BALLISTIC RESEARCH LABORATORIES

REPORT NO. ll83

DECEMBER 1962

THE SIMULATION OF INTERIOR BALLISTIC PERFORMANCE OF GUNS BY DIGITAL COMPUTER PROGRAM

Paul G. Baer

Jerome M. Frankle

Interior Ballistics Laboratory

This do cur to foreigr Approved for public release-approval ( n· t .b t'. . . . Center, AI IS n u IOD IS Unhmited

each transmittal only with prior .rch and Development

RDT & E Project No. 1M010501A004

A B E R D E E N P R 0 V I N G G R 0 U N D, MARYLAND

pg 8

pg 9

pg 14

pg 15

pg 16 and pg 23

pg 19 and pg 24

pg 46

ERRATA-BRLR 1183

m. specific mass of i th propellant, lb-mol/lb 1

R Universal Gas Constant, in-lb/lb-mol-°K

T initial temperature of gun, °K s - -c -c = m.R pi v1 1 n

dx (in Eq(J4)) v (-~-l_c_i_T_o_i __ - T) v2

12 0 38 1. 5 ( 0) ~ c. x • c xm + A i=l 1

Eh = ----------------------------------------------

Bl6

f + (ot:: ::~ sm] vm 2

i=l 1

(1-a )n'+l 0

IF (Yl>O)GOTO(B17)%Yl=O% IF(Y3 etc

J. M. Frankie

BALLISTIC RESEARCH LABORATORIES

REPORT NO. 1183

PGBaer/JMFrankle/mec Aberdeen Proving Ground, Md. December 1962

THE SIMULATION OF INTERIOR BALLISTIC PERFORMANCE OF GUNS BY DIGITAL COMPUTER PROGRAM

ABSTRACT

When non-conventional guns are to be considered or when detailed design

information is required, interior ballistic calculations become more difficult

and time-consuming. To deal with these problems, the equations which describe

the interior ballistic performance of guns and gun-like weapons have been pro­

grammed for the high-speed digital computers available at the.Ballistic Research

Laboratories. The major innovation contained in the equations derived in this

report is the provision for use of propellant charges made uP of several pro­

pellants of different chemical compositions and different granulations. Re­

sults obtained by the method deacr~bed in this report compare favorably with

those of other interior ballistic systems. In addition, considerably more de­

tail is obtained in far less time. A comparison with experimental data from

well-instrumented gun-firings .is also presented to demonstrate the validity of

this method of computation.

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TABLE OF CONTENTS

LIST OF SYMBOLS

INTRODUCTION . . .

INTERIOR BALLISTIC THEORY

Interior Ballistic System

Energy Equation .

Equation of State

Mass-Fraction Burning Rate Equation .

Equations of Projectile Motion

Summary of Interior Ballistic Equations

COMPUTATION ROUTINE

Preliminary Routine .

Main Routine

Options to Routine

DISCUSSION .

LIST OF REFERENCES .

APPENDICES . . • . .

A. Form Function Equations

B. Computation Routine •

C. Input and Output Data .

D. Comparison of Experimental and Predic,ted Performance for Typical l05mm Howitzer Firing

DISTRIBUTION LIST . . . . . . . . . . . . . . . . . . . . .

5

Page

7

11

12

12

13 17 20

21

23

25

25 . . . . . 26

28

29

31

33

35 41

49

. . . . 65

69

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LIST OF SYMBOLS

a acceleration of projectile, in./sec 2

a constant defined by Equation (28a), dimensionless 0

A area of base of projectile including appropriate portion of rotating

band, in.2

bi covolume of i th propellant, in.3/lb

c diameter of bore, in.

c specific heat at constant volume of i th propellant (c is a function

vi ofT), in.-lb/lb-°K vi

dT

mean value of specific heat at constant volume of i th propellant

{over temperature range T toT ), in.-lb/lb-°K oi

mean value of specific heat at constant pressure of i th propellant

(over temperature range T toT ), in:-lb/lb-°K oi

initial weight of i th propellant, lb

initial weight of igniter, lb

diameter of perforation in i th propellant grains, in.

incremental time, sec

incremental temperature, °K

dx incremental distance traveled by projectile, in.

dz.. -1 ~ mass fraction burning rate for i th propellant, sec

dt

Di outside diameter of i th propellant grains, in.

Eh energy lost due to ·heat loss, in.-lb

E kinetic energy of propellant gas and unburned propellant, in.-lb p

E energy lost due to bore friction and engraving of rotating band, in.-lb pr

fi functional relationship between Si and zi

Fa resultant axial force on projectile, lb

7

Ff frictional force on projectile, lb

F. "force" of i th propellant, in.-lb/lb -~

FI "force" of igniter propellant, in.-lb/lb

F propulsive force on base of projectile, lb p

F gas retardation force, lb r

g constant for conversion of weight units to mass units, in./sec

G functional relationship between p and x r

K v

K X

M

n

n'

burning rate velocity coefficient, in. sec in./sec

burning rate displacement coefficient, in. sec-in.

length of i th propellant grains, in.

specific mass of i th propellant, lb-mols/mol

mass of projectile, slugs/12

number of propellants, dimensionless

ratio defined by Equation (28b), dimensionless

2

number of perforations in i th propellant grains, dimensionless

p

Q

r' i

space-mean pressure resulting from burning i propellants, psi

pressure on base of projectile, psi

pressure of gas ·or air ahead of projectile, psi

space-mean pressure resulting from burning of i th propellant, psi

igniter pressure, psi

breech pressure, psi

resistance pressure, psi

energy released by burning propellant, in.-lb

linear burning rate of i th propellant, in./sec

adjusted linear burning rate of i th propellant, in./sec

8

t

T

u

v

v m

v

v c

v 0

w

w p

X

X m

y'

functional relationship between ri and p

surface area of partially burned i th propellant grain, in. 2

2 surface area of an unburned i th propellant grain, in.

time, sec

0 mean temperature of propellant gases, K

0 adiabatic flame temperature of i th propellant, K

0 adiabatic flame temperature of igniter propellant, K

0 temperature of unburned solid propellant, K

two times the distance each surface of i th propellant grains has receded

at a given time, in.

internal energy of propellant gases, in.-lb

velocity of projectile, in./sec

velocity of projectile at muzzle of gun, in./sec

specific volume of propellant gas, in. 3/lb

volume behind projectile available for propellant gas, in.3

volume of an unburned i th propellant grain, in. 3

volume of empty gun chamber, in. 3

external work done on projectile, in.-lb

weight of projectile, lb

travel of projectile, in.

travel of projectile when base reaches muzzle, in.

fraction of mass of i th propellant burned, dimensionless

fraction of mass of igniter burned, dimensionless

burning rate exponent for i th propellant, dimensionless

burning rate coefficient for i th propellant, in. - 1 sec io: ps

effective ratio of specific heats as defined by Equation (27a), dimensionless

ratio of specific heats for i th propellant, dimensionless

ratio of specific heats for igniter propellant, dimensionless

Pidduck-Kent constant, dimensionless

density of i th solid propellant, lb/in.3

10

INTRODUCTION

The interior ballistician must frequently predict the interior ballistic

performance of guns. In some instances, it is sufficient to calculate muzzle

velocity and maximum chamber pressure for a conventional gun f~om a knowledge

of the propellant charge, the projectile weight, and the gun characteristics.

This calculation is usually referred to as the classical central problem (l)*

of interior ballistics. When non-conventional guns are considered or when

detailed design information is required, it is necessary to know more than

these two salient values. For the more demanding problems, complete interior

ballistic trajectories may have to be calculated. These trajectories consist

of displacement, velocity, and acceleration of the projectile and chamber pres­

sure, all as functions of time.

The literature of interior ballistics contains descriptions of many methods

for solving the problem of predicting the performance of guns. (l) (2 ) Methods,

varying from the purely empirical to the "exact" theoretical, have been devised

in tables, graphs, nomograms, slide rules, and simplified equations solved in

closed-form. Some of these methods require data from the firing of the gun

being considered or from a very similar gun. All of these methods require some

simplification of the basic equations of interior ballistics.

To eliminate the restrictions imposed by assumptions made only to facil­

itate the mathematical solution of the problem, the interior ballistic equations

have been programmed for high-speed electronic computers. Both analog and dig­

ital computers have been used to calculate detailed interior ballistic trajec­

tories. There are advantages and disadvantages associated with each type of

computer. Several years ago, (3) the interior ballistic equations were pro­

grammed for the digital computers** available here at the Ballistic Research

Laboratories. Since that time, considerable use has been made of this program

for studying gun and gun-like systems and for routine calculations.

* Superscripts indicate references listed at the end of this report. ** Although the interior ballistic equations were originally programmed only

for the ORDVAC, (4) they have been recently reprogrammed in more general

form (5) for the ORDVAC and the newer BRLESC. ( 4)

11

The computer program described in this report has been deSigned to solve

a set of non-linear, ordinary differential and algebraic equations which simu­

late the interior ballistic performance of a gun. In this method, the usual . . set of equations which pertains to the burning of. a . single propellant has been

modified to account for the burning of composite charges, i.e., charges made

up of several propellants of different chemical compositions and different . *

granulations. The computer program may be suitab~ modified to study non-

conventional guns and gun-like systems. A number of these optional programs

** have been devised and used extensively.

INTERIOR BALLISTIC T.HEORY

Interior Ballistic 8ystem

The basic components of the . interior ballistic system for a conventional

~ are shown 1n Figure 1. A set of equations can be formulated which math­

ematically describes the distribut~on of energy originating from the burning

CHAMBER TUBE

~: /·- 11 ' I I ' / /

... ' . ' , .. ,

PROPELLANT· IGNITION SYSTEM

Figure 1. Basic Components of the Interior Ball.istic System for a Conventional Gun

* The present ·program can be operated with as many as five different types of propellant charges for each problem.

** See Section entitled Qptions to Routine.

12

propellant and the subsequent motion of the components of the system. In the

development which follows, two major assumptions are made to account for the

behavior of composite charges:

1. The total chemical energy available is the simple sum of the chemical

energies of the individual propellants.

2. The total gas pressure is the simple sum of the "partial.pressures"

resulting from the burning of the individual propellants.

