Winged Rocket Performance Analysis Citation Shonerd, David Edwin (1949) Winged Rocket Performance Analysis. Engineer's thesis, California Institute of Technology. doi:10.7907/34ZH-KG02. https://resolver.caltech.edu/CaltechETD:etd-02052009-092504 Abstract A method is presented for estimating the effects of various parameters on the performance of a winged rocket. A program for studying three specific parameters, i.e., wing areas, reduced thrust cruising programs, and trajectory climb angles, and their effect on the horizontal range of a winged rocket is presented. Complete calculations are carried out for one combination of these parameters. An analysis of the Lift, Drag and Stability characteristics of a long, slender rocket with trapezoidal fins and wings is made. The stabilizing effectiveness of delta and trapezoidal fins is compared. Simplified approximate methods of integrating the trajectory equations by step-by-step method are presented. Item Type: Thesis (Engineer's thesis) Subject Keywords: Aeronautics Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Nagamatsu, Henry T. Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1 January 1949 Record Number: CaltechETD:etd-02052009-092504 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-02052009-092504 DOI: 10.7907/34ZH-KG02 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 511 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 05 Feb 2009 Last Modified: 29 Mar 2023 17:20 Thesis Files Preview PDF (Shonerd_de_1949.pdf) - Final Version See Usage Policy. 13MB Repository Staff Only: item control page

WINGED ROCKET PERFCRMANCE ANALYSIS Thesis by David E. Shonerd In Partial Fulfillment of the Requirements Yor the Degree of Aeronautical Engineer det Fropulsion a Galifornia Institute of Technology Pasadena, California 1949 ACKNOWLEDGMENTS The writer ie deeply indebted to Mr. Henry Nagamateu for his con- tinued interest and helpfulness during this investigation. Invaluable assistance was alse rendered by Miss Laura andrews and Mrs. Melba Nead in computing trajectories and by Miss Patricia Line in typing and assembling the final manuscript. ABSTRACT 4 method is presented for estimating the effects of various para- maters on the performance of a winged rocket. A program for studying three specific parameters, iee., wing areas, reduced thrust cruising programa, and trajectory climb angles, and their effect on the horizontal range of a Winged rocket is presented. Complete calculations are carried out for one combination of these parameters. fn amalysis of the Lift, Drag and Stability characteristics of a long, slender rocket with trapezoidal fins and wings is made. The stabi- lizing effectiveness of delta and trapezoidal fins is compared. Simplified approximate methods of integrating the trajectory equa- tions by step-by-step method are presented. TABLE OF CONTENTS List of Tables... . ccc cco nce eves esc cec ce eccececoees List of Figures... ccc ecco cccocsccocescecorres List of Symbols. cece ccc cece eee eee ese eesceeneee le Tutroduction. ..c.ccce cere ccc ccecesces it. Trajectory Considerations..-......-+. Iii. Stability Analysis. ..ccccccecsececess TV. Drag Analysise...-+.ceeee es ecee es ccece Ve Analysis of Lift and Lift-Drag Ratio Vi. Thrust Veriation with Altitude....... VII. Equations of Motion and Trajectory Calculation Procedure. VIII. Discussion of Results.....ccescccaves Appendix I; Summary of German A-9 Projects... REferenCeS. coccccccceessesseevceceesetenccuee TeblLese cc cccacccc cece ee seeseuccetenvcseereses PLeUre sec ccc cece ecco ccc cers shoes eneee soceeens e e ° eeeee eeeee eeeee eee em * ee6e0e%# OR 4268 B® eee eee e ee eee Oe eee 6 ee eee 866 e e666 ¢ bebe Gs e ° 13 16 18 20 22 24 28 3h LIST OF TABLES fable Page Z Design Characteristics of Rocket Gonfigurations......... 29 it Rocket Power Plant Characteristics... ccc cece cccccccce 30 Figure an ns OO wm 5 WO 45 20 ~ if - LIST OF FIGURES Page Rocket Configurations. ...... a 31 Rocket Trajectories... ..0-2ccccceeceers cee e ee sceccececeurce 32 Aerodynamic Forces on Rocket... 2.2... cee eee cee cere eee neee 33 Trapezoidal and Delta Wing Areas Required for Stability versus Mach Number.........-....0.00 2 34 Genter of Pressure Movement with Mach Number......-....ccee. 35 fail Slope of Lift Curve versus Mach Number............. coos 36 Body Slope of Lift Curve versus Mach Numbere............+0-- 37 Base Pressure Coefficient versus Mach Number,..... ........ e 38 Body Drag Variation with Mach Number... .ccee cere ceeee eeroee 39 Wing Drag Coefficient versus Mach Number.e..c.... +e eee cece Lip Total Drag Variation with Mach Number - Configuration A..... UL Total Drag Variation with Mach Number - Configuration B..... 42 i fotal Drag Variation with Mach Number - Configuration C..... 43 6 i Total Drag Variation with Mach Number - Configuration D..... 44 ing Lift Goefficient versus Mach Number for Angles Of Attack. ccc cece ee cc ccc cee cence eet ecen cess seecece 4s Wing Angle of Attack for Maximum Lift-Drag Ratio.........0:- 46 Maximum Lift-Drag Ratio versus Mach Mumber............. ...- 47 LifteDrag Ratio versus Mach Humber at Angles of Attack Configuration A.e..... ee cceeccossesce etceres eee cceees hg Lift-Drag Ratice versus Mach Number at Angles of Attack Configuration Bes.ccsscecsseccecccccccevees Lececeees kg Lift-Drag Retic versus Mach Number at Angles of Attack Configuration €.......