Engineering Design Handbook - Design of Aerodynamically Stabilized Free Rockets

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AD NUMBER AD840582

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TO Approved for public release, distribution unlimited

FROM Distribution authorized to U.S. Gov't. agencies and their contractors; Critical Technology; JUL 1968. Other requests shall be referred to Army Materiel Command, Attn: AMCRD-TV, Washington, DC 20315.

AUTHORITY usamc ltr, 14 jan 1972

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UNCLASSIFIED

AD NUMBER AD840582

NEW LIMITATION CHANGE

TO Distribution authorized to U.S. Gov't. agencies and their contractors; Specific Authority; JUL 1968. Other requests shall be referred to Army Materiel Command, Attn: AMCRD-TV, Washington, DC 20315.

FROM Distribution: Further dissemination only as directed by Army Materiel Command, Attn: AMCRD-TV, Washington, DC 20315, JUL 1968, or higher DoD authority.

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AMCP 706-2$0

AMC PAMPHLET

0 a

KJ:

ENGINEERING DESIGN HANDBOOK

DI DSIGN OF AERODYNAMICALLY

STATEMREN' #2 UNCLASSIFIED ±i&.t'

doaci.leL

is

su!ject to special export

1

controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of: Army Materiel

Command,

L---

ntvrQ

Attn: AMCRD-TV, Washin,. on, D.C. _

20315

JULY 1968

HEADQUARTERS, U.S ARMY MATERIEL COMMAND

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.~-.--'-~.-.--.41

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PREFACE The Engineering Design Handbook Series of the Army Materiel Command is a coordinated series of handbooks containing basic information and fundamental data useful in the design and development of Army materiel and systems. The handbooks are authoritativt, reference books of practical information and quantitative facts helpful in the design and development uf Army materiel so that it will meet the tactical and technical needs of the Armed Forces. 0; t

This handbook pro -ides extremely useful data for the engineer prmarily interested in the preliminary design of aerodynamically stabilized free rockets. The data are arranged in a convenient format-tables, graphs, and solution guides-which permits ready access and easy application in order to make possible the rapid respons- rq--ired of preliminary design activities. As a bonus, the chapter arrangemetnt provides each technical area having responsibilities in the preliminary design phase with an appreciation for the data renuirements and applications of the supporting technical areas. The pieparation of this handbook was initially an in-house effort of the U. S. Army Missile Command. The organizatica of the text, data, and much of the written material originated with that agency The Chrysler Corporction Space Division, Hunthville, Alabama, under subcontract to the Engineering Handbook Office of Duke University, prime contractor to the Army Research Office-Durham for the Engineering Design Handbook Series-with the 2ontinuet -issistance of the U. S. Army Missile Command-completed the handbook. The Handbooks are readily available to all elements of AMC mcl, ding personnel and contractors having a need and or requirement. The Army Materiel Command policy ir, to release these Engineering Design Handbooks to other DOD activities and their contractors, and other Government agencies in accordance with urrent Army Regulation 70-31, datei' 9 September 1966. Procedures for acquiring thpse Handbooks follow a.

Activities within AMC and other DOD agencies should direct their

rpntipqt on an official form to:

Publications Distribution Branch Letterkenny Army Depot ATTN: AMXLE-ATD Chambersburg, Pennsylvania 17201 b. Contractors who have Department of Defense contracts should submit their request, through their contracting officer with proper justification, to:

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Director Defense Documentation Center (for Scientific and Technical Information) Cameron Station Alexandria, Virgin'a 22314 c. Government agencies other than DOD may submit their request directly to: Commanding General U. S. Army Materiel Command ATTN: AMCAD-PP Washington, D. C. 90315 or Director Defense Documentation Center (for Scientific and Technical Information) Cameron Statio, Alexandria, Virginia 22314 d. industry not having a Governmont cntract (this includes Universities) must forward their requests to: Commanding General U. S. Army Materiel Command ATTN: ANICRD-TV Washingtor., D. C. 20315 e. All foreign requests must be submitted through the Washington, D. C. Embassy to: Office of the Assistant Chief of Staff for Intelligence ATTN: Foreign Liaison Office Department of the Army Washington, D. C. 20310 All requests, other than those originating within the DOD, must be accompanied by a valid justification. Comments and suggestions on this handbook are welcome and should be addressed to Army Research Office-Durham, Box CM, Duke Station, North Carolina 27706.

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AMCP 703-280

TABLE OF CONTENTS PREFACE LIST OF ILLUSTRTIONS LIST OF TABLES

ix xvi

CHAPTER 1. INTRODUCTION CHAPTER 2. ATMOSPHERIC DATA Paragraph

3-1 2-2 2-2.1 2-2.2 2-2.3 2-2.4

Page

Introduction .. Atmospheric Properties . Atmospheric Density, Temperature and Pressure Winds, Upper Level Winds, Lcwer Level Regional Annual and Seasonai Density Models

2-1

References

2-4

2-1 2-1 2-1 2-2 2-2

..

CHAPTFR 3. SYSTEM DFSICN S3-2

3-1 3-2.1.1 3-2.1.2 3-2.1.3

z

General Cla.-ses of Rockets Artillery Infantry

Air Defense

3-2.14

Armor

3-2.1..

Aviation

3-2.1.6

Logitz-ic

3-2.1.7 3-2.2

3-2.2.1 3-2.2.2 3-2.2.3 3-2.2.4 3-2 2.5 3-3 3-3.1 3-3.2 -, -3-3.3 3-3 4 3-3.5 3-3.6 3-3.7 3-4 3-4.1 3-4.2 34 .2.1

..

3-1

.

.... ..

...........

....

..

..

..

3-2 .....

3-2

3-2 3-2 3-2

.....

,.

General .. ......... Meteorological .................... iigh Altitude Sounding .. ...

..

Satellites . .................... Dispensing ... ........ Operational Modes ................ General ..... ... .. ..... .. ..... . Ground-to-Ground ... ..... Ground-to-Air ....... ... .......... Air-to-Air ....... Air-to-Ground .... ....................... Underwater-to-Air ............................ Surface/Air-to-Underwat.. Launching Methods. . ........ ................. General .3.............. . Single ....

3-1

.....

.........

Support Research Rocket Systems

Rail Launchers

3-1 3-1

.

...........

........................................

...

3-2 3-3 3-3 3-3 3-3 3-3 340 .

.

3-4

3-4

.

3-4 3-4 3-4 3-4 3-4 3-4

3-5 III

Vf

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TABLE OF CONTENTS (cont) Pagr

Paragraph 3-4.2.2 3-42.3 3-43 3-4.3.1 34.3.P 3-4.3.3 3-4.3.4 34.3.5 3-4.3. 3-4.4 3-4.5 3-4.5.1 3-4.5.2 3-4.5.3 3-4.6

Systetra nts ................................. 3-8 3-8 ................ General Rocket ... ........ ..................... . . .. 3-8 Warhead Design....................... -3..3-8 Motor ....................... .. 3-9 Structure ............ ..... ........... . 3-15 .. 3-i Launcher .na.c........................... '-13 Anillary Equipment ............................ Concept Selection . ............................ 3-13

3-6.1

3-13 Requirements .... ....... ....... .......... Constraints .................................. 3-14 ............ ................. 3-14 Parametrics sYste.m Selection . ............ 3-14 .... .............. 3-14 Preliminary Design Payload .. ........................... ..... .. 3-15

3-7.2 3-7.3 3-7.4

Populsion on .... ......... ....... Aerodynamics .. ......................... ..... . . Dynamics . . ........... ....... .............

3-15 3-15 3-15

3-7.5 3-7.6 3-7.7 3-8 3-9

Strueturet. . Performance Estimates Auxiliary Devices ..... ....................... Design Optit -ft tion ., . . . . .3-16 System Integration .. .... ...............

3-15 3-15 3-15

4-i0

..... .................... Ceotn......... ............ ... . Static Testhig Flight Testing ... ............. ........ Structural Testing . .. .... Aerodynamic Testing . ..... ..... ......... Envirowtnental Testing ....................... Cost Effectiveness ......... ......................

3-18 3-17 3-17 3-17 3-18 3-18 3-18

3-10.2 3-10.3 3-10.4 3-10.5 3-11 IV

3-5 3-5 3-5 3-5 3-5 3-5 3-6 3-6 33-6 3-6 3-6 3-7 3-7 3-7

3-5 3-5.1 3-5.2 3-5.2.1 3-52.2 3-5.2.. 3-5.3 3-5.4 3-6 3-6.2 3-6.3 3-6.4 3-7 3-7.1

Il.3-10.1

.......... Multiple Helical .................................... ... ...... Tube Launcherm ......... Single ............ . Multiple ... Open Breech ... Close B.~lreech . .. ............. Restricted Breech ............. .. Gaffing Other Launcher Types . ................. .. .. Variations Autospin .. ............ ............... ........ Prespin, Automatic Dynamic-Alignment (PADA) Spin-on-Straight-Rail (SOSR) Meth ods of Transport . . .. ......