Energy Equation

Application of the law of conservation of energy leads to the energy equa­

tion of interior ballistics. This may be written as:

Energy.Released by Burning Propellant

Internal Energy = of Propellant Gases

+ External + Secondary Work Done Energy Losses on Projectile (1)

or:

Q = U + W + Losses {la)

In Equation (la) the energy released by the burning propellant (Q) is assumed

to be equal to the simple sum of the energies released by the individual pro~

pellants as previously stated. Therefore:

n T

Q = L [ciz1 /Oo1 cvi dT] 1=1 (2)

Because of gas expansion and external work performed in a gun, the gas temper­

ature is less than the adiabatic. flame temperature (T ). The internal energy oi

of the gas (U) is then:

n

(3)

The external work done on the projectile is given by:

{4)

13

Substituting Equations (2), (3), and (4) into Equation (la) gives:

n T n

L (c1z1 J 01 cv dT) = L (c1z1 f cv dT] +A Jxpb dx +Losses i =1 . 0 i i = 1 0 i 0

which may be rewritten as:

n T

L[cizi J oi cv i=l T i

X

dT] = A fo Ib dx + Losses

As the c do not vary greatly over the temperature ranges from T to T , vi oi

they can .be replaced with mean values (c ). vi

Integration of Equation ( 5)

gives:

n

. I i;::;1

and solving for T:

T =

X

= A J pb dx + Losses 0

Losses

Next, .the "force" of each propellant is defined by:

F1 = m RT i oi

and the well-known relations:

·and: =

are introduced.

14

(5)

(6)

(7)

(8)

(9)

(10)

Combination of Equations (9) and (10) gives:

c (r - 1) = m R vi 1 1

Substitution of Equation (11) into Equation (8) gives:

F. =---.;.;..~--

(11)

(12)

Finally, substitution of Equation (12) into Equation (7) gives Resal's equation

in the form:

n

I 1=1

X

( p dx - Losses Jo b

(:yi - l) T 01 (13)

For most problems, it is convenient to assume the igniter completely burned

(z1

= 1) at zero-time. Equation (13) may be restated as:

[I FiCizi] + FICI A j pb dx - Losses

i=l r1-l r1-l 0 T =--------~-------~-------------------

[ f FiCizi ] + , FI.CI

i = 1 ( r i -1 )To ( r I -1) T oi . i

)

(14) X . .

The terms A . J ~ dx and Losses of Equation (14) can now be considered 0

in more detail. The work done on ~he projectile results in an equivalent gain

in kinetic energy of the projectile except for losses. Including these losses

under the general category of energy losses:

·x

A f pb 0

(15)

15

According to Hunt, (2) the energy losses to be considered are:

{1) kinetic energy of propellant gas and unburned propellant,

(2) kinetic energy of recoiling parts of gun and carriage,

{3) heat energy lost to the gun,

(4) strain energy of the gun,

(5) energy lost in engraving the rotating band and in overcoming friction

down the bore,

and

{6) rotational energy of the projectile.

For discussion of each type of secondary energy loss, see Reference (2). Types (2), (4), and (6) are estimated to be less than one percent for each

category and have been neglected here.

The kinetic energy of propellant gas and unburned propellant can be repre­

sented by (6)

The energy losses resulting from beat lost to the gun can be ·est1mated by a

semi-empirical relationship described by Hunt:(2)

. 0. )8C 1.5

~+ 0.6c 2.175 d 2

vm ( r ct) 0.8375 i:;:;:;l

2 v

(16)

(17)

At the present time, the introduction of a more sophisticated treatment of heat

loss, with its attendant complexity, does not seem to be warranted. Such a

substitution can be made if and when it appears desirable.

16

The final energy losses to be considered here consist of those resulting

from engraving of the rotating band, f riction between the moving projectile

and the gun tube, and acceleration of air ahead of the projectile . Individual

estimates of these ·are difficult to make, so they have been grouped as resis ­

tive pressure i n the f orm:

X

E = Afpdx p r r 0 (18)

The p versus x function is discussed in greater detail in the section concern­r

ing forces act i ng on the projectile.

Substitution of Equations (15), (16), (17), and (18) into Equation (14) computer program:

dx-~ Lf Pr

T = -------------------------------------------------------------------~o _______________ __

(19)

Equation of State

The pressure acting on the base of the projectile can be calculated from

a series of equations, once the temperature of the gas i s determined from the

energy equation. Generally, · the equation of state for an ideal gas takes the

form:

(20)

where V 1 = t he volume per un1 t mass of i t h propellant gas .

Now, define V , the volume behind the projectile which i s available for pro-. c pellant gas, as:

17

Volume Available for Propellant Gas

=

n

Initial Empty Chamber Volume

Volume Occupied by Unburned Solid Propellant

n

+

or: vc = vo + Ax - I ci 1=1 pi

(l-z1

) -I Cizibi . i=l

B.y the definitions of Equations (20) and (21),

v c

Volume Resulting from Projectile Motion

Volume Occupied by Gas Molecules (covolume) (21)

(22)

(23)

Substit ut ing Equations (8) and (23) into Equation (20) and rearranging gives :

(24)

If the b1

are assumed ·to be constants over the temperature range from T to

T , and if the total gas pressure is taken as the simple sum of the "partial 0

pr~ssures" resulting from the burning of the individual propellants as pre-

viously stated, then:

n

I i=l

As before, if it is assumed that the igniter is completely burned (z1

= l)

at zero-time, Equation (25) may be restated as:

T -P= .v c

18

(25)

(26)

-The space-mean pressure, p, given by Equation (26) is used in the cal-

culation of t he fraction of propellant burned at any time. This relationship

is discussed in the section concerning burning rates. There is, however, a

pr essure gradient from the breech of the gun to the base of the projectile

which must be considered in developing the equations of motion for the pro­

jectile . This pressure- gradient problem was first considered by Lagrange and

is commonly referred to as the Lagrange Ballistic Problem. Later studies in

this area were made by Love and Pidduck, (7) Kent, (8) and other s . For this

computer program, the improved Pidduck-Kent solution developed by Vinti and

Kravitz (6) has been used:

-p

1 + w 0

p (27) *

I n addition the breech pressure , p ., is calculated by the method contained 0

in Reference (6) . This is the pressure usually measured in experimental inte-

rior ball i sti c studies:

p = o ( )-n•-1 · 1 -a 0

(28)

2 (n'+l) where: +

(28a)

* In Reference (6), t he determination of B depends on t he ratio of specific beats, 7• For composite char ges , an effective value n is used for. t his purpose. '

[..; cir i r ' = .;;;i=_l.;;;_ __ _

n

(27a)

19

1 and n' =

)'I -1 (28b)

Mass-Fraction Burning Rate Equation

Both the energy equation (Equation (19)) and the equation of state (Equa­

tion (26))·are algebraic equations whose solutions depend upon the solutions

of several non-linear, ordinary differential equations. The mass-fraction burn­

ing rate equation expresses the rate of consumption of solid propellant and

hence the rate of evolution of propellant gas. This may be written as:

dz. l si J. = r.

J.

dt v gi

(29)

where: ri R. (p) J. (30)

and: (31)

For most gun propellants, Equation (30) may be quite satisfactorily stated as:

For certain propellants, including those plateau ~d mesa types used in

solid-fuel rockets, Equation (32) will not suffice for gun calculations. In

these cases, it is preferable to make use of a tabular listing of r. 1 s and J.

corresponding p 1 s (Equation ( 30 )) and to interpolate for the desired r i. The

r. 1 s calculated by either Equation (30) or Equation (32) are closed chamber J.

burning rates. As discussed in later sections of this report, these burning

rates may be increased by addition of factors proportional to the velocity and

displacement of the projectile in the following manner:

(32a)

20

Similarly, the form function described by Equation (31) may be stated

in one of several ways. In many interior ballistic systems, the form function

is chosen for convenience of analytical solution. Where routine nun~rical

computations are handled by use of a high-speed digital computer, the geomet­

rical form of the propellant grain may be used to obtain the functional rela­

tionship, fi, between Si and zi. For the usual grain shapes encountered, these

equations are given in Appendix A. This Appendix also contains the method for

handling such equations in the computer routine. To extend these equations to

include propellant slivering see Reference (9).

Equations of Projectile Motion

The translational motion of the projectile down the gun tube may be cal­

culated from the forces acting on the projectile. Figure 2 shows the axial

forces considered in determining the resultant force.

Figure 2. Axial Forces Acting on Projectile

The propulsive force, F , is that resulting from the pryssure of the p

propellant gas on the base of the projectile according to:

F = p A p b

where-pb is obtained from Equation (27).

(33)

The frictional force, Ff, is the retarding force developed by resistance

between the bearing surfaces of the projectile and the ins.ide of the gun tube.

This is usually the resistance between the rotating band and the rifling of

the tube and includes the force required to engrave the rotating band. It ~

be expressed as:

F = p A f r

21

(34)

The determination of p is difficult in most cases. Many · interior r

ballistic solutions use an increased projectile mass (approximately 5~) to

account for its effect. There are several disadvantages inherent in such a

treatment. Although the muzzle velocity may be calculated reasonably well,

the detailed trajector,y Will be altered considerably. It is not possible to

simulate the case where a projectile lodges in the bore (see Reference (10) for experimental trajectories for this condition). For this computer program,

experimental data of the type given in Reference (11) may be used by insert­

ing a tabulation of the function:

p = G(x) r

(34a)

The gas -retardation force, F , is that which results from the press ure :r of air or gas ahead of the projectile, stated as:

F = p A r g

(35)

where p is small enough to be neglected except for v~ry high velocit y systems, g

light gas guns, and other special applications. In the discussion of the Energy

Equation in the Interior Ballistic Theory Section, p was considered a part of p . g r

or:

The resultant force in the axial direction is then:

F=F-F-F a p f r

F = A(pb - p - p ) . a g r

The acceleration of the projectile, by Newton's second law of motion,

is: A(pb - p g - pr)

or:

a = -------------

a=

M

Ag (p - p - p ) b g r

w p

22

(36)

( 37)

(38)

(39)

Since a= dv dt and v =

t

v = J a dt

0

dx dt' the velocity of the projectile is given by:

and the displacement of the projectile is given by:

t

X = f V dt

0

Summary of Interior Ballistic Equations

(40)

(41)

The equations which are used in the computer program are now summarized

for ease of reference.

