Tb MR A Yb k. E^xlO"8 V, \16 20 1+3*

8395108

1.563.2U1+.7810.0

2,1+058,038

11,1+7918,300

1+8,1+61220,1+66323,1+661+89,000

70.161+.62.862.0

.01+1+

.39^

.7631.811+

510780781781

1,3652,5322,8003,119

16 1+8 56*81+96

108

1.883.335.010.0

3,5938,38512,07118,553

72,691+169,551+237,219333,571

1+2.335.633.632.1+

0.088.1+06.805

1.828

80110111021+1026

1,7062,221+2,1+252,621

16 60 67*8395108

2.263.21+4.7810.0

5,0308,12311,70618,611+

92,937139,010186,315258,262

30.025.723.521.5

0.156.375 7^51.816

1017116312091228

1,8232,061+2,11+52,1+58

16 70 70*82108

2.1+03.164.5810.0

5,^677,812

11,21+018,1+71+

91,502116,15611+6,66519^,613

20.217.815.1+13.2

179.3^2.679

1.769

lll+O1323i'+5i+1576

1,81+31,9751,9862,150

2k 10 55*8395108

1.853.21+ij.,7810.0

3,5308,015

11, ^2318,101

8l+, 985236,0803^8,703530,669

80.078.1+77-977.5

.090 397.765

1.809

629703703703

1,7812,6322,93'+3,396

2k 20 59*8395108

1-973.21+1+.7810.0

3,97^8,031+

11,1+61+18,120

96,975226,01+2332,126502,000

70.067.566.665.5

.110

.395

.761+1.811

680732733733

1,8962,581+2,8633,31+7

2k ko 68*81+96

108

2.33-335.010.0

5,17^8,312

11,91+618,361+

121,1+21200,512287,1221+12,507

1+9.8V7.71+6.11+5.0

.173

.1+10

.8061.819

8168I+1+81+78I+7

2,0602,1+61+2,9272,987

2k 6o 83*95108

3.21+4.7810.0

8,051+11,63618,525

160,51+8216,651+303,506

30.028.526.6

.376

.71+71.813

98310011005

2,3232,1+232,670



76/176


77/176

29

TABLE IV (Continued)n. = 3-01

TV um A MR \_ \_ 4 E^xlO-8 VD VG1 16 14*

707890

1.163-334.5310.0

9998,60311,37018,648

6,606197,650266,817417,702

74.261.560.760.0

0.0070.4340.7321.873

14.1942944945

4172,0782,3062,607

1 24 20*688090

1.253.125.010.0

1,4488,03112,25718,700

13,350168,876262,685378,000

66.153.051.550.5

0.0150.377O.8361.867

84.81,0511,061l,06l

6221,8982,2022,439

1 4o 30*707890

1.433.334.5510.0

2,l4l8,519

11,33518,714

27,467148,091195,038296,016

50.137.936.635.0

0.032o.4io0.7051.846

4181,3601,3841,393

9011,7411,9012,093

1 52 37*698190

1.593.235.2610.0

2,7168,07212,51518,500

376,444118,215171,985230,000

38.128.226.125.O

0.0490.364O.8381.770

7141,6651,7761,818

1,0501,5831,7391,882

8 10 27*717990

1.373.444.7610.0

1,8958,81611,68218,503

25,477219,839298,606454,492

80.176.O75.675.1

0.0260.4590.7781.858

279895896896

8662,2092,4722,801

8 20 32*688090

1.473.135.010.0

2,2577,97512,163

18,570

34,766184,230290,286423,328

70.164.263.262.4

0.0370.3770.8331.860

451954958959

1,0122,0512,3992,671

8 1+0 42*707890

1.733-334.5510.0

3,2128,55911,36618,718

54,483163,808217,461333,471

50.144.042.74l.5

O.0690.4190.7161.859

7221,1441,1561,160

1,3661,9172,0982,322

8 6o 53*698190

2.133.235.2610.0

4,6818,13512,59918,644

73,877120,154171,518226,562

30.226.824.623.2

0.1330.3690.8491.811

1,1481,4731,5721,607

1,4711,7121,8591,949

Completion of tilt phase. (=


78/176


79/176

30 .