- cece scores ec esc cer ses cee ecee 56 =~ 41 ww LIS OF FIGURES - (continued) Figure ek Shrust Variation with Altitude. .ceccccocccavccsccecsccscrece 22 Altitude versus Time - Vertical Trajectoryoo....ccceee 2% Velocity versus Time ~ Vertical Trajectory... ......2.-eceece au Altitude and Velocity versus Horizontal Range..... @eoveecees - iii - LIST OF SYMBOLS Body cross-sectional area (£7) Wiseile acceleration along flight path (£t/sec®) Speed of sound (ft/sec) Drag coefficient Coefficient of induced drag Coefficient of skin friction drag Coefficient of parasite drag Coefficient of nose wave drag Thrust coefficient Coefficient of lift Coefficient of wing lift Wing or tail chord length (rt) Drag force (lbs) Rocket diameter (ft) Thrust (lbs) Velocity thrust (lbs) Bogzle exit area (in®) Nozzle throat area (in?) Acceleration due to gravity (ft/sec®) fotel impulse (1b secs) Specific impulse (secs) Effectiveness factor Lift force (lbs) ~ Lia - LIST OF SYMBOLS (continued) Preliminary value of missile Lift computed for Ge equal to zero (lbs) dt Distance from base of rocket to center of pressure (ft) Distance from base of rocket to center of gravity (ft) Distance from base of recket te nese Lift vector (ft) Distance from base of rocket to tail lift vector (ft) Distance from base of rocket to wing lift vector (ft) Free stream Mach number Body base pressure (1b/£t7) Rocket chamber pressure (1b/in®) Nozzle exit pressure (1b/in*) Atmospheric pressure (1b/in®) Dynamic pressure, § 9 V (1b/ft*) Radius of the earth (ft) Wing area (¢t) Time (secs) Burning time of propellant (secs) Wing thickness (ft) Rocket effective exhaust velocity (ft/sec ) Weight of rocket (lbs) Initial gross weight of rocket (lbs) Weight of propellant (lbs) Burning rate of propellant (lbs/sec) Og Or Ow On ()s Op - lili - LIST OF SYMBOLS (continued) Morizontal coordinate in trajectory (ft) Vertica] coordinate in trajectory (ft) Rocket angle of attack (degrees) Wing angle of attack (degrees) Nozzle area ratio, angle of trajectory relative to horizontal (degrees) Cone half angle of body (degrees) Ratio of gross weight to cross-sectional area (1v/re*) Ratio of propellant weight to gress weight Air density (slug/ft-) Sea level air density (slug/ft-) Ratio of air density at altitude to sea level air density Relative to body Relative to tail Relative to wing Relative to nose Values at summit of trajectory Values at end of burning | I, INTRODUCTION an analysis of the general performance problem of a winged rocket mist consider separately each of the many parameters which will effect the horizontal range, speed and altitude of the rocket. Three of these factors were investigated in this study. With other conditions fixed, wing areas, reduced thrust cruising periods, and trajectory climb angles were varied; their effects on the horizontal range of a missile of the type shown in Figs 1 were considered. fhe general flight problem of winged rockets is becoming of increasing importance as more of the questions associated with the sounding rocket performance are being answered (Cf. Ref. 1). The number of parameters of interest increases manyfold when a vertical trajectory is changed into one of the many possible types of inclined trajectories, end when a vehicle which uses aerodynamic forces only for stability is changed into a vehicle which uses aerodynamic forces to sustain flight. One type of rocket is considered in this investigation (cf. Fe 1 and Table I). It has a long, slender, round body with a conical nose and trapezoidal shaped fins and wings. It is assumed to have sufficient con- trol effectiveness to approximate the types of trajectories investigated. Principle aerodynamic control is achieved through changing of the wing angle of attack. fhie investigation was resolved into two parts. First, an analysis of the variables of rocket aerodynamics, rocket thrust and trajectory was made. Second, these variables were used in step-by-step integrations over trajectories to determine the effects of the variable parameters under discussion. Accordingly, this report of the investigation is divided into the individual analyses of trajectory, missile aerodynemic properties, and trajectory calculation procedure, and finally the conclusions reached. The following general assumptions were made in this study: 1) 2) 3) 4) 5) Aerodynamic properties at very high Mach numbers (4 to 8) may either be calculated by linearized theory, or may be obtained by extrapolation of results obtained for lower Mach numbers. Modifications to conventional supersonic theory due to rarified gases may be neglected. Atmospheric properties are as specified in the NACA Standard Atmosphere (Cf. Refs. 2 and 3). The problems associated with the aerodynamic heating of the rocket will not be considered. Possible limitations caused by structural or control problems will not be considered. Other specific assumptions and approximations are discussed in the sections following. II. TRAJECTORY CONSIDERATIONS A rocket trajectory can be logically separated into three parts, i.e., the launching phase, the powered phase, and the gliding or ballistic