Energy Equation

where:

n

F. c 2 ~ I ci) I I v -- - 2 W + i=1 - A 1 - 1 g P T

I I

v 0.3.8c 1. 5 (xm + Ao)

6 2.175 o. c

1 + " -----0.8375

23

2 v m

(19)

(17)

Equation of State .

p" ~c [(tl :;:izi) +

where: V =V +Ax­e o

-p P-o= n

2: c 1+

i=l 1

w 8 p

p =---0 (l-a )-n'-1

0

Mass-Fraction Burning-Rate Eqaations

dz1 1 =

si ri crt v gi

ri = 131 (p) a. ~

or:

r 1 = r + K v + .K x i i V X

(26)

(22)

(27)

( 28)

(29)

(32)

(32a)

Equations of Projectile MOtion

a = Ag (pb - pg - pr)

w p

v = / a dt

0

x= / v dt

0

(39)

(40)

(41)

CO~UTATION ROUTINE

The set of non-linear, ordinar,y differential and al~ebraic equations,

summarized at the end of the previous section, simulates the interior ballistic

performance of a gun or gun-like system. A numerical computation. routine has

been devised for t he simultaneous solution of these equations. The general­

ized flow-diagram for the routine is presented in Appendix B. Using the FORAST

language, (5) the solution has been programmed for the ORDVAC and BRLESC com­

puters.

Preliminary Routine

To reduce computation time and conserve meJOOr,y space, a preliroinar,y routine

has been introduced. Here all data required for the computation are read into

the computer, constant groupings (e.g.,

' ' ci

, etc., are calculated and stored

for subsequent use, and data to permanently identify the computer run are print­

ed out. A complete listing of required input data may be found in Appendix C.

25

Main Routine

The main computational routine is presented in the generalized flow­

diagram of Appendix B. To follow the procedure, consider the three sequential

phases of the problem:

Phase I - From time of ignition until the projectile starts to move.

Phase II - From time of initial projectile motion until all propellants are consumed.

Phase III - From time of propellant burnout until projectile leaves the gun muzzle.

At the time of ignition (Phase I begins), it is assumed that the igniter

is completely burned (zi = 1) and none of the other propellants have started

to burn (all z i = 0). The space -mean pressure, cons:l.sting only of the igniter

pressure, is calculated from:

p = p = I

FICI

v-c (42)

Equation (42) is derived from Equations (19) and (26) by means of the simpli­

fying ignition assumptions stated above.

The linear burning rate for each propellant can now be determined from

either Equation (30) or Equation (32) in combination with Equation (32a). If

the interpolation indicated by use of Equation (30) is selected, the general­

ized interpolation sub-routine* is employed. The mass-fractions burned, zi•s,

during a small time interval, dt, are determined by integration of Equation

(29). The surface areas of the unburned propellant (see Appendix A) are used

in this initial calculation. The Runga-Kutta method of numerical integration,

as modified by Gill, (l2) is commonly used for the solution of sets of or­

dinary differential equations and has been employed here.

Calculation of the temperature, T, from Equation (19) and the volume

available for propellant gas, V , from Equation (22), will allow the calcu-c

lation of the new space-mean pressure, p, at time, dt, from Equation (26). The

surface areas of the now partially burned propellants are computed from equa­

tions presented in Appendix A. All results of interest are printed-out at this

time ** and these results used as initial conditions for calculations during

* See Reference (18) for interpolation by divided differences.

** See Appendix C for listing of output data.

26

the ensuing time-interval. Those terms in Equations (17), (19), and (22) which

involve velocity or displacement are zero during this phase of the computation.

This calculation-loop is continued until the space-mean pressure exceeds a

pre-selected "shot-start" pressure and the projectile starts to move. Phase I,

which has been arbitrarily defined, ends at this time.

Phase II requires the addition of the equations of motion to the sequence

followed during Phase I. Equations (27), (39), (40), and (41) are used to cal­

culate the values of the acceleration, velocity, and displacement of the pro­

jectile at the end of each time interval. Integration specified in Equations

(40) and (41) is again performed by the Runga-Kutta-Gill method. Values of

velocity and displacement are now available for use in terms of Equations (17),

(19), and (22). To compute values for E- = A [0

x p dx, which is one of · -~r r

the terms in Equation (19), the generalized interpolation sub-routine must be

used to obtain p from the tabular information described by Equation (?4a). r

This integration is performed by use of the Trapezoidal Rule.*

As time is increased by the addition of small time-interVals, calculations

during Phase II are continued around this expanded loop with print-out of appro­

priate results at the end of each time interval. One at a time, the propellants

are completely consumed and this phase is ended. A series of switches has been

incorporated in the program to circumvent the necessity of introducing propel­

lants in any special order. In fact, it may not always be ~ossible to predict

the exact order in which several different propellants will be burned out. The

combination of the propellant switches and the start-of-motion switch makes it

possible to handle problems where one or more propellants burn out before the

projectile starts to move.

With all propellants consumed, Phase III begins. The mass-fractions burned

have all become unity and the equations concerned with burning (Equations (29),

(31), and (30) or (32)) are eliminated from the loop. As in the other phases,

* Although the Trapezoidal Rule is a relatively crude method for numerical integration, the accuracy of the pr versus x data available does not warrant

a more accurate and hence more complex method.

27

results are printed-out at the end of each time-interval. A continual check

is made of the displacement of the projectile to determine whether or not it

has reached the muzzle of the gun. When the projectile passes the muzzle,

Phase III has ended and the program is stopped.

It is possible for the projectile to reach the muzzle (and the program

stopped). before Phase II is completed. This would simulate a gun-firing in

which unburned propellant is ejected from the muzzle. It is also possible

for the program to simulate a firing in which the projectile becomes lodged

in the tube. In this case, Phase III is not completed and the program is

stopped when the projectile displacement does not increase.

At each time-interval after the beginning of Phase II, the breech pressure

is determined from Equation (28) and printed out. This result is not used in

the computational routine but is used to compare theoretical and experimental

results. A continual check is made of the calculated pressures and the maximum

breech pressure is stored with its associated time and projectile displacement.

This information is printed-out at the end of the program. Calculations dur­

ing the last time-interval result in a projectile displacement somewhat greater

than the desired distance to the muzzle. A linear interpolation between re­

sults at the last two time-intervals is used to obtain results exactly at the

muzzle. These results are also printed-out at the end of the program.

Options to Routine

A considerable number of options have been designed and coded for special

problems. These include changes which enable the program to be used for guns,

or gun-like weapons, which are not of conventional design (Figure 1) and changes

which vary the treatment of some of the individual parameters. It is expected

that the number of such options will increase as the program is used for a

greater number and variety of problems.

Typical options for non-conventional guns are those for gun-boosted rockets,

traveling-charge guns, and light-gas guns of the adiabatic compressor type.

Examples of options for varied treatment of individual parameters are those for

adjusted burning rates (previously mentioned), inhibited propellant surfaces,

.delayed propellant ignition, variable time-intervals, constant resistive pres-

sure, and resistive pressure as a function of base pressure.

28

DISCUSSION

No attempt has been made here to present a new and different interior

ballistic theory. The objective was to devise a convenient, flexible scheme

for performing the tedious numerical calculations required to obtain detailed

interior ballistic trajectories. The selection of a program for high-speed . digital computers has made it possible to eliminate most of the simplifications

of theory required to facilitate mathematical solutions by other methods.

The theory presented as the basis for the computer routine is well-known

and has only been modified to account for composite charges. There are several

problems present in all interior ballisticcalculations and these also prove

troublesome here. For example, useful propellant burning rates are not gen­

erally available. It is known that burning rates obtained from experimental

firings in closed chambers are usually low. The results obtained from limited

gun-firings by the authors (ll) indicate gun burning rates may be twice closed

chamber burning rates under certain conditions. As previously mentioned, qp­

tional methods of adjusting closed chamber burning rates have been provided

for in this program. One such approach is to consider the burning rate to be

a function of the projectile velocity (and possibly a function of the projec­

tile displacement) in addition to its known dependence on pressure. This

method results in the use of closed chamber burning rates when the gun chamber

is practically a closed chamber (v and x are effectively zero). When the pro­

jectile is moving at higher velocities and is further down tube, reasonable

increases in burning rates are obtained and used. Other equally important dif­

ficulties are associated with the determination·of reasonable values for resis-

tive pressure and .shot-start pressure.

Considerable versatility has been built into the program. Instead of

stopping the computation at the end of Phase III, a new problem can be auto­

matically read into the computer and solved. This multiple-case feature can

be employed to advantage for any number of additional problems during a single

computer run.

Typical interior ballistic problems were used to compare results obtained

from this computer routine with results from other interior ballistic schemes.

(13), (14), and (15). The agreement was generally very good when the other

29

schemes were fairly sophisticated. I n addition, detailed interior ballistic

t rajectories are produced in considerably l ess time than it takes to calculate

maximum pressure and muzzle velocity by other systems . A typical computer

solution for a conventional gun takes only 10 seconds if magnetic t ape output

is used with the BRLESC.

Results from computer s imulat ions have also been compared to experimental

data obtained from well- instrumented gun firings. To demonstrate the adequacy

of the computer routine, data from a typical 105mm Howitzer firing were pro­

cessed by the method described i n Reference (11) . In Appendix D these experi­

mental r esults are compared with the predicted results obtained from a simu­

l at ion of this firing.

PAUL G. BAER

;o

LIST OF REFERENCES

1. Corner, J. Theory of the Interior Ballistics of Guns. New York: John Wiley and Sons, Inc., 1950.

2. Hunt, F. R. W. Chairman, Editorial Panel. Internal Ballistics. New York: Philosophical Library, 1951.

3. Baer, Paul G., and Frankle, Jerome M. Digital Computer Simulation of the Interior Ballistic Performance of Guns. Ordnance Computer Research Report, VI, No. 2: 17-23, Aberdeen Proving Ground, Apr 1959.

4. Kempf, Karl, Historical Officer. Historical Monograph - Electronic Computers within the Ordnance Corps. Aberdeen Proving Ground, Nov 1961.

5. Campbell, Lloyd W., and Beck, Glenn A. The FORAST Programming Language for ORDVAC and BRLESC. Aberdeen Proving Ground: BRL R-1172, Aug 1962.

6. Vinti, John P., and Kravitz, Sidney. Tables for the Pidduck-Kent Special Solution for the Motion of the Powder Gas in a Gun. Aberdeen Proving Ground: BRL R-693, .Jan 1949.

7· Love, A. E. H., and Pidduck, F. B. The Lagrange Ballistic Problem. Phil. Trans. Roy. Soc. 222: 167, London, 1923.

