digital computers 00 jay n
TRANSCRIPT
-
7/28/2019 Digital Computers 00 Jay n
1/176
A DIGITAL COMPUTER STUDY OF THEFIRST-STAGE TRAJECTORIES OF H\BHINITIAL ACCELERATION ROCKETSGORDON ROWLAND JAYNEandJOSEPH BARBOUR WILKINSON, JR.
-
7/28/2019 Digital Computers 00 Jay n
2/176
LIBRARYU.S. NAVAL POSTGRADUATE SCHOOL
MONTEREY, CALIFORNIA
-
7/28/2019 Digital Computers 00 Jay n
3/176
-
7/28/2019 Digital Computers 00 Jay n
4/176
-
7/28/2019 Digital Computers 00 Jay n
5/176
-
7/28/2019 Digital Computers 00 Jay n
6/176
-
7/28/2019 Digital Computers 00 Jay n
7/176
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
-
7/28/2019 Digital Computers 00 Jay n
8/176
MV*-"
*
-
7/28/2019 Digital Computers 00 Jay n
9/176
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
-
7/28/2019 Digital Computers 00 Jay n
10/176
-
7/28/2019 Digital Computers 00 Jay n
11/176
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.
-
7/28/2019 Digital Computers 00 Jay n
12/176
-
7/28/2019 Digital Computers 00 Jay n
13/176
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
-
7/28/2019 Digital Computers 00 Jay n
14/176
-
7/28/2019 Digital Computers 00 Jay n
15/176
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
-
7/28/2019 Digital Computers 00 Jay n
16/176
-
7/28/2019 Digital Computers 00 Jay n
17/176
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
-
7/28/2019 Digital Computers 00 Jay n
18/176
-
7/28/2019 Digital Computers 00 Jay n
19/176
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
-
7/28/2019 Digital Computers 00 Jay n
20/176
-
7/28/2019 Digital Computers 00 Jay n
21/176
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.
-
7/28/2019 Digital Computers 00 Jay n
22/176
-
7/28/2019 Digital Computers 00 Jay n
23/176
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, rocket- powered 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;
-
7/28/2019 Digital Computers 00 Jay n
24/176
-
7/28/2019 Digital Computers 00 Jay n
25/176
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 constant- attitude 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
-
7/28/2019 Digital Computers 00 Jay n
26/176
-
7/28/2019 Digital Computers 00 Jay n
27/176
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.
-
7/28/2019 Digital Computers 00 Jay n
28/176
-
7/28/2019 Digital Computers 00 Jay n
29/176
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
-
7/28/2019 Digital Computers 00 Jay n
30/176
-
7/28/2019 Digital Computers 00 Jay n
31/176
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)
-
7/28/2019 Digital Computers 00 Jay n
32/176
-
7/28/2019 Digital Computers 00 Jay n
33/176
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 )
-
7/28/2019 Digital Computers 00 Jay n
34/176
-
7/28/2019 Digital Computers 00 Jay n
35/176
8
oo (DH O
-
7/28/2019 Digital Computers 00 Jay n
36/176
-
7/28/2019 Digital Computers 00 Jay n
37/176
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
-
7/28/2019 Digital Computers 00 Jay n
38/176
-
7/28/2019 Digital Computers 00 Jay n
39/176
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.
-
7/28/2019 Digital Computers 00 Jay n
40/176
-
7/28/2019 Digital Computers 00 Jay n
41/176
11
dH
o
CDVa
-
7/28/2019 Digital Computers 00 Jay n
42/176
-
7/28/2019 Digital Computers 00 Jay n
43/176
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
-
7/28/2019 Digital Computers 00 Jay n
44/176
-
7/28/2019 Digital Computers 00 Jay n
45/176
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.
-
7/28/2019 Digital Computers 00 Jay n
46/176
-
7/28/2019 Digital Computers 00 Jay n
47/176
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.
-
7/28/2019 Digital Computers 00 Jay n
48/176
-
7/28/2019 Digital Computers 00 Jay n
49/176
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)
-
7/28/2019 Digital Computers 00 Jay n
50/176
-
7/28/2019 Digital Computers 00 Jay n
51/176
16
Fig. h Simplified diagram of vector quantities associated withthe missile.
-
7/28/2019 Digital Computers 00 Jay n
52/176
-
7/28/2019 Digital Computers 00 Jay n
53/176
17
l 2D/m = (iev%Ag Tu )/W.(Tu-T)i o2L/ra = (lev^LAg^J/WiCVT)
(8)
(9)
-
7/28/2019 Digital Computers 00 Jay n
54/176
-
7/28/2019 Digital Computers 00 Jay n
55/176
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.
-
7/28/2019 Digital Computers 00 Jay n
56/176
-
7/28/2019 Digital Computers 00 Jay n
57/176
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
-
7/28/2019 Digital Computers 00 Jay n
58/176
-
7/28/2019 Digital Computers 00 Jay n
59/176
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.
-
7/28/2019 Digital Computers 00 Jay n
60/176
-
7/28/2019 Digital Computers 00 Jay n
61/176
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.
-
7/28/2019 Digital Computers 00 Jay n
62/176
-
7/28/2019 Digital Computers 00 Jay n
63/176
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.
