U S ARMY MISSILE COMMAND

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1 REPORT NO. RF-TR COPY NO t--... PARAMETRIC DESIGN CURVES FOR SHORT RANGE, r L "HIGH ACCELERATION BALLISTIC ROCKETS c C-- I April 1963 U S ARMY MISSILE COMMAND REDSTONE ARSENAL, ALABAMA RSA FORM JAN 63 PREVIOUS EDITION IS OBSOLETE

2 DDC AVAILABILITY NOTICE Qualified requesters may obtain copies of this report from the Defense Documentation Center for Scientific and Technical Information, Arlington Hall Station, Arlington 12, Virginia. DESTRUCTION NOTICE Destroy; do not return.

3 1 April 1963 Report No. RF-TR-63-7 PARAMETRIC DESIGN CURVES FOR SHORT RANGE, HIGH ACCELERATION BALLISTIC ROCKETS by Herman R. Oswell Department of Army Project No. I-A A-22 AMC Management Structure Code No Advanced Systems Laboratory Future Missiles Systems Division Directorate of Research and Development U. S. Army Missile Command Redstone Arsenal, Alabama

4 ABSTRACT This report is a compilation of design curves which will permit the rapid estimation of the effects of range, warhead weight, diameter, specific impulse, and propellant weight fraction on the weight and length of short range, high acceleration ballistic rockets. ii

5 TABLE OF CONTENTS Page I. INTRODUCTION... I II. SCOPE... I III. ASSUMPTIONS... I IV. APPLICABLE EQUATIONS... 2 V. METHOD OF COMPUTATION OF GROWTH FACTORS VI. PRESENTATION OF GROWTH FACTOR DATA... 4 VII. VERIFICATION OF ACCURACY OF THE METHOD OF COMPUTATION VIII. ROCKET LENGTHS... 5 IX. LENGTH-DIAMETER TRADEOFFS... 7 X. LAUNCH ANGLE TRADEOFFS... 8 XI. CONCLUSIONS LIST OF ILLUSTRATIONS Table I List of Constants for Curve Fit of Range Tables.. 3 II Verification of Accuracy of the Method of Computation Figure 1 Warhead Cylindrical Length Versus Wwh/d' for Various Densities Motor Length Parameter Versus Growth Factor Growth Factor Versus Propellant Weight Fraction tii

6 LIST OF ILLUSTRATIONS (Concluded) Page Figure 19 Weight and Length Tradeoffs for Range = 2 km Weight and Length Tradeoffs for Range = 3 km Weight and Length Tradeoffs for Range = 4 km Comparison of Range at 3*.E. to Range at 45" Q. E Effect of Range on Growth Factor iv

7 LIST OF SYMBOLS Symbol Units Definition C lb/in. Ballistic coefficient of the rocket at burnout d in. Rocket diameter g ft/seca Gravity constant Isp sec Propellant specific impulse i Drag form factor related to BRL type 2 projectile, i = drag drag of type 2 k 1, k 2 LD Constants in curve fit of range tables Volumetric loading fraction of motor 1 in. Length PWF Propellant weight fraction, Wprop Wto - Wwh Pwh lb/in.' Average warhead density PP lb/in. 3 Propellant density Q Growth factor, Wth Wwh Q. E. deg Quadrant elevation or launch angle YB Booster mass ratio, -t Wbo Vactual ft/sec Boost velocity including effects of drag and gravity V

8 LIST OF SYMBOLS (Concluded Symbol Units Definition Videal ft/sec Boost velocity neglecting drag and gravity Wbo lb Rocket weight at booster burnout Wprop lb Propellant weight Wto lb Rocket weight at takeoff Wwh lb Weight of warhead section vi,

