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((

REPORT NO. 1183 DECEMBER 1962

THE SIMULATION OF INTERIOR BALLISTIC PERFORMANCE OFGUNS BY DIGITAL COMPUTER PROGRAM

St1

Paul G. Baer Jerome M. Frankle

RDT & E Project No. IM01050IA004BALLISTIC

RESEARCH LABORATORIES

ABERDEEN PROVING GROUND, MARYLAND

A

ASTIA AVAILABILITY NOTICE Qualified requestors may obtain copies of this report from ASTIA.

The findings in thl.s report are not to be construed as an official Department of the Ar-y position.

SBALLISTIC

RESEARCH

LABORATORIES

REPORT NO. i185DECEMBER 1962

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

Paul G. Baer Jerome M. Frankle

Interior Ballistics Laboratory

RDT & E Project No. 1MO1O051AO04

ABERDEEN

PROVING

GROUND,

MARYLAND

m2l

BALLISTIC

RESEARCH

LABORATORIES

R~EPORlT NO, 1183

PGBaer/JMhFrankle/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 ut of several propellants of different chemical compositions and different granulations. those of other interior ballistic systems. tail Results obtained by the method descrjbed in this report compare favorably with In addition, considerably more de-

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.

3

TABLE OF CONTENTS

SLIST

OF SYMBOLS A.. INTRODUCTION .

. . . . ..

... .. . ..

. .,

....

.

. . . . . .

7lii

INTERIOR BALLISTIC THEORY

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

. .

12 L............. 12

Interior Ballistic System Energy Equation ..............

.......

. ....

Equation of State ...........................Mass-Fraction Burning Rate Equation ........ Equations of Projectile Motion ........ ............... ..................

......

13 1720 21

Summary of Interior Ballistic Equations ....... COMPUTATION ROUTINE ................. Preliminary Routine .......... Main Routine ......... .......................... ........................

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

23 25 25 26 .. 28 . 29 31

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

Options to Routine ................. DISCUSSION .................

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

LIST OF REFERENCES ................. APPENDICES ..................... A. B. C. D.

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

........

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

3335 41 49 65

Form Function Equations .............. Computation Routine ................ Lnput anCd Output Data ..........

Comparison of Experimental and Predicted Performance for Typical 105mm Howitzer Firing ........... ...........................

DISTRIBUTTON LIST

69

5

rI

II

LIST OF SYMBOLS in./sec2 dimensionless

a a0 A

acceleration of projectile,

constant defined by Equation (28a),

area of base of projectile including appropriate portion of rotating band, in.2

bi cc

covv..ume of i th propellant, in.3 /lb diameter of bore, in. is a function

specific heat at constant volume of i th propellant (Ccv of T), in.-lb/lb-K

-

c

vi

mean value of specific heat at constant volume of i th propellant (over temperature range T to T ), in.-lb/lb-K i

-

mean value of specific heat at constant pressure of i th propellant iP (over temperature range T to T ), in,.-lb/lb-K i 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, 0K in. sec

Ci C d dt dT dx dz i dt Di E E E

incremental Cistance traveled by projectile,

mass fraction burning rate for i th propellant, outside diameter of i th propellant grains, in. cncr~o lost due to heat los, ino-lb

ppr

kinetic energy of propellant gas and unburned propellant, in.-lb energy lost due to bore friction and engraving of rotating band, in.-lb functional relationship between Si and zi resultant axial force on projectile, lb

fi F'

S~7

FfFi FI F Fr g G K v Kx

frictional force on projectile, lb"force" of i th propellant, in.-lb/lQ "force" of igniter propellant, In.-lb/lb lb

propulsive force on base of projectile, gas retardation force, lb

units to mass units, in./sec constant for conversion of weight functional relationship between pr and x burning rate velocity coefficient, in. sec in./sec in. sec-in.