TABLE IV (Continued)n. = 3.0

T m Tb MR vb Yb


80/176


81/176

31

19

18

Vb x 10"3(fps)

17 -

16

I r r t r , , ,-

i \\ni = 2.0 \ n. = 3.0 -- -

i i

TV = 1 sec

1 1

n = 2.5

..J 1 1,

-

30 6oU (cleg)

90'm

19

vb x 10(fps)

18-3

IT

16

I I t r I 1 1 I

ni " 3.C"

/ n. = 2.5 -

n = 1.5 Ty= 8 sec n. = 2.0i -

I i J I l 1 1 130 6o 90

Um (deg)

Fig. 5 Variation of V^ with Um for various values of n. at aT of 1 and 8 seconds.


82/176


83/176

32

19

V, x 10b

18-3

17

16

_

^^ ? " r- r i 1 1 1 ' ' . _ " ^rij_=3.0^ /^ "^ X1! 22 - 5>/ \ \ "- \ n.=2.0\ l- -

-

n. - 1.5J 1

T 16 secv ~

1 1 ' ' '

30 U (deg)m6o 90

19

13

V, x 10" 3b(fps)

IT

16

i rr^ = 3.0

I I i

30 60Um (deg)

90

Fig. 6 Variation of Vh with U for various values of n. at aTv of 16 and 2H- seconds.


84/176


85/176

33

20 hoU (deg)m

Fig. 7 Variation of Y, with U for various values of n. at aT of 1 second? m 1


86/176


87/176

3^

U (deg)m

Fig. 8 Variation of Y, with U for various values of n at aT of 8 seconds.v

rn


88/176


89/176

35

700

600 -

500 -

Y^xlO(ft)

-3

400

300 -

200

1 1 1

\ n =1.5\ i

'""

-

\ \ ni = 2 *-

-

\^vi = 3'^^

T =V

\\n = 2.5V n. i

= l6 sec-

\

-

1 1 i i1

-

20 40 6oU (deg)m

Fig. 9 Variation of Y^ vith U for various values of n. at aT of 16 seconds. 1v


90/176


91/176

36

700

6oo

500

Y x 10"b(ft)

-3

4oo

300

20020

T = 24 secv

40 60Um (deg)

Fig. 10 Variation of Y^ with Ura for various values of n. at aT of 24 seconds.


92/176


93/176

37

1.9

1.8

1.7-8E^ x 10D

(ft-lb/slug)1.6

1.5

i 1 i i 1 r

J L30

J L6o

U (deg)90

in

Fig. 11 Variation of K with U for various values of n atT of 1 second.


94/176


95/176

1.9

1.8

E. x 10D

1.78

(ft-lb/slug)1.6

1.5

o

1 r

J L

T = 8 secv

J L30 6o

38

" i i r

1

90U (de )

Fig. 12 Variation of E. with U for various values of n at aT of o seconds.


96/176


97/176

39

1.9-

1.8-

1.7--8

F^ x 10(ft-lb/slug]

1.6-

1-5-

1 r

30

T -r

T = 16 secv

J

6ou (deg)m

90

Fig. 13 Variation of F^ vith Um for various values of n. at a Tof 16 seconds. 1


98/176


99/176

ko

30 60Um (cleg)

90

Fig. Ik Variation of E, with U for various values of n. at a Tof 24- seconds. v


100/176


101/176

kl

i 1 r i r

90

60

Kb(cleg)

30

30 60Um (deg)

Fig. 15 Variation of ^f h with U_ for various values of n, atTv of 1 second. m


102/176


103/176

k2

i >^ i^ i^ I i r

w b(deg)

Ty = 8 sec

Um (deg)

Fig. 16 Variation of )f b with Um for various values of n. at aTv of 8 seconds.


104/176


105/176

^3

T 1 1 P i r

90

60 -

(deg)

30

Tv = l6 sec

Fig. IT Variation of ^b vith Um for various values of n atT of l6 seconds,v


106/176


107/176

I 1^ 1^ I ~

kk

(deg)

Tv = 2k sec

30 60Ura (deg)

90

Fig. 18 Variation of 0^ with Um for various values of n.of 2k seconds. t^ at a T^


108/176


109/176

7

.6

.5

.h

vdCda

.3 -

.2

.1

T = 1 sec

I I

^5

J I90 60 30 o

^(deg)

Fig. 19 Drag velocity loss as a function of #*, for variousvalues of n at a Tv of 1 second.


110/176


111/176

h6

7

VDCDAW.