8. Kent, R. H. Some Special Solutions for the Motion of the Powder Gas. Physics, VII, No. 9: 319, 1936.

9. Frankle, Jerome M., and Hudson, James R. Propellant Surface Area Calculations for Interior Ballistic Systems. Aberdeen Proving Ground: BRL M-1187, Jan 1959.

10. Frankle, Jerome M. Special Interior Ballistic Tests of ll5mm XM378 Slug Rounds. Aberdeen Proving Ground: BRL M-1266, May 1960 (Confidential).

11. Baer, Paul G., and Frankle, Jerome M. Redu~tion of Interior Ballistic Data from Artillery Weapons by High-Speed Digital Computer. Aberdeen Proving Ground: BRL M-1148, Jun 1958.

12. Gill, S. Runge-Kutta-Gill Numerical Procedure for Solving Systems of First Order Ordinary Differential Equations. Proceedings of the Cambridge Philosophical Society, ~' Part I: 96, Jan 1951.

13. Hitchcock, Henry P. Tables for Interior Ballistics. Aberdeen Proving Ground: BRL R-993, Sep 1956 and BRL TN-1298, Feb 1960.

14. Taylor, W. c., and Yagi, F. A Method for Computing Interior Ballistic Trajectories in Guns for Charges of Arbitrarily Varying Burning Surface. Aberdeen Proving Ground: BRL R-1125, Feb 1961.

31

15. Strittmater, R. C. A Single Chart System of Interior Ballistics. Aberdeen Proving Ground: BRL R-1126, Mar 1961.

16. Scarborough, James B. Numerical Mathematical Analysis. Baltimore: The Johns Hopkins Press, 2nd ed., 1950.

17. Baer, Paul G., and Bryson, Kenneth R. Tables of Computed Thermodynamic Properties of Military Gun Propellants. Aberdeen Proving Ground: BRL M-1338, Mar 1961.

18. Milne, William Edmund. Numerical Calculus. Princeton, New Jersey: Princeton University Press, 1949.

32

APPENDICES

A. FORM FUNCTION EQUATIONS

B. COMPUTATION ROUTINE

C. INPUT AND OUTPUT DATA

D. COMPARISON OF EXPERIMENTAL AND PREDICTED PERFORMANCE FOR TYPICAL 105MM HOWITZER FIRING

33

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APPENDIX A

Form Function Equations

35

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FORM FUNCTION EQUATIONS

Geometrical Equations

1. Initial Volume of a Propellant Grain

V = rr (D 2 - N d 2)L 4 l.. i i gi

where: V =volume of an unburned propellant grain, in. 3 gi

Di = outside diameter of grain, in.

Ni = number of perforations, dimensionless

d. = diameter of perforation, in. l.

L = length of grain, in. i

2. Volume of a Partially Burned Propellant Grain

where: z = mass-fraction of i th propellant burned at a i given time, dimensionless

(A-1)

(A-2)

u = two times the distance each surface has receded i at a given time, in.

3. Initial Surface Area of a Propellant Grain

[ ( Di + N. d. ) ( L . ) l. l. l.

+ (A-3)

where: S = surface area of an unburned propellant grain, in. 2 gi

4. Surface Area of a Partially Burned Propellant Grain

S = ·rr i

where: s1

= surface area of partially b~rned i th propellant grain at a given time, in. •

37

Equations for Newton-Raphson Method* for Finding Approximate Values of

the Real Roots of a Numerical Equation

1 . Rear r ange Equation (A-2) to set f(u.) = 0 : ~

- [ 2Li(Di +Nidi)+ (Di2 - Nidi2) ] u1

+ Li (Di2- Nidi2)} · - Vgi (1-zi)

2. Differ entiate Equation (A-5) with respect to u1

:

f ' (ui) o d [ f(ui) l o ·~ { 3(Ni-l)ui2 du

1

- 2 [ L1(N1-1) - 2 (D1 . + Nidi)] u1

) . The value of the root of Equation (A-2) is then :

f(u1

)

(A-5)

(A-6)

u1+1 = ui - f 1 (u1

) (A-7)

Procedure

wher e: ui+l = the improved value of the root , where the first esti mate of t he r oot i s ui.

1 . For each propellant, · determine zi by integration of Equation (29) . In the

initial calculation of each zi , Equation (A-3) is used to compute each

Si (Si = S when ui = 0) . For subsequent calculati ons of each zi, Equation gi

(A-4') is used with ui determined as descr ibed below.

* See Reference (16) for a· di scussi on of this method.

2. The zi 1s obtained from Equation (29) are used to compute the ui's from

Equation (A-7) and then the new Si 1 s are computed from Equation {A-4).

In the initial calculation of ui, the first estimate of its value is zero.

Equation {A-7) is used to calculate the improved value, ui+l" With ui+l

as the estimate, Equation {A-7) is used again to calculate a further-improved

value, ui+2• This procedure is continued until the improvement is less than

10·5 inch.

39

Page intentionally blank

APPENDIX 'B

Computation Routine

1. Generalized Flow Diagram

2. FoRAST Listing

41

Page intentionally blank

INTERIOR BALLISTICS PROGRAM FOR GUNS GENERALIZED FLOW DIAGRAM

COMPUTE R PRINT OUT COMPUTE SPACE- MEAN HAVE

. EAD II 1--...al DENTIFYING 1---41 I STORE PRESSURE INDIVIDUAL No All DATA DATA CONSTANT r---. FROM IGNITER PROPELLANTS ~

GROUPINGS AT TIME • 0 / BURNED OUT ? (Eq(42l] .

YES

INCREASE TIME TO T + dtl T NO

COMPUTE AT TIME: T COMPUTE PROPELLANT COMPUTE LINEAR SURFACE MASS-

BURNING f----t AREAS f.--- FRACTIONS RATES [FORM BURNED BY

[Eq (301 Of' Eq (32l] FUNCTION INTEGRATION SUB-ROUTINE (Eq(29)]

PRIWT our CONDITIONS lmRPOLATE HAS COMPUTE COMPUT£ HAS aT uaxluuu FOR PROJECTILE TEMPERATURE VOLUME SPACE- MEAN "' - 111 111 · ~ CONDITIONS .'!!! REACHED OF AVAILABLE FOR No PRESSURE PRj~RE AT GUN PROPELLANT 1-- PROPELLANT ._:.:.:::.... EXCEEDED AT MUZZLE MUZZLE MUZZLE? GAS GAS SHOT- START 1 j [Eq(19l) (Eq(221] PRESSURE?

NO 1 YES CHECK FOR 1 1 YES

N:i~0cl~ SPAC~~iEAN HAS PRINT OUT AND STORE COMPUTE COMPUTE OR ~ PRESSURE .!!2. PROJECTILE - R~~L~sTEfr ~ PR~~~~RUEM l - SPARGE- MEAN BREECH

SroP PROGRAM STOPPED MOVED ? ASSOCIATED P ESSURE PRESSURE IF LAST CASE INCREASING? TIME•T CONDITIONS [Eq< 26>l (Eq(28l]

1 COMPUTE

BASE PRESSURE (Eq(27)]

1

Page intentionally blank

81

81.1

82

82.1

B3

83 . 2

I 84 84.1

85 85.1

Interior Ball istics Program for Guns FORAST LISTING

8LOC!01 • 0 71Cl · CSi r 1· f51GAt-GA5>C0V l • CO V5 1T Q1 -T 051RH01 · RHOS18ET1 · 8ET5) 0005 RLOC<OCll · DCZ 5lAN1-ANl h4l 1 5 CON T A LP 1-~ L P5 101•USI<OP 1-0P51L , · L5>NPj-NP~IXCt-~C? O IHCt•HC51 0 0 06

E.NTF:R < ~. R [AD I A N l 1131l 7 ~EAO•F ORMATIU l i•(WPIXMIVOIAPIPEIOEt. IPPMAXII 6 R E A 0 • F' ( I Q MAT I 0 ' I • ( C I l F I I GAT I "f 0 1 I I 1 0 0 8 R~ AO-F' O RM A T(011•<Dll N11KV>KXIOIEVP)I SET<J=Ull 9 P 0 0 • F 0 Q MAT ( 0 1 I • ( XC 1 • J I tl R l , J l I C Ott NT ! ? 0 I IN ( JIG 0 T 0 ( A 1 • 1 I I 1 0 E~TEP<INTEGE~INllNliSET< J=Oll INTINCP=-2 • N>I 0011 R E 1\ D • F (J q M A 'I ( 0 1 I • ( C 1 , ,J l F 1 , ,Jl G A 1, d > C 0 V 1 • J I T 01 , J > J.l H 0 l , J I I 1 '-REAO • F O ~MATC 0 l>•<8ET1,JIALPl•JID 1 ,J>nP1,JJL 1 ,JINP1,JII 1012

COttNT( ,NI J N (JIGOTO!B?.I~SqtJ= O II OIJ1~ ENTEH!A,PUNCHIANlll ll ENTERIA.PUNCH> ANa911l l 14 ENTERCA,PU NCH)AN'II"lll 15 PU NC~-FOP~A T !O ? I · <l>(WPIXMlVUIAPIDELIPfiPPMAXI<A>I 16 ENTE~IA,PUNCHIAN891111 ENTERIA.PUN CHIAN17 )2ll 17 PUNC ., -FO~"'ATIO .~l - <l>CCI If" I IGAJ ITO! l<TGN!T£RI\>s 11117 PUNC ~ -FOR ... ATI04l - (j)!Cl•Jlf1 , JIGA1 , JICOVl,JIT01•JlRH01,JI< ~ >• 18 C 0 II N 1 I , N I l N I J I t; 0 1 0 ( 8 ~ ) I S E T( J = ll ) I E NT f Fl ( A • PU NCH I A N 8 9 I 11 I 1 9 ENTE H!A , PU NCHIA N3 31111 20 PU NC~- F OR ... ATIO SI - (l>IRET l,JIALPt. J JD1.JlDP 1 . JILl .JINP1,J)( ft )l 2t I