9 IF-. PARAMETRIC DESIGN CURVES FOR SHORT RANGE, HIGH ACCELERATION BALLISTIC ROCKETS I. INTRODUCTION The preliminary design of a rocket to satisfy a given set of performance requirements usually involves laborious calculations of the effects of various design parameters on the physical characteristics of the rocket. The data presented in this report will allow the rapid estimation of the effects of range, specific impulse, propellant weight fraction, payload weight, and diameter on the weight and length of short range, high acceleration ballistic rockets. II. SCOPE The data presented in this report cover the following ranges of parameters: Range Specific impulse 1 to 4 kilometers ZOO to 26 seconds Propellant weight fraction. 5 to. 9 Payload ballistic factor, Wwh/idZ. 5 to 2. III. ASSUMPTIONS The boost acceleration is assumed to be high enough so that the burning distance is very small compared to the range of the rocket. This allows the use of the Ballistics Research Laboratories range tables*, which are based on zero burning distance. *Exterior Ballistics Tables for Projectile Type 2, BRL Memorandum Report No. 196, Aug

10 Boost phase drag and gravity velocity loss are assumed to be five per cent of the ideal velocity given by Equation 1. Videal = Ispg in \Wbo)" M(1) *l The actual burnout velocity is therefore given as Vactual ` -95 Isp g In MON " IV. APPLICABLE EQUATIONS A. Booster Mass Ratio The booster mass ratio (Wto/Wbo = YB) required for a given velocity is obtained from Equation 2 as Yreq'd = (VB reqld/o 9 5 Isp g) (3) B. Growth Factor The growth factor of the rocket (ratio of takeoff weight to warhead weight) is derived from the definition of takeoff weight given in Equation 4, Wto = Wwh + WprFp (4) PWF and the definition of burnout weight given in Equation 5, Wbo = Wh + Wprop ( - (5) The growth factor is given as W to YB x PWF' 6 Wwh = B PWF -B + 1. (6)

11 C. Ballistic Coefficient The ballistic coefficient used in this repont is defined as C = W (7) idl where i is the ratio of the drag coefficient of the rocket under consideration to the drag coefficient on which the range tables are based (BRL Type 2). The ballistic coefficient can also be defined in terms similar to those in Equation 6 by introducing a parameter which is called the warhead ballistic factor, Wwh/id'. ( w PWF C=(Ž (YBX PWF - YB + 1.) D. Range Table s form The BRL range tables have been fitted by an equation of the V req'd = k, + k. (9) C Values of the constants k, and kj for Q.E. = 45 are given in Table I for ranges up to 4 kilometers. Equation 9 does not accurately describe the range tables for greater ranges. Table I LIST OF CONSTANTS FOR CURVE FIT OF RANGE TABLES Range k, k 2 1 km (C < 2.) 65 1,7 1 km (C > 2.) 1, km (C < 2.) 7 4,5 2 km (C > 2.) 1, ,125 3km(C < 2.) 1, 6,2 3 km (C > 2.) 1, ,775 4 km (C < 2.) 1, km (C > 2.).1, ,575 3

12 E. Length Relationships given as The relationship of warhead length, weight, and diameter is " 7rPw 536h + ogive cyj.j (1) for ogive lengths of 2. 5 to 4 calibers. This relationship is plotted in Figure 1 for a 4-caliber tangent ogive nose shape. The motor length to diameter ratio is (/ o r Wwh[(Q - 1)(PWF)( : I--- LD PP j ( \d/motor =DdT_ where LD is the volumetric loading density of the motor, or V prop /V motorp. pp is the density of the propellant. Figure 2 shows the effect of Q and PWF on the ratio of (l/d)motor to Wwh/d 3. V. METHOD OF COMPUTATION OF GROWTH FACTORS The method of computation of growth factors is an iteration procedure using Equations 2, 3, 6, 8, and 9. The computations were made on an IBM 162 digital computer, and the iteration was allowed to run until the burnout velocity given by Equation 2 was within.5 per cent of the required velocity given by Equation 9. VI. PRESENTATION OF GROWTH FACTOR DATA The missile weight data are presented in Figures 3 through 18 in a dimensionless form (growth factor) in order to present the maximum amount of information on a minimum number of graphs. 4

