Amcp 706-140 Trajectories, Differential Effects, And Data For Projectiles

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AMCP 706·140

AMC PAMPHLET

THIS IS A REPRINT WITHOUT CHANGE OF OROP ZO·UO

RESEARCH AND DEVELOPMENT

OF MATERIEL

. ENGINEERING DESIGN HANDBOOK

TRAJECTORIES, DIFFERENTIAL EFFECTS,

AND DATA FOR PROJECTILES

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HEADQUARTERS. U. S. ARMY MATERIEL COMMAND

AUGUST 1963

HEADQUARTERS UNITED STATES ARMY MATERIEL COMMAND WASHINGTON 25, D. C.

30 August 1963

AMCP 706-140, Trajectories, Differential Effects, and Data for Projectiles, forming part of the Army Materiel Command Engineering Design Handbook Series, is published for the information and guidance of all concerned.

SELWYN D. SMITH, JR. Brigadier General, USA Chief of Staff OFFICIAL:

R. O. DAV Colonel, G Chief, Adm nistrative Office

DISTRIBUTION: Special

/ 'd.

PREFACE This pamphlet. one of the series which will c >mprise the Ordnance

Engineerin~

Design Handbook. deals with the subjp.ct of Exterior Ballistics, covering Tra­ jectories, Differential Effects and Data for Projectiles.

c,

The pamphlet pertains

to that phase of projectile life starting at the gun muzzle (or, for a rocket, at burnout) and ending at the point of burst or impact. The subject is covered in a manner intended primarily to assist the ammunition designer in recognizing and dealing with the design parameters of projectile weight, shape, Inuzzle velocity, yaw, drag and stability, in designing for optimum results in range, time of flight and accuracy. It is hoped that it will also prove helpful to other personnel of the Ord­ nance Corps and of contractors to the Ordnance Corps. Because of its complexity, it is possible to cover the subject only briefly in this volume, but an attempt has been made t~ incorporate references which will permit the designer or the student to explore in more detail any phase of the subject. The bal1istics of bombs and of intermediate and long range missiles are not covered in this pamphlet.

Neither' are the problems associated with Interior

Bal1istics or of Terminal Ballistics. pamphlets.

These subjects will be covered by other

TABLE OF CONTENTS

PAGE PREFACE .........••••••••.••••••••••••••••••••••••..•••.•••••••••••••

SVMBOLS

iv

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B.... LLISTICS ••••..•.•.••..•••.••.•••••••.••

1.

INTRODUCTION TO EXTERIOR

2.

VACUUM TRAJECTORIES ................••••..•••••.••••••••••••••••.

. .

1

1

2

AIR RESISTANCE •••••••••••••••••••••••••••••••••••••••••••••••••••

2

a. Equations of Motion b. Charact~ristics of Trajectori~s 3.

Missil~

.

2

(1) Drag ..................•................................

2

a. Fortts Acting on a

Moving in Air

(2) Crosswind Fortt

b. c. d. e. f. g.

4.

. (3) Magnus ·Force , . Simplified Assumptions ; . Drag Coefficient . Drag Function ..................................•............ Form Factor ............................•.........•.......... Ballistic Coefficimt . Air I>ensity : .

DESCRIPTION OF TVPICAL PROJECTILES .•.•• " •••••••..• , ••••• '..•••.•••

a. Proj~cti1e Type 1 b. Projectile Ty~ 2 c. Proj~il~ Type 8 d. HEAT Shdl. 9O-mm.

5.

:

. TIOB

.

( 1) Ballistic (2) Graphs

Tabl~s

Projectil~s

4

4

5

5

5

. .

8

.

( I) Tabular Functions

4

4

. .

SIACCI TABLES •.••..•••••••••••.•••••••••••••• '• •••.••••••.••.••••••

a. Spin-Stabilized

3

3

5

5

5

TRAJECTORIES IN AIR ••••••••••••••.••••••••••••••••.•••••.•..••••••

a. Equations of Motion b. Charact~ristics of Trajectori~s

6.

.

..................•.......

2

2

2

2

.

5

9

9

9

9

9

(2) Ground Fire at Low EI~vations . (a) Formulas ..........•................................ (b) Example . 10

(3) Antiaircraft Fir~ ...................•..................•.. 10

(a) Formulas .......................•.................. 10

(b) Exampl~.....................•...................... 11

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'­ TABLE OF CONTENTS (continued) PAGF: ":--"

7.

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(4) Aircraft Fire............................................ (a) Formulas (b) Example (c) Interpolation _ . . . . . . . . . . . . . . . . . . . . . . . .. b. Fin-Stabilized Projectiles......... . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (I) Tabular Functions....................................... (2) Appli<'ation.............................................. (a) Formulas (b) Example

II

II

DJP.·ERENTJAL EFFECTS.............................................

15

13

13

14

14

14

14

15

a. Effects on Range........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (I) Height of Target (2) Elevation (J) Muzzle Velocity ; ( 4) Ballistic Coefficient....................................... (5) Weight of Projectile (6) Air Density : (7) Air Temperature (8) Wind h. Effects on I>eflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (I) Wind : (2) Drift c. Ballistic Density, Temperature, and Wind

15

15

16

16

16

16

16

16

17

17

17

17

17

REFERENCES. • • • • • • • • . • • • • • • • • • • • • • • •• • . • • • • • • . . • . . • • . . • . • • • • • • • • . • • ••

18

GLOSSARy •••••••••••••••••••••••••••••••••••••••.••••••••••••••••••••

19

TABLES

I Maximum Range for Projectile Type 1. 6,7

II Maximum Range for Projectile Type 2. .. . . . . . . . .. . . .. . . .. .. .. . .. .• 7

III Maximum Range for Projectile Type 8............................. 8

FIGURES

1. Projectile Shapes " 21

2. Drag Coefficient vs Mach Number 22

J. Time of Flight for Projectile Type 1. .. . .. .. .. .. .. . .. .. .. . .. .. . .. 23

4. Time of Flight for Projectile Type 2 , 24

5. Time of Flight for Projectile Type 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25

6. Trajectory Chart for Projectile Type 2 (12 graphs) 26 to 37

7. Trajectory Chart for Projectile Type 8 (12 graphs) J8 to 49

iii

SYMBOLS

Z-Linl'ar deflection

A-Azimuth, measured from the direction of mo­ tion of the airplane (in aircraft fire) A-Altitude function (Eq. 63) A-Axial moment of inertia B-Drag coefficient in Ibjin 2 -ft (Eq. 19) C-Ballistic coefficient (Eq. 22)

Z-Zenith angle 1--Velocity of sound niameter of the largest cros~ section of the

projectile (in Gbre drag law, Eq. 23)

c'-Damping coefficient due to yawing moment

:lnd crosswind force (Eq. 51)

D-Drag D-A ngular deflection F-Temperature in degrees Fahrenheit G-Drag function (Eq. 20) H-Ratio of air density at altitude y to that at the surface (Eq. 48) H-Height of target I-Inclination function (Eq. 62) K-Coefficient (See Eq. 16 and 34) L-erosswind force (lift) M-Mach number (Eq. 18) M-Pseudo-Mach numbeT (Eq. 65) M-Overtumin~ moment about the center of gravity M-log1oe=O.43429 ~ approx.)

c"--Damping coefficient due to drag (Eq. 33) d-Caliber, Or maximum diameter of projectile

f--Drag coefficient (in the Gavre drag law, F.q.

• 23) g-Gravitational acceleration h-Logarithmic rate of decrease of air density

with altitude: 0.000,031,58 per ft

h-Yawing moment damping factor

i-Form factor (Eq. 21)

k-Logarithmic rate of decrease of sonic velocity

with altitude 0.000,003,01 per ft

k--Ratio of American to French drag function

(Eq. 24)

01- 'lass

of the projectile

m-Slope of the tangent to the trajectory

n-l.ogarithmic rate of change of muzzle velocity

with projectile weight

p-Space function (Eq. 28)

p-Projectile weight

ql-Indination fttlll~tion (Ell. 30) q-Altitude function (Eq. 31)

s-Stability factor

t -Time of fli~ht (someti;nt's the symld t, is used

to distinguish the time of flight from the

Sial'd time function)

t -Time function (Eq. 29)

u-Velocity of the projectile relative to the air

u-Component along the line of departure of the

velocity relative to the air: the argument of

the Siacci functions

v-Velocity of the projectile relative to an inertial

system

N-Spin

P-Performance parameter (Eq. Z7)

P-Siacci range: the distance ~sured along the line of departure to a point directly above the projectile (Eq. S4) Q-Drop of the projectile (Eq. SS) R-Reynolds number (Eq. 17) R-Range S-Space function (Eq. 60) T-Total time of flight T -Time function (Eq. 61) V-Upper limit of integration of Siacci functions V-Velocity

W-Wind speed X-Total range of a trajectory Y-Altitude of an airplane above sea level iv

_. -­

w-True air speed of the airplane in aircraft fire x-Horizontal distance from the origin of the trajectory y-Vertical distance Irom the origin of the tra­ jectory

I-Pertaining to, or relative to, Projectile Type I 2-Pertaining to, or relative to, Projectile Type Z 8-Pertaining to, or relative to, Projectile Type 8 .I-First reviaion .2-Second revision

I-Drift

C-Due to I % increase: in ballistic coefficient

~

(capital delta) or ~'-Incnment of (foUowed by a variable) 6-Density of air (in Givre d.ng law, Eq. 23)

D-Drag H-Due to height of target

R-Range component of

T (gamma)-Factor inversely proportioaaI to the ballistic coeffic:ient (Eq. (6)

e-Effective

a (delta)-Yaw

p-Due to projectile 'weight

(epsilon)-Angle of aite (Eq. 72) " (theta)-Angle of indinatioa of the trajectory 1t (bppa)-Crosswind force damping factor 1& (mu)-VilCOsity of the air p (rho )-Density of the air or (tau)-Temperature of the air , (phi)-Angle of departure .. (omep).....Angle of fall I

s-Summital: at the summit of a trajectory, where

y=o s-Standard: pertaining to standard conditions t-Pertaining to, or relative to, some Typical Pro­ jectile v-Due to I unit increase in muzzle velocity w-Due to wind z-Cross component of

S",wsm,'s

a-Yaw

One dot (example: xl-Derivative with respect to time.

or-Due to increase in the temperature of the atmosphere

Two dots (example: y)-Secoad derivative with respect to time.

Stlbsm," o-At t

= 0, Y= 0,

or

t-Due to an increase in angle of departure where y = 0 on the descending branch of the trajectory

I l l -Terminal:

a =o.

v

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TRAJECTORIES,

DIFFERENT~AL

EFFECTS, AND DATA

FOR PROJECTilES

is the muzzle velocity and angle of dep:lrt I .1- is approximately the elevation of the w~tJOn (fictitious values of these quantities may ~ used in the case of a rocket that bums OlOLSide the launcher) . A bomb is released with the speed of the airplane. generally in a horizontal direction. If a gun is mounted in an airplane or other moving vehicle, the initial conditions arc r" ~ct~ by the motion of the vehicle. In any case. it is custonw:y to compute trajectories for a fe" se­ lected values of initial velocity and elevation. Certain standard conditions are specified for a nonnal trajectory: not only the initial velocity and angle of departure, but also the weight of the projectile and a factor that depends on its shape: the density, temperature, and lack of motion of the air, and the height of the target relative to the projector. In order to determine the effects of variations from these conditions. it may be con­ venient in some cases to compute non-standard trajectories; however, they can usua))y be csti­ rnnted in other ways, as will be explained later.

1. INDODUCTIolf TO EXTDloa BALLISTICS

BaUiltics is the science of the motion and effects of projectiles. Exterior Ballistics deals with that part of the motioa between the pr0­ jector and the point of burst or impllCt. .11Je projector may be any IOrt of weapon: a sling-shot. a bow, a mortar, a rifle, a rocket launcher, an airplane, etc. A projectile may be any body that is projected by such a weapon. However. in this p;unphJet, the term "projectile" wiD be restricted' to those missiles that are in free ftight through the air after beiDg propelJed by a gun or a rocket motor. This term does not in­ clude bombs, which are released from airplanes; they wiD be considered in another pamphlet. motiC?n of a missile in ftight is raJly quite complicated. Even a fin-stabiJiud pr0­ jectile or bomb that is not spinning about its Joncitudinal axil is swinging about a transverse its motion. The nose of a a-cis, and this spi?ning shelJ PftCe5SeS about the tangent to its traJectory, and ita attitude gives rise to forces that not oaly its motion in a ftrtKaJ plane but aJao move it out of this plane. To simplify the ~Ia~, ~wever, it is usually assumed that a misSIle ads hke a particle subject to two forces: (1) gravity, which is directed vertic:a11y down­ ward'. ~ (2) drag. which is directed opposite to the direcbon of motion. Gravity i. assumed con­ stant : until 1956, the gravitatioaaJ acceleration was usually taken to be 32.152 ft/-r (9.80 m/secI); and was used for aD tables rdell eel to in this teXt ; now, the lstandard value is 32.174 ft/-r (9.80665 m/secS). The drag depends 00 the -size of the missile, its velocity, the density of air, 3nd other factors. Under these conditions, the motion t'ol­ lows a smooth. planar path. which is called an ~~ ..Uistic trajectory,- or Htrajectory" for

'!he

urc:ets

urea

'I?e

2. V ACUUII a. EtpUltioM of M oliota

TItA]ItCTORIES

A rough idea of an exterior ballistic trajectory may be pined by neglecting the air resistance, considering only the effects of gra~ity. Then if x is the horizontal distance from the origin, y i. the vertie:aJ distance from the origin. t is the time of flight, r is the gravitational acceleration, and ~ots ~ derivatives with respect to time, the differential equations of motion are

i=O, y= -g.