-
7/28/2019 Digital Computers 00 Jay n
64/176
-
7/28/2019 Digital Computers 00 Jay n
65/176
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. (
-
7/28/2019 Digital Computers 00 Jay n
66/176
-
7/28/2019 Digital Computers 00 Jay n
67/176
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 (
-
7/28/2019 Digital Computers 00 Jay n
68/176
-
7/28/2019 Digital Computers 00 Jay n
69/176
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)
-
7/28/2019 Digital Computers 00 Jay n
70/176
-
7/28/2019 Digital Computers 00 Jay n
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
* Completion of tilt phase. (
-
7/28/2019 Digital Computers 00 Jay n
72/176
-
7/28/2019 Digital Computers 00 Jay n
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
* Completion of tilt phase. (
-
7/28/2019 Digital Computers 00 Jay n
74/176
-
7/28/2019 Digital Computers 00 Jay n
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
* Completion of tilt phase. (
-
7/28/2019 Digital Computers 00 Jay n
76/176
-
7/28/2019 Digital Computers 00 Jay n
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. (=
-
7/28/2019 Digital Computers 00 Jay n
78/176
-
7/28/2019 Digital Computers 00 Jay n
79/176
30 .
TABLE IV (Continued)n. = 3.0
T m Tb MR vb Yb
-
7/28/2019 Digital Computers 00 Jay n
80/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
82/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
84/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
86/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
88/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
90/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
92/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
94/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
96/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
98/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
100/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
102/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
104/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
106/176
-
7/28/2019 Digital Computers 00 Jay n
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^
-
7/28/2019 Digital Computers 00 Jay n
108/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
110/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
112/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
114/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
116/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
118/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
120/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
122/176
-
7/28/2019 Digital Computers 00 Jay n
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
-
7/28/2019 Digital Computers 00 Jay n
124/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
126/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
128/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
130/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
132/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
134/176
-
7/28/2019 Digital Computers 00 Jay n
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,
-
7/28/2019 Digital Computers 00 Jay n
136/176
-
7/28/2019 Digital Computers 00 Jay n
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.
-
7/28/2019 Digital Computers 00 Jay n
138/176
-
7/28/2019 Digital Computers 00 Jay n
139/176
6o
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.
-
7/28/2019 Digital Computers 00 Jay n
140/176
-
7/28/2019 Digital Computers 00 Jay n
141/176
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
-
7/28/2019 Digital Computers 00 Jay n
142/176
-
7/28/2019 Digital Computers 00 Jay n
143/176
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
-
7/28/2019 Digital Computers 00 Jay n
144/176
-
7/28/2019 Digital Computers 00 Jay n
145/176
63
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.
-
7/28/2019 Digital Computers 00 Jay n
146/176
-
7/28/2019 Digital Computers 00 Jay n
147/176
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
-
7/28/2019 Digital Computers 00 Jay n
148/176
-
7/28/2019 Digital Computers 00 Jay n
149/176
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.
-
7/28/2019 Digital Computers 00 Jay n
150/176
-
7/28/2019 Digital Computers 00 Jay n
151/176
66
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.
-
7/28/2019 Digital Computers 00 Jay n
152/176
-
7/28/2019 Digital Computers 00 Jay n
153/176
67
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 .
-
7/28/2019 Digital Computers 00 Jay n
154/176
-
7/28/2019 Digital Computers 00 Jay n
155/176
68
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.
-
7/28/2019 Digital Computers 00 Jay n
156/176
-
7/28/2019 Digital Computers 00 Jay n
157/176
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.
-
7/28/2019 Digital Computers 00 Jay n
158/176
-
7/28/2019 Digital Computers 00 Jay n
159/176
70
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.
-
7/28/2019 Digital Computers 00 Jay n
160/176
-
7/28/2019 Digital Computers 00 Jay n
161/176
71
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.
-
7/28/2019 Digital Computers 00 Jay n
162/176
-
7/28/2019 Digital Computers 00 Jay n
163/176
-
7/28/2019 Digital Computers 00 Jay n
164/176
-
7/28/2019 Digital Computers 00 Jay n
165/176
73
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
-
7/28/2019 Digital Computers 00 Jay n
166/176
-
7/28/2019 Digital Computers 00 Jay n
167/176
7^
800 900 1000 1100Speed of Sound (fps)
1200
Fig. 33 Variation of speed of sound with altitude.
-
7/28/2019 Digital Computers 00 Jay n
168/176
-
7/28/2019 Digital Computers 00 Jay n
169/176
75
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.
-
7/28/2019 Digital Computers 00 Jay n
170/176
-
7/28/2019 Digital Computers 00 Jay n
171/176
-
7/28/2019 Digital Computers 00 Jay n
172/176
-
7/28/2019 Digital Computers 00 Jay n
173/176
-
7/28/2019 Digital Computers 00 Jay n
174/176
-
7/28/2019 Digital Computers 00 Jay n
175/176
-
7/28/2019 Digital Computers 00 Jay n
176/176
thesJ37A digital computer study of the fi rst-st
3 2768 002 10049 7DUDLEY KNOX LIBRARY