13 VII. VERIFICATION OF ACCURACY OF THE METHOD OF COMPUTATION Several point mass trajectories have been run to verify the accuracy of the growth factor data. At each of the four ranges considered, two design points were chosen for verification on the point mass deck. Table II gives the results of these runs and a comparison with the values obtained from the method used in this study. Table II VERIFICATION OF ACCURACY OF THE METHOD OF COMPUTATION Range, km Vburnout Vburnout Range, Range (Predicted) h ft/sec ft/sec km error, idz (Predicted) (Point mass) (Point mass) percent ,359 2, ,596 1, ,889 3, ,89 2, ,536 4, ,557 3, ,789 4, ,884 3, VIII. ROCKET LENGTHS Because there are so many parameters affecting the length of a rocket (range, warhead weight, diameter, specific impulse, propellant weight fraction, motor volumetric loading fraction, and propellant density), it is not practical to present parametric length data. The relationships presented in Section IV. E., however, will aid in the estimation of the length of a rocket with a given set of performance parameters. A sample problem will best illustrate the method of estimating the rocket length. Assume the following parameters: 5

14 Range Warhead weight Diameter 3 km 2 lb 1 in. PWF.5 Isp i (drag form factor) sec Warhead density. 55 lb/in. 3 Propellant density. 6 lb/in. 3 LD.5 (1/d) ogive 4 1. Compute warhead density parameter. Wwh 2 d 3 - = 3 = Obtain length of cylindrical portion of warhead from curve. (Dcy = 2.5 (From Fig. 1). 3. Compute warhead ballistic factor. Wwh ida (.)(12) = Obtain growth factor from curve. Q = 3.2 (From Fig. 12). 5. Compute rocket takeoff weight. Wto = Qx Wwh = 3.2 X 2 =64 lb 6

15 6. Obtain motor length parameter from curve. (-motor = 58 (From Fig. 2). 7. Compute motor length in calibers. ti-) = 58 X.2 = 11.6 kdlmotor 7. Compute rocket length in calibers. rkt give cyl otor 9. Compute rocket length in inches. I rkt = 18.1 X 1 = 181. IX. LENGTH-DIAMETER TRADEOFFS In the absence of diameter restraints, such as minimum diameter of warhead devices, the selection of the optimum diameter for a given rocket mission is the result of tradeoffs between the length and weight of the rocket. For fin stabilized rockets, the range of length to diameter ratios usually considered runs from 8 to 2, with the lower restraint imposed because of stabilization considerations, and the upper limit.imposed for structural reasons. Figures 19, 2, and 21 show length and weight tradeoffs for ranges of 2, 3, and 4 kilometers, based on a PWF of.65 and an Isp of 24 seconds. Similar tradeoffs can be made for other values of propulsion efficiency using data presented in this report. These figures show the relationship between length and weight of rocket for various fixed warhead weights. It can be seen that there is an asymptote at each end of the constant payload curves, showing that there is a minimum rocket weight regardless of how much the diameter is reduced. There is also a minimum length, regardless of how much the diameter is increased. 7