(1)

The initial 'UKlitions. denoted by the suhscript 0,

are

=

X. = v. cos a.. j. V o sin .a,,, Xo = 0, y. = 0, where v is the velocity. .a the angle of inclination.

By integrating equations (1), we find that at

any time, ,

initial conditions for the trajectory of a

particular missile are its initial ft1oc::ity and angle

of departure.· For a projectile, the initial velocity en. tenD _gk of til,.."." and the tenD ,,--,. . u 1Ded ia this pamphlet repI t the aacie from the 1IoNotIIU to the line of departwe tbad'ore: woa1d be ~ aceuntcI7 .... ClllllPIeIeIy ten8ed ....., -.k of

=

=

:it v. cos BOo Y v" sit. 8 0 and by integrating (2),

.,.".".

1

-

gt,

(2~

=

=

posite to that of the motion of the center of gravity. This is the only component acting on a non-spinning projectile that is trailing perfectly. (2) Cro.~mf'ld Forct in the component per­ pendicular to the direction of motion of the center of gravity in the plane of yaw. The yaw is the angle between the direction of motion of the center of gravity and the longitudinal axis of the missile. This component is the cause of the drift of a spinning shell. (3) M ag"vs Force acts in a direction pe". pendicular to the plane of yaw. This is the iorce that makes a spinning golf ball or baseball swerve.

x (Vocos .so)t, y (vo sin .so)t - gt2/2. (3) b. Charculeristies of Trajectories From Equations (3). we find that the time at any horizontal range is (4) t = x/vo cos.so and hence the ordinate is y x tan .so - gx 2 /2v 0 2 cos 2 .so. (5) This is a parabola with a vertical axis. The total range X is found by letting y =0 in ( 5) and solving for x: X:= (2v 0 2 jg) sin.socos.s o (v 0 2 /g) sin2.s 0 • (6) This shows that the range is a maximum when .so 45°. The total time of flight T is found by substi­ tuting (6) for x in (4) :

=

=

b. Simplified ASS14mptiofls

=

The motion caused by these forces is explained in detail in the Ordnance Corps Pamphlet, "De· sign for Control of Flight Characteristics.''! In general, the. motion of the center of gravity is affected by the motion about the center of gravity and occurs in three dimensions. To simplify the calculations, it is usually assumed that the only forces acting on the missile in free flight are drag and gravity. The crosswind force is implied in the drift, but this is detennined experimentally as a variation from the plane trajectory. This approximation is satisfactory except at extremely high elevations, where the summital yaw is Yery large.

T = (2vo/g) sin -80' (7) The components of the tenninal velocity are found by substituting (7) for t in (2):

x = Vo cos .so. Y = -

sin " •.• Hence the terminal velocity is

v.=

Vo

(8) (9)

V o•

and the angle of fall (considered positive) is

"0'

= (10) By differentiating equation (5) with respect to x, we find that the slope of the trajectory at any point is (a)

:~ =

tan ". - gx/vo' cos' -8..

Dr~g Coefficintt The drag encountered by a projectile moving in air is defined b)' the equation of motion

c.

(11)

dv . mdt"= - D -mg SID

At the summit, the slope is 0; hence the horizontal range to the summit is x. (v.lII/g) sin.s o C06 -80. X/2. (12) By substituting this for x in (5), we find the max­ imum ordinate

=

y.

= (vo' /2g) sin

2

-80

=

m-is the mass of the projectile, v-the velocity of the projectile relative to an inertial system, t-the time, D-the drag, g-the gravitational acceleration, -8-the angle of inclination.

(1 - cos 2-80) v.'/4g

(13)

By substituting (12) for x in (4), we find the time to the summit (14) to = (v./g) sin.s o = T /2. 3. Ala

(IS)

where

=

= (g/8) T'.

III

v,

The drag coeffieind is defined by the relation

K D = D/pd'u', (16) where Kn-is the drag coefficient, p'-the air density, d-the caliber or maximum diameter, u-the velocity of the projecttle relative to the air. The drag coefficient is a function of the Reynolds

RESISTANCE

a. Forces Acti"g Oft G Missile MOfIiflg" Air A missile moving in air encounters other forces besides gravity. The principal components of the force produced by motion through air are drag, crosswind force. and. for a spinning projectile, Magnus force. ( 1) Dra,g is the component in the direction op­ 2

-­

ORDP 20-140 number R and the Mach numher M. Reynolds number is defined as

The non-dimensional drag coefficient K D for the 9O-null H EAT Shell T 108 is tabulated as a func­ tion of Mnch number. (See References 3, 4. 5 and (I, which may be obtained from the Ballistic Re­ search Laboratories, Aberdeen Proving Ground. Maryland). d. Drag Function The drag function G is computed by the formula

The

R=udp/", (17) where", is the viscosity of the air, and the Mach number as

=

M u/a, (18) where a is the velocity of sound in the surround­ ing medium.* The viscosity is related to the friction of the air; the velocity of sound, to its elasticity. At extremely low velocities, the Reynolds number causes greater variation, but the effect of the Mach number is larger when it exceeds 0.5. The angle between the direction of motion of the center of gravity and the axis of symmetry of the shell, called yaw, also affects the drag coefficient, but this is usually stripped out of the resistance firing data. Figure 2 (p. 22) is a graph of drag coefficient vs Mach number for the four typical projectiles whose shapes are shown in Figure 1 (p. 21). These projectiles are described and the hasic firings explained in Section 4. It should be noted that subscripts are used to distin~ish the typical projectiles to which the drag coefficients pertain.; a second subscript is sometimes added after a decimal point to denote a revision. It is evident that all the drag coefficients increase rapidly in the vicinity of Mach number 1. The main differ­ ence between curves is in the ratio of the maxi­ mum, which occurs between Mach numbers 1.0 and 1.5, to the minimum subsonic value. Although the drag coefficient has t>een determined and tab­ ulated for several other projectiles, these four represent the principal shapes for artillery pro­ jectiles. K D is a non-dimensional coefficient, and a con­ sistent set of units should be used in Equation ( 16). However, it is customary to express the density as a ratio, the caliber in inches, and the velocity in feet per second; then, since the drag is in poundals, the drag coefficient is expressed in pounds per square inch per foot and is denoted by B. Until 1956, the standard air density was taken as po 0.07513 Ib/ft3 5.217 X 1O- 4 1b/in2-ft.* The practical drag coefficient, (19) B = 5.217 X 10- 4 K D , for Projectile Types I, 2 and 8 is tabulated as a function of velocity for the standard velocity of sound, llo 1120.27 fps. *

=

G= Bu (20) with u in feet per second. For convenience in computing trajectories, this has been tabulated as a function of u2/100 with u in meters per second and either feet or yards per second. for Projectile Types 1, 2 and 8. (See References 7, 8 and 9.) e. Form Factor It is not practical to determine the drag co­ efficient as a function of Mach number for all projectiles. Instead, the drag coefficient of a pro­ jectile whose shape differs only slightly from one of the typical projectiles is assumed to be pro­ portional to the typical drag coefficient. If Kilt is the drag coefficient of a Typical Projectile, the form factor of a projectile whose drag coefficient is K D is the ratio it

= KD/K Dt .

(21)

The form factor may be determined from the time of flight at short ranges, or from the range to impact or the position of burst at a few eleva­ tions. If it is desired to estimate the form factor of a projectile without firing it, the Ordnance Corps Pamphlet, "Design for Control of Flight Char­ acteristics"2 should be consulted. It explains the effects on drag of variations in length and radius of head, meplat diameter, base area, length of shell. and yaw. . Generally, the form factor of a projectile with a square base and an ogive less than 1.75 calibers high should be referred to Projectile Type 1 ; that of a projectile with a square base and an ogive more than 1.75 calibers high, to Projectile Type 8; that of a projectile with a boattail, to Projectile Type 2; and that of a finned projectile, to the TI08 Shell. However, there are exceptions to these rules: e.g., K Dt is used for fin-stabilized mor­ tar shells at low velocities, and K D2 fits the calcu­ lated drag coefficient for some longer fin-stabilized shells. The drag coefficients of bombs are treated elsewhere. to

=

=

0.076,475 Ibl ft" and the velocity of sound is 1I16.89 fps; also the viscosity is 1.205 x 10- 1 Ib/ft-sec, but the effect of Reynolds number is not considered in computing tra­ jectories.

---.--rn 1956, the Department of Defense adopted the stand­

ard atmosphere of the International Civil Aviation Organ­ ization.' In this atmosphere, at the surface, the density is

3

ORDP 20-140

f.

Ballistic Coelficient The ballistic coefficient relative to the tabulated drag coefficient is defined by the formula

C t = m/i t d2 , (22) where m is the mass of the projectile in pounds. d is the caliber (or maximum diameter) in inches. g. A ir Density The air density is a function of altitude. The exponential function adopted by the Ordnance Corps· is tabulated in References 7 and 11. The ratio of the density at an altitude y to that at the surface is H(y).

were used: one was the same as the Russian (or the results reduced to the 'same), but the others probably had longer ogives. In France, a com­ mission of the naval artillery conducted many such experiments at Gavre Proving Ground from 1873 to 1898 at velocities from 118 to 1163 m/s; they fired ogival projectiles with three semi-apical angles: 45°55' (1.64 CRH), 41 °40' (1.98 CRH). and 31 °45' (3.34 CRH). For further details, see References 12 to 16. In analyzing the results of the experiments,IT the Gavre Commission of Experiments assumed that the drag could be expressed in the form D

=~ a 2V2 f(V), g

(23)

4. DF-SCRIPTION OF TYPICAL PROJECTILES This pamphlet contains data that pertain spe­ cifically to four types of projectiles: Types I, 2 and 8 and the ~-mm HEAT Shell T lOR They can he applied to other shapes by using a form factor, as explained in paragraph 3e. a. Projectile Type 1

where D-is the drag (kg), ~-the density of the air (kg/m ll ), g-the gravitational acceleration (m/sec2). a-the diameter of the largest cross section of the projectile (m), V-the velocity (m/s).

Most of the service projectiles used by the European artillery during the last third of the nineteenth century had square bases and ogival heads of various apical angles. In Russia, during the years 1868 and 1869, General Magenski con­ ducted some experiments at St. Petersbur~ to de­ termine the law of air resistance of projectiles with a semi-apical angle of 48°22' (lA9-ealiber radius head) at velocities from 172 to 409 m!s. In Eng­ land, from 1866 to 1880, Bashforth conducted sim­ ilar experiments with projectiles of the same form (it may be that the tables were reduced to this form) at velocities from 148 to 825 m!s. In HoI­ land, Colonel Hojel conducted a few such experi­ ments during 1883; the exact form is uncertain, but the ogive was evidently somewhat longer than those of the Russian experiments. In Germany, the Krupp factory conducted a large number of air resistance experiments at Meppen Proving Ground, starting in 1881 (the principal results were reported by Siacci in the "Rivista" for March 1896); the velocities varied from 368 to 910 m!s, and ~hree different types of projectiles

The drag was treated as a function of velocitv without taking account of the velocity of sound, although the dependence of drag on the velocity of sound was strongly suspected: the coefficient f was plotted as a function of Valone. After studying the results obtained with projectiles of various ogival angles, the Commission confirmed the law deduced by Helie in 1888 that f(V) is proportional to the sine of the ogival angle, at least for values from 40° to 90°, within the experi­ mental errors. In 1917, Chief Engineer Garnier of the Gavre Commission published a table 18 of log B and the corresponding logarithmic derivative, with ve­ locity as argument: this B is equivalent to f(V) for an ogival projectile with a 2-ealiber radius head. This ideal shape is what we call Projectile Type 1 (see Figure I). The Ordnance Depart­ ment of the United States Army then prepared a table of the logarithm of the drag function 1'

* The tabulated trajectory data based on the old stand­ ard atmosphere give results that are nearly the same as those based on the new atmosphere providing a suitable change is made in the ballistic coefficient. For example, a 280-mm shell, fired with a muzzle velocity of 1795 fps, at an elevation of 45°, attains a range of 15,790 yds.; the ballistic coefficient is 3.62 with ICAO atmosphere and 3.SO with the old Ordnance Corps atmosphere. This func­ tion was used in computing trajectories before 1956, when the Department of Defense adopted the standard atmos­ phere of the International Civil Aviation Organization.