CO !I Nll ,NifNIJ)r,OTOt8 ,3,tl:li SETC .J =Oll fNTF.R(A.PUN CH)ANP-9)1)1 ;(? £N f[k(A , PUNCHIAN4112 ll 23 PUNCH-FOR~Ali06)·<l>!XC1 , J>PR1, JI< ~ >s 24 COUN I <20I TNI JlGOl O<B3 . 21 1 SET I J= O I~ FNT ER <A.PUNCHIAN891111 25 £NTEf.t iA, PUNCiiiAN'::I712 >1 SET!SioiP:8jR . 1LIP= OISTUCIC=Bl8.5ll ~6 PUNCh -fOR~ATC071 - <l><DTIN1l~VI~Xlf.VPlOI<I\>1 2 7 i BC I • ~I • CI/!GAI-111 A C J:HCI; TOI~CC t:rT•Cl/TO II EVM= EVP •1 21 00 35 HCl • J =Ft, J •Ct, J/ (GA l , J - 11 1 AC 1.J=ACj , J /T01·JI CC1 , J=F1 ,J•C t . J/ 0066

CONT TOt ,JI 00 37 DC1 • J = C 1. JI~HQ1 , J l ECl · J =ct,J•COVt.JI FC1·J=NP1•J-l l 00 38 GC1 , J =L1, J INPt ,J·1 1- 2 ( D1,J • NP1,J • DP1 , Jll 003 9 HC t , J=~ • Ll,J(DJ • J •NP1 ,J•UPl,JI+(Ol,J •• 2-NP 1,J•DP1 ,J••211 00 4Q ! Ct , J = l 1, J!Dl , J ++ 2 -N P1,J • OP1 , J• • 2>~ 0041 Jc t , J c 3 , 1416 •T Cl • JI4~COUNr< , NITNCJIG0TOCR4.1l i ~ETtJ=Oll 00 42 CT=O~ T P1 = 01 004 ~ ! 'f p p:Cl.. J•GA11J+TPh CT=l!LJ • Ch CO i tNTC,NltNCJIG0TO<I:IS. 111 0044 ; GAP•1P1/C T15ET<Jz OIIGAf=GAP/IGAP•1 IIEP:CT/WP I 0045 ; EP1• l •EP/OELJTPl=1;(GAP-l>%TP2=1/C<2~TPt+311DEL•C2tTP1+~)/[Pll 0046-1 MC T D:WP~C T 10EL•TP4=UlAGW=AP•38~. 4/wP• 0047 i EP 2•EX P I RAF +L OGI1 - TP?I>• TP l =ExP<1.5 +LoncD> >• t P 2 • EXP<2.17~•LO& oo 4e I

------~C~O~N~T~< ~DTI~I~~~~·~·~~~~~--------------~----------------------~0~0!!-J 1P3 • F. Xp(. e ;nS• LOG<CT>ll OO~n l

86.1 86.2 87.1

TP4•C1 ,J•T 01, J +TP41 COuNT(,N)INIJlGOT0 !8S,?.ll SET< J • Ol l OO St HC t. • l. 3 q 2• Pl( 1'1-tV O/A )(T 4/CT - ?.98))1((1+0 ,6+ P . /T OU?

CONTEVM• • 211 00 53 C[(ARc1>NOS , AT<~lll C[£ARc7>NOS.Arcy3JS C(EAR(7JNOS ,Ar c0i>s 0054 PR : PR• P8R •Xl•ALP•I NTPR• 01 T5D T% ~5 CLEAF1!51NOS ,A T<U111 ~LST= O I PRLST=Os 00 '::16 Y3,J=1.11 COUNTISllN I JlGOTO<B6.111 SET<J=Oll 00 57 y3,J= 01 COUNTt .Nli NCJ )G 0TO C86. 2 ll SfTCJ• OII uo5e IF - ! NT <N•11G OT0 (87.Sll SETCSW3• DR3 .1 II OD59 IF -I NTIN •2>GOTO(B7.6> 1 SETISW4•0R3,tll 6 lf" - lNTIN•31GOT 0 187 , 71J SETISW5 =DR3 . l IS 006t

45

87,6 SETCSW4aAt4liQOTOC88ll 0064 _a7~·~'~----~S~~~T~I~S~W~5~•~~1~4LWli~GuO~T~O~C~B~8~l~I------------------------------------------~0~0~6~5~

87.8 SETCSW6•814l1GOT0CB8ll 0066 88 TPt• DI 0067 88.1 TPt•DC1,J+TP 11 CO UNTC,NIINCJIGOTOC88,1ll St.TCJ=Oll 0068

B9 · 89.1 810 810.1 811 811·1

. 812

813·1 814

814.2

814,4

814.5 ORl

DR3

OR3.1 DR4

DR!:i · OR9

815 815.1 816

816.1 817

. 817 .1 I 817 .2

' 817.3 818 818.1

818.3

PT:C ! tFI/CyO·Jplll SETCSWl:R151SW8:R14,5ll PM4X•PTI 0069 ENTEkCR,K. fi , l OTl~, N l~91Yll~llQ111 GnTnc,SWtl• 0070 IEcY 3>=1lGOT0<99.1!1SfTISW11•61 01 !=Ol!GOTQ<DR111 00 71 Y3=1~K ~ • U ISt=o~Hl= O i u 3: 0 1SETCSw11=B1 0 lJ•OliQOTOCDR3.1ll 7 ~ , IE!Y4):1!G OTO ! Al 0 ,J p<sETCSWll=Bll l .J•1llGOTOCQRl )I 0073 ' Y4:t~K4a O I S 2=0~H2• 0 I ~ 4: 0 1SfTISW11:B11l J•1liGOT O C,SW3ll 74 !EcYS>=1!GOTO!Al1,1l'SEIISW11=R121J=?I•GOTOCDR1ll 0075 Y5=1~K~ = ~ IS3=0\H ~ = o i U 5=0ISETISW11=81~lJ•2liGDTOC,SW4)1 76 IEIY6>=1l~OT 0CA l 2 .ll1SEICSW11=Bt3l J•!liGOTO!OR1ll 007 7 Y6:1\K6a O IS4:0,R4: U I ~ 6: 0 1SETCSW11aB13l J:3liGOT O C, SW 511 78 lf!Y7>=1lGQT018lJ , I I~SETCSW11•R141J~~l~GOTOCpRl)' 007 9 Y 7: U: K 7 • 0 I SS :z 0% R !>: 11 " 0 7: 0 IS ETC SW 11 = 8 3 4) J~ 4 I I G 0 T 0 ( , S W 6 l I 8 fl SETIJ= G is TP1 =0~ 0081 i TP1 =DC1 , Jtt -Y3,Jl+EC1, J •Y3,J+TP11COUNTC,NIINIJIGOTOCB14.1l% 008 2 VC=V u+AptX J-TpliS[l( J :OliTpl•ACl" 008 3 TPt:BCl, J •Y3,J+lP1sCOij NTC,NllN1JlGOTOCB1 4.2ll OO h4 SE1 1J =0 )IJP2=ACJI 0 08 5 TP2•A Ct , J •Y3,J•TP21COu NTC,NliN!J)GOTOCB14,31" 008 6 SETC J : O II T[ ~P=IT~1-A L Pl/TP21TP1:CCll 008 7 TPl•CCl ,J•v3,J+T P1s COUNT !, NIIN!JIGOTO!R14,4)1 00 b8 • SETC J =O)IPT•TEMP• TP \IVCI GOTQ!,SW6JI Ou8 9 I EN~EHC R ,K,GOII 009~ R 1 , J = e u 1 , .1 • e: x P c ALP 1 , J • LoG c P T 1 1 suo= u 1 , ,, 1 H 1 = F C1 , J s H 2 = G c 1 , J" o o 9 2 H3:Hfl,JIH4•ICl,JIH5:JCl,J!1•Y3,JliH6:L1,JIH7:D1,JI 0093 H8=DP1,JIH9=NP1•JIH1 0 :DC1,JIH1l•JC1,JIGOTQIGA~?II 0094 Rt,J=Rl, J +KV.K ? •KX•XII 1 09 4 i.

K3,J=St, J •R1•J/DC1,JI 95 GOTQ(,SWitll 1 09 5 PB=PT/EP11PRR:p8/EP21 0096 K1:AGW!~8-PRIIK2=Y11Xl=Y21 0097 IF XI<XC ?O 0 OCURS)I PH"PR2 01 GOT O R9 I 009 9

l NTPR •<D£LX• SUMll/2+1NTPHIXLST:XIIPRLST:PRI 0099 ALP •<McTD•K2•• 2177 ? ,8l+AP•INTPR+HcL•K?.t•21 GOT O C614 . ~ll OlUO t r!P l <PEIB OT O!Blti.tll SETCS~8sOR4lS~1•B16l' 01 01 PRR•PTI PB=P T! 0102 IFCY1> 01GO TOIB1711IFIY3>•11AND<Y4>=11ANDCY5>=11ANO!Y6>•1l 103

CONTANDCY7>•11GOTOC816.111GO~OC817l~ 1103 SETCSTUCK•82211 . 21 03 XF=XII12%V2Y 1/121AE•K1112'SETc J •O lST= nl 0104 DCZ1,J•K3,J• t.J~COU JC, l!N(JlGOTO<B17.1l1StTCJ= Oll 01 05 ST:Sl,J+S T xCOuNTC, N llNIJIGOTOCA17,2l~SET!Ja O ll 01U6

, IF!~BR<PPMAXIGOl0C B 17,311 ENTERCA, PUN~HlAN731111 GOTO!NEWRNll 11 06 lf!PMAX>PBRIGOT0(818l• PMAX&PSRI XP~AX=Xll TPMAX=TI 107 GOTOC ,SWPII 1 08 tNTERCA,PU NCHIAN111ll 109 [NTER(A,PUNCHIANB9lll1 TM•T•1000I 110 PUNCH·fORMATC08l·~1>iTMlXIlPBRlPTIPBlVlAf l<h>l 111