16 The selection of the optimum diameter for a given payload weight must therefore be based on consideration of the following factors: 1. Rocket length to diameter ratio. 2. Maximum allowable length. 3. Maximum allowable weight. 4. Special warhead considerations limiting the diameter. 5. Storage. 6. Boost acceleration requirements (acceleration is inversely proportioned to motor l/d because of interior ballistic limitations). 7. Accuracy requirements (free flight dispersion due to meteorogical effects are reduced by high ballistic coefficients, which correspond to long, slender configurations). X. LAUNCH ANGLE TRADEOFFS In certain cases it may be desirable that a system be designed to achieve a certain range at a launch angle less than the optimum, in order to reduce time of flight and to decrease the sensitiveness of the rocket to meteorological effects. Figure 22 shows the effect on range of launching at 3 Q. E., compared to launching at 45 Q. E., which is near optimum. Since this figure only shows the range reduction at the - lower Q. E., it is necessary to know the relationship between range and growth factor to compute the effect of Q. E. on growth factor required to achieve a given range. Fortunately, the relationship between range and growth factor is nearly linear, as illustrated by Figure 23. As a first approximation, the percentage increase in growth factor required to achieve a given range at a Q.E. other than 45 will be equal to the percentage range reduction given in Figure 22 for that Q. E. An example will illustrate how this correction can be made. Assume that it is desired to estimate the weight of a rocket with a range of 3 kilometers at 3 Q. E. This rocket has the following characteristics: Wwh 15 lb Diameter = 1 in. 8

17 PWF =.5 Isp = 22 i (drag form factor) = 1. I. Compute warhead ballistic factor lb/in.' idz (1.)(12) 2. Read growth factor required for 45 Q.E. from Figure 12. Q45' = Compute rocket weight at takeoff and burnout. Wto -- Q X Wwh= (3.7)(15) = 555 lb Wbo Wto - PWF (Wto - Wwh) = (555-15) = lb 4. Compute ballistic coefficient. Wbo ida (1.)(12) 5. Compute burnout velocity. 555 VB actual =.95 Isp g ln (YB) =.95(22)(32. 2) ln 555 = 3,15 ft/sec 6. Obtain range reduction for 3* firing compared to 45* firing from Figure 22. E19 =. 86 (14% reduction) R49

18 7. Compute increased growth factor (assuming a linear relationship between growth factor and range). _45* 37 = 4.3 Q3" o Compute rocket weight required for range of 3 kilometers at 3 Q. E. Wto =Q X Wwh = 4.3X 15 = 6 451b XI. CONCLUSIONS The data presented in this report will permit the rapid estimation of rocket weights and lengths for given sets of performance requirements. The data are especially useful in the determination of weight-lengthdiameter tradeoffs, and the assessment of the relative importance of the propulsion efficiency parameters, Isp and PWF. This report is intended for use in making "first-cut" approximations only, and should be followed by more detailed studies on each design point selected. 1

19 8 4-caliber tangent ogive nose shape 4 7 4, Average 6 warhead density S---.5_ U 5 5 u /. 6 U 4 z 4 /.4 ;1 /OF2. 4 JJ W wh /dý, pounds per cubic inch Figure 1. WARHEAD CYLINDRICAL LENGTH VERSUS Wwhde -FOR VARIOUS DENSITIES.

20 Propellant density =. 6 lb/in. 3 Motor loading fraction =. 5 (Prop. vol. /motor vol.) PWVF, va Growth factor Figure 2. MOTOR LENGTH PARAMETER VERSUS GROWTH FACTOR 12

21 1. 4Q BRL type 2 drag coefficient 45 Q. E. 9. Burning distance = 5% velocity loss during boost Range = 1 km 1 2 sec 8. sp U 5. o 4. Wwh/idz * Propellant weight fraction Figure 3. GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 13

22 1. 9. BRL type 2 drag coefficient 45 Q. E. Burning distance = 5% velocity loss during boost Range = 1 km 8. Is = 22 sec ' J S idz ,,., 2.o Propellant weight fraction Figure 4. GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 14

23 1. BRL type 2 drag coefficient Q. E. Burning distance = 5% velocity loss during boost Range = 1 kn 8. Isp =Z4 sec A 7., 6. U i idz o _AJ.5 o Propellant weight fraction Figure 5. GRIOWTH FACTORI VERSUS PROPELLANT WEIGHT FRAC TION. 15

24 B. 8. BRL type 2 drag coefficient 45 Q.E. Burning distance =ý 51o velocity loss during boost Range = 1 km 8I. Z6 sec S7. 6. 'U S.C ' C Propellant weight fraction Figure 6. GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 16