G = kBV, (24) with V expressed in meters per second, and the factor k to take account ot differences in defini­ tions and standards. Below 600 m/s, k = 0.00114; but at higher velocities, it is variable. G was called the Gavre function; later, the symbol was changed to G 1 to denote that it pertains to Pro­ jectile Type 1. In the table of log G, the argument, V2/100. varies from to 0 to 33,000 (equivalent to 1817 m!s or 5961 fps). G has been tabulated for V2/100 up to 25,000 (1581 m/s or 5187 fps).

in th~ larg~ Spark Range of the Exterior Bal­ listic Laboratory at Aberdeen Proving Ground. The dl'2( coefficient (defined in paragraph 3c: was fitted with a series of analytical expressions. which were used in tabulating it as a fWlCtion (\'~ Mach number irom 0.1 to 2.7.

~. tabulated as a fomction of v in feet per second up to 6000 ipS.1O

·'The product Gv has also b. PrDjeclik

T~lP!

l

A sketcb of ProjectIle Type 2 is shown in Figure L This projectile has a 6- boattail ~ caliber ionr and an ogivo-eonical head 2.7 cali­ bers long. The eXpet'imental 4.7-inch Shell E1

5.

had thi. shape. However. the resistance firings "'ere actually conducted with two types of 3.3-inch shell: Type ISS with a S° boattail and Type 157 with a 7" boattail. The results were so dose that thq. were grouped together to represent one tY.Pe. T~\e projectiies were fired from 1922 to 1925 at veJonties from 655 to 3187 ips. The results of these nnngs were fitted ;by the tabulated drag function .~ (this is defined in paragraph 3d). Later. it was extrapolated to 10,000 ips by means oi theoretical considerations. For the sake of greater smoothness and accuracy. two revisions have been made: Gu • tabulated to 6000 ips, and Gu • to 7000 ips. Gt . 1 is approximately the same as Gt • but Gu is lower at velocities above 2000

TU]ECTORIItS IN AIR

a. Eqll4lions of Jot Vi"vn Under standard conditions, the tion are .. GH. " GH.

x=-c- x,

equatio~lS

y=-c- r -

('

,1­

g•

where x is the horizontai rangt. y-the altitud~, H-the air density ratio, g-the gravitational acceleration (~~.1:2 ft/sec 2 ), and dots denote derivatives with respect to timL The initial conditions are Xo

=0, Yo = 0, Xc =

VO

CO~ ~, .. ~'~ =

\'0

sin .,)01

where Vo is the initial velocity,

.a.o-the angle of departure.

These equations cannot be solved analyticaHv. but they can be solved by a method oi numeric.a.i integration explained ;n References 21, 22. 23 and 24. b. CJuvoderistics. of Tra;ectone.i ( 1) Ballis'k T abies. Trajectory data have been tabulated ior Pro.iectile T~s 1 &1;0 2 ane for some bomb shape:>. The "Exterior Ballistic Tables Baseci on Nt.·· merical Integration" (Reterences 7, 25. and 20; pertain to Proiect;;e Type 1. The merer ~s toe unit of lengtt.. Tne tr"J.ectones were c,:;.i1pute~ iorwarci and t.ackwa!'c· [r(Jl"!1 the swnrr.i.: and cut off at several .ltitude·. 1i c.. is the '·cummita.: ballistic c~fficien::" ci .. ~rajectory and y. the max­ imum ordinate of a partia.; trajectory, the baliisti.c:: coetncip.nt C 0; this parti4 1 trajector;.· -"1.usfies '.i,e relation loglOC = loglcC. - O.<XXl.045 y.. (26) For each Y.. C•• and summita1 velocity v •• Volume ! gh'es the horizontal range, time, and velocity component~ 3.t the beginning and end of the partial trajectory. If the muzzle vel.xity. angle oi departure, and ballistic coefficient a.·e knO\lo"1\, Volume Il mo:.y be used to find the time of fli(;lu: tG the summit, te. the velocity at the summit, the horizontal range to the summit,,,,,,. and the maximum ordinate. Then C. can be com­ puted and trajectory data can he found in Volume I for the i-ven C. aile: v, at various altitudes.

Ips. c. PrDjectik Type 8

A sUtch of Projectile Type 8 is shown in Fig­ ure 1. Tnis projectile has a square base and a IItCant ogive 2.18 calibers high. Its ogival radius is 10 calibers. which is twice that of a tange~t oglve of the same neignt. The drag function G 8 was calculated from data obtained 'oy the British Ordnance Board from t(\ftt'source!>: (j) !\:ieuurements (If a model in the National Pnysica; Laboratory wind tunnel at Maen. numoers from 0.22 tr.. O.~~. (~ awsunce lirings ot ~-pounder Sc'1eE at Macn numilers itOm 0.6S to ~.a. f 3) Resistanee firings oi 3-inch shell at ~aci. numbers from 0.3:; to :+. L. V. was .irst tabuiatett to SOOO ips. Later. a revise
d. HEAT SMI1, 90-• •, TI08 A sketch of the 9().mm High Explosive Anti­ tank Shell Tl~ is shown in Figure 1. This pro­ ~ecti1e hu tim ~ caliber in span at the end of a long boom and a conical ogive about 2 calibers high. The drag of this projttti.e WllS determiaed at velocities from 700 to 2700 fops or free flight firings

s

purpose,.

The table of range for an elevation of n(~r1y' the maximum ran~e. is rq>ro­ duced iX're (T;,ble 1). Tht: "Extcror Ballistic Table for Projectile T.vpe 2" (Refuence 28) consists or three parts: Tahle !. Sumnital and Tenninal Values; Table I I. Coordinates as Functions of Time of Flight;

ann a lo.r,-{e gra?h of Ma..'l.imum Range vs. Bal1is:ic

Coefficient. The tables have been extended to

include data for everv 15° from 0° to 90°. T~le

muzzle velocity varies irom 1200 to 5200 fet"t ,. ,­

second. and the ballistic coefficient from 0.50 ;0

10.00. The coordinates are given in yard". i:h~ time in seconds. and the velocity in feet per second. Tahle I gives the time to the summit. the ho:-i­ 1.Ontal range tn the summit. the maxim·,.m ordinate. the time of flight. the total range. l'I,­ angle of fall. and the terminal velocity. Table I l gi"es the horizontal r:mge and altitude at e,,("ry 5 seconds of time. Unfortunately. the velocity

However, terminal values can be found more con­ veniently from Volume III. which gives the hori­ ;:ootal range X. time of flight T. angle of fall w. and terminal velocity Vw; the arguments are the same as in Volume II: the muzzle velocity varies from 80 to 920 meters per second. the angle of departure from 0° to 80° (at intervals of 1° f,om 0° to 30°. and 2° from 30° to 80° I, the hallistic coefficient from 1.000 to 17.78 (at inter­ .. als of 0.05 in log C). The "Brief r:xterior Ballistic Tables for Pro­ wrtik- -ivr>e i" (Reference 27) cOlltains values I hM wf'r~' ohtaineej hy interpolation in Volume II I am! conversiOll to English units (yards and ft't'T j~r "t'rond ,. The muzzle velocity varies from 200 rf, :)()(X) f("~T I>f'; second in 100 foot per seconrl 1'll'remf'ntS. The ilngle of departure varies from ISO to 75° in 15° increments. Obviously, inter­ Vlbtion for angle of departure in this table is difficult: nf'vertheless, the table is useful for many

45°, whKh is

./

TABLE I

MAXDJUM RANGE FOR PROJECTILE TYPE 1

ELEVATION 45°

Range in yards. l,og C, C,

0.000 i O.o.'iO ;, 0.100 1.000 i 1.122 ,1.259

0.100 1.413

0.200 1.585

,=;

_yo.

~~;

:."X,:· 300 400 ;'00

397: 1447, 14H1 ~ 71J77:

R5t\. 1441 i 2121 ~

____ .

i

i

27"11: 21162 j 700 :to'):.! 1 . :jt~, loll 10 4:!47 44J 1, 'NIO '4W,tf, 51~i liOO

I

399 R64 1462 2162

400 1\71 I41l1 2201

2934 3749

3003 3M.'i6 4724 '!),!j56

1499

'r1:37

·~.')70

,">349

I

7314 7689 ~25

\lro .1 17m i 1)4(1. :1 t9110 ,i

7~1 7."ifi9

Rti.'ll 1\\112 9t"'7 94.'\4'

:i 'I

R2'j')l

II

AAflf\

2too

2'_'110 I' :l:jtlfl l4 II I,

4!5tJO

Htllli

I'

7ti."it;

~ HH~

-2fD\-1' 9642 2700 9092 I 9823 2'ltlO 9247', 10001 2!I!JO 9m ,tOI77 3(0) 91>49 J0351

!!

'I'

0.400 10.450 2.512 2.818

0.500 3.162

152R

2300

I

1541 1 15.~ 2328 I 23M

7176 7671 8087 8460

74S2 8027

'1

1l320

9141

3.54tI

~

gm

1564

1573

2378

'2399

1.'\H2 2418

3322 4372

3.161 44:~ N'19.'i 67!CJ 7919

8115 9172 10073 10872 11616 11329

94M

9M2 !lit'll \1975 10U18

99721106.';5 10225 110942 IG473 11225 10711' 11.'iO;' 10961 117M2

10397 10603 1(M)7 11010 11210

llIDI 11438 11672 11903 12132

12056 13237

HO'mj

!'I4IM fl634

7714

77f.l4 x3Ml 88m 9353 9787

8.166

8644

8733 9077 9292 9693 9804' 10258 10287 10792

9417 10092 10711 11301

8913 9749 1~ 11165 11814

9fJ64 110200 t0021 I 105!-l9 10368 '1109'<9 1m07 113'71

10747 11302 11193 11798 1163C '112288 12059 12771

11869 12421 12967 13.'i05

12442 13056 13862 14262

1t372 I 12121 1\6971124AA 1:.'011\ 12X53 t2336 1321.'> 1:.!t\52 1:S57.'\

12903! 133Hl I

137:':2 1414:1 14552

14567 IMM 15096 116049 15624 116644 16152 17239 16679 117834

l~

13931 14285 141l.'Ji

1Z.'i95

13275 13SR2

12AAI 13123

J~

I

14\l~7

14187

Ii

15334

6

0.000 3.9S1

IO.6flO 4.4fr.

=1==004 409 [ 409

910 913

lfillO lW7

24.16 24.'>2

I

---1--- - - - - - - - - - - - -1---­

31111 3232 I 3279 4(1)3 I 4140 4222 4301 500H :\141 ;'2f.7 53"4 5947 fila.1 6..110 647f. 6792 7037 _.~=-1_749f\_ 312f.

8488 8905 9294

O.5:lO

3397 4500 5690

0023

13021 13701 ~'kt\27 1l7fi7 14374 ll:,~ "'H6.1 tlXl&l 15041 M!l", 0093_~~ 10364_!~~=-i 1174:'~_,~_11403711485D 15706 lll:i4 7x73')\.~

k447 !lOHI\ AAt2 I \1274 1r174 I 94W

89351

0.350 2.239

3()67 3958 4S70 .'i755 654=-

_1.~-' fi«79-1~:'I_~~:" ~~I_~ 77f>2

1514 2270

--- ---

1100 .. .'i925 i 6~ ti.";!'h1 1200 il 62.'\1 6603 i 6\1.~ 1300 Ii 6521 i 69051' 7293 1400 67.'')9 I 7172 7.~

~"}_':_7113~,

I

0.250 0.300 1.778[1.1195

==1===1== 401 403: 404 405· 406 407 I 40fI tl,77 AA3 ~ '892 I 897 000 1
~OlJO_1: --..:~H~:_.'i;!)~ _~l:~ ~I

7JR.'i·

Muzzle Velocity (v o ) in fps

~s8

15362 15764 16165 11)563.