phase (Cf. Fig. 2). The launching phase has important problems in low speed stability and in complexity of launching equipment required, The powered portion of the flight must be oriented so as to convert as much as possible of the power plant energy into usable kinetic and potential energy of the rocket. The glide or ballistic part of the trajectory must be established to convert rocket energy into the longest possible horizontal range. The calculations of vertical vacuum trajectories give performance data which are useful in the preliminary estimates of design conditions. Values of » the ratio of gross weight to rocket cross-sectional area, indicate how closely the vertical vacuum trajectory of a wingless rocket will corre- spond to an actual trajectory; for high values of “¢ , the drag effect in an atmospheric trajectory is small, Since the values of for these con- figurations are approximately 3,000 (Cf. Zable I), the drag effect for a wingless vehicle would be small. With wings, however, a vacumm trajectory may vary appreciably from an actual trajectory. 4 vacuum trajectory for the sea level thrust conditions of Case I (Cf. Table II) was computed. It approximates the conditions existing during a vertical launching phase and indicates the order of magnitude of the Mach numbers to be achieved at the end of propellant burning. ‘The following equations were used; —- Met, L5f2 J= AGP (I- eG) logli-v-) + vet -49t raf | Zz vertical coordinate where oY Ve fhe following results were obtained for Ve equal to 6,440 feet per second, effective exhaust velocity i] ZT equal to 0.7 and Te equal to 40 seconds: After ten seconds of flight: Altitude 5,500 ft. th Velocity 900 ft./sec. it Mach number 0.8 ti After 40 seconds of flight: Altitude = 121,700 ft. Velocity = 6,440 ft. /sec. 6.1 Mach number 88 At the summit of vacuum trajectory: Time = 240 sec. Altitude = 766,000 ft. This data differs from the actual trajectory data by the magnitude of the drag term and the effect of increase of thrust with altitude. & vertical launching phase will be assumed for all trajectories. Launching and equipment required for rockets of the size of these configura- tions is simple for vertical flight, but becomes complex for launching angles tilted from the vertical. Vertical flight will be maintained until sufficient velocity is attained to give, first, adequate stability and, second, the control effectiveness required for changing the flight path angle and, third, sufficient aerodynamic lifting forces to hold the rocket on a nonevertical flight path. Data from the vacuum trajectory indicates that a flight time of 10 seconds will be sufficient for the boost phase of all trajectories. The high burning rate of propellant of Case I will be used during this initial launching period. The second phase of the trajectory has its objective in positioning the rocket for an optimum coasting path with a minimum expenditure of energy. Energy is absorbed during the powered flight by drag forces, propellant inértial forces and inertial forces of the rodcet empty weight. The posi- tion of the rocket for an optimum coasting trajectory is dependent on the aerodynamic characteristics of the vehicle but generally must be within the atmosphere (Cf. Ref. 5). Each of the sources of energy absorption during the powered trajectory may be minimized by proper choice of thrust and flight program. fhe ree quirements are contradictory, however, and an optimum solution represents a balance of compensating effects. The drag forces are a minimum for the shortest paths and lowest velocities in the atmosphere. The propellant inertial forces are a minimum for the most rapid burning rate of the pro- pellent; this results in highest velocities at relatively low altitudes. Effects of inertial forces due to rocket empty weight are minimized in the vacuum trajectory case for angles of the flight path inclined initially at 45 degrees to the horizontal. | Three trajectory angles, @, of 20, Lo, and 60 degrees relative to the horizontal will establish the influence of this factor on rocket horizontal range, The vehicle can be maintained on a constant climb angle trajectory by appropriate control of the aerodynamic surfaces. The third portion of the trajectory is identified by zero thrust from the rocke¢ motor, If the rocket is above the atmosphere, the trajectory is ballistic. If the rocket is in the atmosphere, the trajectory is some type of a free or controlled glide path. It is assumed in this investi ge- tion that the vehicle glidee at maximum liftedrag ratic for this portion of the trajectory. Tr. Walther (Cf. Ref. 5) discusses the oscillations which occur in a maximum lifte-drag ratic glide if the glide path is not begun at an altitude for which the aerodynamic forces are in equilibrium st 4 constant glide angle. If the rocket is toc high at the beginning of the glide, it will drop rapidly into denser air and approach a constant glide angle in an oscillatory manner, It was concluded thet maximum range would be obtained by controlling the vehicle so that it begins the glide at maximum lift-drag ratio after esfew oscillations as possible. The following assumptions are made with regard to the three phases of the trajectory: 1) No energy loss cecurs in turning from the vertical launching path to the inclined climb path. 2) he change from initial thrust to reduced thrust occurs in- stantenecusly. 