.6

T = 8 sec

J I I I I I L90 60 30

Fig. 20 Drag velocity loss as a function of X for variousvalues of n, at a T of 8 seconds,

i v


112/176


113/176

kj

7

.6

.5

.k

3

.2

.1 -

T I 1 I

Tv = 16 sec

90 60}fb (deg)

30

Fig. 21 Drag velocity loss as a function of 0, for variousvalues of n. at a T of l6 seconds,i v


114/176


115/176

k8

5

.k

vpCpAW.

.3

.2

r i 1 1 r

T = 2k secv

J I I L90 60 X

J L

(deg)30

Fig. 22 Drag velocity loss as a function of # for variousvalues of n. at a T of 2k seconds.i v


116/176


117/176

^9

\

l.k

1 .2 -

"I 1 1

T = 1 sec1 .0

.8

.6

.2

V

i r i r

30 6o 90U (deg)m

Fig. 23 Variation of maximum lift load factor with U for variousvalues of n. at a T of 8 seconds.


118/176


119/176

50

1.4

\

1.2

1.0n = 3.0

T = 8 secv

30 6oU (deg)

h =2.0

90m

Fig. 2k Variation of maximum lift load factor with U for variousvalues of ^ at a T of 8 seconds. m


120/176


121/176

51

n_

1.4

1.2

1.0

.8

.6

.4

T = 16 secv

30 6o 90U (deg)m

Fig. 25 Variation of maximum lift load factor with U for variousvalues of n^. at a T of l6 seconds. m


122/176


123/176

52

1.4

1.2

1.0 -

n

= 2kv sec

30

T 1 1 r

6oUm (deg)

90

Fig. 26 Variation of maximum lift load factor with U for variousvalues of n. at a T of 2k seconds. m1 v


124/176


125/176

53

70

60

50

Uo

Tilt Time(Sees)

30

20

i r

K = 30 cb

MR = 10

10

J I

1.0 2.0 3.0 k.O

Fig. 27 Variation of tilting time with n. for various values of Tto reach a fr, of 30.


126/176


127/176

5h

1.9-8E x 10

b(ft-lb/slug)1.8

Um(deg)

T(sec) 8 ~

Fig. 28 Values of U and Tv required to obtain a specified %munder maximum energy conditions for n. of 2.0.


128/176


129/176

55

1.9-8

EL x 10(ft-lb/slug)

1.8

1.7

8o

6o

4oum(deg) 20

(sec)

24 -

16 _

tf b (deg)

Fig. 29 Values of U_ and T required to obtain a specified tfm v t)under maximum energy conditions for n of 2.5.


130/176


131/176

56

90 6o * (deg)30

*ig. 30 Values of U and Ty required to obtain a specified Xunder maximum energy conditions for n. of 3*0.


132/176


133/176

57

700

6oo

500Y x 10" 3

(ft) 1+00

300

200

19

Vb xlO(fps)17

16

1 r

n. = 2.0

1 1 t r

n. = 3.0

r^ = 3-0

J I 1 L90 60 * b (deg)

30-a

Fig. 31 Values of Y, and V-, at maximum energy.


134/176


135/176

58

TABLE V

COMPUTER RESULTS FOR REPRESENTATIVE TRAJECTORIES

A. n. = 3-01 T = 1 sec U = 20mT V Y If E VDsec fps ft dee; ft-lb/slUR fps4 265.78352 525.70761 87.002181 52234.583 . 206446838 5^6.99767 2129.7715 80.949374 2l8l26.6l 2.2177488

16 1152. 0408 8609. U335 70.776125 940599.34 27.85009817* 1227.7526 9726 . 6649 70.160493 1066634.4 37.68545421 1535.2739 11*867.757 69.635561 1656889.0 87.56034229 2087.3356 28129.005 66.455353 3083509.0 331.3074937 2731.9199 45316. 107 63.757048 5189695.9 610.932394? 3626.1614 67522.151 61.496942 8746984.1 802.4488753 4799.5585 96429.217 59-667247 14620399-0 925.233536l 6325.6918 13^027.09 58.208353 24319383.0 982.4321869 8308.6788 182846 A5 57.055556 40399983.0 1000 . 478077 10972.134 246406.84 56.151720 68121766.O 1005 . 509885 14886.326 330622.30 55.454918 121438810 1006.289090 18668.867 398806.33 54.922312 187094520 1006.2980* End of tilt phase.