3-16

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TABLE OF CONTENTS (cont) Paragraph

KCHAPTER

,

V 1

V

4 ,

"

Page

4. PERFORMANCE PAPAMETRICS 4-1 4-2 4-2.1 4-2.2 4-2.3 4-3 4-3.1 4-3.1.1 4-3.1.2 4-3.1.3 4-3.2 4-3.2.1 4-3.2.2 4-3.2.3 4-3.3 4-4 4-41 -4.1.1 4-4.1.2 4-4.2 4-5 4-5.1 4-5.1.1 4-5.1.2 4-5.2 4-6 4-6.1 4-6.1.1 44.1 2

Symbols Introduction Performance Parameters Performance Factors Propulsion System Factors Aerodynamic Considerations Appr,,ximation Techniqi~es and Applicable Equations Estanation of Velocitq Requirement Indirect-Fire Systems Direct-Fire Rockets Sounding Rockets Estimation of Rocket Motor Requirements Sperific Impulse and Booster-Mass Ratio Propellant-Weight Fraction Growth Factor Summary Parametric Performance Data for Indirect-Fire Systems Delivery Techniques Trajectory Profile Energy Management Techniques Parametric Performance Data Parametric Performance Data for Direct-Fire Systems Delivery Techniques Trajectory Profiles Energy-Management Techniques Parametric Performance LDa Parametric Performance Data for Sounding Rockets Delivery Techniques Trajectory Profile Energy-Management Techniques

4-1 4-2 4-2 4-2 4-2 4-2 4-3 4-3 4-3 4-3 4-5 4-5 4-5 4.5 4-5 4-5 4-8 4-8 4-8 4-8 4-8 4-12 4-15 4-12 4-12 4-13 4-15 4-15 4-15 4-15

4-t;.2 4-7 4-7 1 4-7.1.1 4-7.1.2 47 4-8

Parametric Performai.ce Data Parametric Pe'formance Data for Surface-to-Air Rockets Delivery Techniques Trajectory Profile Energy-Management Techniqu2s Paramtr.ic 0-.i ...... Data Numerical Examiple

4-15 4-16 4-16 4-16 4-16

CHAPTER 5. 5-1 5-2

5-2.1 S'Q

5-2.1.1

4-18 4-19

PRCPULSION

Symbols ... .... ......... General ................... Nozzle .... Thermodynamic Rclations .... Ideal Flow

5-1 5-2

5-3 5-4 5-4 V

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TABLE OF COITENTS (cant) ,1

Paragraph

5-2.1.2

Pay'e

Real Mow

5-7

...............

5-2.2 5-2.3 5-3 5-3.1 5-3.l 5-3.1.2 5-3.1.3 5-3.L4 5-3.2 5-3.3 5-4

Noz,,e Contours ... Nozzle Erosion Propellants Grain .. Chemical Compositions ... .... .. Configuration Geometry Burning Rate Erosion . ................ Ignition .......... .. Hodling ....... .................. Internal Ballistics .. ...

5-5 5-6

Scaling of Solid Propellant Motors

.

....

5-7 5-7 5-7 5.7 5-10 5-10 5-11 5-12 5-12 5-13 5-13 5-13

.15

References ................................. CHAPTER 6. 6-1 6-2 6-2.1 6-2.2 6-2.3 6-3 6-3.2 6-3.2

641 6-42 6-403 6-4.4 6- .5 6-5 qu-'

6-6.1 6-6.L1 &S6.L 2 6-6.13 6-6.1.4 6-6.1.5 "2 6-6.3 6-6.4

6-7

Ct

5-16

STRUCTURES

Symbols ..................................... General ........ Weight and Balance .. ............. Mass and Center of Gravity 'Sstimation . Pitch Inertia . . ........................ Roll Inertia ..................... ............ Loads ...... ............. .......... Transport and Handling Loads Flight Loads .............................. FlgStrea .................................

6-1 6-2 6-2 6-2 6-2 6-7 6-10 6-10 6-11 6-13

Beams ......... .. . 6-13 Columns .... .... ....... .... 6-15 etP Vessels .......... ............. .... 6-15 Plates ............................... . 6-15 Joints ..... ..... .... . ......... 6-15 Safety Factors ........ 6-19 n~auing ................ . ............... b-19 General ...... 6-19 Conduction Heat Transfer ........ 6-19 Radiation Heat Transfer .................... 6-20 Convection Heat Transfer ................. 6-21 Conibined Heat Transfer 6-21 Transient Heat Transfer 6-21 Combustion Chamber Heating ............... 6-2 Exhaust Plume Heailng .... 6-22 Aerodynamic Friction Heaing . ......... 6-22 Tting ....... 6-24 Refer e es .. '.. ........... . . ....... . ........ 6-2

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TALF OF CONTENTS (cont) Paragrapi

Pagr CHAPTER 7. ACCURACY

-

Introduction

7-2 7-3 7-' 1

7-2 7-2 7-2

7-7.1 7-7.2

Definitions of Error Sources Design Considerations Influencing Accuracy Design Considerations Associated With Speed Change Errors Design Considerations Associpted With Angular Errors The Effect of Aerodynamic Stbdilty The Effect vf Wind The Effect of Thrust Mala.inment The Effect of a Slow Spin Tie Effect of Dispersion Reduction on the Optimum or Prelaunch Error. Aiming Errors Errors Due to Variations in Meteorological Conditions Calculation of Angular Errors. Launch Phase Errors Angular Velocity Translational Velocity Dynamic Unbalance Propulsion Phase Errors Nonrotating Rucket Dispersion Reducton Technques

7-7.2.1 7-7.2.2 7-7 2.3 7-7.2.4 7-7.2.5 7-7.2.6 7-8 7-.1 7-8.2 7-8.2.1 7-8.2.2 7-8.2.3 7-8.2.4 7-8.2.5 7-8.2.6 7-8.2.7 7-8.2.8 7-8.3 7-8.3.1 7-8.3.1.1 7-8.3.1.2 7-8.3.2 7-8.3.2.1

Constait Spin Rate Constant Spin Acceleration Slowly Unifinily Decreasing Sri (SUDS) Spin-Buck Prespin Automatic Dynamic Alignment (PADA) Variable Acceleration Ballistic Phase Errors Forces Acting on the Project!c Sources of Error Errors Due to Winds Change in Drag Nonstandard Condition,; Malalignment of Fins Static Unbalance Dynamic Unbalance Curvature of the Trajectory Fuzing Errors Calculation of Dispersion Launch Error. Malaim .. Mallaunch Propulsion Errors Wind •. •

-7-7

•

C,

7-1

7-1

7-3.2 7-3.2.1 7-3.2.2 7-3.2.3 7-3.2.4 7-3.25 7-4 7-4.1 7-4 2 7-5 7-6 7-6.1 7-6.2 7-6.3

--. -

Symbols

"

7-2 7-3 7-3 7-3 7-3 7-5 7-7 7-7 7-7 7-8 7 7-10 7-10 7-10 7-14 7-14 7-14 7-18

7-18 7-20 7-20 7-20 7-24 7-24 7-25 725 7-25 7-26 7-27 7-97 7-27 7-27 7-27 7-27 7-27 7-27 7-28 7-28 .... 7-29 7-30 7-30 vii

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TABLE OF CONTENTS (cont) Pagr

Paragraph 7-8.3.22 7-8.3.2.3 7-8-.3.3 7-8.3..1 7-3.3.32 7-8.3.3.3 7-8.3.4 7-8.3.5 7-9 7-9.1

f

7-9.1.1 7-9.1-2 7-9.1.3 7-92 7-9.3 7-10 7-10.1 7-10.2 7-10.3

Thrust MiJalignmerit impulse Vae-iatio.i Balliatic Errors .7-j0 Density Buflistic Wind Z&a3istic Coefficient Tabulation o, Result', Additional Reference Grapl's Statistical Method, Measures of Di.,persion for One Error Source Variance Standard Deviation Probable Error Measures o1 Dispersion for Several Error Sources Use of Figures 742 and 7-43 Computation of Accuracy Range Probable Error (RPE) Deflection Probable Error (DPE) Circular Probable Error (CPE) References CHAPTER 9.

Vill

7-31 7-31 7-31 7-31 7 31 7-32 7-32 7-32 7-32 7-34 7-34 7-34 7-35 7-35 7-35 7-92

AERODYNAMICS

8-1 8-2 8-2.1 8-2.1.1 8-2.1.2 8-2.1.3 8-21.4 8-2.2 8-2.3 8-2.4

General Design Considerations Stability Characteristics of Rockets Bodies of Revolution Nose Cylinder Boattail Conical-Flare Afterbody Oversize Head Configtxations Fins Tailof Complete Configuration ,8-22 Ring Stability

8-1. 8-3 8-4 8-4 8-4 8-5 8-5 8-16 8-17 8-22

8-2.4.1 8-2.4.2 8-2.4.3 8-2.4.4 8-3

General Fin-Body Inter-erence Fin-Fin Interference Sample Calculation Sheet Drag

8-22 8-34 8-37 8-42 -

SymboLs

vii

7-30 7-30 7-30

8-3.1 8-3.1.1 8-3.1.2 8-3.1.3 8-3.1.4 8-3.1.5 8-32 8-3.3 8-3.3.1

.....

...

Wave Drag Nose Wave Drag ..... Boattail Wave Drag Flare Wave Drag Fin Wave Drag Ring Tail Wave Drag Friction Drag Base Drag Body-of-Revolution Base Drag, Rocket Jft Off

3-44 8-44

8-45 b-6 8-43 8-48" 8-51 P-51 8-51

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TABLE OF CONTENTS (cont) Paragraph

3-3.3.2 8-3.3.3 8-3.4 8-3.4.1 8-3.4.2 8-4

Page

Body-of-Revolution Base Drag. Rocket Jet On Fm Base Drag Drag Characteistics of Complete Configurations Interference Effects-Fin on Iase Compuitational Table Aerodynamic Testing References Index

F-63 8-65 8-65 8-65 8-66 8-67 8-71 1-1

LIST OF ILLUSIRATIONS Fig. No.