3) ‘The rocket maintains zere angle of attack relative to the flight path through proper controls. Variation of aerodynamic lift is obtained by varying the incidence angle of the wing. 4) Wing angles of attack are limited to the values required for. maximum Liftedrag ratie at a Mach number of 6.0. III. STABILITY ANALYSIS The basic problem of aerodynamic stabilization of a rocket is to maintain the rocket center of pressure behind the center of gravity for all flight Mach numbers, The stabilizing moment is proportional to the distence between the center of gravity and center of pressure locations. In general it should be relatively small since the control apparatus must overcome this moment for maneuvering, Accordingly, it is desirable to have a center of pressure which remaing in a fixed position a short dis- tance behind the center of gravity. The atability analysis established a rocket configuration which was stable up to the highest Mach number determined in the vacuum trajectories, i.e., a Mach number of 6.0. ‘The rocket center of gravity position was assumed constant during the trajectory since a complete weight analysis would be necessary for a more accurate determination of center of gravity location. Wing and fin planforms were selected after comparing the stabilizing effectiveness of delta and trapezoidal planforms., <A final determination of center of pressure travel for all flight Mach numbers was made for each of the vehicle configurations. Selection of fin and wing planforms was based on the considerations of liftedrag ratio, weight of structure, center of pressure movement with Mach number and areas required to give positive stability and design lift. As a first step in the selection of fin and wing planforms, a study was made of the fin areas required to give positive stability of 0.5 caliber for Mach numbers as high as 6.0. ‘wo fin planforms were compared by writing the stability equation for a rocket at a small angle of attack. The forces acting on a rocket and the planforms considered are shown im Figs. 3 and 4, The stability equation is Pry by + Ppl = (Py +P) &c.p. (1) or (2E),egd2hy + (C&) wgd?Z =|(¢%),+ + (2E) Jug? Cp. sinplifying (SE), Ly + ($2), b =[(25), +(2@) flee (2) Rocket stability is indicated by the relation between Lee. and Lege For positive stability, hep must be less than hog . Equation (2) may be expressed in terms of Mech number and fin area in the following manner 3 So) Fyn (Mach) (Cf. Fig. 7) (Sa),2 —t— dx) ~ YZ] dG, 7 ie. 6 (Se), Fn (tiech) (Cf. Fig, 6), for planform 1 for planform 2 = 55 feet assuming the nose lift acts at the tip N +$CGe, for planform 1 be 55 Ce, for planform 2 14.7 feet (Cf. Table I) k Lep. hog - $ caliber =» 13.4 feet S’m o CE, 567 Sr, Ce = 0.93 St2 Equation (2) becomes, after substitution: For Planform 1: IG AG) _ de 35 (SB )y + $6e, Bu (A),= (Boy + GH), Se] 134 Simplifying: St, V5 720.9 Sa, +207 (Bt ae, = O (3) aod) 7 For Planform 2: AC <i S; AC. JS" Se) + an SB x 55Cee = (ee nt ean Sa li34 Simplifying: Sr. T> 7 Ta HERS” Se = (4) 2VSt2 ~ 2503 Sta t69R.5° YM21 = =O Equations (3) and (4) are plotted in Figure 4 as the fin area required for stability with no body effects considered, The effectiveness of the delta wing is increased greatly if the Yconsidered, Spreiter has outlined a method body Lift effects are (cf, Ref. 6) whereby body effect can be included by use of an effective- ness factor such that: (Ce _ K (SE) ao /fin-+ Gooy AX] Fin ALONE | ~ 10 ~- whers the projected areas of the fin plus body and fin alone are equal. For fin areas of from 10 to 15 square feet, and for the delta fin planform shomm, the values of the effectiveness factor, K, range from 0.85 to 0.86, A constant value of 0.36 was used in this analysis. After corrections are made for reference areas, the equation becomes: a _ (~[an Meer oF Fin + Booy A Jems gooy = 0.86 (Sz ein * dé. dG = 0.66 ¢ O55 S7 + Vr ) (Sz Fin ALone (5) ~ This equation is applicable only to fin plus body configuretions which lie well within the Mach cone, Accordingly, for the delta planform under consideration, the results are not reliable beyond a Mach number of 3,0. The stability equation for planform 1, using the increment of lift dus to the body, becomes: dc) | _ ; E 5. | | (8) S77, V S7, 20.97 57,+ 317 i ‘ ' ace VST, * OSS St, Solutions to Equation (6) ere plotted in Fig. 4. The trapezoidel fin planform was selected for the final configura- tions because, first, the comparison indicatss thet a smaller fin area is required for stability at high Mach numbers, and second, the long root chord required for a delta tail of sufficient area takes up the space on the body that must be used for a wing. The loeation of the center of pressure for each of the missile con- figurations was determined from the stability equations as follows: Fr 4 + Pulw + Bor = (A+ Ret P, )Le.p. (7) (26 Sey Ge +S) wo Bd, br = (SEH LS) HES), | Substituting: ~f_ SE x. 55°Ce +t, Sw 4.7 + (AS) 13S = peep A Vez] oe Ss da yes 2 + 7 Be + (Sety} Lee. Simplifying: St Nsy - 2S. 3S7 = ~-63.54/72. ULE) - —2.F%Sw (8) Bquation (8) was first used to establish the fin area required