1371f;

141~

\4655 15121 15585

i

16372 17038 17706 lR375 Il1048

3-::!f\ I 4556 1 5m 7054 1l299\

I

9420

1~

I

3461

4fJ06

/i.'\,W

7174

1I4~~

9(>5M

10ti91

11253111626

12065 12508

12845' 13360 13005 14356 15099 15839 165llO

14194

15019

15839

16659

17480

17323 1 18m6

lfOl8 119136

18820 119975

19.'')74 2m21

~i30 21675

t~

l004S j 17205 18430 1972S-Z1095 16510 117731 I 19027 20405 21Hfi6' 1&.:<70 18257· 1~25 21091 22645 t7429 lR782! 20225 21778 23428 C:....." l.5jI930312OH23 22464 24213

23412

24294

25184

20083

-

TABLE I (Continued) MAXIMUM RANGE FOR PROJEcrlLE TYPE 1

ELEVATION 45·

Range in yards. Muzzle Velocity (v.) in fps

ILoc C, .. .O.a)

0.7~

0:100

0.&110 4

0.800

0.800

1-..;:;...,:'1'11_.. 3••:.1-t..·.46'7_ .6.'.01.2+6.·.e23_.6••3.1-j0. : . : . 6UU

1690 2'36 M28

-tOO

SlO a) 700

409

410

m

m

410

1.000

8.913

10.00

410

m

411

411

m

~

~I

412

m

412

~

8_

7389 8786

9a)

96M

1001M 10296 10484 10660 10826 101J7V 11120 11252 11268 11153& 11794 l203R 12268 12487 12891 12882 1 3020 13339 13&t4 131136 1~131,"m 12344 12688 13375 13796 14207 141lO4 14991 1l53lM 15722 11108'1 14381 14881 15314 It&7 16331 16"110 17235 1'7887 1 16370""- -1-6636-- -1-71-og--l'-1-7ff1-4 -1-8227- -18-786-- 19291 16357 1'7030 1'1702 18371 19031 19682 20324 . . 17344 18108 18875 19643 :Dt08 21162 21913

10891 11628 12llO8 13380

1700 1<&M6 1800 '1MD9

1~19

15839

1100 DQ

15839 1661lO

16859 17480

17491 18399

18338 19340

19198 2029G

2100 2200

17323 180118 U183>

2400

19674

18306 1913& 19975 :Il821 21ff15

19315

2300

20353 21380 22421 23478 24563

21416 22561 '&mYT 24883 3lI083

~-11i06 ~

3mo

~

21177 22124 23082

21096 22539 21886 23412 22M5 24294 23428 26184 24213 1 211083 I

-

~

1.1~ l5~ ~7~'

1.100 12 58

. 14.1

72lU 8634

10388 11263 1~ 12845

"­

412

1614 1619 1624 1627 1630 2491 ~1 2511 251\l I 2526 1

36M -35M-- --as-7-.'\-1--35Q-1·!-3lI05--1 4733 4766 4800 4829 4865 60&4 6120 6172 6219 6200 7484 7573 7662 7725 rnn 8928 9080 9180 9292 93lM

9882 10984 llV90 1~ 13812 14'183 1a186 166118

2SX)

1.0.'50

1611 24llO 3513 4693 0>1

~-'-----I----1·--"---

1100 13JO 13)1) 1400 UlOO

'::1i

1604 2467 :MaD 4651 5934

51TT 10M I

m

1

0.960

7.943

1597 2462 3461 4606 68llO 7174 8472

46&a

!1M)

900 1000

409

_

m

D

1

0.900

240M 2ro4S 2lI04O 27049 2lDlO •

2S&t6 28763 2'nIIlO

'rrs1I 28M4

129849

29037 31183 30217 i 32Ii01

1634 I 1837 2533\ 2MO 3619 3631 4879 I 4899 6298 6333 7852 I 7'9O'T 94891_9.575

,- ­

~il'

1639 2M6 3&60 4918 6364

2'7873

~ d'.; ..-043

~, 25-,o;s

I ZolA'ti I

eoo.'\!

30'&8 31910 330UO 349+1

~5A

I

'm4 I'

1l7~ :_~~.~

!

11'81

13228 16702 18474

j

'JI:t8!f1

'II13..'qt1

:[:~ II

IIAt7~ 1fW'1

I

18852

20756

26217

2ml

2D3'72

31028

---I--~·--.,---l·_--I---+---i

29048

I,

4934. 4951 6394 I !l4JI\.

11373' 13082 14m 18394 180lIO 19?98 21659

~

2S59fl

24972 26385 27832 29321

•

7968' 9663

D)83 21270 20936 22253 ~2C711S1O~0· _ _ ~2M28-- ~20967

---1---+--+---1

22497 23748 'JS1'r1 26334

41.3 ! ' , 3

413

~

I - - - I

:llll61 32748 32446

34091

3S197

37558

I

i

i

TABLE II

..

M j

MAXIMUM RANGE OF'PROJECfILE TYPE 2

Range in Units of 10 Yards

vcoat,.

fPl

i 13JO............

1

10>.......•....

I

2000..•••.......

2400.....•.••.. ,

• •

I 2800............ !, 33JO.•..•••..•.. 3lIOO.••.•.•.. " . ~

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

·, +m............ (

4800•...........

~200 ....

.......

.

851

1001

1143

1131

1078

1182

1287

139t

1284

1426 1571 173)

1218

1299

1384

Il50t 1617

17315

1874

2Qst

3)37

3342

37V6

-C.2

....

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

.... .... .... 7119 819

m

934

0

992

1051

1112

•••

'"

905

I 1061

983

I ,

,

i~

-0.3

0.0

I

709

2220

I

components are not tabulated. The graph has a cune of maximum range vs. ballistic coefficient for every 400 feet per second muzzle velocity; it also lists the angle of departure for maximum ~ A table f.- the maximum ranee is included

I I

I

0.4

0.0

0.8

861

1088

1322

UlOI

lOOt 1335

ItIllO 207l

1806

r62

2179

2928

...

1227

182l 2499

3286

2335

262G

I. I I

I

0.2

M38 4Oll8

4834

li846

7196

1121

1l5&l

an

1298

40'74 01 6a:5

7911

9838

12091

LO

3XJ)

2lI8O 31183

~

M89

I

S3ll9 7006

7201

9106

11356

11335

13888

16752

14019

1701.'\

9034

I

I

I

~

in this pamphlet (Table II). Some trajectories have been computed for Pro­ jectile Type 8 at an angle of departure of 45·. The range. which is nearly the maximum raace. is tabclated here (Table III).

1

TABLE III

MAXIMUM RANGE FOR PROJECTILE TYPE 8

ELEVATION 45°

Range in Units of 10 Yar:ds

II

MuuJe Velocity Jpe

1200...................

. .......... 2800 ... ......... ....

2000 ....

3900 ...................

4-400 ...................

5200 ...................

Los c. -0.2

0.0

0.2

0.4

0.6

621

6& 923

815 1223 1650 2145 2726 3424

151U 2287 3162

961

1091 1934 3124

680

R22

9lW 1108

12li8

1168

1432 1720 ~

When bombs were dropped at low speeds, tra­ jectory data based on the Gavre drag function G J were accurate enough for calibrating the sights. In fact. the descending branches of the trajectories in Volume I of the "Exterior Ballistic Tables" could be used for this purpose. Actually, some "Bomb Ballistic Reduction Tables" were pub­ lished, which were more convenient for this pur­ pose. However. at transonic speeds, the drag functions of modem bombs differ considerably ft'om G 1 • Drag coefficients have been detennined for several shapes. but three of them have been selected as bases of "Bomb Ballistic Tables" (Ref­ erence 10): drag coefficient A was detennined from wind tunnel tests of a shape designed by the :-:ational Advisory Committee for Aeronautics, and is probably the lowest drag coefficient that any oomb will have; drag coefficient B was determined from range bombing of the 3000-pound Demoli­ tion Bomb M119. and is suitable for other mem­ hers of the series including the 75O-pound Bomb MilS and the IO,OOO-pound Bomb M121; drag coefficient C was determined from resistance fir­ ings of the 75-mm Slug Mark I, and is probably the highest coefficient that any bomb will have. The arguments of the "Bomb Ballistic Tables" are release altitude (1000 to 80,000 feet above sea level), true air speed of release in level flight (250 to 2(XX) knots), and perfonnance parameter (0 to 1.5). The performance parameter P is inverse­ ly proportional to the ballistic coefficient; it is de­ fined by the fonnula P

=

lOOdJ/m,

4470 660(

4865

7492

11693

----0.8

1.0

- - ---1201

1288

2403

2826

4231

6936

10884 16582

5251 87401

13661 20319

m-the mass of the bomb in pounds. The tables give the horizontal range from release to impact (feet), the time from release to impact (seconds), the striking velocity (ft/sec), and the striking angle (degrees) . The dng coefficients are tabulated as functions of the Mach number in the appendix; they are graphed in the intro­ duction. (2) G,.a!'hs. Figures 3, 4 and 5 are graphs of the total time of Right for an elevatioTl of 45°. The time of flight is useful in estimating wind effects. For' antiaircraft firing and for firing to the ground at different altitudes, points along the tra­ jectory are needed. Figures 6 and 7 show the trajectories ComPl~ted with G2 and G, at angles of departut:e of 45° and 60° (there is a different graph for each muzzle velocity, each elevation, and each Projectile Type). Trajectories based on G 1 are not included because no antiaircraft pro­ jectiles have a blunt head. The TI08 HEAT Shell is usually used only in direct fire at short ranges. Trajecto:,y data for other elevations can he estimated from those at 45 ° or 60° by assuming that the slant range is the same for a given time of flight. Unlike the vacuum trajectories, in air, the horizontal range to the summit is more than half the total range, and the time to the summit is 1e4s than half the total time; however, the maximlJ4l ordinate is approximately (g/8)Tt. In air, the tenninal velocity is less than the initial velocity and the angle of fall is greater than the angle of departure. At a given elevation, the total flIlgt! increases less rapidly than the square of tilt muzzle velocity; at a given muzzle velocity, the

(27)

where d is the maximum diameter of the bomb in feet.

8

-­

:,\

\ ..'

range is a maximum at an angle somewhat differ­ ent from 45° : lower for small ballistic coefficients, higher for large ones. However, the range at 45° is nearly as much as the maximum. 6. SIACCI TABLES G. Spirt-stabilized Projecliles (1) Tabllw Fllftctiofts. In 18M, the Italian Col. Siacci introduced the Space, Time. Altitude, tnd Inclination Functions. (Reference 29), which simplify the calculation of trajectory data under certain restrictions. In particular, their use will be explained for three conditions: ground fire at low superelevations, antiaircraft fire (where the superelevation is small), and aircraft fire. These and some auxiliary functions based on Glo Gu (the second revision of G2 ) and Gu (the first revision of G.) have been tabulated for the Air Force (References 30, 31 and 32). Since these Siacci tables were arranged primarily for use in computing aircraft firing tables, they were called Aircraft Tables; however, they can be used for other purposes, as explained below, in lieu of other forms. In these Aircraft Tables the argument is the component u along the line of departure of the velocity relative to the air, which is assumed mo­ tionless. The following functions and some of their derivatives with respect to u are tabulat,ed: Space

du G(u)

(28)

Time

du uG(u) ,

(29)

Inclination·

ql

ru

=J

u

x

gdu u2G(u) ,

_ (U Qldu

Altitude·

q- J u

Velocity increment l1'u

G(u)

= KD8 G(u), Z

(31)

(32)

(KDS = 16.4 per rad'), Damping coefficient c" = G(u)/2u.

03;

Tht: upper limit of integration, U, is ;A;\) iI'" tor Projectile Type 1 and 7000 fps for Tyr,es l. and 8. KDS is the yaw-drag coefficient, defined by the formula

KD = Kn" (1

+K

D

32 ),

(34)

where K Do is the drag coefficient for 0 yaw, a-is the yaw in radians.

(2) Grm.,.d Fi,.e Gl Low Eln1Glions (a) FoJ'MlI1as. Although the Aircraft Tables are not accurate in computing complete trajectories at high elevations, they can be used to obtain tra­ jectory data at superelevations below 15· (for the derivation and application of the following for­ mulas, see Reference 33). Here, we shall take p/ P.~s the ratio of air density to normal, as­ sumed constant for each problem, a/a.-the ration of the velocity of sound to normal, assumed constant for each problem.

If F is the temperature in degrees Fahrenheit, a/a.

(30)

=\/(459 + F)/518.

(35)

Then the horizontal range is

= (Cp./p) cos 8. fp(ua./a) -

p(u..ao/a»),

(36)

the time of flight is tr = (Cp..a./pa) [t(ua./a) - t(u..ao/a »),

(37)

the altitude is y

= x tan -8

0 -

(Cp..a./pa)' [q(ua./a) - q(u..a./a»)

+ x(Cp.,a,,/pa) sec -8. [Ql(u"a../a»),

(38)

and the siope of the trajectory is

tan 8 = tan 80 - (Cp.....'/pa') sec 8 0 [ql (ua./a) - q. (u.a./a)]. The angle of departure for y sin 8

= Cp.,a"

',a

= 0, may be found by formula

[qp(uao/ (ua./a) a) -

If pip. ~ I and a/oa.

=

(39)

q (u..a./a ) _ fa)] . ( p(u..ao/a ) ql u..a.

I, let p = p(u), p. = p (Uo). etc.

• Ia most places, the altitude and incliDatiOll fuDctiom arc dcfioed a. twice tbac iDte,raJ••

9

(40)

Then these formulas can be expressed more simply: (36a) x C cos 8. (p - Po), (37a) tr C(t - to),

NOT£: Since u is the component of the velocity along the line of fire, the tangential velocity is v u cos ,s./cos ,.

=

= =

y

=

x tan

C sec tan ~

= tan -8

fo~ y ~

ind,

.s. - CS(q .s. x QIO)

=-

sin -9,

0 --

q.)

In this example, v 2(XX).5 fps.