2

burning rate displacement coefficient,

Li mi M n n' N

length of i th propellant grains,

in.

specific mass of i th propellant, lb-mols/mol mass of projectile, slugs/12 dimensionless dimensionless

number of propellants,

ratio defined by Equation (28b),

nnumber of perforations in i th propellant grains, dimensionless

p Pb p Pi Pi PO Pr

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

Qri r'i

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

8

Ri S! S t T T oi T 01 T ui

functional relationship between r

and p ta,22

grai surface area of partially burned I th propellant grain, in. surface area of an unburned i th propellant time, sec mean temperature of propellant gases, OK adiabatic flame temperature of i th propellant, OK adiabatic flame temperature of igniter propellant, OK temperature of unburned solid propellant, OK

two times the distance each surface of i th propellant grains has receded at a given time, in.

U

internal energy of propellant gases, in..-lb

"v velocity of projectile, in./sec "vV Vc 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

Vgi volume of an unburned i th propellant grain, in.3 V0 W W p volume of empty gun chamber, in.3

external work done on projectile, in.-lb weight of projectile, lb

"x travel of projectile, in. "Xmzi zI ai i travel of projectile when base reaches muzzle, in.

fraction of maza of i. th propollant burned, dimensionleas fraction of mass of igniter burned, dimensionless burning rate exponent for i th propellant, dimiensionless burning rate coefficient for i th propellant, in

in. "

1

1 psia

Y'

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

9

y. *i 8 Pi

ratio of specific heats for i th propellant,

dimensiomless dimensionless

ratio of specific h.eats for igniter propellant, Pidduck-Kent constant, dimensionless

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

10

4

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 from a knowledge the projectile weight, and the gun characteristics. problem (* calculation is usually referred to as the classical central This of the propellant charge, of interior ballistics. When non-conventional guns are considered or when is necessary to know more than complete interior These trajectories consist

detailed design information is required, it these two salient values.

For the more demanding problems,

ballistic trajectories may have to be calculated. 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. in tables, graphs, nomograms, closed-form. (1) (2) Methods, varying from the purely empirical to the "exact" theoretical, have been devised slide rules, and simplified equations solved in All of these methods require some Some of these methods require data from the firing of the gun

being considered or from a very similar gun.

simplification of the basic equations of interior ballistics. To eliminate the restrictions imposed by assumptions made only to facilitate the mathematical solution of the problem, the interior ballistic equations have been programmed for high-speed electronic computers. Both analog and digital computers have been used to calculate detailed interior ballistic trajectories. computer. There are advantages and disadvantages associated with each type of 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 insiicate references listed at the end of this report.

** Although the interior ballistic equations were originally programmed only

they have been recently reprogrammed in more general 4) form (5) for the ORDVAC and the newer BRLESC. for the ORDVAC, 11

H

~

j

o;

The computer program described in this report has been designed to solve a set of non-linear, ordinary differential and algebraic equations which simu. In this method, the usual

late the interior ballistic performance of a gun.

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 suitably modified to study nonA number of these optional programs

conventional guns and gun-like systems. have been devised and used extensively.

IN TERIOR BALLISTIC THEORY Interior Ballistic System The basic components of the interior ballistic system for a conventional gun are bhown in Figure 1. A set of equations can be formulated which mathematically describes the distribution of energy originating from the burning

CHAMBER

POETL VPROJETILE

TUBE

ZPROPELLANT- IGNITION SYSTEM

Figure 1.

Basic Components of the Interior Ballistic 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 Options to Routine.*

12

SiIpropellant and the subsequent motion of the components of the system.behavior of composite charges: 1. The total chemical energy available is the simple sum of the chemical

In the

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

energies of the individual propellants. 2. 9he total gas pressure is the simple sum of the "Ipartial'pressures"

resulting from the burning of the individual propellants. Energy Equation Application of the law of conservation of energy leads to the energy equation of interior ballistics. Energy Released by Burning Propellant or: U + W + Losses (la) This may be written as: + External + Secondary Work Done Energy Losses on Projectile (1)

Internal Energy of Propellant Gases

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 propellants as previously stated. n T Therefore:

il[cizi

f0

c

dT]

Because of gas expansion and external work performed in a gun, the gas temperature is less than the adiabatic flame temperature (T of the gas (U) is then: ). The internal energy

n

T

i=1The external work done on the projectile is given by:x

(I)

W =A

Pb 13

(4)