for 0.5 caliber stability at a Mech number of 6.9 for each of the wing areas. Fin areas as shown in Table I were obtained, For these fin areas, the stability equation was next used to determine the position of the center of pressure at flight Mach numbers, For each configuration, the stability equation (8) becomes: Configuration A 292 + 35° VMz = |v2e + Yn] (22) Léep Configuration B 493 + 3S YT) (See Ww =[a27 + VIET (ZE)y | -~ 12 - Configuration C 692 + 35 (er (La), = |r 74 arH(ZE), [Le p. The solutions of these equations are shown in Fig. 5, where center of pressure and center of gravity positions are plotted for flight Mach nun- bers. The problem of missile stability for wing angles of attack other than zero is simplified by the fact thet the wing center of pressure is assumed to be located at the missile center of gravity. The added lift due to wing angle of atteck will move the center of pressure forward from the position showmm in Fig. 5. The center of pressure will not move forward of ‘the center of gravity due to this added lift, however, since the lift is acting at the center of gravity. Accordingly, the missile will be stabls or unstable as shown by Fig. 5. the stabilizing or destabilizing moment will ba less than that showm by an amount dependent upon wing angle of attack. -~13 = IV. DRAG ANALYSIS The drag for each of the rocket configurations was assumed to be made up of the following components: Body Drag: a) Base b) Nose (Wave) c) Friction Wing Drag: a) Friction b) VWave c) -Induced Tail Drag: a) Friction b) Wave Bach of the drag components was evaluated in the supersonic case as follows: I. Body Drag: The base drag was obtained from the base pressure coefficient, aa , defined by iz dé Base drag = $)e ie This coefficient was considered from the viewpoint of Refs. 8, 9, and 10. <A comparison of values obtained by each method is shown in Fig. 3. Values directly from Ref, 9 were used in the range of Mach numbers from 1.25 to 4,0. Yor higher Mach numbers, this curve of the base drag coefficient versus Mach number was extra- polated as shown in Fig. &. The values of wave drag coefficient were obtained from Ref. 11, in which tabulated values are given as functions of Mach num- ber and cone half angle,Q@s. ‘the cone half angle for this rocket - 1. configuration is 7.1 degrees and interpolation in the tables between cone half angles of 5 degrees and 7.5 degrees estab- lished the values of wave drag coefficient as plotted in Fig. 7. Skin friction drag was obtained by use of a constant skin friction coefficient of .002 based on total surface area. Therefore, , Friction drag = .002 Ssurface x II. Wing and Tail Drag Skin friction drag was evaluated by use of a skin friction coefficient of .002. Wave drag and induced drag were obtained from Ref. 12 which gives the following formula for the ‘total drag (without friction) for thin circular are airfoils at angles of attack: Cow = $e 36 (Sef where, m= VFRaT Applying this formula to a 6 percent thick airfoil Ch = 102. 4 00122 od? V7 21 VrT2—/ for ¢< in degrees. This formula for wing drag is based on wing area. It applies to wing and tail surfaces. Fig. 10 shows the total drag coefficient for the wing and tail surfaces. In the subsonic region, induced drag was assumed zero since the pro- posed trajectory called for zere wing deflection during the initial verti- -15- cal flight. ‘The wave drag coefficient was assumed to be replaced by a form drag coefficient which was arbitrarily set at .OOI based on wing projected area. Base drag was estimated from results available on the German A~4 and A-9 missiles (Cf. Ref. 13). The total drag coefficient for the rocket at various wing angles of attack and var ious Mach numbers was obtained from the following equa- tion; Cp = COogit “Dor 7 COow * COew where each of the components were obtained as follows: CDog = Cor + COn +r Se _ 16 [tw CDor = CO¢ + Sor CY) CDow =Cor + u O22 €hF COcw = )72—1 These values of drag coefficient were converted to a reference grea equal to the square of the body diameter and were plotted in Figs. 11, 12, 13, and 14, ~ This drag analysis does not consider the following factors: a) Body-wing interference effects b) Wing-tail interference effects c) Change of base drag for jet burning (Cf. Ref. 13) d) Change of skin friction drag with Reynolds number Since drag effects are comparatively unimportant for rockets with high values of << (Cf. Ref. 1), the assumptions made in this analysis will have less effect on rocket performance than any of the other approximations mentioned in the infroduction. ~ 16 - V. ANALYSIS OF LIFT AND LIFT-DRAG RATIO It has been assumed that the rocket remains at zero angle of attack throughout the flight and that the wing is rotated to give lift during the inclined portion of the trajectory. Since the rocket is at supersonic speeds throughout the inclined trajectory, analysis of the lifting force of the wing is necessary only for supersonic speeds. fhe lift coefficient, Czw, for the wing is determined from : Cow = where Q/w is measured in degrees and the lift force is based on wing projected area. The curves of wing lift coefficient versus Mach number of Fig. 15 were obtained from this equation. The liftedrag ratio of each configuration is of interest since this ratio determined the angle of glide for the glide path portion of the trajectory. Values of the Liftedrag ratio as function of Mach number and wing angle of attack are useful in the trajectory calculations. The angles of attack for maximum lift-drag ratio for each configura- tion were obtained by minimizing the equations for the ratio of drag co- efficient to lift coefficient as follows: Cp = Cron * CDor + CDow + COiw = CDdo + Cdiw .O122 & VF VAI 2—/ 2 C2 é “ | ex), | Cew “76 Cp = CDo9 + -l7- ‘Substituting Cp = -00305 YZ GA, + Do Co = (00305 V7 / CLw + £2 w Cuw ‘Differentiating Co, = .00305" 2, — +e Co For minimun cw _ a Cpg= . 