+ (3&)

C sec~. (ql - qJ.),

(39a)

[~qlO ] P-PO


(4Oa)

(h) Exam~l,

nata: Tank Gun. 76-mm, MIA2; Shot, AP, M339. Caliber 3.000 in. Mass 14.58 lb. Muzzle Velocity 3200 ips Form Factor 1.164 on G~.l A.ir Density Ratio 1.000 Sound Velocity Ratio 1.000

Let m

x p (ua./a)

.so = 23.26 mils,

.so =

cos

t,

=

=

tan ~

.s =-

=

f

dx

(43)

-,

u .

o

.99974, sec.•• =

x m

.02284

1.392 ( UXXl27)( .03886)

= sec..5 o

=m. -

= .05411

fgSU:~·

dx,

(44)

o

-.03127 (Eq. 39a) x

31.84 mils x 1.392 (.99974 )(76542) 10652 h (Eq. 36a) t,= 1.392 (3.0]35) = 4.19 sec (Eq.37a)

=

=

=

v

=[mdx.

=

Check: x tan 8. 10652 (.02284) - C:. 1 (q - q.) = (1.392)1 (327.4) Ce.1 sec -8 X q1. 1.392 (Ul0027) (10652) (.02637) 0

y

=

=

(45)

o

=

-

=

2432... 634.40 391.11 0.00 (Eq. 3&)

10

\

)

--

x

HXX rel="nofollow">27

tan ~.

= p(u.a./a) + sec -8. [~ dx, (42) o

= 14.58/1.164 (3.000)' = 1.392 (Eq. 22) =

=tan ~, m. =tan ••.

Then the following trajectory data can be found by Simpson's rule or some other method of nu­ merical integration: .

Solution: From the Aircraft Table Based on GI.1 : u p tq ql 2000 23079.3 6.5132 489.0 .06523 3200 15425.1 3.4997 161.6 .02637 Dift'. 7654.2 3.0135 327.4 .03886

sin -8" 1.392 (327.4;7654.2 - .02637) .02283 (Eq. 4Oa)

(41)

y - x tan·-8..

Problem: To find the range, time of fti(ht, angle of departure, and angle of fall where the velocity is 2000 fps.

Ca.1

-

<3) A ntiaircraft Fir,

0,

C

=2000 (.99974)/.99951 =

(b) Ex. .,,,

SIlo"

D.ta: Auto_do (AA) 01lD, 4O-mm, M2Alj AP. 1481 YuuIe V_ClUy : 2870 fpl lWIillde ~ : 0.574 OIl 0 ... .up 01 D.,.nun : 12lO ... Air : B-.... Ilow1d V. .~,. Rado: 1.000 Problem: To ftDd lhe 11_ of elcb~ loAd aldWde for bori_tal ranI" flOln 0 to 2400 n. Solution: (See _ u below 1oAd_ t.be AinnI~ Table B.-d on 0 •.,).

eo....

o-a,,. 8&.

, .-

:It

f&

I .­

1

.00lm

eoo

.00908 .01., .02lI80

am

900 ID»

a

::1 2400

1.0000 .9'T14 .9M3 .933'7 .il:a8 .891i .8718 .8621 .8328

.03UT3

1~

j

--

u

p(u)

2613.1 u

fpa

2 0

[BAh

B

Mh,.

.04geI'I

.Q5GllO

.0llG63 .07MS

4 0.00 •

0

1

6

7

8

2870.0

0.9105

0.000

3>109.1

2~.4

1.0681

0.694

ii4li:7e .... ...

DIO.6

i>_i

. ......

22860.1 0

••

•

.. ....

0

•••

•••••

1716.3

1.5225

2198.26

27424.2

1413.7

1.8484

..... .

..... .

.... . .... . 1.295 .... . 2.132 .... .

3.143

I

9 0.8290

..... .

1.ld

10

11 112 2.4142':0

591

I 2.1"il52 ; 1+13

... .

....

......

.2593

I~:~ 2.3:1)8

3.4166

43lt

2.2765

0

•••

1.1Ol1

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

f~

0

2.31~

••

I I

1416 ...

.

.. ..

0

•••••

.

..

~

....

4282 . .. 6664 ---J

Col. 9. The square of Col. 7.

feet.

=

f(26~3.1rol

y

Col. 8. Eq. (43). Simpson's rule gives 3.131 for the last value, but the trapezoid rule was used in order to obtain intermediate values.

Col. 1. The interval of x should be chosen small enourh so that the integrations may be performed accurately by Simpson's rule: not more than 600

Col. 10. Note the factor 1()1 in these values. Simpson's rule gives 4259 for the last value, but the trapezoid rule was used so as to obtain more values.

=

Col. 2. M loglO e 0.43429. h 0.000,031,­ 58 ft_a. The estimate of y = x tan ~o. A slirhtly smaller factor would be better at the 10nrer ranges.

Col. 11. Computed by Eq. (44) with I1lo = tan ~o, g = 32.152 ft / secJ.

=

Col. J. For c:onvenience, the formula H anti·' colorlo Mhy was used. The density function H(y) has been tabulated (Reference 11) ; it gives the same values. The ICAO "Standard Atmos­ phere-Tables and Data"l gives slightly different values.

Col. 12. Computed by Eq. (45). Simpson's rule gives 5666 ft for the final value. (4) Aircraft Fire (CJ) F or",ulas. These tables are especially

useful for aircraft fire (for th~ derivation and at>' plication of the follo.... :ng form uas, see References 34 and 35). Before using them, the initial con­ ditions must be determined. Let A-be the 4Zimuth. measured from the direc­ tion of motion of the airplane, Z-the zenith angle, measured from the upward vertical, vo-the muzzle velocity (relative to the gun),

w-the true air speed,

Uo-the initial velocity relative to the air,

3,,-the initial yaw relative to the motion of the air.

Col. 4. Simpson's rule was used, but the trap­ ezoid rule gives 2193.30 for the last value. Col. 5. The first value was found in the Aircraft Table for Uo = 2870 fps. The other values were computed by Eq. (42) with sec ~O/C •. l 2.6131/0.574 4.5524.

=

. .....

250118.6

Remarks:

=

.. ....

11111.96

...... .

(261:.1),

-

m

lee:

17431t.e

•••••

188••

~f

­

=

Col. 6. The values of u after the first were found in the Aircraft Table by inverse interpolation. usinr the first two terms of Taylor's series.

=

Col. 7. sec.9. 2.6131. This is multiplied by 1000 to avoid writing zeros.

11

If h 15 rhe 'yawing tMmmt damping factor• ...-the c«):;,. wind fora: damping factor. c'= (h+ ...)/2u. (51) the )()sS in velocity due to yaw is

~u- !ola (s.-I,12) A'u (u./a) (52)

a. (s" - 1) Cc' c" (u.a./a) Then the effective initial velocity i. u. = u. - Au. (5:t) Now we can find the distance measured aloni

the line of departure to • point directly above the.

The initial velocity and yaw may be found by the formuJu

u.1 =v.1 + wi + 2wv. sin Z COl A, (46) 1 1 l a. = (1 - sin Z co" A) wi/u. (rad)l. (47) In jDrfl1tll'd 6riag, A = 0; Z = 1600 mil., u. = v.

+ w,

+

(4lla) (47a)

A

= 3~ mil., Z= 1600

u.=v.- w,

(4&)

B.=O.

(4i'b)

projecti1e.

P

The ail' density ratio H and the sound velocity ratio a/a. may be taken u coastanta. Approxi­

mately, if Y is the altitude of the airplane above lea

=e-u , h = O.<xx>,031,S8 ft-I.

a/a. =

e-Ilr• k

=O.<XX>.003.01 ft-I.

(C/H) [p(ua./a) - p (u.a./a) 1.

(54)

Q = (Cae/Ha)' [q(ua./a) - q(u.a./a») - (Cae/H.) Pql (u.a./a). (55)

and the time of Sight is,

t, = (Cae/Ha) [t(ua,./a) - t (u.a./a»). (56)

The horizontal and vertic:a1 compoaenta. if de­

sired. may be found by the fonaulu

x = P cos ie; (57) y P sini.- Q. (58) with the angle of departure detenniDed by the formula

left!.

H

=

the drop of the projectl1e.

(48) (49)

If .. is the standard stability factor. which correspoads to the muzzle velocity. muzzle spin, and standard air deDIity. the initial stability factor is /a.. H. (SO)

=

sun.e

.. =..."••

tanS'. = (aiDZainA)2+coal(aiDZcosA+w/v.). Z

'

(59)

-'

01'

eDt".

= (tan Z sin A)I +

(taD. Z COl A +

wIT. cos Z)I.

(59&)

.

"

'.

12

~.

. ... ,

,

..-

.

,

.. ~:/

\2e = 3409.3 - 3)6.5 = 3202.8 fpe (Eq. 53)

(b) E.IGllltI.

Data: Azimuth Zenith Angle Muzzle VeJoc:ity Air Speed

lZO mila S)() mila JOOO Epa lcm fp. 30,000 It 3.66

Altitude Stuldard Stability FlIdor

Dunpiar Factors h

c..1/H = 0.235/0.3871 = 0.6061 c..la./Ha = 0.6061/0.9137 = 0.6633 (c..1a./Ha)1 = (0.6633)1 =0.4400

+a

PH/~1=IZO/O.~I=I~.9ft

u.II./a = 33)2.8/0.9137 = 3505.3 fIN Find p(u.a./a) in Table, tbeIl p(. ./a) by Eq.

54.

aec- 1

10.6 at JOO) fpa 0.235 on Ga.1

BaDiatic Coefficient

ua./a

t q \h 3.548 165.7 .L..~...ti

13600.6 2.9~5 118.4 .C'~l1"

1979.9 0.593 47.3 .057:

t, = 0.6633 x 0.593 = 0.393 sec (Eq. 56) 0.4400 x 47.3 = 20.8 0.6633 x 1~ x 0.02114 = 16.8 Q = 4.0ft (Eq. 55)

3174.3 3.505.3

8icbt aDd drop for a SilIcci . . . . (P) at lZO it.

Problem: To 6Dd the time of

Solutioa: (Ule the Aircraft Table Baled on Ga.1)' COl

A = 0.38268

sin Z = 0.70711

(c) l,,'er,o~. The derivatives listed in the Aircraft Tables should be used for iDter­ polating backward or forward from the nearest tabular value of the argument. For instance. in the last ~ple, it is desired to find p for u = 3.505.3 and then to find u for p 1S~.5. Fant, the table gives

Z COl A = 0.27060 2wv. = 2 x 1000 x 3000 = 6,000,cm v.1 = (JOOO}I = 9,000,000 wi = (1000)1 = 1,000,000 2wv. sin Z cos A = 1,623,600

u.S 11,623,600 (Eq. <46).

SiD

=

=

_."

u = 3510,

at

3409.3 fIN ..I = 0.72940 x 1,000,000/11,623,600 = 0.06275 (Eq. 47) Ie = 0.2505 rae! (255.2 mils), (1 rad = 1018.6 mils). bY =0.000,031,58 x 30,000 = 0.9474 H = 0.3871 (Eq. 48) kY = 0.000,003,01 x 30,000 = 0.0903 a/a. = 0.9137 (Eq. 49) Ie = 3.66 oX 9,000,000/11,623,600 x 0.3871 = 7.31 (Eq. 50)

c' = 10.6/2 x 3000 = 0.00177 ft- 1

~/a = 3409.3/0.9137 3731.3 fIN

~'u(u.a./a} = 1.4003 sec- 1

(from Aircraft Table)

c"(u.a./a) 2288 x 10-1 ft- 1

(from Aircraft Table)

-!-= -

=

u.

= 0.05733

o.OO2,d~~.ooo,023

5.923

=

p

= 1S606.2, ~ = -

Heace, aacI

=

6.31 x 0.235 x

P = 13572.81,

~u = - 4j, ~ = + 27.84, Hence, aad, at u = 3.505.3, P 13600.65. Thea we have P = 1S~.5 aad the table gives

=

~a = 0.06275 x 0.9137

p

1S~.5

::~1~~000,022,88 3)6.5 fps (Eq. 52) 13

~p

6.044

= - 2Sj

at

u = 3170.

60= +4.3, u =3174.3.

b. F"'-stabilued Projectile. R~erence

6 ("ontains a tahl~ of modified Siacci functions baaed on the Cirag coefficient for the S().mm High Explosive Antitank Shell T1OB, which has rigid fins. These tables can be used for other similar projectiles. although they are not suitable for all fin-stabilized projectiles. The following functions and their derivatives are tabulated as functions of Mach r.umber M: (I) Tabtdor FllfletW,tS.

2.7

Space

S=

[ M

dM MK»(M) ,

dM [ MlIK»(M) , M

T=

ut m be the mass of the projectile in pounds,

i-the fonn factor of the projectile, d-the maximum diameter of the projectile in feet, v-the velocity of the projectile in feet per second,

(60)

2.7

Time

( 2 ) AI'plkGliofi (a) Fo""ula.s. The following formulas are applicable for computing trajectory data at super­ elevations below 15·.

(61)

v.-the muzzle velocity in feet per IIeCOIld, a-the velocity of sound in air in feet per second, p-the density of air in pounds per cubic foot, a-the angle of inclination of the trajectory. a.-the angle of departure. Abo let

:lo = 1120.27 ips, p.