2-1 2-2 2-3 2-4(A) 2-4 (B) 2-4(C) 2-4(T) 2-4(E) 2-4(F) 2-4(G) 4-1 4-2 4-3 4-4

*

4-5 4-6 4-7 4-A 4-9 4-10 4-1 , 4-12 i" 4-13

Title

Maximum Speed and Associated Shear Stror -st Wind for Temperature Range Density Deviation Versus Altitude-Worldwide, Annual Subtropics and Tropics-Annual Density Model Temperate Zone--Annual Density Model Polar Zone-Annual Density Model Subtropics-Seasonal Density Models 2ropics-S-.azonal Density Models Temperate Zone-Seasonal Density Models Polar Zone-Seasonal Density Models Effect of Ballistic Coefficient on Burnout Velocity Effect ut Ideal Burnout Velocity on Booster-Mass RR.'io f , -zt f Growth Factor cr Ideal Burnout Vekoc,#.y Indirect Fire-All-Boost; Ffect of Thrust-toWAVeigbt Ratio on Optizium Launch Quadrant Blevatio:. Indirect Fire-Boost 'Sustain; Efect ,ofImpulse Ratio on Optimum Latuch Quadrant Elevation Indirect Fire-All-2oost; Effeec of Range on Growth Factor Boost/Sustain Engine; Variation of Specific Impulse With Thrust ~ Tndliragpt Fire -B^ost,'""Indirect Fire-Boost/Sustain, Effect of Range on Growth Factor Indirect Fire-All-Boost; Effect of Propellant Weight Fraction on Growth Factor Indirect Fire-All-Boost; Ffect of Ballistic Coefficient on Growth Factor Direct Fire-Boost/Sustain; Effect of impulse Ratio on Time to Target Direct Fire--All-Boost; Effect of Growth Factor on Minimum Time to Target

Page

2-3 2-12 2-13 2-19 2-20 2-21 2-22 2-23 2-24 2-?5/,2-26 4-4 4-6 4-7 4-9 4-9 4-10 4-10 A$n41..1.. A2~

4-11 --12 4-2.2 4-13 4-14

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LIST OF ILLUSTRATIONS .cont) Fig9. No.

Title

Direct Fire-Ail-Boost; Fffect of Ballistic Coefficient on Growth Factor 4-15 Direct Fire-All-Boost; Effect of Propellant Weight Fraction on Growth Factor 4-16 Sounding Rocket-All-Boost and Boost/Sustain; Effect of Growth ractor on Summit Altitude 4-17 Sounding Rocket-A)l-Boost; Effect of PrGpellant Weight Fraction on Growth F'actor 4-18 Sounding 'Rocket-All-Boost; Effect of Ballistic Coefficient on Growth Factor 4-19 Surface to-Air-All-Boost; Effect of Time to Altitude on Growth Factor 4-20 Surface-to-Air-All-Boost; Effect of Propellant Weight Fraction on GrowLh Factor 4-21 Surface-to-Air-All-Boost; Effect of Ballistic Coefficient on Growth Factor 4-22 Flow Diagram 5-1 Schematic of a Case-Bonded, Unrestricted-Burning Solid-Prpellant Rocket Motor 5-2 Subsonic Flow Through a Converging Nozzle 5-3 Flow Through a Supersonic (DeLaval) Nozzle 5-4 Supemonic Nozzle Area Expansion Ratios 5-5(A) Thrust Coefficient Versus Aita Ratio for y = 1.3 5-5(B) Thrust Coefficient Versus Area Ratio for y 1.2 5-6 Examples of Grain Cros i-Sections 5-7 Typical Grain Installations ..... 5-8 Typical Igniter lVolumes of Cones . 6-2(A) Ratio of Volume of Ogiv to Cone Versus 1/r at V,"riour 1id's 6-2(B) Ratio of Volume of Ogive to Cone Versus Z/r at Various I/d's 6-2 (C) Ratio of Volume of Tangent Ogive to Cone With Identical t/d 6-3 Surface Area of Cones 6.4(A) Ratio of Area of Ogive to Cone With Identical lid Versus /r at Various lid's Less Than or Equal To 0.5 C-4 (B) Ratio of Area of Ogive to Cone With Identical l/d .5 A . Th-', ....t t Vijc17,, aVeI 64(C) Ratio of Area of Tangent Ogive to Cone With Identical lid 6-5 Axial Loads on Free Flight Rocket 6-6 Concentrated Bending Loads on Free Rocket 6-7 Circamferential Loads on Combustion Chamber 6-8 Beam-Sction Load Distribution 6-9 Flat Plate Stress Width-to-Length Ratio Parameter 6-10 Riveted and Bolted Joints 6-11 Welded Joints .. . 6-12 Bolted Joints 6-13 Plane Conduction Heat Transfer Medium 6-14 Cylindical Conduction of Heat Transfer Medium

Page

4-14

X

4-14 4-15 4-16 4-17 4-17 4-18 4-18 4-19 4-20 5-2 5-3 5-5 5-6 5-8 5-9 5-11 5-11 5-12 6-5 6-6 6-6 6-7 6-3 6-9

6-10 6-11 6-12 6-13 6-14 6-17 6-17 6-18 6-19 6-23 6-20

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LIST OF ILLUSTRATIONS Fig. No. 6-15 7-1 7.2 7-3 7-4 7-5 7-6 7-7

,

7-8 7-9 7-10 7-11 .

7-12 (A i 7-12 (B) 7-12 (C) 7-13 (A)

:ont)

Title

Page

General Geomet ry of Rocket Exhaust Plumes Variation of Angular Dispersion and Wavelength of Yaw During Flight Effect of Wind on an Aerodynamically Stable Rocket Effect of Wavelength of Yaw on Angular Dispersion Due to Wind Effect of a Thrust Malalignment on an Aerodynamically Stable Rocket Effect of Wavelength of Yaw on Angulhr Dispersion Due to Thrust Malalignment Optimum Wavelength of Yaw for Minimum Total Dispersion Growth of Angular Dispersion for a Rocket With a Thrust Malalignment and No Spin Growth of Angular Dispersion for a Rocket With a Thrust Malalignment and a Slow Spin Effect of Spin on the Build-Up of Angular Dispersion Due to Thrust Malalignment Aiming Errors Definitions of Sign Conventions for the Rocket Equations of Motion Angular Dispersion Due to Mallaunch-Initial Angular Rate Angular Dispersion Due to Mallaunch- -initial Angular Rate Angular Dispersion Due to Mallaunch-Initial Angular Rate. Angular Dispersion Due to Mallaunch-Initial Translational

6-23

Velocify

7-12

7-13 (B) Angular Dispersion Due to Mallaunch-Initial Translational Velocity 7-13 (C) Angular Dispersion Due to Mallaunch-Initial Translational Velocity 7-14(A) Angular Dispersion Due to Wind 7-14(D) Angular Dispersion Due to Wind 7-14(C) Angular Dispersion Due t( Wind-Initial Translational 'velocity . .................... 7-15 (A) Angular Dispersion Due to Thrust Malalignment-Zero Spin 7-15 (B) Angular Dispersion Due to Thrust Malalignment-Zero Spin 7-15(C) Angular Dispersion Due to Thrust Malalignment-Zero Spin 7-16 Effect of Constant Spin on Angular Dispersion 7-17(A) Constant Spin 7-17 (B) Constant Spin 7-17(C) Cor.,-.nt Spin 7-18(A' Constant Spin Acceleration 7-18(B) Constant Spin Acceleration 7-18 (C) Constant Spin Acceleration 7-19(A) Slowly Uniformly Decreasing Spin (SUDS) 7-19(B) Slowly Uniformly Decreasing Spin (SUDS) 7-19(C) Slowly Uniformly Decreasing Spin (SUDS)

7-4 7-4 7-4 7-5 7-5 7-5 7-5 7-5 7-6 7-8 7-8 7-11 7-11 7-12

7-13

.7-33

2 •

__

_X!

7-15 7-15 7-16 7-16 7-17 7-17 7-48 7-19 7-19 7-20 7-21 7-21 7-22 7-22 7-23 7-23

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LIST OF ILLUSTRATIONS (cont) Fig.N,.

Page

Title

Effect of Wavelength of Yaw on Buck Distance for .. ...... Zero Angular Dispersion ... Effect of Buck Distance on Dispersion Reduction 7-21 Action of Winds on a Free Rocket 7-22 7-23 Initial Velocity Versus Maximum Range 7-24(A) Unit Effect, Range/Departure Angle Versus R/Rma--Impact Fuze 7-04(B) Unit Effect, Range/Departure Angle Versus R/Rmax--Impaet Fuze 7-24(C) Unit Effect, Range/Departure Angle Versus R/Rmax--Impact Fuze 7-25(A) Unit Effect, Range/Velocity Versus R/Rmax-Impact Fuze 7-25(B) Unit Effect, Range/Velocity Versus R/Rmax-Imp~ct Fuze 7-25(C) Unit Effect, Range/Velocity Versus R/Rmax-lmpact Fuze. 1-26(A) Unit Effect, Range/Density Versus R/Rmax--npact Fuze 7-26(B) Unit Effect, Range/Density Versus R/Rmax--Impact Fuze, 7-26(C) Unit Effect, Range/Density Versus R/Rmax--Impact Fuze. 7-27(A) Unit Effect, Range/Wind Versus R/Rrnax-Impact Fuze.... 7-27(B) Unit Effect, Range/Wind Versus R/Rmax-Impact Fuze. 7-27(C) Unit Effect, Range/Wind Versus R/Rm"x--Impact Fuze 7-28(A) Unit Effect, Deflection/Wind Versus R/Rmax-Imract Fuze.. 7-28(B) Unit Effct,Deflection/Wind Versus R/Rmax-Impact Fuze 7-28(C) Unit Effect, Deflection/Wind Versus R/Rmax-Impact Fuze.. ............ 7-29(A) QE Versus R/Rmax--Impact Fuze 7,29(B) QE Versus R/Rmax-Impact Fuze .. ............. .. ............. 7-29(C) QE Versus R/Rmax-Impact Fuze . t Fuze .......... 7-30(A) Time of Fight Versus R/Rmax-Impac 7-30(B) Time of Flight Versus R/Rmax-Impact Fuze .. 7-30(C) Time of Flight Versus R/Rma-Inpact Fuze ............ 7-20

....