. PUNCH-FOPMATI09l•<t><TMlXIIXFITEMPlVC!PRISTl<A>I 112 PUNCH~rO~MATCOlOl·<l><TMlXIIYJ,JIPCZ1,JlR1,JlS1•JI<A>s 1112

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01 02

ca 11 NTI15 NCP llNt.IP> Gnra r Bh 4H spcsw P - H1fl 1 1 IP- n >nnrot S l !!CKIS :s,, SF.:T ( SWP s f:l t 8 , e 1 ~GO T O ! • STUCK 1 4112 T= T+O H; 11 3 I F<Xl>XM IGOTO C B~ll~XILST=XI I VLS T= VSP~I ST =PBS 011 4 E~IE R ! R , K . G ll! . 0115 V"1 AX =IIXM•X!L STIIV-VL STI/CXl·XIL ST l)•VtS U 011 6 Pf3M AX:t(XM-XI LSTIC PA-P BLST >IIXJ•XI ! Sfli +PBLSU 0117 TPMAX=TPMAX+l O~O ~ 1117 fNIERCA . P!INCI'I ! ANI:i9!1'S! E NTF.~ (A,PliNCH>ANR1 HI• Ull PU NCH ·FORMATI Ol 1>-<1>C yM AX) PMAX)X pMAY >T PMAXIPBMAXI<A>S 119 GOTOINF.~PN lS 1119 ~NT[R ( A , PUNCHJ AN89 )1l~ENT£R (A.PUNCH ! AN97)1)1[NT(N(A .PUNCH!AN81 !1 JS 1 ~0 VMAX=O%P8MAX=Q~TPMAX= TPMAX + l000~ 121 PUNCH • FOHMA TC0111·<1>C VM AX!PMAYIX PM AX>T PMA XI PBM AXI<A>S 122 GOTO<Alll 1 23 Tl=H7 · UOIT?=H8+U0 01~ 7 FU=, 785 4< lJQ U3•H l • !J0 .. 2 +H 2· UO • H3 • H4 I •H 'il 0128 FPU = .7854 ( 3 + U 0•+2+ Hl - 2 + U0 + H2·H~It 0 1~9

1 : lJO -F I FP . 01 3 IF' -A BS I CU01 - U(p <=. IJ0001 l GQTOIFF41S IIQ :lloU 0131 GQ J OIGAM21! 01 32 S l =3 .1416 ( ( H6·110 l ( H7 -UQ+H9 C HB• UO ll +. S• T 1 + +?. -. 'HH9 0133

CONTtT 2••2>~Sl~J=SI •HlO/h11~Ut~J;UQ~GOTO!pR31S 01S 4 FO R~<1 0 • 10>1 - 7> 132 f' Qf;M ( p - ;; - 9 )1. • 1.)l2-3•1 1t )l -:0 3 - ?!12-1 • 6) ;~-1 l 1 2-6 • fjl :~-3 1 1 2-6-f•O I 133 FORM<12· 2 · ~ )1 2- 7 -91 3·2 >1~-1- 7 l ~ - 1311? · 4 · 6~t5l 134 03

04 f' O~ M<t 2• ?·9)12- 7 -913-2>1~-l- 713 • 4 1 12 · ? -71~ -211c-4· 613-4 lt 2 - 0 •8-201 1S5 ' OS 06 07 08

F OR M< 12•913•1 1 12-6> 3 ·4> 12-ol 3 - 4l1?-6>~-41 1?. · 1-7>~ - 511 2·1·3~ 23 1 136 FORM < ~-2 0 1 12·3-613 - 11112-5-7 · 3? 1 137 FORMC12 • 713• 5 l12•1-313- 4>t2 ·9> 3- 1 ! 12•91 3-5!l?·4 - 613 · 5112· 1- 7 · t7 l 138 FORM(J?•?- 8 13•1 l 1?- 3•8l3·21 12-6· tot 1? • 6·1 n>J? · 6-10ll2- 5· 913-1l 139 CONT12-4-10• 9> 1139 · '

09

01 0 011

e

EOR HC12• 2- RI 3- t 1 1~-3 · 8>3-2 1 12 · ? • 8 1 ~ · 2 1 12·4-813-2132 · 5 - 1011 2·4-713•31 14 0 CONT12-5·9·1 b> . 114 0 FORMI12• 2 · 8 )3• J l 1?· 3-8)3 ·2) 12·J • 7 ) 3•31 12•4-9 13 · 1112·3·8 ) 3• 2112•5•9· 20 l 1 41 fORMc1?• 5· 8 13• 5 11 2-6 -8>3 - 4)12-3-8 >3· 2 1J? - ?-713 - 3 >1 2· 6 - 9 - 2.4J 142 END GOIO IBll~ . 0148

105 MM HOWIT ZE R RO 76 5 b lPROJ I WI. CHAMBER kORE AREA P- K SS PRESS MAX G~N PRE SSUR E 6 1 . Ml PROPEL LANT 6 1 CHAH GE 1 BETA 1 1 1 1 1

or

lMUZZl.E

EO~C£ GAMMA COvOL uME FLAME TEMP DENSITY n ALPHA 0 , 0 , GRAIN OJA, PERF GR.LF.NG.T H NO. PERf, A

RESISTANCE 0 PRQJ , TR AVEL . PRESSURE d

M SCELL ANE'OUS 6 NO , PROP, KV !(X ES T I Mll7. VEL . DIAMETER ll

P GRE ATER THAN DESIRED ~ AX PRESSU RE n VEL, MAX, PRESSuRE X AT PMAX T AT PMAX MUZ PRESSuRE ts

i

i

'

,.,

tJ 1~---------------------------z~~~~~~~~----------------------------~~ 1 PROJECT IL E STOPPED 33 . 81· 153. t3. 77 46 00 ~ ,0429 11520 00. 1.25 20 u0 , 1• 03 2. 0 0

.so

4 500 I

4 50 0 . 450 0 • . 4 500. 45 0(1.

47

3 •. 024

15 00 . I 1

?. I 3 f

soo oo.

4 i 5 I 6 7

I 8.

···-·····-- ------·--------------------:""-.....-------

2.0 0 450 0 . 10 3, 50 5 ll • 1 4, 00 28 0 (1 . 1 2 41 ,25 2~ 0 ll . 1 3 4.50 2 3 Slt . 14 s.oo l QO l) , 1 5 5 .25 1 6 50 . 1 6 s.su 140 (1. 1 7 () ,0U 1.00 0 , 1e

1 0.00 100 11 . 1 9 3 0 .0 0 1000. 2o 4 0 0 1.1 • c so.oo 10 0 lt . 22 6 0 .0 0 1 no (t • 23 ,6325 367 lt15o. 1· 21\ 4 31· 08 243 3· .Q 56 7 24 . so79 · o3 ,8497 , 0 4 / 8 , Ql94 . 24 53 1 . 2 5 2,1 3!:>6 367 (1 1 50 , · 1. ?.6 4 31 . 0FI 24 3 .~ . • 056? 26 . 5079- 0;:1 .8497 .1 344 . 014 2 • 3 p 7 7. 2?

PROS

APPENDIX C

Input and Output Data

1. Input Data

2. Output Data

3. Sample of Output Format

49

Page intentionally blank

1. INPUT DATA

Gun Constants

Weight of Projectile

Length of Gun Tube

Empty Volume of Chamber

Cross-sectional Area of Bore

Shot-Start Pressure

Pidduck-Kent Constant

Resistive Pressures

Travel of Projectile Corres.ponding to each of 20 Resistive Pressures

Diameter of Bore

Propellant Physical Constants

Weights of Propellants

Weight of Igniter

Densities of Propellants

Outside Diameter of Propellant Grains

Diameter of Propellant Perforations

Length of Propellant Grains

Number of Perforations per Grain

Number of Propellants

Propellant Thermo~c Constants*

Forces of Propellants

Force of Igniter

* See Reference (17) for these data.

51

Units

lb

in.

in. 3

in. 2

psi

dimensionless

psi

in.

in.

lb

lb

lb/in. 3

in.

in.

in.

dimensionless

dimensionless

in. -lb/lb

in.-lb/lb

Program Symbol

WP

XM

vo

AP

PE

DEL

PRl,J

XCl,J

D

Cl,J

CI

RHOl,J

Dl,J

DPl,J

Ll,J

NPl,J

Nl.

Fl,J

FI

Program Units Symbol

Ratios of Specific Heats of Propellants dimensionless GAl,J

Ratio of Specific Heats of Igniter dimensionless GAI

Covolumes of Propellants in. 3/lb COVl,J

Adiabatic Flame Temperatures of Propellants OK TOl,J

Adiabatic Flame Temperature of Igniter OK TOI

Burning Rate Coefficients in. 1 BETl,J --sec psi a

Burning Rate Exponents (a1 s) dimensionless ALPl,J

Burning Rate Velocity Coefficient in. K:-1 sec in./sec

Burning Rate Displacement Coefficient in. K.X sec-in.

Miscellaneous Constants

Time Interval sec DT

Estimated Muzzle Velocity ft /sec EVP

Maximum Allowable Breech Pressure psi PPMAX

52

2. OUTPUT DATA

Identifying Data

The complete list of input data is printed out to permanently identify

the computation.

Trajectory Data

Time

Travel of Projectile

Travel of Projectile

Breech Pressure

Space-mean Pressure

Base Pressure

Velocity of Projectile

Acceleration of Projectile

Temperature of Propellant Gas

Volume behind Projectile available for Propellant Gas

Resistive Pressure

Total Surface Area of Propellants

Mass-fractions of Propellants Burned

Mass Burning Rates of Propellants

Linear Burning Rates of Propellants

Surface Areas of Propellants

53

Units

millisec

in.

ft

psi

psi

psi

ft/sec

ft/sec2

0 K

psi

. 2 l.n.

dimensionless

lb/sec

in./ sec

2 in.

Program Symbol

'1M

XI

XF

PBR

PT

PB

v

AF

TEMP

vc

PR

ST

Y),J

DCZl,J

Rl,J

Sl,J

Page intentionally blank

OUTRJT FORMAT

~J. WI. A~RRfl cHAMRER 80RF AREA P-K SS PRESS I'IAX BIIN PPBSURF l3.o ooo o a1. ooooo 1S~. o ouo o t3.77 ooo 3, 02 4 00 46Q Q, SO OQO,

Ct1ARGF.: .0429 0 ,6 :S 25 o

2. 1 ~56 0

BE TA .oooso79 ,000!) 079

F" ClRCE 11520 0 0 , 3 6 7 01 50. 367 0150,

ALPHA . 849 7 ,8497

Mi PROPELLANT GAMMA COyOLUME f LAME TEMP DfN S I TY 1.2 50 0 ' 2 000 . lo2640 u.oao 243 ,~ I . 0 5670 1. 264 0 31. 060 243 3 , . 0 567 00

g ,O. GRAIN UJA, PERF GR,LENGT H NO. PERf, . 047& . 0194 ·2453 1· ,1344 .0142 ,J127 7.