00305 VAIF7 Czy, The curves of wing angle of attack for maximum lift-dreg ratio (Cf. Fig. 16) were obtained by graphically solving the equation for total parasite drag equal to wing induced drag at flight Mach number. Values of the lift-drag ratio versus Mach number are shown in Figs. 18, 19, and 20. Maximum liftedrag ratios are shown in Fig. 17. - 18 - VI. THRUST VARIATION WITH ALTITUDE For each of the three thrust programe under consideration, an approximation to the change of thrust with altitude must be made. Selection of sea level values for specific impulse and burning rate specify the values of thrust at sea level, since tw . F sea level isp Wp fhe decrease of atmospheric pressure with altitude, however, causes an increase in thrust over the sea level values. The magnitude of this in- crease is dependent upon nozzle configuration. The variation of thrust with altitude may be computed from the following formula FF = velocity thrust -+ pressure thrust fhe values of velocity thrust are determined from epecified sea level values of total thrust and atmospheric pressure and from design values of exit pressure and exit area. Nozzle design determines f2 and. fe and thus, the value of pressure thrust at altitude. It may be shown (Cf, Ref. / ) that maximum thrust at any altitude is obtained when the nozzle throat area to exit area ratio, © , is established to give optimum expension, i.e., zero pressure thrust since Ba and BB areequal for optimum expansion. Therefore, selection of a single design nozzle configuration gives maximum thrust at one altitude only. If this design configuration is selected to give maximum thrust near the midpoint of propellant burning time, the thrust decrement for higher and lower altitudes will not be great. Further design conditions on nozzle shape are the rocket motor chamber pressure determined by _ E Fo = Cr f¢ and the missile outside diameter which limits the maximum size of the nozzle exit area, Selection of an exit area, fe , of 700 in.® throat area, fy , of 70 in? and a gives rocket motor characteristics as shown in Table II. Thrust variation with altitude as shown in Fig. 21 were obtained from the following equations Case I; F = 58,000 + (6.3 - P,) 700 Case ITI: F = 41,500 + (4.2 - P,) 700° Case III: | F = 26,800 + (1.7 - Pg) 700 ~ 20 - VII. EQUATIONS OF MOTION AND TRAJECTORY CALCULATION PROCEDURE The following equations of motion were utilized in the trajectory analyses (Cf. Ref. 4): Along the flight path W oy = F-D-WsmNe Normal to the flight path Jo _ _ _ VSe = gcos 9 + GJ On fixed coordinates gx = Vcos 8 rh = Vsin O ake oie These equations are general and were applied to the boost, powered and glide phases of the trajectory. The rocket trajectories were computed by a step-by-step integration of the equations of motion. The following relations were used to deter- mine quantities appearing in these equations: W = Wo- Wp \ o 4 5° < Ny oe \) D Z= Ce Tse vy? d* where Co and Cc. are found for the Mach numbers and wing angles of ° -~21- Two types of corrections must be applied to rockets which reach very high altitudes or long ranges, The first is due to the variation of gfavity with altitude. This correction is given by eR 2 sy = 9 (zs) where dy is the acceleration due to gravity at the altitude y . The second correction is due to the difference between the horizontal range and the range measured along the surface of the earth, This correction 4s also a function of the altitude of the trajectory (Cf. Ref. 15). Since the trajectories being investigated have summit altitudes of less than 1,000,000 feet these corrections are of small magnitude and are omitted in the step-by-step integrations. - 22 » VIII. DISCUSSION OF RESULTS Four vertical trajectories were computed by the step«by-step integration nethod. The results, i.e., altitude and velocity, are shown plotted against time in Figs. 22 and 2}. Configuration D achieves higher altitudes and velo~ cities than configuration A because of the decreased drag and weight of the wingless rocket. Higher performance is achieved in Case III for either rocket; in Case III, the thrust phase is made up of a 10 second boost and 78 second cruise period. Increases of the summit altitude, Yc, of approxi- mately 9 percent resulted for each configuration with the two stage thrust period. The increase in altitude resulting from the two stage thrust period is a net effect of three factors. The primary cause is an increase in total im- pulse; the total impulse, i.e., integral of the thrust time curve, is approxi- mately 15 percent higher for Case III than for Case I. Secondary factors