7-31(A) 7-31(B) 7-31(C) 7-32(A) 7-32(B)

QE Versus R/Rmax-Time Fuze ............... QE Versus R/Rmax--Time Fuze .................. QE Versus R/Rmax--Time Fuze ................... Time of Flight Versus R/Rmax-Time Fuze Time of Flight Versus R/Rmax-Time Fuze

7-32(C) 7-33(A) 7-_° .I R I 7-33(C)

. ... Time of Flight Versus R/Rmax-Time Fuze Unit Effect, Range/)ensity Versus R/Rmnax-Ti.e Fuze T~n~t ify Versus Fu5e R/Rmax-Time F,.v' Versus R/R----Tirre Unit r~ffrt, Effect, Ranae/De, Range/Density

77- (A) Unit Effect, Range/Velocity Versus R/(riax---Time Fuze 7-34(b) 7-34(C) 7-35(A) 7-35(B) 7-35(C) 7-36(A)

01

Xii

Unit Effect, Range/Velhcity Versus R/Rmba-.- Time Fuze. Unit Effect, Range/Velocity Versus R/Rmax-TimeFuze Unit Effect, Range/Wind Versus R/Rmax-Time Fuze Unit Effect, Range/Wid Versus R/Rmax-Time Fuze Unit Effect, iRange/Wind Versus R/Rmax- Time Fuze Unit Effect, Range/Departure Angle Versus R/'max-Tute .... .................................. Fuze

..

7-24 7-25 7-26 7-29 7-36 7-37 7-38 7-39 7-40 7-41 7-42 7-43 7-44 7-45 7-46 7-47 7-48 749 7-50 7-51 7-52 7-53 7-54 7-55 7-56 7-57 7-58 7-59 7-60 7-61 7-62 7-63 7.-M 7-65

7.66 7-67 7-68 7-69 7-0 7-71 7-72

A

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K

AMCP 703-260

CAjLIST

OF ILLUSTRATIONS (cont) Fig. No.

Title

Pager

7-36(B) Unit Effect, Range/Departure Angle Versus R/Rmax--Time

0

Fuze 7-36(C) Unit Effect, Range/Departure Angle Versus R/Rn,ax-Time Fuze

7-37 (A) 7-37 (B) 7-37 (C) 7-38 (A) 7-38 (B) 7-38(C) 7-39(A) 7-39(B)

7-73 7-74

......

Unit Effect, Deflection/Wind Versus R/Rmax--Time Fuze Unit Effect, Deflection/Wind Versus R/Rmax-Time Fuze Unit Effect, Deflection/Wind Versus R/Rmax-Time Fuze Unit Effect, Altitude/Density Versus R/Rmax-Time Fuze Unit Effect, Altitude/Density Versus R,/Rmax-Time Fuze Unit Effect, Altitude/Density Versus R/Rmax-Tme Fuze Unit Effect, Altitude/Velocity Versus R/Rmax-Time Fuze Unit Effect, Altitude/Velocity Versus R/Rmax-Time Fuze

7-75 7-76 7-77 7-78 7-79 7-80 7-81 7-82

7-39(C) Unit Effect, Altitude/Velocity Versus R/Rmnax-Time Fuze 7-83 7-40(A) Unit Effect, Range/Time Versus R/Rmay.--Time Fuze 7-84 740(B) Unit Effect, Range/Time Versus R/Rmax- i:me Fuze 7-85 740(C) Unit Effect, Range/Time Versus R/Rmax- Tune Fuze 7-86 7-41 (A) Unit Effect, Altitude/Time Versus R/Rmax-Time Fuze 7-87 741 (B) Unit Effect, Altitude/Time Versus R/Rmx--Time Fuze 7-88 7-41 (C) Unit Effect, Altitude/Time Versus R/Rmax-Tiie Fuze 7-89 7-42 Ratio of CPE to a, la y for Elliptical Distribution 7-90 7-43 Chart for Determination of Circular Probable Error 7-91 7-44 Variation of Range Probable Error With Range-Impact Fuze 7-91 7-45 Variation of Deflection Accuracy With Rang--Impact Fuze 7-91 7.46 Variation of CPE With Range-Impact Fuze 7-91 8-1 Apparent Mass Factor ... 8-4 8-2 Normal Force Coefficient Gradient for Tangent Ogive-Cylinder Configurations 8-6 8-3 Center of Pressure for Tangent Ogive-Cylinder Configurations 8-7 8-4 Normal Force Coefficient Gradi mt for Cone-Cylinder Configurations ... ........ 8-8 8-5 Center of Pressure for Cone-Cylinder Configurations .... 8-9 8-6(A) Normal Force Coefficient Gradient and Center of Pressure4-Caliber Tangent Ogive With Varying Afterbody Length . .8-10 8-6(B) Normal Force Coefficient Gradient and Center of Pressure7.1250 Cone With Varying Afterbod'y Length 8-11 8-6(C) Normal Force Coefficient Gradient and Center of Pressure-1/2-Power Nose With Varying Afterbody Length 8-12 7.--1 A I -. Fr..! Cccfficc... .dn ,G. Ce ntr. o -r--

U.

f

8-7 (B)

Varying Tangent Ogive Nose Length With Constant Afterbody Length of 6 Calibers Normal Force Coefficient Gradient and Center of PressureVarying Conical Nose Angle With Constant Afterbody Length of 6 Calibers ...

8-7(C) 8,-8

_

_

_

__

....

Normal Force Coefficient Gradient and Center of PressureVarying n-Power Nose Shape With Constant Afterbody Length of 6 Calibers Normal Fo:ce Coefficient Gradient for a Boattail ..

8-13

8-14

8-15 8-16

_

_

_

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At=' MC-23L

LIST OF ILLUSTRATIONS (cont) Fig.No.

S8-10(D)

j

"'tle

Page

8-9

Center of Pressure for a Boattail

8-10(A) 8-10(B) 8-10(C) 8-10(E)

Incremental Normal-Force Incremental Normal-Force Incremental Normal-Force Incremental Normal-Force Incremental Normal-Force

.......

8-17

Coefficient Gradient for a Flare Coefficient Gradient for a Flare Coefficient Gradient for a Flare Coefficient Gradient for a IFiare Coefficient Gradient for a Flare

8.,7 8-18 8-18 8-19 8-19

8-11 8-12(A) 8-12 (B) 8-13(A)

Subsonic Fin Normal-Force Coefficient Gradient Normal-Force Coefficient Gradient for Rectangular Fins Center of Pressure for Rectangular Fins Fin Normal Force Coefficient Gradient at Supersonic Mach Numberq 8-13(B) Fin Normal Force Coefficient Gradient at Supersonic Mach Numbers 8-13(C) Fin Normal Force Coefficient Gradient at Supersonic Mach Numbe's

8-24 8-25 8-26

..

....

8-27

8-14(C) Fm Center of Pressure

8-28

.....

Fin Normal Force Coefficient Gradient Correction Factor for Sonic Leading Edge Region

8-16

8-23

...

8-14(A ) Fin Center ot Pressure 8-14(B) Fin Center of Pressure 8-15

8-20 8-21 8-21

.

.........

8-29

Normal Force Coefficient Gradient arid Center of Pressure for Rectangular Fins

..........

.......

8-17(A) Incremental Normal Force Coefficient Gradient for a Ring Tail Mounted on a Cylindrical Afterbody ... 8-17(B) Incremental Normal Force Coefficient Gradient for a Ring

8-29 8-30

Tail Mounted on a Cylindrical Afterbody

8-30

8-17 (C) Incremental Normal Force Coefficient Gradient for a Ring Tail Mounted on a Cylindrical Afterbody

8-31

....

8-17 (D) Incremental Normal Forke Coefficient Gradient for a Ring Tail Mounted on a Cylindrical Afterbody

.

..

8-17(E) Incremental Normal Force Coefficient Gradient for a Ring Tail Mounted on a Cylindrical Afterbody. ......... 8-17 (F) Incremental Normal Force Coefficient Gradient for a Ring Tail Mounted on a Cylindrical Afterbody

8-18 8-19 8-20 8-21(B) 8-21(C) 8-21(D) 8-21(E) 8-21(F) 8-22

Lift of Fin (Body) for at/m = 0.2 (Clipped Delta Fin) Lift of Fin (Body) for at/m = 0.4 (Clippl Delta Fin) Lift of Fin (Body) for at/m = 0.6 (Clipped Delta Fin) Lift of Fin (Body) for at/m = 0.8 (Clipped Delta Yin) Lift of Fin (Body) for at/m = 1.0 (Delta Fin) Normal Force Coefficient Gradient of Multiple Fins

8-23

Fizx Geometry

8-24

Body Geometry

at Supersonic Speeds

......

8-35 8-36 ..

..

8-38

8-33 8-39 8-39 8-40 8-40

8-41

.......

................... ..

8-32 8-32

..