RESISTANCE PRQ ,J . TRAVE L PRESS u RE

.ooo 4 500 . , 0 0 II .200 4So n • • 350 45 n .suo 450 0 , . uo o 45 )

2. ooo ~5oo . 500 45 II

4.0 0 0 280 0 , 4 . 25 0 2~ u 4.~0 0 2350 . s.o oo 190 0 . 5.250 1650 •

• suo 400 6.000 100 0 .

10. 0 00 100 0 . !t,OOO 100 0 , 40 . 000 1 000 , so.ooo 1oon. 60.000 1000

MISCELLANEOUS OT NO, PROP, KV KX EST, MUZ, vr. L, DIAMETER

.00010 2. .0 000000 .ooooooo 15 00 4.1~40

.55

IGNITER

U!i ~114 ~0·•11 TZFR RFl 76!i

*TM•.t ooa XI•. 0 0 0 PM• 5S9 I 3D pr .. ~29 1 3 0 f'&'ll 5591 3 Q V• . no AF= I 0 0 0 c TM·~u o o X'I•, 0 0 0 ~r•.ooou~~o5~.?5 . vc ... 1 n 4 • 1 4 7 Pit• '0 6T• 4 018,93 J'Mo. 1 l) 0 0 x:rw~!IOO 't!l•. n a 1 5 DC~I• 9 o61ll fZI•. 1 a 2 $\-: 1 661 • 8 :s '1ll1•· 10 0 0 l!U. .... noo ~ ... 0006 OC.~1·13 • 618 ~a .. 11 0? ~2•23 57.10

.2000 • 0 0 ll 6'.)1.34 651.34 61)L 34 • 0 0 • 0 0 0 ()

.2000 I (j 0 0 • 0 Q 0 0 t~o97.81 104,112 • 0 4039158 ,2000 • 0 0 0 ,0032 101945 ,116 166l.,54 12000 I li 0 () I 0 OJ 3 15,5H I 11 ~~ 23~8.04

.3000 • ll 0 u 756.33 7~6.33 776~ 'B • 0 0 • 0 0 Q 0

.3ll00 • 0 0 u • 0 u u 0 ~137.31 104~U72 I 0 4020131 • 3 fJ 0 0 1 no o ,0051 12,444 ,132 1661.20 ~3uUO • no o ,U0£>2 17.671 ,132 2359,10

.4000 • 0 0 0 875166 El75.66 875.1\6 • 0 0 .oooo

.4000 In o o • on o o LJ1?;dl1fl 1041026 I 0 4021.14 I 4 0 0 0 I() 0 0 10073 14.U2 1150 tfl60,82 ~4l!UO • 0 0 () • 0031 201055 . 1 5 () 2360,31

.5000 • !i 0 0 1010.97 1010,97 lol0,97 • 0 0 ,0000

.5000 • 0 0 0 I 0 0 0 1.) 2202.89 1031975 • 0 4022.07 1SQUD 1 n D 0 ,0026 l ~I 21\3 1lZO 366013~ 15000 • 0 0 0 ,0041 22.706 ,l 7 0 23611fl6

16000 I 0 0 0 1164102 1164,02 1t64102 I 0 0 I 0 0 0 0 .600Q • Q Q u I D u 0 Q 2229189 1031917 • Q 4Q23113 ,6000 • 0 0 0 ,0126 18.015 ,191 1659191 .6i.lOQ • 0 0 Q IOQ53 2,.648 ,191 2363,22

.7ooa I Q 0 Q t3~~.74 1.336.74 1336174 I Q Q I 0 0 Q (l

.7000 • 0 0 0 • 0 0 0 0 2253161 103.851 I 0 4024,32

.7000 t Q Q !J 10126 201283 123!! 1659136' ~7uoo Ill 0 0 10066 2~.907 .216 2364196

~euoo • no o 1531o26 1531.26 1531.1?6 I 0 0 10000 .aoua I Q Q 0 I 0 0 0 Q 2224!43 1Q3~zz:z I Q 402Sdi6 .eooo • 0 0 0 10193 22.785 .242 165R~74 1eouo .ooo 100!U 32.513 124? 23!!6191

.9QOO 1 0 0 0 1H9.~9 1719.89 1H2dl2 I Q Q I Q D D 0 ,9000 • I) d 0 • 0 () 0 IJ 2292169 1031695 I 0 4 027.15 o9UOQ I 0 0 0 • 0233 2~.542 1272 165B 1 Q5 19000 • 0 0 0 • 0 09~ 36,496 127? 2369.10

1.oooo • 0 0 0 1995116 1995.16 1995-16 • 0 0 I 0 0 0 Q 11QOOQ I Q Q Q • D Q Q 0 23Q8~7Z 1Q3.1'1Q? I Q 4028183 1.0000 I U 0 0 • 0278 28.574 ,304 1657128 1.0000 100Q 1 0117 40.6~9 I:~ 04 ii137l.22

1.1000 I 0 Q Q 2269.8~ 2269,85 2269.85 1 Q 0 • g 0 Q (l 111000 • 0 0 0 .• 0 0 0 0 2322.79 10~S.499 I 0 4030170 1·1UQO I 0 Q Q 10328 3lo9Q3 1~40 1656142 1.1000 • 0 0 0 I 0138 451729 • 340 n74. ?6

---~---

I *See list of Output Data for Program Symbols and Units.

56

i 115 Crl1!4 t40 I IT lF:R RD U.!io

1,2\)liO I II u ll ;:!577. 111 2~"17.01 2577~01 '[) 0 I o o a p 1.2UOO I (!I)[) , 0 (I 0 U tl3.35115 to~~3A.3 I 0 4032178 1.2uoo 1 (I Q (l I IJ3fl.S 3!>,'5'52 1379 1A52~46

1~2uoo I(! 0 0 lll1.6c 511055 1379 2:sn~-~2

~.3000 I (I(} 0 2919197 29.19,97 2919.97 • u 0 1 0 0 0 0 1 I :s Ull 0 • (l 0 0 • 0 !l 0 iJ (34C> .1)2 103.255 I 0 4035~09

1,3iJOO ILl 0 () ,U445 ,S9.54d ,4?.? 1654.:~8 1 I ,$1j (JI) , o u n , !l 1 A 8 56.911 1422 2;~80170

1.4tJOO • (I 0 0 Hn2.3A 3302.3!:! 33n?.3A 'Q 0 .ooon 1.4000 I Q 0 \) .oooo 235!>.58 1[)3,11?. • 0 4037166 1. 4fl 00 • t) 0 0 • u 51 4 4.Se918 ,469 1653.J9 1.4000 1000 • 0217 6..S.344 1469 2384,46

.1. 51.!0 0 • 0 0 I) 3726·28 3728.28 3728.?8 • 0 0 I 0 u 0 0 1,5LJOO • 0 0 0 • 0 0 0 0 2;~64,00 1021953 I 0 4040150 1. 50 0 0 • 0 0 0 • 0 59 0 48.690 ,52(j 1651.87 1.5000 • (J 0 0 ,ll?50' 70.407 .520 238A,62

1.6UOO • 0 0 0 4202o09 4:!02.09 4202.09 • 0 0 • 0 0 0 0 1.6000 • 0 0 0 I 0 0 0 lJ 2371.43 10?.777 • 0 4043.64 1.6U00 • 00 0 10674 S.S,898 1 57(, 1650.41 1.6000 • () 0 0 .0286 78.156 ,':J7f., 2393.23

1.7000 • 0 0 0 4 726 ··68 4/;i8 .68 4728.68 1 0 0 • o o a o 1.7000 • 0 0 0 • 0 I) 0 0 2371.98 10?.58? • 0 4047.11 1.7UOO • 0 0 0 ,0767 591574 ,fl37 1648.79 1.7000 • 0 0 0 .0326 86.655 ,637 2398.32

1.8000 • 0 0 0. 5384.73 5:H2. 49 5169.30 1,49 8990,2031 1o80(J0 I 0 0 0 • 0 0 0 0 2383.69 10?1:~82 4500.0 4050193 1.8000 • 0 0 0 .0870 65.753 .704 1647.00 1.8000 I 0 0 0 ,0371 95.972 .704 2403.93

1.9000 .004 6040o79 5959.75 5798.89 2. 77 17452.175 1.9000 In 04 .ooo3 2388.72 1Q?,H2 45oo.o 4055.14 1.9000 ,004 ,0983 ]';.1457 .777 1645.03 1.9000 .004 .0420 106.156 • 777 ?410.11

2.oooa • (l 09 6765.89 6675.12 6494.96. 4.94 26804.618 2.oooo t009 ~aoo7 2393.04 1011946 450010 4059.76 2.0000 .oo9 .1106 79.730 ,856 1642185 2.oooo .oo9 , U4?4 117 I 296 • 8 56 241tL91

2·1000 ·016 7565o33 7463,83 7262.38 a.os 37115.875 2o1ll00 .016 .oo14 2396.65 1o,.n9 45oo.o 4064.83 211000 .016 .1245 87.594 ,942 1640.46 211000 .016 • 0 5.34 129.452 ,942 2424.37

2.2000 .o28 8444·01 8330.73 8105.88 1::'.30 48449.250 2.2000. .o28 .oo24 2399.56 101.556 45QO.o 4070138 2.2000 .028 .1395 96.069 1,035 1637.83 2.2uoo .o28 ,0600 142.686 1. 03 5 2432,55

57

1 II 5 r!M HD•t I IZE Q f;Hl 76\'i

2.3UOO op4t\ 94it6 ·1 (J \J(BQ. tl tJ 9Q;)Q,t;;l 17.7 0 60859.,6~

c,3llUO ol 46 • (I !!39 <:'4 ,j 1. 7 4 1 I) 1 • 41 I) 45on.n 40lt..44 2.3Ll00 • 114(:- • 15~ 0 1 ,, ., • 1 ,., 9 1.1 ~·! 1,., .~ 4. ~ 4 c,3UUO • ;, ~ fi • ,,, 7 2 t5/.o52 1.1~4 /44l. r;o

2,4uoo 'I 11 11)4')~.21 tOH4,94 to n .< fl • ~ 4 ('4,38 74:~89. 969 ;2,4UOO ,1'71 , I) (I !:19 ~4u-'•U 1 n 1 , :<1? 11 4'50U,'l 4U83,05 2o4tJOO • [, 7t o174 0 11.4·1:!96 1..;.>4;.;' 1t'd1o78 2,4UOO • lJ 71 , II 7 <, tJ 1/t.S9H 1 • ;.;4? £45j. 28