B. n. 2.5 Tv = 1 sec um 26

T V Y 1 E VDsec fps ft de ft-lb/slug fps4 199.15440 393-99420 86.681749 32507.627 .115416018 409.79788 1593-1459 80.054237 i35225.ll 1.2274019

12 632.93575 3600.1614 73.392714 316135.60 4.683211116 870.24830 6387.4618 66.923305 584176.57 11.73884418* 992.92178 8071.9079 64.713429 752652.80 18.59828022 1244.3838 12031.649 63.844218 1161^52.4 39.66350330 1705.8846 2213.8303 58.869145 2167300.1 180 . 9026138 2151.1125 34644.219 54.543873 3428287.3 425.5958346 2721.8130 49616.932 50.715587 5300510.7 647.4864154 3476.8072 67782.124 47.445993 8224919.4 808.1228162 4427.3496 89914.139 44.721666 12693614.0 924.5263570 5617.1983 116914.58 42.479115 19538075.0 994.2929878 7100.7933 149892.68 40 . 646909 30033288.0 1026.765686 8964.8019 190264.90 39.157532 46305429.0 1040.646094 11391.601 240036.72 37.954541 72607238.0 1045 . 6924102 14785.568 302500.64 36.996442 119039180 1047.0206108 18585.407 361322.79 36.152776 184333900 1047 . 0806* End of tilt phase,


136/176


137/176

59

TABLE V (Continued)

c. n = 1.5 Ty - 16 sec U = 18mT V Y tf E VDsec fps ft deB ft-lb/slug fpsk 67.000877 132.68853 89.999999 6513.68628 137 . 9621+2 5I+I.2I+719 89.999999 26930.929 .06025196112 212.91698 121+1.5799 89.999999 62613 . kjk .3177309^

16 291.88676 22^9.6993 89.999999 111+980.88 .9301+75 1+620 37^.09575 3580.1953 88.110768 185163.20 2 . 38271882k 1+59.5^522 5236.3599 82.72l)-083 27^65.82 1+. 891167028 551.81331 7208.5201 76.258076 381+176.26 8.2929821+31* 626.5981+2 8900.2976 72.77^338 1+82671.1+1 11.33580735 732.62606 lli+5l+.l+3^ 71.827600 636906.OO 15 . 98271+5^3 961k 122^ 17603.6I+8 65.55221^5 103111+6.7 31.15052551 1207.21+51 250I+5 . 312 59.270032 153^529.5 72.761+79759 11+50,8226 33523.212 53.090988 2131020.6 157.1391567 1706.2258 1^2720.706 11-7.055876 2830101.1+ 278.0803775 2027.7^50 52525.718 1+1. 291939 37^581+0.0 386.3095983 21+23.2376 62935 ^90 J5.98500I+ I+960929.9 1+78. 6681+191 2898.1+151 73921.632 31. 228^59 6578763.I+ 55^.2591599 31+52.0813 851+22.589 27.038602 8706823.I 619.8I+83I+107 1+089. 971+8 973^9.100 23.3800U9 Hl+96062.0 676.75^58

115 1+821.1+522 109613.97 20. 200? 1+0 1511+9926.0 72I+. 967^0123 5658.59^ 122137.53 l-(,kkkOkl 19939505.0 765.06602131 662I.I699U 13^856.15 15.057^69 26262321.O 79^.20610135 715^.6835 1I+1271.78 13.987885 3011+0033.0 807.78328ll+7 9009.8838 160685.28 11.208726 I+5758899.O 8I+I+.95I+17155 10523.81^8 173759.37 9.6691722 6096623 3.0 867.32021163 1235^.673 186986.81 8.3^6301+9 82335093.0 888.652I+7171 1^652.533 2001+90.62 7.21791+99 11379896.0 910.07619180 I8l8l.017 216305. 1+7 5.8239900 17223I+II.O 936.87I+76* End of tilt phase.