Values of Lift Ratios Based on Slender-Body Theory Interference Effects of Fin on Body . .8-35 Lift Factors-Influence of Fin on Body

8-21(A) Lift of Fin (Body) for at/m = 0 (Rectangular Fin)

8-31

.

8-42 8

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AMCP 703-280

LIST OF ILLUSTRATIONS (cont) Fig. No.

': I'8-26(A) :. Tt

[

Al

8-25

Title Configuration and Design Data for Numerical Example Effects of Mach Number and Nose Fineness Ratio on

Wave Drag 8-26(B) Effects of Mach Number and Nose Fineness Ratio on Wave Drag 8-27 Wave Drag Coefficient of Optimum Secant Ogive Cylinder at Transonic Speed 8-28 Wave Drag Coefficient of Slender Ogives at Transonic Speeds 8-29 Wave Drag Coefficient of Cones and Ogives at Supersonic Speeds 8-30 Wave Drag Coefficient of Conical Boattails at S:personic Speeds Wave Drag Coefficient of Parabolic Boattails rt Supersonic 8-31 Speeds 8-32 Wave Drag Coefficient of a Boattail at Transonic Speeds 8-33 (A) Wave Drag Coefficient of Conical Flare 8-33(B) Wave Drag Coefficient of Conical Flare 8-33 (C) Wave Drag Coefficient of Conical Flare 8-33(D) Wave Drag Coefficient 3f Conical Flare 8-33(E) Wave Drag Coefficient of Conical Flare 8-33 (F) Wave Drag Coefficient of Conical Flare

Page 8-43 8-44 844 8-45 846 8-46 8-47 8-47 8-48 8-49 8-49 8-49 3-49 8-50 8-50

8-33 (G) Wave Drag Coefficient of Conical Flare

8-50

8-33(H) Wave Drag Coefficient of Conical Flare 8-33(I) Wave Drag Coefficient of Conical Flare 8-34 (A) Wave Drag Coefficient of Fins at Supersonic Speeds

8-50 8-51 8-52

8-34(B) Wave Drag Coefficient of Fins at Supersonic Speeds

8-53

8-34 (C) 8-34(D) 8-34(E) 8-34 (F) 8-34 (G) 8-34 (H) 8-35 8-36

Wave Drag Coefficient of Fins at Supersonic Speeds Wave Drag Ccefficient of Fins at Supersonic Speeds Wave Drag Coofficicnt o' Fins at Supersonic Speeds Wave Drag ',oefficient of Fins at Supersonic Speeds Wave Drag Coefficient of Fins at Supersonic Speeds Wave Drag Coefficient of Fins at Supersonic Speeds Wave Drag Coefficient of Fins of Various Sectional Shapes Wave Drag Coefficient of Rectangular Fins at Subsonic and Transonic Speeds

8-54 8-55 8-56 8-57 8-58 8-59 8-560

8-37

Wave Drag Coefficient of Delta Fins at Subsonic and

8-38 8-9 8-39 8-40

Transonic Speeds Wave Drag Coefficient of a Double Wedge Fin at Transonic Speeds Flat Plate Average S'in Friction Coefficient Reynolds Number as a Function of Flight Mach Number

8..61

and Altitude

8-63

8-41 8-42 8-43 8-44

L

8.61

-62 8-62

Base Pressure Coefficient of Cylinders, Boattails, and Flares With Rocket Jet Off 8-64 Effect of Rocket Jet on Base Pressure 8-64 Base Pressure Correction for Boattail and Flared Aft'-rbodies 8-65 Base Pressure on Cylindrical Bodies 8-66

____

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AM1CP 72SO

LIST OF ILLUSTRATIONS (cont) "itlr

Pagr

Base Pressure Coefficient ot Fins at Supersonic Speeds Base Pressure Coefficient of Fins it Transonic Speeds Comparison of Test Results on Full Size and Scale Model Artillery Missile Relationship Between Maximum Allowable Modil Dimensions and Test Section Dimension in a Specific Wind Tunnel Facility Aerodynamic Force Components

8-66 8-67

Fig. No.

8-45 8-46 8-47 8-48 8-49

I

8-69 8-70 8-71

LIST OF TABLES Tablr No.

S2

Tith

Page

2-1 2-2 2-3 2-4

U. S. Standard Atmosphere 1962 Resultant Wind Condition s in the Northern Hemisphere Extreme Annual Wind Speeds Density and Temperature Profiles A. Key to Code Numbers of Density and Temperature Profiles B. Density Profiles for Tropic.; and Subtropics (kg/im) C. Density Profiles for Temperate Zone (kg 'm 3 ) D. Density Profiles for Polar Zone (kg/m 3 ) E. Density Profiles for Tropics and Subtropics (% Dev. SA) F. Density Profiles for Temperate Zone (% Dev. SA) G. Density Profiles for Polar Zone (% Dev. SA) H. Density Profiles for Tropics and Subtropics ('K) T. Dersity Profiles for Temperate Zone (*K) J. Density Profiles for Polar Zone ('K)

2-2 2-5 2-11 2-14 2-14 2-14 2-15 2-15 2-16 2-46 2-17 2-17 2-18 2-18

6-1 6-2

Geometrical Properties of Typical Rocket Sections Geometric Properties of Airfoil Sections

6-3 6-16

7-1

Error Budget for Indirect Fire Rocket With Impact Fuze, Launch Quadrant Elevation 45 Degrees

7-28

8-1

Computational Table Ding Force Calculati-,n Sheei'

e 74 8-96

'Ff

4[

A!

xvi

-

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L AMCP 7M,-280

CHAPTER 1

0.

INTRODUCTION

I ' 0

*

x1

)

This handbook, written for the engineer interested iii the preliminary design of aerodynamically stabilized free rockets, has a twofold purpose: a. To provide the preliminary design engineer with specific, "best available" design information and data devised to allo, the rapid response rcquired of preliminary design activities, and b. To provide each technical area having responsibilitzeE in the preliminary design phase an appreciation or "feel" for the data requirements and data applications of other specific technical areas. The term free-flight rocket implies the absence of an active guidance system. Such a rocket is guided or aimed by a launching rail or tube and can be classified into one of two categories, a. Spin-stabilized b. Aerodynamically stabilized The spin-stabilized rocket, as the name implies, depends upon a high rate of spin and resulting gyroscopic moments to oppose disturbing moments and forces. Conversely, the aerodynamically stabilized rocket depends upon the moments generated by a flare or fins placed aft of the center of gravity to oppose disturbing moments and forces. The aerodynamically stabilized rocket generally employs some spin history to minimize dispersion due to nonstandard conditions (body malalignment, fin malalignment, etc.). The data

Finally, there are three major propulsion systems that could be applicable: a. liquid propellants b. solid propellants c. hybrid propellants (combination of liquid and solid) The applications of this handbook are limited to solid-propellant motors, used almost exclusively in free-flight aerodynamically stabilized rockets. The basic handbook is organized into chapters, each self-contained and applicable to a particular technical area w'th which preliminary design is concerned. These ai,-as are" tmospheric Data, Systems Design, Parametric Performance, Propulsion, Structures, Accuracy, and Aerodynain:cs. Chapter 2, Atmospheric Data, presents climatological data partinent to free rocket design. Chapter 3, Systems Design. discusses the factors affecting design, considering each technical area from prliminary design to actual hardware. Chapter 4, Paiametric Performance, presents data describing the performance of various design concepts, with variations that permit consideration of trade-offs to maximize range for given mass or mass for givin range. Chapter 5, Propulsion, presents concepts and data necessary to predict propulsion system nprformance, o well a- imp.rtnt aspects to consider in conceptual and preliminary design. Chapter 6, Structurcs, presents data and methods

and concepts presented by this handbook are limited to aerodynamically stabilized free-flight rockets. The rocket is assumed to be a rigid body, i.e., the elastic properties of the structure have been neglected. However, for some configurations (primariay long, slender bodies) the dynamic iau&. uf uCillaion may be of sufficient amplitude to warrant detailed investigations,

pertinent to structural design. Chapter 7, Accuracy, considers both burning-phase and ballisticphase errors, the effect of these errors on rocket accuracy, and techniques necessary to estimate accuracy. Finally, Chapter 8, Aerodynamics, presents design curves and formulas that will permit the prediction of stability (force and moment) and drag characteristics for practically any conceivable aerodynamic body or combinations of bodies.

-111-2

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0

AMCF 703-2e0

CHAPTER 2 ATMOSPHERIC DATA 2-1 INTRODUCTION

Strolled - :

;

Atmospheric information is a ve.y important .ons,"eraton in the prelimmary design of rvckets. particularly those that are free-fl.ght or unconafter launch. Since atmospheric data are Lime- and space-dependent, and therefore widely variable, statistical processes are normally used for their presentation, analyss, and utilizatin.

,.

'

C~ ,

Because :' some types of statistical data are subject to considerable controversy, one can hardly hope to develop standardized actual climatological data profiles, and perturbations to them, that w.ll receive universal acceptance. Instead, meteorologists have developed synthetic .tandard profiles, independent of physi-al location. Means and extremes of these profile- were then developed. dependent upon rather general geographic lOCdtion. Finally, models of microclunatologivei (1ocalized) conditions can be accomplished to meet specific conditions. In rocket design, the prelumary design phase is concerned primarily with macrocl.natolog. (large-scale conditrons). Density, temperature, and presure profiles (variation of these factors with altitude) must be presenteo to the trajectory wnalyst so that configuration performance can be determined. Wind profiles and wind shear information are necessary for structural design as vell as for accaracy studies. Finally, considerat,on of extreme cundition is necessary to ensure complete system integration and operation.

2-2 ATMOSPHERIC PROPERTIES 2-2.