2. su lj 0 • 1 0., 1. l5C;3. u tt4S7,59 11t:.-8,RQ :if.>. 46 t;9(16fi,99e; 2.St100 • j i) ') • 0 (.88 :!4o.L7o 1o1.:~n4 4?0 n. D 4()90.?4 0!,51)00 • 1 0 s , t9.HI 12!)o('41 1.. >;57 t6?.il. :3 2 <:!,5llUO .105 • ill\,, u t89 •. ~ "i6 1, .~57 24ft1 ,92

2.6000 ·1 s [) 12A;;>Oo27 t?64R.27 123t•ft. fiQ 4(',()5 104894.83 2.6000 • 1 50 • 01 25 1.4uS...SH lfl1. • :~fl.~ 4.,on.o 4098.04 2,6UOO ,.1.,11 • <-'15(1 U6, t 17 1,471:1 1tl?4,55 2.6UOO .150 , u9.~6 20 I d36 1 't+ 7 A ?4 H, 4d

2.7uoo • fl 0 7 14U4. 6Q 13Y45,Q6 1 ~51'-I'I.MI 5~.?6 121848.41 2.7uoo • ? 0 7 • () 1 1'2 ?41)2.1?. 10~.579 45uii.O 41o6.47 2,7uoo • '(' Q.7 ,i:'.382 1,4/,66!) 1. Ft a7 t6tJo,4!> 2.7ll00 • ? 0 l .1Q41 22b.53? t.6fi"l 24Hft,01

2.8000 ,;,;78 15t;~1•68 1532~.1111 149r9.7? 6(:.. 1.9 13986b.91 2.8uOO .27A ·0?32 "3 9 9. fl:~ 101.91ft 4'ill (l, (1 4115.55 2.8000 .?7P • 26.~3 1'3J~oftfl2 1.14? 1 (:. 1". 01 2.8000 .:na ·11'55 24o,A9':1 1.'14? ?.499,55

2.9000 .MJ6 17()il3o23 161/5.11 1.6.~2:?. ~C) 8(1,9.3 151:if14 7. 23 c.9uoo ,366 • 0 .~ 0 'j ;2$96.46 1n~.4?.1 45()0.0 4125 • .31 2.9000 ,366 ,29ulf 171.973 1,882 1611.19 2.9000 .~66 .1?.1<0 268. :~48 1, hR? 2514.12

3oOLIOO ,473 1H5:~7 • 66 t82HA,95 l77t~s.~q 97. 58 1 7A638. 5r' ;s.oooo ,-.·n • Q.~ 94 ?...i91.94 1n3.l?;j 45oo.o 4135.75 :s.ouon ,47~ .. ~t96 184.593 2.fl?7 ll'l0ft,01 :s.oooo • 4 7 ~ .1.415 291). 76 'l 2. fJ?7 25?.9.7,

3,1000 • 6 01 211U9.3u 19849 • .P 1931:.L6,~ 11fl.20 l99Q31LM 3,1000 • tl!) 1' .o?ol ?,586 .?li 104,1)5(1 4500,0 41.46,87 3.1-)00 • 6 0 l ,35n8 19/, HB 2.175 160Q,43 .J,ltJUO • 6 01 olS61 .313.984 2.17? 2541'1,44

3.2000 , I 5.~ 217;.>8.4~ 214.s6.9:s 2n8 "iB. ~<; 136,82 :?19793. 7t-. 3. 2tl () 0 ,75.:1 • 01'.28 2379,:19 11)5,?.:35 4500,0 4158,~f.l 3.2ooo • 15.~ .3840 210.040 2. ;~?3 1594,47 .So2UOO • /53 ·1719 ,i;~7.7HQ 2. :~?3 2564.18

3•3000 • 931 23341•69 23028.53 224116.98 1 59. 46 240601·50 3.3UOO • 931 • n776 ·r.H0.87 1 (16. 710 45on.o 4171.09 3.3000 .931 ,4t93 222.so6 2.471 1588,12 3,3000 ,931 ,llltlB 361,1'190 . 2,471 258?,96

58

1. il i M~ ~0·' JTU:R Rl=l 7~ !i.i

_,1.4000 1 .1 37 24932.4.S 2459719.~ V9.S4 I D3 184110 261119.19 3.4000 1 ,13 7 .Q947 2361.21 tol:l •. 51o 45ou.o 4164 .1·2 3.40UO ] • j 3 7 14566 23'L 232 21616 151:111.39 3.4000 1.ol37 ,t!Qf>B 386,Q06 2.6H 260?.74

3.sueo 1 ! 6 7.5 26471 .• 99 26116.83 25411.94 2UI. 6~ 2B0976,6l 3·, so no 1,373 . 11 4 4 1350123 11n (o,f'..7 4500,0 4197,72 3.5000 1. ~n3 ,49'58 245.905 2,75? 1574,29 ;s.sooo 1! 3 73 ,(260 409.784 217"il') 26V I 44

316000 1 ! 6 4 .~ 279,q}.86 2755n,p 26812 •. ~9 (39. 13 ?99793.2A 3,6000 1,643 ! 1 .~69 2.33'7,93 113,::>17 450010 4:?H, 82 3.6000 1.64.5 ,1)36/ 256.41.7 2,f18(o, l566,A3 .3.6000 1.. 6 4 ~~ • '?.464 432.862 :?,A8f- 2644,99

3.7000 j,948 292/:10.3? ?.81387.48 2A107.81 ?69,U 317198.79 317000 11946 1 jfi 2 1~ ,) .~?.4. 36 116119? 45(1 01 0 42?61:~5

:s.7ooo 1! 948 .5792 ?65.874 3. n n A t552.u5 3 17UOQ 1 1948 ,2678 454,872 3,1101:1 2M7 u

:s.euuo ?.?911 3Ll4Y 4 • 1 n :~ouH4.9A 292/2,98 SOl, u6 332854.33 3~8000 ? • ;;9 [) 11908 23o2.6o 11.9.~?'5 45oo~o 4241.23 3.81)00 2! /)9(J 1 62.~2 274,JQ8 3 I 1l7 l55o~97 3,auoo ?,290 ,t19n2 47!:>,458 3,117 2690.26

.,S,91JOO ·?.,671 3154'i·.96 31126.67 3U;>H6,56 334, i:'O 346472.95 ~~9000 2,671 ·22?6 22931/5 123,545 45oo.o 4256,38 3.9000 ?,671 ,6683 ?.8U,9R5 :L2t2 1542,63 319000 2. ell 1 ..s 1 •~ 5 4941300 ~~~21>l ?.713.75

4,QUOO :3 1 0 92 324~0.96 3l'J95,85 3:1.1. •~ 2128 36RI50 35'}836,18 4! 0 \) 0 0 3.092 ·2~77 2276.91 1?7.9fl1 45oo.o 4271.70 4,0000 3! 1''92 I /144 286 I 417 3129~ l534~n7 4.0000 3.09;,! ! 3377 5111126 3! ?9 :~ 2737.£..3

4,1UOO 3,555 3:~1?6,3:? 326t:!1.18fl 317Ci9oflf"i 403,76 3668Q!:i,04 4.1000 3,:,55 .;>963 2259123 13?,9'>6 4353.8 4281.1.0 4o1U00 :3 I !:> ':) !:i ! /6j2 ?.9Ud58 3.-'>57 1525.33 4.1000 3.'>55 ,36?6 5;?!:>.7?6 3.~57 2761.77

4,2000 4! 1161 .366:~4.57 3311):~ 132 32?87.70 440,69 38!;.840,0Q 4,2lJOO 4. il61 1331:14 2241.04 13R,493 2710.R 4302,49 4,2000 41 [16]' ,808!; 292.834 3,401'. 1516,45 4.2uoo 4. (161 • -~882 53!:l~oo2 3,406 2786,05

4.3uuo 4,614 33945,68 3349Q,25 32'5fl6.:15 4'19,72 403930o04 4,31J[)Q 4,614 .384!; 2?21.95 144.62£.. 224119 43p.ao 4,3000 4,614 ,t156l. 29.L805 3,437 1507,47 4, 3 o o ·a 4,614 ,4142 54./, 733 3,437 281.(),33

4.4000 s.?.13 34o63.oo 33605199 3:?.698,97 51.9,69 412771.20 4.4000 5o213 • 4344 .. ~2Qto25 15:1..390 1688,0 4332.93 4.-4000 s.2u ,90:'16 296,326 3 ! 4 5:? 1498,44 4.4000 5.2U .44n6 554,862. 3,452 2834.49

59

1 I! 5 MM HOW T TZEQ ~(:) 765

4.5000 l'i,Hfll 3H47,55 B241 I 4(1 3?, ,( 6. 1 4 '>60146 42fJ08?.5j 4.,uoo ",!<61. . .481:!4 ?.1Be.oA 1~A.~1n 109719 4347.8:? 4.5000 <;,1:!6] ,95(11,1 291.497 .L45( 148911\1 4~5Ull0 511:!61 I 4613 55914.50 314c:i? 2~?A,42

4.6000 61':>58 3:~71'>2·85 3.S~09.87 ~2410oA4 6 01 .• 7 .s 423888.9(1 .4,61100 6.558 1546!:1 216115? 161'-,904 6<;8.5 4:3fl?,4f) 4.6000 6.':>SR ,997/ ?.8~.442 3,43f.- 1480140 4,6000 6,.,58 ,4941 561.5<'9 3143l'\ 281.1?,UU

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APPENDIX D

Comparison of Experimental and Predicted performance for

T,ypical 105mm Howitzer Firing

Page intentionally blank

0 0 Q 0

----~

pg 8

pg 9

pg 14

pg 15

pg 16 and pg 23

pg 19 and pg 24

pg 46

ERRATA-BRLR 1183

m. specific mass of i th propellant, lb-mol/lb 1

R Universal Gas Constant, in-lb/lb-mo1-°K

T initial temperature of gun, °K !5

- -c -c = m.R v. 1 n pi 1

(~c. To. dx (in Eq(14)) . 1 1 1=

V n 12x0. 38cl. 5 {x 0) ~ c.

E = m +A i=1 1

h f 0 6 2.175 ] 2 1 + . c ( ~ C.)0.8375 vm i=l 1

p = ----=-0 (1-a )n'+1

0

1

Bl6 IF (Y1>0}GOTO(Bl7)%Yl=O% IF(Y3 etc

-T) 2 v

J. M. Frankie