Substituting Equations (2), nV

(3), and (4) into Equation (la) 8iven:n ,dT

'

ri

T

fi c, ICv dT

+A

lb dx + Losed

which may be rewritten as: n T =I P"Z[,zi f 0i dT]=

A

X Pb dx + Losses

(5)

ii

As the c

v

do not vary greatly over the temperature ranges from T to T 0).

they can be replaced with mean values (_Cv

Integration of Equation (9)

gives: n Cizic (Toi T)= A

Pb

dx +

Losses

(6)

i=land solving for T: n Czi

0

1.lCv 1i n

-

A0Pb 'C)

dx

-

Losses

(7)

nCizi

Next,

the "force" of each propellant is defined by:

F.i RT 0 F, = M iTiand the well-known relations:

(8)

and: i

c Pi

cviare introduced.

(ao)

14

I

.

.

.

... .

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

ov (7

-1 )

=miRinto Equation (8) gives:

(x-r

vii

Substitution of Equation (ll) Fi

T-

0

1 (i

vi

(12)

Finally, substitution of Equation (12) into Equation (7) in the form: i_ =1 Tl " - A

gives Rbsal's

equation

f

Pb dx " Losses

TY

i=For most problems, (zI it = 1) at zero-time. Equation (13) may be restated as:

(13 1%

is convenient to assume the igniter completely burned

niC Zi[] +

F IC-A

x Pb dx Losses

T= [x The terms A in more detail. f

1

0 (if')iToI)

Fi i

pb dx and Losses of Equation

(14) can now be considered

The work done on the projectile results in an equivalent gain Including thse losses

in kinetic energy of the projectile except for losses. under the general category of energy losses:

xpb

W Pdx= 011/2 -P

22(15)

IIAccording 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) (4) heat energy lost to the gun, strain energy of the gun,

(5) energy lost in engraving the rotating band and in overcoming frictiondown the bore, and (6) Types (2), rotational energy of the projectile. (4), and (6) are estimated to be less than one percent for each For discussion of each type of secondary energy loss, see Reference (2). category and have been neglected here. The kinetic energy of propellant gas and unburned propellant can be represented by (6)n

E

2g-

(16)

The energy losses resulting from heat lost to the gun can be estimated by a semi-empirical relationship described by Hunt:.2) n

i=l

[18C

+ (

0l 2.175 ' .6c CC087 0.6

2

At the present time, the introduction of a more sophisticated treatment of heat loss, with its attendant complexity, does not seem to be warranted. substitution can be made if and -hen it appears desirable. Such a

16

L

cX.~~-~.---

The final energy losses to be considered here consist of those resulting from engraving of the rotating band, friction between the moving projectile and the gun tube, and acceleration of air ahead of the projectile. estimates of these are difficult to make, tive pressure in the form: Epr Individual so they have been grouped as resis-

=

A

jx

.r dx

0.(18)

The pr versus x function is discussed in greater detail in the section concerning forces acting on the projectile. Substitution of Equations (15),

(16),

(17),

and (18)

into Equation (14)

results in the form of the energy equation used in this computer program:[z

i =-'I + Fi FiCi7-]

7i -i

2g p L+

I=l I~C C~ -A

x

Pr dx - E

T=df

-- 4 T + (I'1) T0Equation of State

(19)

The pressure acting on the base of the projectile can be calculated from a series of equations, once the temperature of the gas is determined from the energy equation. form: pvI where V Now,= mI RT

Generally, the equation of state for an ideal gas takes the

(20)

= the volume per unit mass of i th propellant gas. the volume behind the projectile which is available for pro-

define Vc,

pellant gas, as:

17

IVolume Available for Propellant Gas Initial Empty Chamber Volume Volume Occupiedby Unburned Solid Propellant n or: V V +Ax n Cizioi

Volue Resulting+

from Projectile Motion Volume Occupiedby Gas Molecules (covolume) (21)

it-l I

1

(22)

By the definitions of Equations (20) and (21), Vc

Ciz1Substituting Equations (8) and (23) FICsz T into Equation (20)

(23)and rearranging gives:

Pi V TCO(a)

e 0o1

(24)

If T

the bi are assumed to be constants over the temperature range from T to , and if the total gas pressure is taken as the simple sum of the "partial

prissures" resulting from the burning of the individual propellants as previously stated, then: nSPi

T

n

FiCizi

i=iAs before, if it

c

i=-

01)

(25)

is assumed that the igniter is completely burned (zi

at zero-timep Equation (25) may be restated as: T 1 Fi i i I

o

J

(26)

18

The space-mean pressure, p,

given by Equation (26)

is used in the cal-

culation of the fraction of propellant burned at any time.is discussed in the section concerning burning rates.

This relationship

There is, however, a

pressure 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 projectile. This pressure-gradient problem was first considered by Lagrange and Later studies in For this

is commonly referred to as the Lagrange Ballistic Problem. this area were made by Love and Pidduck, (7) Kent,

(8) and others.

computer program, the improved Pidduck-Kent solution developed by Vinti and Kravitz (6) has been used:

Pb

n

1+ WB In addition the breech pressure, in Reference (6). rior ballistic studies:

(27)*p0 , is calculated by the method contained

This is the pressure usually measured in experimental inte-

PbPO (1-a) (28) 2 n1+3 where: i/a=+

2 (n'+l) Sn C/WPiul

(28a)

Tn Rpfevenr'n A)

the det mrlmnatinn of B dppan&; on the ratin of

AepenIffe

heats, 7.

For composite charges, an effective value

nn

is used for this purpose.

i=1 19

(270)

Iand n' =-

(28b) Mass-Fraction Burning Rate Equation Both the energy equation (Equation (19)) and the equation of state (Equation (26')) are algebraic equations whose solutions depend upon the solutions of several non-linear, ordinary differential equations. The mass-fraction burning rate equation expresses the rate of consumption of solid propellant and hence the rate of evolutiun uf propellant gas. This may be written as: dzi-

S

1 V

S 1

r ri (29)

dt where: ri=

Ri

(G)

(30)

and:

Si

fi (z)

(31)

For most gun propellants, Equation (30) may be quite satisfactorily stated as: ri= 131()Ci(32)

For certain propellants,

including those plateau and 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 ri's and corresponding i's (Equation (30)) and to interpolate for the desired ri . The r is calculated by either Equation (30) or Equation (32) are closed chamber 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: ri= ri + Kv v + K x (32a)

i020!

Similarly, the form function described by Equation (31) may be stated in one of several ways. is In many interior ballistic systems, the form function chosen for convenience of analytical solution. Where routine numerical

computations are handled by use of a high-speed digital computer, the geometrical form of the propellant grain may be used to obtain the functional relationship, f,, between S, and z,. equations are given in Appendix A. For the usual grain shapes encountered, these This Appendix also contains the method for To extend these equations to

handling such equations in the computer routine. include propellant slivering see Reference (9). Equations of Projectile Motion

The translational motion of the projectile down the gun tube may be calculated from the forces acting on the projectile. Figure 2 shows the axial forces considered in determining the resultant force.

~~FfFr

Figure 2.

Axial Forces Acting on Projectile

The propulsive force, Fp, is that resulting from the prqssure of the propellant gas on the base of the projectile according to:

F = pbAwhere.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 inside 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. be expressed as: It may

Ff pof Pr 21

(34)

The determination of'p

is

difficult in most cases.

Many interior

ballistic solutions use an increased projectile mass (approximately 5%) to account for its effect. treatment. There are several disadvantages inherent in such a It is not possible to Although the muzzle velocity may be calculated reasonably well,

the detailed'trajectory will be altered considerably.

simulate the case where a projectile lodges in the bore (see Reference (10) for experimental trajectories for this condition). ing a tabulation of the function: (34a) is that which results from the pressure For this computer program, experimental data of the type given in Reference (11) may be used by insert-

Pr = G(x) The gas retardation force, Fr'

of air or gas ahead of the projectile, stated as:

Fr = pA

(35)

where pg is small enough to be neglected except for very high velocity systems, light gas guns, and other special applicatioilb. In the discussion of the Energy was considered a part of Pr* Equation in the Interior Ballistic Theory Section, p