( iiard

ATMOSPHERIC DENSITY, TEMPERATURE AND PRESSURE

ard Atmosphere The major reason for revising standard atmospheres in recent years has been the ubserved ozbit pertu rbationb of artificial satc!.lItes due to atmospheric drag. This sub,--t is beyond the present scope of Anteres-t. See RIeference 2 for complete tables. Table 2-1 presents a useful summar. of atmospheric properties taken from Reference 2.

2-2.2 WINDS, UPPER LEVEL The problem of Lelecurg wind profile infcrmatiun fur ute a-s design Lriteria led to development of dri ebtirmated synthetic profile ,%hich presented th- 1 perment probable % knd speed and associated shedr at thE mu%. Lr, ital altitude, and speeds of other altitL des ty pica, for such wind fields. Sub_equent investigation revealed that, if accuracy in ,he calculated risk is desired, the use of synthetic wind profils is hazardous. However, logically developed synthetic orofiles are useful in prelimm.ry design. Fig. 2 1 is a synthetic wind profile that was developed in 1954 to cletermme %ehicle responses that wuuld bl. exceeded during only 1 percent of the windiest season of the year in that area of the U. S. where tropospheric wind streams were considered the strongest. It is tea sunable to shift thE curve upward or downward by as much as 5000 ft to make the peak wind speed coincide ,i ith the altitude of maximum wind influence. Ideally, in missile design and accuracy studies. the designer must know mean wind velocity and standard deviation for both hemispheres. Tab..e 2-2 gihes the resultant wind direction, vector mean wid velocitv and the standard vdctor deviation for the No:thern hemisphere between 20°N and 80 0N at altitudes from 10,000 to 100,000

The U. S. Standard Atmosphere (USSA) is based upon the International Civil Aviatimn Or-

ft for winter and summer. See Reference 4. the llandbool of Grophyjz;cs

ganzatin e (ICAO) Standard Atmosphere to 20

and Spa(# Entiranmrnts, 1965, Chapter 4. for the

kin altitude. and upon the proposed ICAO exten,,n from 20 km to 32 kma. Data for the first 20, krDevelopment are in ag.reement with the Air1959 Research Command (ARDC) Stand-

mean wind speed, standard dev:ation, and corre lation between levels for a series of altitudes for vector component at specific stations duxing each the winter season.

7

24

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SA

IC? 7C3.2 I. TADLE 2-1. U. S. ST,,DARD ATrOSPHERE, 12

,3

f,

Clo

Geornetril Altitude, km

De-sity, kN

Temperature,

Prewure. ,, %m

ActceJ. of Gravity, m/seet

Speed of Sound, m,sec

0 1 2

12250 11.117 LWM,

283 !50 281.651

10325 89376.2 79501 4

9.80G6 9.8036 9.3055

340.294 336,A35 312.53

0.0=5

2a.059

70121.1

9.7974

328.&R3

4

0.81935

=52.1C4

61650.4

9.7943

324.5S9

5 6 7 8 9 10

0.7343 0.0,.1 0.5002 0.52579 0.467063 0.41351

255.678 249,187 242.700 23G215 229.733 223252

540482 47217.6 41105.2 35651.6 303007 26499.9

9.7912 9.7882 97851 9.7820 97789 9.7759

11 12

0=,I4' 0 0,31194

210 774 216.650

22699.9 19399 4

9.7728 9.7697

320.545 316.452 312.306 303.105 303.848 299.532 295.154 295.069

13 14

02CC00 0227eS 0.19475

216.6,O 216.C50 216.650

1679.6 14170.4 12111.8

9.7G67 9.7636 9.7605

205.0,39 295.069 295.069

0.1647 0.1230 0.12165 0.10400 0.0m3810 0.075715 0.064510 0.055006 0.010233 0.040034

216.650 216.650 216.650 216.650 216.650 217.531 218.574 219.567 220.5.0 221.552

30352.8 8849.71 756522 f47.4-8 5529.30 4728.93 4047.49 3466.86 2971.74 2549.22

9.7575 9.7544 9.7513 9.7483 9.7452 9.7422 9.7391 9.7361 9.7330 9.7301)

295.069 295.069 295.09 295.069 295.059 295.703 296.377 297.049 297.720 293.389

17 1 19 20 21 22 23 24 2

2-2.3 WINJDS, LOWER LEVEL

the coldest temperatures, where winds are 50 percent of the indicated values.

Chapter 4, Section 4.1 of Reference 4 contains information concerning mean wind as a function of height. This section gives approximate equa;ons to compute the mean wind speed. It also gives infornnatioti and tabulated data concerning

wind-direction shifts and directional variation,

L

and low-;eve] jet streamrs. Labie 2-3 gives the number of years of record, mean, and standard deviation of the extreme wind speed at 50 ft above the ground for various stations in the northern hemizphere. Fig. 2-2 shows the strongest wind for temperal.ure rangre observed during a five-year period. Wind speeds are, in general, for 40 to 100 ft abo.,e the surface. Speeds at 10 feet are approximtely SO percent of the values given, .xcept for 2-4

2-2.4 REGIONAL ANNUAL AND SEASONAL DENSITYMODELS Evaluation of flight performance iud rocket

design necessarily includes the consideration of

distrib'.."rn uf aiospheric environment parameters, Our knowledge in this field has rapidly expandecd beyond the status of a "Standard Atmospheric Model", which can only describe the atmospheric environment as a first approximation, under limited circumstmces. In recognition .)f these limitations, the Committee on thp Extension of the Standard Atmosphere (COESA) has recency adopted supplemntmy atmospheres vprii,,o

(Reference 2).

9

.

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AfoI~r 70'-2f 0 S~.uch wi approach is .er* attri'.ctive because it

130-

0

comprezzes a large amount of tatistical informauon into aife-w Altitude cuirve,. However, it co~mignores vie aitiiude relrt4cn:lup of meteorological elements, whicti is admittedly cormplex and cannot be expressed in simple tern,.7,. it is logical to establish any probabil~y

120.pletely '10300Although

thi eshold separately at ary altitude level, the 0 -

~viate 70for

co

=1XvMA SIA ft scf' por 1000 ft

~ ro45 Z40 :n 20 10-

iMo IEO MO: 2W Z.4)I WEED (ft 8ft-)

zo

Mm'marn speed 01M0 ft see -) =n' sciated ,.wj. mum shli! (45 ft sec-I per 1000 ft) are likely to be exceedeZ only 1% of thz win-tertime over the northeastern USA. Wind

speeds &j'dshears at levels aL.Ye and 1below the levels ot mgaxlinra speed aud shear azo tp5-sl of those likely to be encounterted for tl±~ cod~ton.0. 1%Proal~lit

Fir,... 2.1. Maximum Npeed and Associated Shear Fro~rdandbook of Oeot-hysics zd Spar~e Fnrnruients,by Shea L. Valley, Air Force Cambz.tLgc Research Laboratorics.

Used by permlission of copyright owner, McGraw-Hill Book

Co., Inc.

These suppements to the standurd airrncsphere, al~hough wvelcomied 'by th, Jesigm euarneer, still cannot fully describe the status of the atmnosph~ere and are not intendeti to do -so. Eebid is ha 'ing ref erence models )f mean coniduiorib, th, designer *

Cerror

F,

naot

infnrmatann tn eApeibh

as a consequence of this negligence

Although the -ntroduction of more realistic profiles may incre-se some of the engineers' coraputational work by adding a few profiles or by making his computations more complex, hopeV11l3 the engineer will thereby realize a gain for his system by avoiding inpicoper deskri. Ideally, investigations would employ sample profiles, The disadvantage of this approach is the tremendous amount of calculations that result if individuai

___________________

50

4,_~'~ are combined to form a stigle -ert.-cal totally from realistic profiles, par-Licularly density. Fig. 2-3 illustrates this point. unprope. design of the rocket system may result

the Ae~atmmn%~ from

the~e preset models, usually in the fo'-m of some iuaction or bta:-Ie1rd deviations. It has be'-n cEmitomary to assume a certain devjation, e.g., th,. plus-one-sign- a threshold, at every height le vel, and then to connect these points fronm ont 1tvel to the next. The resulting synthet-c altituie relationship will be call!.1 here the plus-or& sigara kvnvelope (to be disar.guished from profiles). A..ic, co;-istants, su h &_~ a 1~ percent deviation for density, ha~ve been utilized,

atmsospheric conditions are used xis inputs from random data selections Some of the advantages of realistic data can be retained, with a reasonablte amount of compua~ation, by using carefully selected, sinall but representative samples. One such selection has been prepared by Dr. M. Essenwailger, Chief, Aerophysics Branch, Physical Science Laboratory, Army Missile Comn-rind, Redstiane Arsenal. His collection consists ..,f 800 ropresentative individual profiles, a.-rangetd i

g-,set

f10poie;tost

o

aho

eg f10poie;tost o aho four stations represent summer and winter conditions. Fig. 2-3 is a selection from this collection. TVhe entire collection, on panched cacds. rtaa, be obtained fromn Dr. Essenwanger. The x~ext best, and also simpler, approach would be to use a representative profile. Their use -vill keep computational (cffort relatively simple, but will provide realistic estimates of actual atmt,slahtpnc ccmneiitions. Data obtained ti1 &uuwh use' of these profilerz. will provide inforirrPt~on on missile behavior adequate for preliminacy design. Such a set of representative profiles is the 20 sets of air densi %yand associated temperatures presented in Table 2-4 and Fig. 2-4. These pro. files may be used to represent nu~an and onesigma coraditionf, for four zone:, by summer, v..r ter, and arknual referenice peruids. The ri.uper profile jor each conditi.,n is oresented on the first page of Table 2-4, where the numbers refer 2-3