are the drag decrease and inertial force increase for the two stage case. Drag forces are decreased slightly because the rocket has smaller velocities in the low altitude portion cf the trajectory. Inertial forces are increased by the slower burning rate of the propellant. Since the altitude is not in- creased proportionately with the increase in total impulse, it is evident that the decrement due to increase of inertial force is of greater magnitude then the increment due to decrease of drag force. The maximum range of a vacuum trajectory is approximately twice the summit altitude of a vertical trajectory (Cf. Ref. 4). accordingly, approxi- mations to the maximum ballistic ranges for the configurations would be as follows; - 23 - Configuration A; Case I 1,600,000 ft. (300 mi.) Case III | 1,760,000 ft. (334 mi.) Cenfiguration D CaseI 1,744,000 ft. (330 mi.) Case III 1,910,000 ft. (360 mi.) One horizontal trajectory was computed for Configuration A with the constant thrust program designated Case I. The altitude, velocity and horizontal range for this trajectory are shown in Fig. ou. The range at impact is 1,784,000 ft. or approximately 262 miles. The maximum velocity reached is 6360 feet per second at the end of burning of the propellant. This velocity-corresponds to a Mach number of 6.5, This horizontal trajectory is of interest because it indicates the general nature of the powered, ballistic and glide path trajectory of the winged rocket of Configuration 4. To obtain the maximum horizontal range, various values of the flight path angle, © , must be used. The effect on horizontal range of adding a wing to a rocket could be obtained by comparison of the maximum ranges of Configurations A and D as computed by step-by-step integration methods. The effect of wing area could be evaluated similarly by comparison of the performance of Configura~ tions A, B and C. - oh. APPENDIX I SUMMARY OF GERMAN A-9 PROJECT General “The German A-9 missile was a winged version of the A-4 long-distance rocket (Cf, Ref. 13). Development was not completed due to the end of the war, but considerable data was collected on the aerodynamic cheracter- istics of various wing plus tail configurations and on the theoretical per~ formance of the A~9 as a single stage missile or as the glider stage of a tworstage missile. As a further development it was intended to convert the 4-9 into a crew carrying missile with manual steering, retractable landing gear and special aerodynamic aids to reduce the landing speed to approximately 160 kilometers per hour. This crewecarrying version was to go 600 kilometers in approximately 17 minutes. The single stage Av9 missile was developed in an effort to increase the range of the 4-4 from 270 kilometers to 450 kilometers. Sixteen wing and tail configurations on the standard #-4 fuselage were tested in sub~ sonic and supersonic wind tunnels, and aerodynamic studies were continuing at the end of the war. Several trajectories were computed in an effort to ascertain the maximum range possible based on assumed aerodynamic properties. The tweestage version of the se9 was considered because the range of either the piloted or pilotiess versiong could be increased consider- ably if the 4-9 propulsion unit could be started after the rocket had reached a certain initial velocity. The first stage, a large assisted~- take-off rocket designated the A-10, was designed to give the A~-9 an initial velocity of 1200 meters per second. - 2 - The A-9 separated at the end of the burning period of the A-10 booster, and continued in a combined power and glide path trajectory. Under these circumstances, the A-9 would have a range of approximately 5000 kilometers. The proposed design of the A~10 booster stage specified development of 200 tons thrust and installation of air brakes and a para~ chute to permit reuse after water impact. Development of the Singie Stage Version The first step in the development of the A~9 miesile was the deter- mination, from wind tunnel test results, of the liftedrag ratio and the center of pressure movement for each of sixteen wing plus tail configura=- tions at subsonic and supersonic velocities and for an angle of attack range of 0 to 8 degrees. The configuration finally selected was the Auy12/c shown in Fig. 1. This model had the highest lift-drag ratio of all models tested, but also had excessive shift of the center of pressure in the transonic speed range. After selection of the basic configuration, studies were continued in an effort to reduce movement of the center of pressure during transi- tion from subsonic to supersonic speeds, Data on effects of span of swept wings, wing position and airfoil section were being obtained at the end of the war. Several trajectories were computed for the single stage version. In the first, the missile was launched vertically to propellant cut-off and continued to a maximum height of &0 kilometers. At this height, the air density was too low to support gliding of the alvyi2/c and the missile fell in a ballistic trajectory and oscillated after entry into the lower -~ 26 - atmosphere. Load factors and control requirements were excessive for this trajectory. <A flat-path trajectory with a summit altitude of approximately 40 kilometers