Inclination

(62)

1=

=

M. = v./a,

M= 2.7

Altitude

I(M)dM

A= [

MK»(M)

0.07513 Ib/ft',

v cos a/a cos a..

(66)

Then the horizontal range in feet is

x = (p./n) (S(M) - S(M.)]. the ti~ of flight in seconda i.

The factor in the inclination function is g/a. 32.1S4/112n.27 = 0.0287 sec-i.

=

t, = (P. sec a./apl) (T(M) ­ the altitude in feet is

'" ­ &oP sec l y = x tan a. - :loP.I2 seclI ~i [A(M) - A(M.)] + x ·J a Pl

ap

:loP.:ee' a. a Pl

(6S)

1 = p.idl/m.

(63)

M

tan a = tan a. -

(64)

a•

(67)

T(M.)]. (68)

I(M.).

(69)

{I(M) - HM.)].

(10)

The angle of departure for y = 0 may be found by the fonnula sin 2a. = 3oP. [" A(M) - A(M.) _ I(M.)]. allpl S(M) - S(M.) Ifp=p. and a=a.. let S=S(M), S.= S(Mo), etc. Then these formulas can be ex­ pressed more simply:

= (l/l)(S - 5.), t, = (sec. a./a.l)(T x

(67a)

T.),

(6&)

·y=xtana.- secJ~. (A-A.) +x seela. Ie a.1 "l

(69a)

(71)

. '"

tan a = tan a. - seeI a. (I - Ie). ('tIa) and for y = 0,

'2a. = 80l2[A-A.] S_ S. Ie •

sm

(11a)

..

..

•.j"

~.

,

",

'

..

(b)

=

E.~,~

Data: Gun, 9O-mm; SbcIl. HEAT. TI08E1D

= -78.36ft

Caliber

: 0.29525 It

Ma.

:

Maule Velocity

: 2400 fpe

Form FKtor Air Dmaity

: 1.00 on Gn. : 0.07513 Ib/&-

SoaDd Velocity

: 113).27 fpe

142.

I. _ J(XX)( 1.CXX>10) (0.QOSS1)

"T 0.51667

49.42 ft

y 28.92 + 49.42 - i'8.36 om ft

x sa::' '.

=+

=

of c:Ieputure. aad ucte of faD at a of 3000 feet.

7. DIFl'EUNTIAL 'EPnc:n Some of the eft'ects on ranee and defi~<m caUled by ~ from staadard CIODditioas will be ctilCllEsed and explained Mcft detailed exp1aDatioaa are giftll in R.efaences 2' 22, 23 and 24.

ranee

Solutioa: From the Table of the Siaa:i F.,....,... hued 011 SbeIl. 9O-mm, HEAT. T.J.OJ.

5

Ij3650 3.orJ196 2.14234­ 1-'0836

T

A

1.427502 0.109983

1.J836OM' Oj11519

Dif.

.02537

CJ.

.01923 .Q0851

.Q067O

-.01867 .01072

the target with respect to the lUll: OII~ by meaoa to the aagle

1Il&:-1

0.000,238.29 ft- 1 1Il&:-1

-S - S.

= =

. l1li». =

2 [0.01867 ] 0.51667 1.38360 - ().(QISl 0.01928 (Eq. 71&)

TX 0.CXX>.-t61,2( 3(00) 1.38360 (Eq. 67a)

=

=

•=

21.= 0.99995, 19.64 mils, ' . =9.82 mila. em'" = = 1.00005,IIItJl:' •• = 1.00010

to

- 0.51667 (om072)

tall' = • =v

te

11.32 mils,

ancte

(72)

obtaiDi:Ic

,=,.+a.

0.00964

=

taD-I (H/R.),

the angle of departure ...

ree:teC.

1Il&:'.

tall' . =

1.00010

Of deputure.

(a) For ~ variatioaa in height, the tra­ jectory chart should be ued. The trajectory that passes throucb the point dctermiDed by the cMn horizoDtal raIJIe aad altitude can be found. The total ~ of this trajectory is the ~ where it croaes the axis. The diJlaeDtt bdweaa the gival ~ aad the total ranee is the eh:t of the height of target. For greater accuncy, the a­ terior ballistic tables may be used in lieu of the charta. (b) For small ~tioaa in height, an ~ proximate effect may be fouacl by adding the aacIe of site,

O.ooo,-t61,2 ft- I (Eq.66)

= U3).27(O.ooo,-t61,2) =0.51667 ..y* = 0.51667(O.ooo.-t61,2) =

011

of the trajectory chart, uad another by n co~

= 0.07513(1.00) (0.29525)'/142 =

"T

E.8«b

RtMg. (1) H,;gIU of Targd. Two methods may be used for determining the e&ct of ~ height of

I

- Me = 2«)()/113).27 = 2.14234 (Eq. 64) 'T

=-

(Eq.69&).

Problem: To fiDei the ftIoc:ity. time of flicbt. angle

II

=

a.eck: x taD'" 3000 (.00964) 28.92 ft

~ '.(A - A.) 1.CXX>10(O.O~867)

..y* = - O.CXX>,2J8.29-­

the cor­ (73)

Here, H is the height of target.

Rr-tbe map range,

-0.02015

..--the angle of departure corTeSporu'ing to the map raaee in the ~.or Ballistic Tables,

- 0.01111 (Eq. '0&)

em' =0.9994

= 1120.27 (lj36S0) (0.99995) / (0..99994) = 1945.37 ips (Eq. 65)

,.......the angle of departure cone:spwdiug to the COl iected

nmce R.

Then the effect 011 rauge due to height of target q

= ~= (OjI7519) = 1.3S911l&: (Eq.6&)

A.R = R. - R. 15

(74)

(2) Eletnuiots.

Since the ExIJcrior Ballistic give the range as a faoc:tioa of eJe.vation for n.tant values of muzzle Ydocity aod baDistic cocfficient, the effect of aD inc:rc:ax in elevation may be found from the diftOelaces. A.t short ran~es. tht: effect may be COdlpIfti:ld - IDOr'e ac­ curately by using tht Siaoc:i tables. as aplaincd ... bove Th\~ vacuum {.:mnula (Eq.6) may be usc­ ful in estimating the effect. rspeciaDy for ~ balh>tic ccx:fficients. It shoukl br DOftd that the effect is sr,lall in the '-icinity of the maximum range and, of course, is oegatift at elevations above that corresponding to the olhiodlol rauge. ( 3) MtlZzle VeJocity. SiDce the Exterior Bal­ listic Tables give the range as a f1IIXtioa of DnJZZIe velocity for constant 'faIacs of cIentioo and ballistic coefficient, the effect of an increase in muzzle velocity may be found &om the diflCI'CIICa. At short ranges, the effect may be computed by using the Siacci tables. In YaCUdID, the range is proportional to the square of the 'ftIoc:it:y, as shown in Equation 6; in air. this is a good approximation if the ballistic coefficient is large (4) Ballistic CQe{finewt. Siner the Exterior Balli!;tlc Tables give the ~ as a faoction of ballistic coefficient for c:oostant values of muzzle velocity and elevation.. the effect of an increase in log C may be found from the c1iBt:abkb. At short ranges, the effect may br computed by using the Siacci tables. It is customary to find the efff'.'Ct of a I % increase in C. whidl is eqainImt to an increase of 0.00432 in log C. (5) Weight oj p,.oj.cIile. An iucrase in weight of projectile dcerc:ases the muzzle Ydocity and increases the ballistic c:odIie ient.

the effect

'!- abIes

~C

C nay be expressed

=C(Ap)!a».

(76)

If ~R is the effect 011 nncc of 1 unit incn:ue in muzzle velocity, tJ{ tbr effect 011 ~ of IS incn:ue in ballisti: cocfticicnt. the died OIl raace afan illCfQSe in projectile weight is

6pR

= I1"R(~,,) + 6oR(100 ~C)!C.

(77)

(6) Air DnuU'J. 'The retardation doe to air rcsistantt is prop:Jrtioaal to the air deasity and inversely proporti..w to the baJlistic uwft'">c ieat. as indicated in Eq. (25). Theiefo~ to a first approximation. the effect on ~ of aD iacrase in air density is equal to the effa:t of the 8IIIIl: percelltage decrase in C.

.

(7) Air TerapertJltr.. An increase iD the temperatun: of the atlnosphcn affects the . . . . in two ways: it iocrcascs the density of the air aad the velocity af sound. TIle temperaton: of die air. may affect the taupeiature of "the propcIIdt. which affects the muzzle Ydocity, bat tt.t is aD interior ballistic problem. For a givm pressure and humidity, the deusity is proportional to the absolute temperature. The eifect of density on range has been treated aIJoft. For a given humidity, the velocity of IOIIDd is proportional to the square root of the aheolute temperature. For a given projectile ftIocity, the Mach number is inva-sely proportional to die acoustic velocity. A change in M: prodnas a change in the drag fuDc:tion. A !leW trajec:lory could be computed with the varied fuDction; how­ ever. it is simpler to use an approximate fonnuJa deduced from a semi-anpirical formula daiftd by Dr. Gronwall ~

The effect on muu1c vdocity may be expesaed Apvo = nvo(Ap)!p.

0:1

(75)

~R = 0.1929 [O.OIR - O.OOSvAR

- (I -O.OOO,OlSy.)4oR]A-r,

where V o is the muu1c velocity p is the projectile Jteicbt. n the logarithmic diftaeutial meff· imL

(78)

where R is the ..... ~R -the incrase

in

tonperatare

The value of n may be oIIqjnrd &om iatelior balli!'tic tables. It is appiocincdrly - 0.3 for a rifled gun with multiperfonkd propellant grains. - 0.4 for a riflt!d guu with siacJe-pofuaab!d grains. - ().47 for a smooth bore mortar with flake propellant. and - 0.65 for a J"'Il'OiI1rss ri8e with multiperforated gnUm. Since the ballistic roefficirnt is pruportioaaI to the projectile weight by defirritim. Eqaat.Da (22),

nDge

due to wease .

m

v.-----tbe mazzle veIoc:ifJ' in fed: per ...... ~R-the

effect

OIl

R of 1 fps iDc:reue ill v.

yr-tbe maximum ordinate in feet. AcR-the effect 011 R of 1~ inaeue ill Co A'f-the i ~ in tonperatare in ....ees Fanrenheit.

16

A rear wiad baa tbfte dfccts: it .a.ea the ~ SJ*ID aJDac with it rcJa­ 1M to the: eutb; it iDc:raIa tbe qIe of ck­ ,.mare rdabYe to tbe air; ... it dec:reucs the ...... ftIocity m.tive to the air. The total . . . OIl J'UIIe may be e:xpRMed .. folio. .:

(8) WiU.

A.Jl=W. [

sin, ~R ~R] T+----coa,.-. 4t y.

:l

_ Z

graYity,

x-tbe

Y.

~l

ranee.

Cr-tbc drag fuoctioo.

p-1he air ckasity,

p.-the staDdanI air density (0.07513 Ib/ftI) ,

D.~dWa

(1) WiU. A c:rosswiDd has two effects: it moves tbe coorctinate system with it aud it changes the directioa of motioa relative to the air. The IlDtaI effect 011 IiDear de&dioa may be expressed

e-tbe baDictic ~ Dots dade time daiwatives, and the initial con­ ditioas are

. (80)

z.= O.Ze = O.

r"'eralflre,

where W. is the d'ouwiDcl. The qaIar deSectioa dae to wiDd, in radians, ia

0.= W. (T/R -l/y.cos,):

.

d----«be caliber• u--tbe ftIoc::ity of the projectile rehtiv ~ :0 the air. M-tbc overturniDg moment about the ccoter of

A'R..... dfect oa ranee due to a c:bange of ~ ndiaas in ,. u...... eftec:t 011 raaee due to a c:baaee ~v. in

R/y.cos ,),

(82)

N.--thc iaitial spin.

m-tbe mass of the projectile.

Y.-dIe muzzle velocity,

= W. (T -

I.e ' eravibtional

N-thc spiu.

...... aacIe of deputure.

A.Z

Gpi

::It -

v.-
L-tbe crouwind bu" (lift),

£AV.

wIae W. is !be J'UIIe ~ of the wind (a rear wiad is positive; • !wad wiad is aqatiYe), T..... time of 8ipt.

E'edl _

2qAy.LN.

=&JJdnaIMN.

when: C is the acceleration•

A-the axial iIICJPleOf of inertia,

(79)

f).