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TABLE V (Continued)

D. H --U IT - 16- sec 12*u = 8?mT V Y 1 E VDsec fps ft dep; ft-lb/slug fpsk 132.98571 262.85677 86.01*007 17299.766 .0510839638* Zjk.5kl39 1061.2315 78.231773 71832.250 .537^8828

12 1*25.70506 2ko6.96kl 76.811880 16805 1*. 18 1.551*81*1920 755.87968 678I*. 91*1*9 69.I9187I* 503976.21 8.1*06236928 1118,9996 13^04.599 62.519935 1057360.3 31.113581*36 11*73.1912 2201*8.882 56.628937 1791*51*8.0 117.69071*IA 1797.8701 32228.993 51.2W01 2653105.7 295.5952952 2l81*.0l*12 1*3659.1*80 1*6.21*2080 3789720.3 1*81.81*89760 2687.623^ 56560.978 1*1.736271* 5^31^55.5 629-6861*368 3312.1201* 7120^.977 37.809023 7776023.5 71*6.6579676 i*06l*.6675 87822.151 31*. 1*1*1*157 IIO86355.O 81*0.0391*681+ 1*967.0819 106668.91 31.587081* 15767923.0 907.7^15592 601*5.2513 128059. k2 29.17^501* 22392722.0 951.86607100 7336.8l6l 152389.86 27.11*1*237 318171*3^.0 976.00713

108 889^.5880 180171.00 25.1*1*0375 1^5353679.0 991.09169116 10821.625 212127.15 2k. OI6065 65378775.0 999.1*050512k 13307. ^39 21*9378.99 22.835027 965671*97.0 1003.5360132 16759.873 293825 31 21.51*6871* 11*9900220 IOO5.0832135 l8l*81*.668 312997.36 21. 25093 It I809II87O. 1005 . 2250* End of tilt phase.


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61

CHAPTER 7

DISCUSSION OF RESULTS

The status of the vehicle at burnout is of prime importance.This vehicle status is best described by the burnout quantities, V, , Y^, E^,

and & -k, as displayed in Figs. 5 through 18.These burnout values are not shown for the vehicle where the value

of n. is 1.5 and the Ty is 1 second, due to the fact that the vehicle passesthrough the horizontal and heads back towards the earth before reachingburnout. This result is readily explained since the vehicle is relativelyslow and turns at a low altitude where drag losses are very large.

The vehicle does reach burnout conditions for the other values ofTy , however. For the case where n- is 1.5> the value of V, peaks at a verylow value of U . The larger values of U cause the relatively slow vehiclem & m Jto turn more at a low altitude. This reduces V, due to the fact that thebvehicle operates for a longer period in dense atmosphere where the dragvelocity loss is very large.

As T is increased, the maximum value of V, obtained for an n. ofv * b i1.5 occurs at the higher values of U . This happens as a result of the in-creased velocity and the higher altitude reached before the tilting iscommenced. Drag velocity loss is much less under these circumstances. Itis true that gravity velocity loss increases as T is increased but it doesnot offset the reduction of the velocity loss due to drag. This fact is


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62

further borne out by observing that the curves indicate higher burnout alti-tudes are reached as the value of U is reduced.m

The burnout altitude reached varies inversely with the value of Umand directly with the value of T for the vehicle with an n. of 1.5- Bothof these phenomena are readily explained since the altitude attained is adirect function of the vertical component of the velocity vector which isdirectly affected by these two factors.

The value of tf , of the vehicle for the values of n. 1.5 generallydecreases with increased values of IT, and with an increase in T . A largem vU allows a greater amount of turn of the vehicle, hence a smaller Hi ,. Atthe higher values of T the vehicle has a greater velocity before commencingthe turn, and hence does not get turned as much since the gravity vectorcausing the turn after U is reached is small in comparison to the verticalmcomponent of the velocity vector of the vehicle.

Generally, the burnout energy of the vehicle follows the trend ofthe velocity at burnout. This is explained by the fact that energy is directlyproportional to the square of the velocity.

As the value of n^ is increased the burnout velocity attained be-comes less and less dependent upon the value of Um , which is indicated by thefact that the curves of V-u versus U tend to flatten out as the value of n.D m xincreases. Vertical flight time does not affect the burnout velocity to anygreat degree and at higher vertical flight times the value of U appears tohave less affect on the V-^ reached. The basic reason for these effects isthat with a high n. and T the vehicle is out of the very dense atmospherebefore turning, and hence the drag velocity loss is relatively small.

As n. is increased the burnout altitudes become less and less atithe low values of Um and just slightly more at the high values of U . At


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the higher values of U the high n. vehicle does not get turned as much aso m 1the lower n. vehicle and hence attains a slightly higher burnout altitude.