was calculated to give the maximum range of 450 kilometers for the aly12/c. , Development of the Two-Stage Version Work on the two—stage version of the A~9 was limited to basic design and trajectory studies based on aerodynamic data being collected on the glider ANVl2/c. Basic assumptions for the trajectory studies were as follows; First Stage; A110 Booster Rocket Combustion time; 60 seconds Weight; Payload 16,260 Ke Frame 17,000 Fuel 48,156 Caliber: 4.15 meters Second Stage: A-9 (Glider a4vl2/c) Combustion times 103.8 seconds Weight; Payload 1,000 Kg Frane 3,000 Fuel 11,575 Caliber: 1.65 meters ¢ Your investigations were made in an effort to determine which tra~ dectory would give the longest range withpermissible wing loadings. The first two investigations assumed the second stage combustion to follow - 27 - immediately after the first and that the missile would traverse a skip trajectory after reentering the lower atmosphere. As a result of the first two analyses, it was determined that the whole trajectory should be made at the lower levels of the atmosphere. The third investigation assuned, therefore, that the second combustion stage would begin after the vertex was reached and that the thrust would be utilized for the pull-out and for orienting the missile for gliding. The fourth investigation de- layed the second-stage combustion for 16 seconds after the end of burning of the booster stage. The glider was then turned uniformly 90 degrees during the second combustion period in order to have it at the proper stage. 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) ~ 28 w«- REFERENCES Seifert, H. S», Mills, M. M., Summerfield, M., "The Physics of Rockets", Reprint, American Journal of Physics, Vol. 15 (1947). Diehl, Walter S. "Standard Atmosphere - Tables and Data", NACA Report 218 (1925). Warfield, Calvin N., "Tentative Tables for the Properties of the Upper Atmosphere", NACA Technical Note 1200 (1947). Stewart, H. J., Lecture Notes, Course in Jet Propulsion Systems, California Institute of Technology (1949). Walther, A., and Thiel, &., "Report on Glider Missile", Peenemunde Report, archive 50/8 eK (1942). Spreiter, John R., “Aerodynamic Properties of Slender Wing Body Combinations at Subsonic, Transonic and Supersonic Speeds", NACA Technical Note 1662 (1948). Puckett, A.%., and Stewart, H. J., “Aerodynamic Performance of Belta Wings at Supersonic Speeds", Journal of Aeronautical Sciences, Vol. 14 (1947). von Karman, Th., and Moore, N. B., “Resistance of Slender Bodies Moving with Supersonic Velocities with Special Reference to Projectiles", ASME Transactions (1932). Hill, F. Ke, and Alpher, R. A., "Base Pressures at Supersonic Velocities", Joprnal of Aeronautical Sciences, Vol. 16 (1949). Charters, A. C., "Some Ballistic Contributions to Aerodynamics", Journal of Aeronautical Sciences, Yol. 14 (1947). Kopal, Zdenek, "Tables of Supersonic Flow sround Cones", Massachu- setts Institute of Technology Technical Report No. 1 (1947). Liepmann, H. W., and Puckett, A. B., "Aerodynamics of a Compressible Fluid", wiley (1947). gwicky, F., “Report on Certain Phases of War Research in Germany, AMC Report F-SU-3-RB-(1947). Hayes, T. Je, “Elements of Ordnance", Wiley (1938). Kooy, J. M. J., and Uytenbogaart, J. W. H., "Ballistics of the Foturett McGraw Hill (1946). - 29 - TABLE I DESIGN CHARAOTERISTICS OF ROCKET CONFIGURATIONS Characteristic _ Config. Config. Config. Config. A B Cc D Body length (ft.) 35 35 35 35 Body diameter (ft. ) 2.5 2.5 2.5 2.5 Length/ diameter 14 14 14 14 Nose Gone length (ft.) 10 10 10 10 Nose cone half angle 721° 7.1° 7.19 721° Tail area (ft.) 92 98.6 104.4 85 Tail span (ft.) 43 45 4.6 4.2 Tail root chord (ft.) 6.5 6.7 6.9 6.3 Wing area (ft) 20 ho 60 -- Wing span (ft.) 303 4 4.6 -~ Wing root chord (ft.) 5 7 9.2 ~ Maximum angle of attack of wing 14° 12° 10° -- Sweep back of tail & wing L.E. 60° 60° 60° 60° Wing & tail percent thickness 6 6 6 6 Payload weight (1b.) 500 500 500 500 Structure Weight (1b.) 4000 4200 - 4400 3800 Propellant weight (1b. ) 10,500 10,500 10,500 | 10,500 Gross weight (1b.) 15,000 15,200 15,400 14, 800 Center ete ee bese) “147 14.7 14.7 14.7 Propellant-gross weight ratio 0:7 0.69 0.68 0.71 Ratio of gross weight to 3040 3080 3130 3000 cross sectional area (1b. /ft.*) - 30 - TABLE IT ROCKET POWER PLANT CHARACTERISTICS Characteristic Case Case Case I II IIt Specific impulse: at sea level (sec.) 200 190 180 Burning rate (lbs./sec.) 262.5 180 100 Burning time (sec.) ho 43.7 78.7 Sea level thrust (1bs.) 52,000 34,000 | 18,000 Chamber pressure (psi) 570 375 200 Nozzle throat area (in.?) 70 70 70 Nozzle exit area (in.?) 700 700 700 Expansion ratio, 10 10 10 Altitude for optimum expansion (ft.) 22,000 32,000 45,000 SNO/LE AND ANOD LINDY f FRHN9lY FS ePRO/s 98A 2b o9 Ob o2 005 0/ 00S 0/ 005 0/ 00S OY 008 S/ 006 'G/ 002 S/ O0004/ C ) a4 id ‘DAAVOD OxNOD ‘OY ANOD ‘WANOD OBE _ Y S24 be yan pues 2 IL <i> mam ono, GL GL CII LY FT $b “Ia ON 00s 92 00S 92 | LHSIZANT FO o02GE OOO0E |LHSIZNM SsOx* b-v b-bY NY iW FTA) NYWHT) a SPS m® LO Oe 32 ANVOLIT HL LIA IOLS ZL PHIO YS OVL GLY DAT -L7/ 7 BROMIK YY LY PLE FO/7TID © OFIMWY LNeLEVOD LY GW/7) CF#INQSY ©@ ISUMA LS00OG WIHLLFI GZ 33 LPVIAA NWO DNL SFIS WWYNAWOSFT [ff EO PHILS SY I$ RM eee $/ ae aor a FF She Foe