The difleseutial equation of motion i..;r the drift iaII

c. BoIJisIie DnuitJ. tuJd Wind Iu t:Jatiag the effects of variations in the atmospbrre. it has hem assumed that the percent­ age Yariatioos from stmdard density and tempera­ tuft aDd the wind are coastaat all along the tra­ jedOry. AdaaDy. they vary with altitude, so that weigbtal mcaa nlucs must be used. The wcigbtiag fadon are cktamiaed by the propor­ tioa of the toQI effect that a variation occurring iu each alritade mae woaJd have. Instead of c::omput.iac a set of wei&htiag factors for each tra­ jectory. a few n:piuentative curves are used. The sum of the prodacts of the zonal variations by their weighti:ug f:ietors is called the ballistic deasity. taDperatiIJ'e, or wiud. The results are thea ~ as coastaDts in computing the effects

(81)

(2) Drill. After the yaw due to ioitia1 dis­ t1U'baaca has damped oat. a spUming sheil has a yaw of repose due to the Clll"Yat1lre of the tra­ jectory. This yaw is mainly to the right of the trajer:lory if the sbeIl has rigbt-baDded spin; to the left if it bas left-handed spin. It·may be de­ termined at short ~ by 6riag alternately frau bands with right aud left hand twists of riftmc. or at Ioac ruga by COl letting the ~ served de6ectioo for the effeds of wind. CUlt of tnmnions, aDd rocatioa of earth.

aD

17

nm&e aad defkdion.

REFltIlENCP.s

Inluna,i<Jlla' C,vil Avial;(ft1 OrKaniulion. Standard Atm'''l>h~r~ Tal,!.. , and /)a~ lor Altltud~. to 1.5.IlOO lerl. 1~"Kley Aeron.;autical Lab: NACA H· J 235 2. D.-sill'n O}{DP

17. l.alurw.. I<eport. No•. 1.1. and 1429 of the Gavre como.inioll (AUI. 1897 and July 1898). TheK were the origin.. I sources of information contained in ref.

(I~S51.

I •.

III. Garnier. Calcul des trajectoires par arca. Giwe Comm. (I~17). 19. EXlerior I:allisti..: Tables based 011 Numerical In­ tegratior.. Vol. 1. Ord. Dept. of U. S. Army (1924). Table la: Log G(V). Table Ib: d log G/(V dV). 20. Herrmann, E. E. Ran~ a,nd Ballistic Tables. 'I S. Naval Academy (1926, 19JO. 1935). Table I: Thl' Gavre Re.ardaticn Function Gv. 21. Woulton, F. R. New Wethods in Exterior BaJJistics. Univ. of Chicago Press (1926). zz. Hitchcock, H. P. Computation of Firing Tables for the U. S. Army. Aberdeen Proving Ground: BRL

Control of Flight Characteristics. ~1-246. (Art. Am Series, Ord. Eog. Des. Handbook I (1957). 3. Dng COt:ffic;~nl HI' Aberdccn Proving Ground: BkL ~.

Orag

Fil~

for

!'oj. 1·50

Co.·fIi(l~nl

(1942). B2.

Aberdeen Proving Ground:

BRL File :-/·/-5l il942). Drag Coefficient Bu. File 1\-1-86 (1945). Drag Coefficient Bu. File N-I-IOS

ll946). 5. ; iarringto:l, M. E. Drag Coefficient BI . Aberdeen P·oving Ground: BRL File N-I-69 (1943). Drag -:')efficien, B.. ). File N-I-89 (1945). G. Odom. C. T. Drag Coefficient. KI)f and Siacci Func­ lions for a Rigid-fin Shell Based on the Shell,9Omm, HEAT. TUll!. Aberdccn Proving Groand: BRL MR-882 (Rev. 1956). 7.0rd. Dept., li. S. Army. Exterior Bamstic Tables bHed on Numerical Integration. Vol. 1. Washing­ I'm: War 1Jept. Document 1107 Om). G-Table. A~rdeen Proving Ground: BRL File N·I-40 (1945). Table of G. BRL Fik N-I-92 (1945). (These tables pertain to Projectile Type 1. Doc. 1107 has log G 1 with V in m/s; N-I-
.

;

X-IOZ (!9z.4). 23. Hayes, T. J. Elements of Ordnauce.

New York: John Wiley aDd Sons (1938). 24. Bliss, G. A. W:lthematics for Exterior B.llistics. New Yorl.;: John Wiley and Sons (1944). 25. Ord. Dept., U. S. Army. Exterior Ballistic Tables based on Numerical Integratio" Vol. II. Washing­ ton (1931). 26.0rd. Dr.»t., U. S. Anny. Exterior Bamstic Tables ba.ed on Numerical Inteeration, Vol. III. Aberdeen PraYing Ground: BRL (1940). D. Brief Exterior BaJJistic Tables for Projectile TFPe I. Aberdt.'f':n Proyi. Ground: BRL File N-I.90 (1945). 2&. Exterior .3aJJistic T.ble for Projectile Type 2. Aberdeen PlOviDg Ground: BRL WR 1096 (Rev. 1957). 29. Siacei, F. Corso di balistica. ht ed. Romc (1870 to 1884). 2d ed. Turin (1888). Frenth trans. "" P. Laurent, Balistique extuieure, Paris (1892). 30. Aircraft Table Based -on G.. Aberdeen Provin, Ground: BRL File N-I-Ill (1955). 31. Aircraft "Table Based OIl G, 2' Aberdeen ProYing Ground: BRL File N-I-I21 (i9S5). 32. Aircraft 7able Based on Gu . Aberdeen Proviac Ground: BRL File N-J-I26 (1955). 33. Hitchcock, H. P., and Kent, R. H. Applicatiou of Siac:ci's Methad to Flat Trajectories. Aberdeea Proving GrOUDJl: BRL R-1l4 (1938). 34. Sterne, T. E. The ESect of Vaw Upon Aircraft Gunfire Trajectories. Aberdeen Proving Ground: BRL R-J45 (1943). 35. Hatch, G., and Gr'u, R. W. ANew Form of B.I­ listic Data for Airborne Gunnery. Mus. lnlt. Tech: Instnuncntation Lab R-tO (1953) (Coaf). 36. r.~()n_lI, T. H. An Approximate Formula for the ; l~ Variation due to a Chance in Temperature. ~a5hingto:' :.OrcL Office, TS-161 (1921).

8,. (;2- Table. Aberdeen Proving Ground: BRL File . N ·1-20 (1944). Table of G,.r File N-I-93 (1945). 9. Harrington, M. E. Table of G.. Aberdeen ProYine Ground: BRL File N-I-68 (1943). Table of G u . File N-I-96 (1945), 10. Martin, E, S. Bomb Ballistic Tables. Aberdeen Proving Ground: BRLM 662 (1953). II. 1/ H- Table. Aberdccn Proving Ground: BRL File N-I.... I (1942). H-T'able. File N-I-1I4 (1950). Alti­ tude is in meters in N-I....I; yard. in N-I-114). 12. Charbonnier and GOlly-Ache. .Wesure des Yitesse de projectiles, Part 2, Book III. Wan. de I'art. de la marine, .JO: I (1902). 13. Jacob. Une solution de problme balistique, Chap. II 1. Wen. de I'art. de Ia marine, 27: 481 (1899). 1•. Note sur la ddermination expb'imentale do co­ efficient de 1a risist.alltt de I'air, d'apra les clemiera etudes de Ill. commission d'expa-iences de GiYft. Mem. de rart. de Ill. marine, 27: 521 (1899). IS. Charbonnier. Note sur Ill. resistance de rair au mouvement de projectiles. Men de fart. de la marine, J3: 829 (1895). 16. Ingalls, ]. M. Ballistic Tables, Art. Circular 1l (Rev. 1917). Scc p. III af Introduetioa.

• ...

18

r",-·

GLOSSl\RY

A. ued in this pamphlet tIw::R terms mean "."..., .-gil 01 which is defined &I the angle between the horizontal and the line along which the pro­ jectile leava the projector. ODe of the two initial c:ondition.s 01 a trajectory, the other con­ dibOll beinc iwiIidI wl«il,. For a roeIaet-fired projectile in wbicb the pr0­ pellant bunUac CClIltmua after the rocket leaves the launcher, it is ....C'''Suy to make ~ ·tiona relative to the point at which the rocket becomes a missile in free ¥t aod the velocity

..... of cIepattan. (elnatiaa)

d,,,..,.,.,

.,)nati.aL See

exterior ....Wsdca. See bGllistics. blirizoll.ta1 rage. 11le horizontal UlmflODcn. ,·f the
iJaitia! ftlodty. (muzzle velocity) 1Dc: projccti1e velocity at the moment that the proj~t . ~ :e"1ses to be acted upon by propellingfon:es.

For a gun-fired projectile the initial velocity, expressed in feet or meter$. per second, is called ff&auzle veloci'y or ifliti4l v,loci". It is obtained by measuring the velocity over a distance for­ ward of the gun, and com:cting bade to the muule for the retardation in Bight. For a rocket-fired projectile, the velocity at­ tain~ at the moment w11m roclc:et burnout oc­ curs is the initial velocity. A slightiy fictitious value is used, referred. to an assumed point as origin of the trajectory. The initial v~locity of a bomb dropped from an airplane is the speed of the airplane.

attained at that point.

-

1JIC1e ..f ilaclillau... The angle between the hori­ zontaJ and the tangalt to the trajectory at any point. At the origin of the trajectoly this is the same as the tJ1IIfIlI 01 d,/HJrl1W', apical <_I-apkal) aqIe. In general the angle formed at the apex or tip of anything. As ap­ plied to projectiles, the angle between the ~­ gelts to the Olrft oudining the contour of the projectile at its tip, or for semi-apical angle, the angle between the axis and one of the tan­ gents. For a projectile haYing a conical tip. the cone apex angle.

.w.tica. Branch of applied mechanics which deals with the motion and behavior chancteristics of missiles, that is, projectiles, bombs, rockets, guided missiles, etc., and of accompanying phe­ nomena, It can be conveniendy divided into three branches :-ildnior ballistics, which d"2ls with the motion of the projectile in the bore of the weapon; utnior ballisIics, which deals with the motion of the projectile while in flight; and t"",i'fQl ballistics, which is concerned with the eff«t and action of the projectile when it im­ pacts or bursts. crosswind force. The component of the force of air resistance acting in a direction perpendicular to the direction of motion, and in the plane de­ tennined by the tangent to the trajectory formed by the center of gravity of a missile and the axis of the missile. (This plane is calkd the plane of yaw.) drag. The component of the force of air resistanCe acting in a direction opposite. to the motion of the center of gravity of a missile.

lIfIgu 0/ d.,IJrlw•.

llIapu force. The component of the force of air resist.ance acting in a dir«tion perpendicular to the plane of yaw (see crossuMul force). It is caused by the action between the boundary layer of air rotating with the projectile and the air stream. This force is small, compared with drag and crosswifld force. mu..ule ftlodty. The velocity with which the projectile leaves the muzzle of the gun. See i'"t~ w/oci'y. slant range. The distance. measured along a straight line, from the beginning of the trajectory (point of origin) to a point on the trajectory.

stability factor. A factor which indicates the rela­ tive stability of a proj«tile under given condi­ tions. It is dependent upon the force momertts acting to align the projectile axis with the tan­ gent to the trajectory, and the oYertuming force moments. For the projectile to be stable, the stability factor must be greater b'1an unity. standard atlusphere. An average or representa­ tive structure of the air U$ed as a point of de­ parture in computing firing tabl~ for projec­ 19

tiles. The standard atn~ adopted ill 1956 ,r the Armed Services and DOW known as the l,. S. Standard Atmosphere, is that uaed by the i~.(emational Civil AYiation Organiz.:ation (I CAO ). . This Standard AtmoIpbn-e aaaumes a ground pressun of 7(;1) m/m of mercury. a ~ound temperature of IS° C. air dauity at ground level of 0.076,475 Ib/ft', aDd a taDpc:ra­ ture grallirot to the beginniIIc of the ~ ..phere expressed by the formula

absolute temperature T(OK}

= 288.16-6.SH

where H is height above sea level in Jn1ometa's. ~n the stratosphere (above II kilometers) the temperature is assumed ttIOStaDt at 216.66° Ie. For firing table purposes wiad .,eIOCitir:I of zero are assumed.

...porei~"'_:tiClll. An ;I(~ded positive angle in aati­ aircraft gt:m'~r)' that COR\(l("nSlltes for the faD of the projectile dunng the time of Raght due to the pull of gravity; the angk the: gun or projector

/

must be elevated above the: gun-targd tiDe.

trajectery.· The path followed in aDtel' of gravity of a missile while

by the ill free 8icht.

5pQIC'e

yaW'. 1ne angle between thc direction of moboIl of a projectile and the axis of the projectile, ~ ferred to either as ,.". cr more c.«lIIlpIe.tdy. tJAgk 01 ya;tI1. The angle of yaw incn::ases with time of flight in ~ UDStab1c projectile (see " . bilily lactor) aad dccn:ascs to a aJOstant nlae. caI1cd the jOII1 01 nfHne, or the re~ .."u of yorv. in a stable projectile.

-

-

r..------:s·

28- - - - -.....

.....- - - 1...- - -.....-1.32'-----oIl-.c

PROJECTILE TYPE I

..-:1-.....- - - - - ­ --5J,-----------­

-F

--L.I.. ' - -.....

1. . . . . . . . .

A1 - ----2.

70-------'l~

PRO-ECTILE TYPE 2

......------3.•.---------4.-1 .067 R.

i

\.