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REFERENCES

to the subscripts in subsequent column headings

p

and the various curves of Fig. 24. rt. H. P. Dudel of the Aerophysics Branch, V ,¢ Plysical Sciences Lab, R&DD, U. S. Army Missile ', Command, hr published a report (Reference 7) giving the mean, rinus-one sigra. and plus-one sigma density profiles for the midseason months 0 0 of January and JLly at latitudes of 15 N, 30 N,


IA

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__________ 706-280______________________________________

REFERENCES I. J. B. Rosser,

KMotion

ct 31, Mathematical Theory of Rocket Flight, McGrsw-Hill, Inc., New Yo-k, 1947. 2 L. Davis Jr., et al, Ezterior Ballistic#of Rockets, 1). Va~n Nostrand Co., 1958. :1. R. A. Ra:Jrin, "True Mathematical Theory of the of Rotated and Unjrotated Rockets"', Philosophical Transactions of tihe Royal Society of London, Series A, No. 837, 241, 457-585, Cambridge University Press, London, 1949. s. Rt. R. Newton, Spin Programs Which Cause the D~ispersion Produced bt Thrust Malalignmcnt to I'nish, The University of Tennessee, Knoxville, Tenn., 1955. ~.R. E. Maexker and R. R. Newton, Spin Programs Which Cause Dispersion Prn-d-.zd by Thrust Malalignment to V'anish, Part 11, The University of Tennessee, Knoxville, Tenn., 1955. 6. R. R. Newton, Spin Programs Which Cause Dispersion Produced by Thrust Malalignmemt to J'amish, Part I I , D, nartment of Ph-aics, Tulane University, New Orleans, La., 1956. 7. MJ.W. Hunter, et a.. "Some Recent Aicrodynxinic Techniques in Design of Fin-Stabilized FreeF.light Nitssiles for Mlinimum Dispersion.- Jourmat of the Aeronautical Sciences, 571-7 (June 1936). 8. W. C. NicCorkle. Recent JDevelo,,-,t in High .lecuracy Free Rocket Werapons Systems (U),

Carde Report !; AX!' 8, Rcdstone Scientific Information Center, Redstone Arsenal, Ala., Apiril. 1959. 1). R. ii. Newton, Effect of Variable Acceleration Upon the Dispersion of Fin-Stabilized RocA eta, Report PH-RR-1O, University of Trenn see Djelartwiiit of Physics, Kiioxville, Tenn., April 1. 1953. R.. S. Buringtoii and 1). C. May, Jr., Handbook of Probabilityand Statistics With Tables, Handbook Publishers%, Inc .Sandusky, (hdo, 1953.

II 12.

S. S. Chin. 1Missil- ('onfigurato 1) sign. Nlc Graw-Hili. Inc., 1961. L Davis, Statwsical Methods n Research and New York. P'roduction, Third Edlition. F. t*er 1961.

E. Freud, Modern Elementary Statistwcs, Prentice-Hl-al, Ne w York, 1952. I ;. E. S. Keeping and J, F. Keeney, Mathematics of Stalistics, ]'art One, Third Edition, D. Van ",ostrand, New York, 195 1. 15. G. A. Korn and r'. MJ. Korn~, Math-matical 11amdbook for 'Scientists and Engineers, MeGraw-Hill, Inc., New York, 1931. 16. F. Mosteller, R. B. F. Rourke, and G. B. Thomias, Jr., Probability and Statistics, Addison-Wesley, Readiag, Massachusetts, 1961. 17. E. L. Crow, F. A. Davi-s. and M. W. Maxield, Statisic, Manuel, U. S, Naval Ordnance Test Station, China Lake, Callfornia, 1955. 18. W. Volk, Applied Statistics for Engineers, Urmw-Hill, Inc., New York, 1955. 19. H. S. Scigjent, Required Accuracy of Corporal Fire, JPL Publication 2?5, 15 April 1954. 20. J1.NV. Haughy, taiculation of Missilc Accuracy Fromn .1ctual Test Flight Data, Army Ballistic Niissilr Agency Report No. RG-TN-2-61, 13 Jan 1961. 21. C.E.)'. for 'Correlnted Unequal Standard Deviations, G.M. 41.2-108, Ramo-Woolridge, 28 Oct

13.

22.

24J.

2.

1957. Error Approximating to the Circular Prbal (of an Elliptical Gaussian 1.)stribution 'C), Royal Aircraft Fstablisnliment. Tr. MJ. No. Ci. W. 300. M arci 1957. Pat Maxwell, Jlr., Estimation of the C.E.P. of a Missile Systemn from the Results of Firings, 6;. IN. 'fR-!39. Rtamc-Woolride. a April I957'. . L. Eaton, Accuracy C.E.P. 'Estimates, NAMNTC 'f.fll 105, Bureau of Aeronautics, I .Jan 1957.

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AMCP 706-280

CHAPTER 8 AERODINAM!CS LIST OF SfMBOLS Symbol

"

Me oning

Soic velocity, ft sec

d

. .. factr ,see Fig. 8-23) W.n gccoiietu y-flow parameter (see Fig. 8-21) Area, ft' (general reference area) Jet throat area, ft2 Fin aspect ratio (fin span squared/fin area) Fin sp-'n, it Fin chord, ft Fin mean chord (see Fig. 8-23), ft Cord length Drag coefficient Total skin friction coefficient Thrust coefficient Pitching moment coefficient Pitching force moment coefficiei-t gradient Normal coefficient Normal force coefficient gradient, per rad or per or per deg Pressure coefficient Center of gravity location (axially from nose) Diameter ft; tiferentia1

V f h I

Drag force, lb Fineness ratio (l/d) Vin base thickness (see Fig. 8-46) Correlation parameter for fin-fin interfer-

F.%.

at A 4T .4R b r " cI 'D

C)"

CF CM Cba

11.C C.,

a

CP CG

enc,

sfator (see Eq. 8-1) n

K (f Z, I led m Af n N P

i

zJ1iterlerence zactor Interference factorItsbl Length measure,

Leading edge diameter, ft In Natira! ogarithm Fin geometry-flow parameter (see Fig. 8-21); factor for 1ig. 8-35 'Mach numoer; pitching moment, ft-lb Number of fins; exponent for power law nos Normal force, lb Static pressure lh,/ftl

Sy-bl

MO.,In;

Motor chamber pressure, lb'ft-

P,

q r 1I, R.\ r c

Dynamic pressure, lb,'ft2 Radius, ft Reynolds numbe: Ring trailing edge (the location of ring trailing edge with respect to base of body;

values aft of body base are positive.) S Area, it' (particular reference area) .,eil Effective area (see par. 8-2.4.3), it TE Trailing edge t Fin thickness, it V Rocket velocity, ft sec X; x Axial distance, ft 1 Laterai distance, ft a Angle of attack, rad or deg y y 8

i of specific Ratio s heats h

h

ncremengt Included angle of fin leading edge (see Fig. 8-23), deg Spanwise location of fin mean chord (ste

Boattail half angle, degrees; factor for fin correction (see Fig. 8-15)

0 A A

Fig. 8-23) Cone or flare half angle, deg Fin taper ratio Fin leading edge sweep angle, deg

p

Atmospheric density, slug/ft

ff

3.1116

Subscript:

a,ab b B bo b: c c/2 cyl cp

Aft-rbody Body Base Rocket motor burnout conditions Boattail Cylinder; cone Fin mid-chord Cylinder Center of pressure

t,I

.

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LIST OF SYMBOLS (Cont) Subs

~hf*~

an Inol

raxw valut Flare; fin-, friction IF fb Forebody fe Exposed fin f l Flare g Rocket launch condition IF Interference t-et Jet (or nozzle exit plane) conditions I le LeadingII edge None 4 Pree stream or stagnation eonditions o, p Plwndorm e

3

.11

Subscr ipts:

Meaning

Root chord Exposed root chord Reference condition Ring tail Total Tip chord riigeg riigeg t theory Theortical prediction IV Wave drag; "wetted" conoion Two dimensional considerption 21) .I Perpendic ,!ar measure r re ref rt T t

I.I II

LISTCentF SYBOLS I

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706-260

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8-1 GENERAL DESIGN CONSIDERATIONS This chapter will discuss the aerodynamic aspects of free rocket design and will Lidlcate mean: of predicting the significant aerodynamic coefficients. Simply stated, the usual aerodynamic design goal is to ,/cet an external configuration which provides stable flight with mirimum drag through the desired altitude-velocity range.

length, weight, and structural rigidity. Reduction of the axial force coefficient is more important generally for inairect-fire, artillery-type rockets where the sustain and ballistic flight times are much greater than that of the boost phase. Axial force reduction in this case can result in either a lighter and smaller rocket for a specified maximum range, or increased range for a fixed rocket size and we:ght. In addition,