Vertical flight time seems to have little effect on the burnoutvelocity attained. It appears to be a slight factor at low values of T ,but for values of Ty greater than 8 seconds, vertical flight time has noappreciable effect upon the burnout velocity reached. The value of "tf . de-creases with an increase in n. for the reasons previously explained.

The drag velocity loss of the missile for the various trajectoriesstudied is shown in Figs. 2k through 27. The variation of this loss with # ^and n^ for four values of T is shown. The more prominent indications ofthese results are:(1) High nj_ missiles have the highest drag velocity loss.(2) The drag velocity losses decrease slightly with an increase

in vertical flight time.(3) Drag velocity losses greatly increase as the missile attitude

approaches the horizontal at burnout.

The higher drag velocity losses accompanying an increase in thevalue of n. is a result of the higher velocities attained by the vehicle atlower altitudes. Since drag force is directly proportional to the square ofthe velocity and to the density of the atmosphere, this force, and hence theresulting velocity loss, is large for a high n. missile. Since the dragvelocity loss may be expressed as:

V = So / *it must be realized that a slow, large vehicle would experience a lower dragvelocity loss than would a smaller, faster missile.


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6^

The drag velocity loss decreases slightly with an increase in ver-tical flight time because the missile is at a higher altitude and hence ina less dense atmosphere "before it commences to turn. It therefore spendsless time in the denser atmosphere of low altitudes. This decrease in dragvelocity loss as indicated in the aforementioned plots, is not as great aswould be expected. The drag loss is plotted versus # , , and to get to thesame value of ^-^ for a high vertical flight time as for a low verticalflight time, the vehicle must be turning at an angle of attack for a longerperiod of time since the missile has a greater velocity at the start of thetilting phase. This increased time of tilt with an angle of attack increasesthe drag coefficient due to the induced drag present while this conditionexists.

Nearly the same reasoning applies to the condition of increaseddrag velocity loss for a smaller value of ^ attained for a particularvehicle with a given n. at a certain T . The smaller value of 7$", requiresthat the missile be tilted at an angle of attack for a longer period andhence the drag coefficient is again increased.

The values of n-r which are plotted against U in Figs. 28 through31 show the maximum negative lift load factor encountered during a trajectoryof specified Tv and Um . Lift is always negative in the tilt phase, that is,in the direction of rotation of the vehicle. Lift load factor increasesrapidly with increased U for the high n. trajectories. The slope of thelift load factor curve increases as T is increased, as could be expectedfrom the higher dynamic pressures caused by the higher velocities associatedwith long vertical flight times. An interesting aspect of the higher Tplots is that the maximum nT seems to level off, reaching a maximum of about1.2 g's for an n. of 3.0. In these cases the missile has reached the less


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65

dense portions of the atmosphere, where the extremely low density has offsetthe increased velocity, and the maximum n encountered has become essentially

Li

constant. The large values of nL encountered for high n^ trajectories andthe accompanying bending moments are too great for most contemporary liquidfueled vehicles. Use of trajectories of this nature require the heavierstructural design associated with solid fueled vehicles.

It might be mentioned at this point that the rate of tilt shouldhave a considerable affect on lift. This program uses a tilting rate of twodegrees per second, which is considered a nominal rate for a large rocket-powered vehicle control system to achieve, but a minimum rate to reach themaximum programmed tilt angle within a reasonable time. The two degree persecond tilt rate was selected to keep the lift load factor within practicalbounds. A study of methods for obtaining minimum lift loads through differ-ent types of tilting programs is of interest, but beyond the purposes ofthis paper.

The variation of tilt time necessary to reach a "tf, of 30 forfour different vertical flight times is plotted against n^ in Fig. 27- Thecurves indicate that the tilt time increases as both T and n. increases.v 1

The higher the value of T used, the greater the velocity of thevehicle becomes before the vehicle starts to turn. Since this is the case,it takes much longer to turn the vehicle from the vertical position to a ^ ,of 30 at the higher value of T than at the lower values. This same typeof reasoning may also be applied to the second observation, in that at highvalues of n. the vehicle gains greater velocities sooner than at low valuesof n . . It therefore requires a greater tilt time to reach a ^ , of 30 atthe higher values of n.