The resultant force in the axial direction is then: Fa = F or: (37) - Ff - Fr(36)

P Fa = A(pb -p

- pr). by Newton's second law of motion,

The acceleration of theprojectile, is: A(pb pg Pr)

Mor:a=

(38)Pr)

Ag (pb - pg

wp22

(39)

'.5

dv Since a = and v t v=j a dt

dx the velocity of the projectile is given by:

0and the displacement of the projectile is given by: t x

(4o)

0o

v dt)

(41)

Summary of Interior Ballistic Equations The equations which arc uced in the computer program are now sumarized for ease of reference. Energy Equation n

[T=

7fnF

+ii J

~(p+

>

fPr 0

dx~

_____z

I I

i-l(''l)o 0

(yi-l) T 0n V T=2)

(19)

where:

0.38c 1.5 (Xm+

(i

lCi

v2

jl+

.18375

Iv

m

23

Equation of State c~ i~ c~ Z i

whre

c

0

(1'z0

S=( Pi

i=l

(22)

Pb l+

n c i Wp (27)

PO= (1ao)-n-(28)

Mass-Fraction Burning-Rate Equations dzt di 2.1 gd Sr (29)

r

() ai

(32)

or:r' = r +K v ,+K x (32a)

i

A

Equations of Projectile Motion

a =Ag(P

pb - g Wp

"

Pp)

r

(39)

t

V 0tx=

a dt

(40)

f

v dt

0COMPUTATION ROUTINE

(41_)

The set of non-linear, ordinary differential and algebraic 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 The generalUsing the FORAST

been devised for the simultaneous solution of these equations. ized flow-diagram for the routine is presented in Appendix B. language, puters. Preliminary Routine

(5) the solution has been programmed for the ORDVAO and BRLESC com-

To reduce computation time and conserve meory space, a preliminary routine has been introduced. Here all data required for the computation are read into

the computer, constant groupings (e.g., FiCi (yi'1 Toi Y FiCi (i'-l) I -Ci , Pi etc., are calculated and stored

for subsequent use, and data to permanently identify the computer run are printed out. A complete listing of required input data may be found in Appendix C.

25

Main RoutineThe main computational routine is presented in the generalized fl"wdiagram 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 thegun 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 zI . 0). FII The space-mean pressure, consisting only of the igniter pressure, is calculated from:

pEquation (42)

=

V(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 generalized interpolation sub-routine* is employed. The mass-fractions burned, zi o, 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 Gil., (12) is commonly used for the solution of sets of ordinary differential equations and has been employed here. Calculation of the temperature, T, from Equation (19) and the volume available for propellant gas, Va, from Equation (22), will allow the calculation of the new space-mean pressure, j., at time, dt, from Equation (26). The surface areas of the now partially burned propellants are computed from equations presented in Appendix A. All results of interest arc printod-out at this time**

and these results used as initial conditions for calculations during

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

Bee 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. which has been arbitrarily defined, ends at this time. Phase II requires the addition of the equations of motion to the sequence Phase I,

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 projectile 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 us in terms of Equations (17), pr dx, which is one of (19), and (22). To compute values for Epr = A the terms in Equation (19), the generalized interpolation sub-routine must be used to obtain pr from the tabular information described by Equation (34a). This integration is performed by use of the Trapezoidal Rule.* As time is increased by the addition of small time-interMls, calculations during Phase II are continued around this expanded loop with print-out of appropriate 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 introdacing propellants in amy special order. In fact, it may not always be possible to predict the exact order in which several different propellants will be burned out. The combination of the propellant switches and the btart-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), (3l), and (30) or (32)) are eliminated from the loop. As in the other phases,

* Although the Trapezoidal Rule in 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 coplex methods

27

results are printed-out at the end of each time-interval. Z

A continual check

is made of the displacement of the projectile to determine whether or not it reached the muzzle of the gum. ahas III has ended and the pPhase It When the projectile passes the muzzle is stopped.