For a rocket to possess flight stability, a restoring moment must be produced when its Iongitudinal axis is rotated from the flight dirc-

improved accuracy is achieved by r-duced rensitivity to atmospheric variations during the ballistic flight because it is primarily through the

tion, i.e., when an angle of attack exists. This flight stability is achieved in the case of aerodynamically stabilized rockets by selecting the external configuration such that the center of pressure of aerodynamic forces normal to the longitudinal axis is located farther aft of the rocket's nose tip than the center ot gravity. Since the aerodynamic forces are proportional to angleof-attack (the angle between the flight direction or velocity vector and the longitudinal axis of the rocket), any deviation will produce a moment to restore the axis to its a)i ned condition. When the center of pressure is aft of the center of gravity, the rocket is said to be statically stable. The degree of aerodynamic stability, or the 3tatic margin requirement, varies with the desired ac,racy of each ro:ket and its design ap-

aerodynanic axial force that non-graitational acceleratior.s are t'ansmitted to the rocket. The external configuration of a free rocket can vary significantly depending on the trad--off between aer'odynamic requirements inposed by performance and accuracy considerations, and other system requirements. Some generalization can be made, however, based on past designs. Typically, the rocket's external configuration consists of a pointed body-of-revolution housing payload and propulsion unit. with a stabilizing device attached to the aft section. A circvlar crosssection is preferred because its syrrmunetry about the longitudinal axis makes for simplicity, both in manufacturing in determining aerodynamic cefcet n and asietapoete.T h isoaerodynamioally inra wh od rocet uroctable, various normal-fore-producing devices

proach. For example, a rozket designed for minimum dispersion during powered free flight requiressary

are attached at its aft end to provide the necesmoment rostbilty. Thin ary reti

restoring moment for stability. Thin-profile

over its Mach number ri while a high-acplanar fins, spaced evenly around the circumolerits Mc hich aherregime, hits-aefe ference of the body, are used in many rocket deceleration releas whic achiees msofis signs as stabilizing devices. This type of fin locity prior to release from t..3 launcher requires usually will produce the maximum stabilizing ailiforc t moment with miniumt h and cerwith remain margin only ti.ut the stability axial force and diameter wight minimum tawuppr owe tld bonds Th with omoment f tis penalties. with Waen minimum overall is tane te i o erao esinm na . desian stability band is governed primarily by the reaa dominwit consideration. the ringz-tail quiremeni to mainiain a significant spread be-

and conical ilare become of greater interest.

tween the roll and p'tch-yaw frequencies. Although the static margin is of paramount inttrest to the accuracy of a free rocket during the powered, high-ac Aleration phase, the aerodynamic drag or axial force is a prime factor affecting the accuwracy and performance during the sustain and ballistic flight phases. For an anguided rocket, the angle of attack is nominally zero; therefore, the axial and drag forces ra-e equal. The general goal ;s to keen the axial for,! coefficient as *low as possible, consistent

ring-tail will produce, at both subsonic and supersonic velocities, approximately twice the restoring moment of a cruciform planar fin with equal total span and chord. The conical flare is of interest for restricted-diameter rockets with maximum velocities above approximately 5000 ft/sec. O the basis of projected planform area, a conical flare will produce better than twice the normal force of cruciform fins at hypervlocity speeds. However, the axial force of a flare greatly exceeds that of fins providing equal restoring mo-

with other design considerations such as body

ment.

A

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sion that a "best" aerodynamic configuration does not exist. The great multitude of paramcters affecting he coneguration selection -sually results in its choice being based on past design experience un aesthetic values rather than an opIn order to estimate the aerodynamic stability characteristics for complete rocket configurations rind to provide design guidance, it is necessary -to know the aerodynamic coefficients of each major rocket component and interference between components. The paragraphs which follow will discuss first the stability parameters for various rocket forebody shapes, fins, ring-tails, conical flares, and boattails; and then -will discuss how these component aerodynamic coefficients are combined to arrive at values for the complete configuration. Wherever possible, simple analytical or semi-empirical equations will be prTmkeaerodynamic hats too make logcharts wth senedwith sented along aerodeahec estimates. No effort will be made to provide the theoretical basis or experimental substantiation for the data presented since numerous textbooks and reports exist which discuss these topics detal. dtaild inalya cmpuatinal great detail. Finally, a detailed computational chart wsources and methods of obtaining design data, and will provide a format and check list for design computation.

8-2 STABILITY CHARACTERISTICS OF ROCKETS 4-2.1 BODIES OF REVOLUTION 44 ,tions, .......... . The forebody of a rocket norrmaly consists of

= 2(k 2 -k 1 )

(CN

-

(8-2)

Src

In the above expressions e n CN = normal force coefficient bper b

=

ao

rad or per deg (see par. 8-2.4.1) fa

the longitudinal axis of the rocket and the velocity vector, intad or deg den = apparentma ft l = body length, ft dx = incremental axial disetance, ft SB = body base area, ft' Sref = reference area, ft' The factor (k2 - k ) isthe apparent mass factor acorived y kun iteapren t mas factor by Munk (Reference 1). The values as for derived ellipsoids of revolution presented iWFig. 8-1 rpeetaraoal prxmto o n represent a reasonable approximation for any axisymmetrical body of comparable fineness raio. The center of pressure may be assumed to tinthe centr of se mose assued to act at the centroid of the nose projected area for X,, < I. he fundamental assumptions of the theory are that all second order partial derivatives of velocity can be neglected, and that velocity per-

'

•

turbations along the body axis are small compared to the transverse values. The solution imiplies that adding cylinder length to the nose i no effect, and that there is no compressibility effect due to varying Mach number. However, expLrimental data and more refined theoretical solutoo complex to discuss here, shuw these efects to be significant. The reader is referred to References 2, 3, and 4 for further detals of

a pointad cone, an ogive, or a power series curve, followed by a cylindrical section. The slender-

.I

M 1.0

body theory provides a simple means of express-

ing the stability characteristics of these bodies in terms of the geometric parameters only, as follows: k1) = (k2 s En 2a ... L:-

Se.

%dx/

&"uCI, when integrated from x = 0 to x

gives

14

(8-1)4 =

1,

.6

i,

1

FIFIENESS RATIO, I /d

16

I

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2

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) these solutions.

86

In the absence of more prec ise solutions, the slender- body theory should be tved

\2.. -

in the subsonic-throuh.-sonic Mach number r"age. At supersonic Mach numbers, the normal-force coefficient gradient and center-of-pressure estimates should be obtained, respectively, from Figs. 8-2 and 3-3 for tangent-ogive-cylinder bodies, and from Figs. 8-4 and 8-5 for cone-cylinder bodies. These curves were extracted from Reference 5 end were constructed from empirical data covering a Mach-number range from 1.4 to 7.0 and a nose-fineness ratio range from 3.0 to 7.0. Stated accuracies are = 10 percent for normal-force coefficient and 0.5 calibers for center of pressure. effects of particular nose shapes and cylindrical

rdb I i-

dj dSrf (8-3)

where Sbt

cross-sectional area of boattail at its smallest diameter, A" diameter, ft

=

d = bt = boattail c = cylinder A increment It is recommended that slender-body theory predictions be used for the subsonic-to-sonic Machnumber range since systematic empirical investigations are not available. ence 5) w.'l provide normal-force coefficient gra-

lengths is presented in Reference 6. Aerodynaxnic-stability parameters were deternned at Mach numbers from 0.8 to 4.5 for tangent-ogive, conical, and power-series noses, all with a fineness ratio of four, combined with cylinder lengths of from 4 to 11 calibers. The effect of changing the nose-fineness ratio from 3 to 5 for conical -ud tangcnt-ogive shapes was determined for a cyiindrical afterbody length of 6 calibers. Since the overall study covers body configuratinas of general interest to free rocket design, the pertinent results are presented in Figs. 8-6(A), (B), (C) and 8-7(A), (B), (C). These results should give normal-force coefficient gradients within - 5 percent and center of pressure within ± .1 caliber for the range of test variables. Whenever

dients for conical boattails located behind a semiinfinite cylinder; i.e., tle local flow conditions upstream of the boattail are equal to the free stream conditions. These data, derived from linearized theory calculations and slender-body theory predictions, have not been verified by a detailed comparison with experimental data. Therefore, no statement can be made concerning the expected accuracy. The center-of-pressure for the boattail normal force is located approximately 0.6 16, from the cylinder-boattail juncture at subsonic Mach numbers. The center of pressure at supersonic Mach numbers may be evaluated from Fig. 8-9, taken from Reference 5. It is recommended that the data in this section

the body of interest falls within the range of test variables, it is recommended that the stability parameters be established from these data.

be used for bohttail angles of less than 10 deg and ratios of d greater than 0.8 to avoid flow separation fromcthe boattail.

62.12

5-2.1.3 Conical-Flare Afterbody 9.,

A conical-flare afterbody can be added to nose cylinder configurations to provide aerodynamic stability. The slender-body prediction for 'e incremental normal force coefficient gr',lien of a flared afterbody is

Where the rocket propulsive nozzle is smaller in diameter than the body cylinder, the rocket afterbody may be tapered to form a boattail, which reduces base drag. The normal loading over this boattail negative, thus reducing the total norreal-forceis coefficient and shifting the center of pressure forward. The slender-body theory predicts the normal-force coefficient gradient W be:

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NOTE Values of -Z measured rearward from the 1 eylinder-fiare juncture are negative. The important geometric parameter is seen to be ratio of forebody cylinder -diameter to base

• anf: N,Ithe

01______ ____:

diameter. Flare angle and Mach number do not influence the flare normal-force within the lim-

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f

tations of the slender-body assumptions. To a

Eence

degree, the experimental data of Refer-

-limiied

IA.

7 verify this trend at transonic Mach numbers and for flare angles of less than 8 degrees. The actual normal-force contribution of flared afterxodies is shown by other theories (Refer8 and 9) and by experimental results to be influenced by flow conditions forward of the flare, as well az by flare angle and length. Also, largeknown flare toangles than about deg) cause (greater flow separation at the10 cyiinder-frustunt juncture, which alters considerably

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For preliminary design estinates, however, the W

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incremental normal-force coefficient gradients presented in Figs. 8-10(A) through (E) a.re considered adequate.

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These data were computed

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