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The optimization program undertaken in this paper is based on max-imum specific energy. The combination of T and U which would give maximumburnout energy for a specified burnout angle are determined. Energy is max-imized in this manner for the three higher values of n. . The resulting valuesof E^, U and Tv are plotted versus burnout angle in Figs. 19 through 21.It is not possible to obtain a good optimization for an n. of 1.5 with thedata available. At this value of n-^, burnout energy is extremely sensitiveto Ty and U , and any attempt at optimization requires a large number of tra-jectories. Accordingly, no optimization is included for an n. of 1.5

Good maximum energy points are found for burnout angles below aboutthirty degrees. In the range of thirty to fifty degrees, two approximatelyequal maximum energy points appear. One of these points occurs at a valueof T consistent with the maximum points found in the thirty degree and belowrange, while the other occurs at the minimum T , which is one second. Abovethe thirty to fifty degree range the maximum energy points are found at theminimum values of T and U for the particular burnout angle. These samev m ^ characteristics, in varying degree, are found for each value of n. . Theoverlapping in the plots of Tv and Um versus burnout angle in the thirty tofifty degree range show that in this area maximum energy can be obtained byusing either of two combinations of Tv and U . This effect is undoubtedly

caused by the non-linear action of drag on the trajectories. A point is reachedfor each n. where the beneficial effects of early tilting, and consequent earl-ier alignment of thrust and velocity vectors in the desired direction, areovercome by the higher drag losses associated with large programmed tilt anglesat low altitudes. At this point it is necessary to use a period of verticalflight time to get the vehicle out of the denser portions of the atmosphereduring the tilting maneuver.


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The amount of vertical flight time required to maximize burnoutenergy for low "burnout angles does not increase indefinitely as burnout angleapproaches zero, but tends to level off in the neighborhood of l6 to 20 seconds.This is especially evident in Fig. 21 where n* is 3.0. The high velocityreached at the end of a long vertical flight time causes an increased negativelift load during the tilting phase, which increases the drag coefficient,thereby reducing burnout velocity and burnout energy.

It is interesting to note that at burnout angles of about ten degreesand below, the maximum burnout energies for all three values of n. fall veryclose together. This indicates that for very low burnout angles the advantageof a high initial acceleration rocket is questionable, since the same burnoutenergy can be obtained using a lower initial acceleration with lower aerodynamicloads. At other values of burnout angle the higher burnout energies obtainedfrom high n. rockets are apparent. The point at which maximum overall burnoutenergy is reached starts at a burnout angle in the vicinity of twenty-fivedegrees for an n. of 2.0, and moves in the direction of increasing burnoutangle as n. increases. In this case, the increased drag associated with highinitial acceleration and low burnout angle causes the maximum overall energypoints to fall at higher burnout angles for the higher values of n.

Values of burnout velocity and altitude for the maximum energy condi-tions discussed before are shown in Fig. 22. It logically follows that, sinceenergy is a function of velocity and altitude, these curves are coincident inthe same manner as the maximum energy curves at low burnout angles. At higherburnout angles, burnout velocity increases and burnout altitude decreases asn. is increased.

Each representative trajectory tabulated in Table V is the nearesttrajectory available to the overall maximum energy case for that particular n .


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This is the optimum trajectory for the burnout angle listed only, and it can-not be said that it is the optimum trajectory for any other attitude reachedbefore burnout. Naturally it is not applicable to a burnout angle lower thanthat listed. The question arises whether or not more energy is obtained byusing the tabulated trajectory for the overall maximum energy case until thedesired attitude is obtained, followed by constant attitude thrust untilburnout, than by using the maximum energy trajectory for the burnout anglecorresponding to the attitude desired. It seems that if the vehicle is abovethe denser atmosphere, the energy generated after constant attitude thrustis started would be about the same as that for the gravity turn. A constantattitude thrust program would give higher altitude with lower velocity thanthe gravity turn, although the actual difference between the two programswould depend upon the time of application of the constant attitude thrustprogram. If the vehicle reaches the desired attitude in the early tilt phase,before leaving the denser atmosphere, a constant attitude thrust programwould involve lift loads and additional drag, but it would get the vehicleout of the sensible atmosphere sooner. In either case the non-linearity ofthe problem would require a separate computation for each situation.