25 1. BRL type 2 drag coefficient 9. 45' Q.E. Burning distance 57o velocity loss during boost Range = 2 km 8. Isp 2 sec 7 Wwh ~ 7.ida ' Figure Propellant weight fraction GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 17

26 1. 9. BRL type 2 drag coefficient 45 Q.E. Burning distance 4 5% velocity loss during boost 8. Range = 2 km Isp= 22 sec 7. r Wwh If id U S5.o i.~2.t Propellant weight fraction Figure 8. GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 18

27 1. BRL type 2 drag coefficient Q.E. Burning distance = 5% velocity loss during boost Range = 2 km 8. Isp 24 sec Wwh idz _ Propellant weight fraction Figure 9. GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 19

28 1. o BRL type 2 drag coefficient 9. 45* Q.E. Burning distance = 5% velocity loss during boost Range = 2 km 8. Isp =Z6 sec U 5. id _ Propellant weight fraction Figure 1. GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 2

29 Ir 1., I I Wwh/id' BRL type 2 drag coefficient W4 Q.-E.. k5 Burning distance = velocity loss during boost Range = 3 km Isp =ZOO sec ' o Figure 11. Propellant weight fraction GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 21

30 1. BRL type 2 drag coefficient Q.E. Wwh/id. Burning distance = 516 velocity loss during SRange boost = 3 krn sp S7. 6. U Propellant weight fraction Figure 12. GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 22

31 1. I BRL type 2 drag coefficient 45* Q. E. 9. Burning distance = 5% velocity loss during boost Range = 3 km Isp 24 sec 8. Wwh/id& o " Propellant weight fraction Figure 13. GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 23

32 1. -BRL type 2 drag coefficient 9. 45T Q.E. 8. Burning distance = 5% velocity loss during boost Range = 3 km isp 26 sec 7. S~.5J Wwh/ida. 6. U S5. A o o Propellant weight fraction Figure 14. GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 24

33 1. 5 Wwh/ id 3 9. Wwh/ '4 4 U Ss BRL, typ2drgcefint 45"Q. E. 2. -Burning distance = 5% velocity loss during boost Range = 4 km Isp = 2 sec 1.o + I I I Propellant weight fraction Figure 15. GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 25

34 1. ~.5 S BRL type 2 drag coefficient 45 Q. E [ Burning distance = 9. 5% velocity loss during boost- Wwh/id& Range = 4 km disp = 22 sec S6. II 1. ' Propellant weight fraction Figure 16. GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 26

35 Io. - BRL type 2 drag coefficient 456 Q.E jwwh/jd.i _ Burning distance z 5% velocity loss during boost.5 Range = 4 km Isp 24 sec 7. o U Figure 17. Propellant weight fraction GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 27

36 1b.oo BRL type 2 drag coefficient 45So Q. E. 9. Burning distance = 5% velocity loss during boost- Range = 4 km Isp =26 sec 8. Wwh/ida o.75 4J U Propellant weight fraction Figure 18. GROWTH FACTOR VERSUS PROPELLANT WEIGHT FRACTION. 28

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42 APPROVAL Report jno. RF-TR-63-7 HERMAN R. OSWELL, Author DAVID A. MENj~tL2 Chief, Systems Integration Branch WILLIAM C. McCORKLE, JR. Director, Advanced Systems Laboratory 34

43 DISTRIBUTION Copy In accordance with U. S. Army Missile Command Distribution List A for Technical Reports, dated 11 March Director Ballistics Research Laboratories Aberdeen Proving Ground, Maryland ATTN: Mr. D. O'Neil 99-1 Commandant U. S. Military Academy West Point, New York ATTN: Librarian AMSMI-W 13 -R 14 -RK 15 -RG 16 -RR 17 -RS 18 -RL 19 -RT 11 -RF 111 -RFS RFC 123,124 -RFE 125,126 -RFSI RB RAP 154 -RH

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