-

!='M··'---:j:

Z.III-----Cl-t

PROJECTILE TYPE 8

90-mm HEAT SHELL TIOS ALL DIllE. . . . IN CAL_

Figure 1. Projectile Shapes

N N

00

.05

0.5

•10 tf-1tiMtH-++J=.

.15

KD

10

·1!5

LO 2.0

u VII

MaCH NO.

Figure l. Drag Coefficient

1.5

Mach Number

10

15

,1-,

--

4.0

I

4.5

1-1-1-++ '-:'-­

,:.)

aD

-~

i~~

-

--r~ . Ii' "1.'': . -;-,.-. +T~

•

T:TT­

-'1;;:~(:;tTl rr~

,-~

i-H' i ,, ,~ ..--1-. I

---'+p~~:-~r+1

_~.

100

.

t

10

1-1-

-1

-i

I;

7!

T

~J-

+

~

l'

-

.:. 11::''t:+ ;;f~+-< 1 ~j

.

j

t

"i-

,

H-t-,+ , .-+t~~ . .

•

10

. i+'

~,'~'i

I

~t-H .~

~.

' . ~fl

70

.. ~

lrit.-l . -~

eo

I

i

~50

/

:\ ~

~

..

..,

III

!

40

30

.'

20

10

I:

.2

Figure 3.

.8 LOG Cg

.8

LO

Tim.e of Flight for Projectile Type 1

r.2

/--~

\

i }.

110

140

120

---

80

40

20

o

-0.4

-02

Figure 4.

o

.2

LOG C.

A

Tim.e of Flight for Projectile Type Z

Z4

.8

1.0

-.-'

~ :-+~ t-..+-· looK

180

~

:0'4

;

.

&l

.

120 ,-I • i

I

i:;

.......

"'100

...o

10 -...l-+-!- . ;

,

H

eo t

-

'.'..,.­.....

40 ,

, ,, .....

,

20

o

-002

, !

,

o Figure 5.

02

.4

LOG

C.

oS

08

Time of Flight 'i.')r Pt"ojectile Typ.a 8

zs

1.0

I

IV 0-

....

~

~

'j .

.• mj

i

u

8 10 6 HORIZONTAL RANGE (1000 Yos.)

i~t

.

tmili ~ttltttlt W l~

-<

._ ..

t1

.. ,

12

t

_..... ~

..,.-"r;. ...

14

-;'~'tt~e

(

(

F~gure 6 (1 of IZ). Trajectories lor Projectile Type Z, 45° elevation, IZOO Ips muzzle velocity

("""4

.oJ

~

I&J 0 ;:, t­

-­

0

0 0

c >

--en.

'i

~ "

C;

..... ~)

/"; N

:.

­-

", 'Il

'o.

r:. ...... ,-~

:":.

-,.. ( J)

0

0 0 0

~

;~

...." ~,

,'ll

:> Q) ...... c:;

0

l.lJ

~

Z

4 ct: ...J

~

LC'"

'
N

.

QI

~

f-I

4: 0

........

c:: Q

'-'

N

'r:u

... ..

Q)

C

!~

I-l

....0 III 4)

.... I-l

...u 0

...... Q)

'" .

I-l

~

N .....

'0 t'\l

-..0

"

I-l :j

....

Ql)

~

2.7

(

tv

00

.

'.

q

°0

4

8

''+T~

rt

~

8

nd

n~~

II

"11 ~I

~.tl·I~.

ust

16

·li

M

t+f- '"

!2

~

40

(

(

a,

48

H~

......

d:~--

56

------~s

45° elevation, 2800 fps muzzle velocity

HORIZONTAL RANGE (I000YDS)

lH

Fipre 6 (3 of 12). Trajectories for Projectile Type

~

..J

~

~

'"c

::)

....~

0

>­

0

"

L

n'

I'

IfJ~1

_12

(It

I

!t,r.

~~.;

',­

N

'.(1

>

~:S" e-~f;, .. ~.t~O!l;

IIIII

+>-1'1I'~llHltt

,FigLlre 6 (4 of 12). L'l'ajec:l,orien for T'}rojc,;tile Type 1..

e

----...,

:.,'

::,:,00 £p=, /

Hlftl

­

,. 1·

c' ' r,"

[ffi

• .'t

11.. ·r-:: ',1"

_/

,.

!(

v~!ocit>

~

t

w

0

+.J:.....

1::I't l·i';TI~1 t-'-·r' ·1

.LIlliiil .' :tH

:~:

'It

......

H·

H

•. '

L

...

}l

fiii!.:

ii'i' t'

,

,,

:: ~~

1'.

t- .•

,

"

t

1 " :[

t

t

t '

......,

•

:1

"t

..

.

•

.

t I· t

.

-+

r. -I

.

t

n~:

... •

m

""'il

'j""'"' • •..,.q it

''''

t!~ i· f

~

.•

"t ... ~

.,.

'"'' ''-. ··'":.l.ti 'tlrfF;::,', ·~t+·"H·

-i, ..

.

t....

,.,1 t·+ "1- --

+--... .. ..

'T'

-r~

•......,

'1

,I

o~ o

... ,

f

p. r

-~.

.

.. I

~~

.

. ..

t.. .....

. .

l!t-l , l l ;

" ••"r-M ,""

•

rl

~

¥

'

r

1

20

... ..... f ;.4 ~,.

. ',1'

,L.--t.

0-

~

... .,.J "1:' ~ ~..

ot .. ,

40

r~-i-+

tL. t-1·.

..

ct'

'··o " .f

.1'£...f'

~-r

..

r

-. .

•

I

.:.

~

I

.~ ~lt

;Hi'

.1-.. _+

• ......

t

'

~

f

L

•

• _.

•

J.

"m"' l·:t~~ 1:,"t .t, , H " i ttt.. :Ij!;.tti'1 1 "'-.: • t fH 1+";.,'1 11l\

,-

-+ ••

.. 0..,..

.,.

•• ,

•

•

80

V;T

.•

Itf:!'

_

100

120

rt-t't· . .!-!~ .. ~. . t'.. . j---+t . . .

+::-

_0­

JJ#l.,;r ... · ....

.

·

ttr

140

•

~-

..

• r-- .:

.. 'r 'C'-'" ~U'· r"! • , .. • ~

t-,

:-:::-.

t

.W-+f.t~1 ;a!'l'~~ ":';~~·it~,·:1 ' "., 1.. '+'1+.t i.••.•. >;..;. .• ~.~ ' ~ ++ ~ ...,._ - •• r~:' . -. :Lh:" ..- E ..i"-.-,·;-.. l ,t·· ~ . ' 'f'l ··t:,ll,~", ,... ·i"4~.j.=t':ir-:~; .• ;'::_~"·'. ::, Hr:::: :::: . :ftitt ~ _'M- . ' H. ~+ : -..ttHl •."'~IU2 ,~ .. ••.I'I--' L~ ~ . .

I·· ;.;: 1 '

-t-..,.

• :-:-:­

.••

t,,;., ...

I)..;' .., ...

'H.I ;::

+tt ,t:;

tt :--:;

'"

t.~i, ':: ,..

t

: ~: Ij m.1 II - •

.+++ ........

..

1ft ,", .

• ',. .r·

r .. ·

frnit ,"~'l'" ...... t· itt

... _

tl; IiiI ,tl

lli~1T t ...

+" 1'" ••.. t · ... t

; •.

•

.

tl :;; f· '"

t­

t :~~~ ~~ . .? +~ .... -i. ~.~ ~.::: j ••• "!", .:1, .:lJ: I¥:. -.-... ~dY.-l ,I.~l:\; . .tt.i.t, I'" .•• 11-I,i' q;' ~~n~-~ ~~: :::~ ~. ~~~f';'~-t r.1"~_~ ;.:-r~ "~+H·rt -t1t"' r-tf;::: :::~ ItA 1ft .·~r... ~i_:" 1;~.t rt~ 1~:: '.~ ~ . +tttf+;r~ ::-?: ,trftttttfHm+-. l.i'." ,E~ ... ~ ••. - t.t, t.·~ . thi

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Figure 7 (2 of 12). Trajectories for Projectile Type 8, 45

I ,.-__

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elevati(\n

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- 12

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Figure 7 (4 of 12). Trajectories for Projectile Type 8, 45° elevation, i(IOO fps

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Figure 7 (5 of 12). Trajectories for Projectile Type 8,45 8 elevation, 4400 fpb muzzle velocity

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49

"\ )

ENGfNEERING DESIGN HANDBOOK SERIES The Engineering Detiign Handbook Series is intended to provide a compilation of principles and fund~~ental data to supplement experience in assisting engineers 1n the evolution of new designs which will meet tactical and technical needs while also embodying satisfactory producibility and maintainability. Listed below are the Handbooks which have been published or submitted for publi­ cation. Handbooks with publication dates prior to I August 1962 were published as 20-series Ordnance Corps pamphlets. AHC Circular 310-38, 19 July 1963, redesignated those publications as 70b-series AMC pamphlets (e.g., ORDP 20-138 was redesignated AHCP 706-138). All new, reprinted, or revised handbooks are being published as 706-series AHC pamphlets. Gelleral and Milee11aneous Subjects Number

106 107 108 110 111

liZ 113

114 134

135

136 137 U8 139 lTO(C)

Z70 Z90(C) 331 355

Title Elemenu of A~nt Engineering, Part One, Soutces of Energy Elem_s of Armament Engineering. Part Two, Ball18tics Elements of Armament EngIneering. Part Three, Weapon Syeteme and Components Experimental Statistics, Section I, Basic Con­ cepts and ~lysis of Measurement Data Experimental Statietics. Section 2. Analysis o{ Enumerative and Classificatory Data Experimental Statistics. Section 3. Planning and AnaJ.ysis of Comparative Experiments Expe rimental Statistic s. Section 4. Special Topics Experimental Statistic s. Section 5, Tables Maintenance Engineering Guide for Ordnance Design Inventions. Patents. and Related Matter. ServomechaDiems, Section I. Theory Servomechaniems. Section 2. Measurement aDd Sigw Converters Servomechaniems, Section 3. Amplification Servomechanisms, Section 4, Power Element. and Syetem Deeign Armor and Itl Application to Vehicles (U) Propellant Actuated Devices Warheads--General (U) Compensating Elements (Fire Control Series) The Automotive Assembly (Automotive Series)

AmmW1ition and Explosives Series 175 Solid Propellants. Part One 176(C) SoUd Propellants. Part Two (UI 177 Properties of Explosives of Military Interest, Section 1 178(CI Properties of Explosives of Military Intere.t, Section Z (U) ZIO Fuzes. General and ~echanical Z11(C) Fuzee. Proximity. Electrical. Part One (UI ZlZ(5) Fuzee. Proximity, Electrical, Part Two (UI 2.13(5) Fuzee. Proximity, Electrical, Part Three (U) Zl4 (5) Fuzee. Proximity, Electrical, Part Four (U) Zl5(C) Fuzes. Proximity, Electrical. Part Five (UI U4 Section 1. Artillery Ammunition--General. with Table of Contents, Glossary and Index for Series U5(CI Section 2. Design for Terminal Effects (U) U6 Section 3. Design for Control of Flight Char­ acteristics

U7(C) U8 U9(C)

Section 4. Design for Pro)ection (U) Section 5. Inspection Aspects of Artillery AnunW1ition DeSign Section 6. Manufacture of Metallic Components of Artillery Ammunition (U)

Ballbtic Mluile Series Number 281 (S-RDI

l82 284(C) 286

Title Weapon Syetem EUeCtivenell (U) Propuls ion and Prope ll&JIti Trajectories (U) Structuree

Ballietic8 Serie. 140 Trajectoriee, Differential Effect., and Data for PrOjectile. IbO(51 Elements of Terminal &llbstice. Part ODe. Introduction. Kill Mechaniams. and Vulnerability (U) 161 (S) Elementa of Terminal Ballistice, Part Two, Collection and Analy.is of Data Concern­ ing Targets (U) 162IS-RD} Elements of Terminal Ballistics, Part Application to Millile and Space Targets Carriages and Mounts Series 34i Cradles 342 Recoil Systems 34 3 Top Carriages 344 Bottom Carriages 345 Equilibrators 346 Elevating Mechaniams Traver.ing Mechal1iem. 347 Materials Handbooks AlumInum and Aluminum Alloys 302 Ccvper and Copper Alloya 303 Magnesium and Magneelum Alloys 305 Titanium and Titallium Alloya 306 Adhesiv"e 307 Gasket Materials (Nonmetallic) 308 Glass 309 Plastic s 310 Rubber and Rubber-Like Materials 311 Corrosion and Corrosion Protection of 30 I

Surface-to-Air Missile Series Part One, System Integration 291 292 Part Two. Weapon Control Part 1hr"e, Computers 293 294(S) Part Four. Millile Armament (UI 295(S) Part Five, Countermeaeuree (U) 296 Part Six. Stl"Uctures and Power Sources 291 (S) Part Seven, Sample Problem (U)