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CO

CHAPTER 8

CONCLUSIONS

Determining the optimum powered flight trajectory for a largerocket powered vehicle is a complex problem, which is strongly influencedby the desired trajectory burnout angle and the rocket initial thrust-to-weight ratio. The burnout angle may be considered a design parameter forthe booster trajectory, since different burnout angles are required for dif-ferent missions. This paper shows that a combination of vertical flighttime and initial tilt angle to give maximum energy at burnout can be deter-mined for any desired burnout angle. There is, in addition, one value ofburnout angle, which gives maximum burnout energy, for each value of initialacceleration. In this manner an optimum booster flight trajectory is avail-able for any'desired burnout angle.

Usually the vehicle with the higher initial acceleration will havethe greater burnout energy. At low burnout angles, however, in the zero toten degree range, values of burnout energy for a wide range of initial accel-erations closely coincide. For low burnout angles, therefore, the initialacceleration of the vehicle is of little consequence with respect to maximumenergy optimization. In fact, it can be said that a lower initial accelera-tion is preferable for low burnout angles, since the high lift load factorsassociated with high initial accelerations are avoided.


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The time required to tilt the vehicle from the vertical to a pointwhere the angle-of-attack is zero is also quite high for high accelerationvehicles. The procedure involving a relatively small initial tilt anglefollowed by a gravity turn to the desired burnout angle is no longer feasiblewith high initial accelerations, which leads to the conclusion that the tiltphase for high initial acceleration vehicles must be programmed in itsentirety.


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APPENDIX A

ATMOSPHERIC DATA

The atmospheric data used is based on the ARDC model atmosphereof 1959 as described in Ref . 3 In order to facilitate computer procedures

the density ratio ( f/fi,) and sonic speed data are treated in a simplifiedmanner

Utilizing the atmospheric density data of Ref. 3 a plot of theratio of /q is made extending from sea level to an altitude of ^00,000feet, as shown in Fig. 32. The resulting curve is divided into four seg-ments which are accurately approximated by appropriate exponential functions,The resulting segments of the curve and respective describing exponentialfunctions representing the density ratios are shown in Table VI.

Sonic speed is plotted versus altitude from sea level to an alti-tude of U00,000 feet as shown in Fig. 33- The curve results in a series offive straight line segments. Straight line functions are used to describethese segments of the curve. Table VI displays these functions and theirrespective areas of applicability.


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TABLE VI

ATMOSPHERIC DATA APPROXIMATION FORMULAS

1 Altitude(ft)

Density Ratio,

Speed of Sound(fps)

36,800e-Y/32,000 1120-.001H7 Y

82,5001.65 e -Y/20,800 968.08

120,0001.65 e- Y/20,800 813.78 .00]87 Y

168,000 51e-Y/25,200 813.78 .00187 Y

263,000.51 e"V25,200 1625-.00298 Y

i+50,000 ?T e-Y/l7,000 81*6.5


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800 900 1000 1100Speed of Sound (fps)

1200

Fig. 33 Variation of speed of sound with altitude.


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REFERENCES

1. Prigge, J.S., Jr., Parsons, T., and Berman, L. , A Digital ComputerStudy of the Powered Flight Trajectory of Long Range Ballistic Missiles ,Report ARG-1 , USAF 33(6l6)2392, 1956.

2. Traenkle, C.A., "Mechanics of the Power and Launching Phase forMissiles and Satellites", Wright Air Development Center TechnicalReport 58-579, September, 1958.

3. Minzer, R.A. , Champion, K.S.W., Pond, H.L., "The ARDC Model Atmosphere1959/' Airforce Surveys in Geophysics No. 115 , Geophysics ResearchDirectorate, Air Force Cambridge Research Center, August 1959*

h. Sandorff, P.E., Orbital and Ballistic Flight , Technical PublicationsGroup of the Instrumentation Laboratory, Massachusetts Institute ofTechnology, Cambridge, i960, Chapter 3-

5. Fried, B.D., "Trajectory Optimization for Powered Flight," Chapter kin Space Technology , Seifert, H. , ed. , John Wiley and Sons, Inc.,New York, 1959.

6. Hoult, C.P., GRD Research Notes No. 17, "The Approximate Analysis of

Zero Lift Trajectories," August 1959> Geophysics Research DirectorateAFCRC, ARDC, USAF, Bedford, Massachusetts.

7. Bryson, Ross, "Optimum Trajectories with Aerodynamic Drag," Jet Propulsion28, U65, 1958.


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