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. in the tube. It is also possible

frLh' UIL1JrcJb'a.M to simulate a firing in which the projectile becomes lodged In this case, Phase III is not completed and the program is

:;topped when the projectile displacement does not increase. At each time-interval after the beginning of Phase II, i; det;erminimL from Equation (2M) and printed out. Lhc computatJionaL routine but i ,,;uLt,. t.'t',ch I'C the breech pressure

This result is not used in

used. to compare theoretical and experimental

A continual check is made of the calculated pressures and the maximum . I:; torcd with its associated time and projectile displacement. Calculations dur-

'Ihio infornmaLion is printed-out at the end of the program.tihen the desired distance to the muzzle.

ing the last time-interval result in a projectile displacement somewhat greater A linear interpolation between results at the lai;t two time-intervals is used to obtain results exactly at the muz4zlc. Thsse results are also printed-out at the end of the program. Optiono to Routine A considcrable 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. greater number and variety of problems. Typical options for non-conventional guns are those for gun-boosted rockets, traveling-charge guns, and light-gas gums 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 pressure, and resistive pressure as a function of base pressure. It is expected that the number of such options will increase as the program is used for a

28

II

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. troublesome here. erally available. There are several problems present in all, interior ballistic calculations and these also prove For example, useful propellant burning rates are not genIt is known that burning rates obtained from experimental The results obtained from limited As previously mentioned, Qp-

firings in closed chambers are usually low. gun-firings by the authors

indicate gun burning rates may be twice closed

chamber burning rates under certain conditions.

tional methods of adjusting closed chamber burning rates have been provided for in this program. tile displacement) One such approach is to consider the burning rate to be This

a function of the projectile velocity (and possibly a function of the projecin addition to its known dependence on pressure.

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 projectile 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 resistive pressure and shot-start pressure. Considerable versatility has been built into the program. stopping the computation at the end of Phase I11, matically read into the computer and solved. Instead of

a new problem can be auto-

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 vas generally very good when the other

29

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 ballistic calculations and these also prove troublesome here. erally available. For example, useful propellant burning rates are not genIt is known that burning rates obtained from experimental The results obtained, from limited As previously mentioned, opindicate gun burning rates may be twice closed

firings in closed chambers are usually low. gun-firings by the authors (I-)

chamber burning rates under certain conditions. for in this program.

tional methods of adjusting closed chamber burning rates have been provided One such approach is to consider the burning rate to be a function of the projectile velocity (and possibly a function of the projectile 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 projectile is moving at higher velocities and is further down tube, reasonable increases in burning rates are obtained and used. tive pressure and shot-start pressure. Considerable versatility has been built into the program. Instead of Other equally important difficulties are associated with the determination of reasonable values for resis-

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

In addition, detailed interior ballistic takes to calculate A typical computer magnetic tape output

trajectories are produced in considerably less time than it maximum pressure and muzzle velocity by other systems. solution for a conventional gun takes only 10 seconds if is used with the BRLESC.

Results from computer simulations 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 processed by the method described in Reference (11.). lation of this firing. Tn Appendix D these experimental results are compared with the predicted results obtained from a simu-

PAUL G. BAER

30

LIST OF REFERENCES 1. Corner, J. Theory of the Interior Ballistics of Guns. John Wiley and Sons, Inc., 1950. Hunt, F. R. W. Chairman, Editorial Panel. New York: Philosophical Library, 1951. New York:

2.

Internal Ballistics.

3.

Baer, Paul G., and Frankle, Jerome M. Digital Computer Simulation of Ordnance Computer the Interior Ballistic Performance of Guns. Research Report, VI, No. 2: 17-23, Aberdeen Proving Ground, Apr 1959. Historical Monograph - Electronic Kempf, Karl, Historical Officer. Computers within the Ordnance Corps. Aberdeen Proving Ground, Nov 1961. Campbell, Lloyd W., and Beck, Glenn A. The FORAST Programming Language for ORDVAC and BRLESC. Aberdeen Proving Ground: BRL R-1172, Aug 1962. Vinti, John P., and Kravitz, Sidney. Tables for the Pidduck-Kent Aberdeen Special Solution for the Motion of the Powder Gas in a Gun. Proving Ground: BRL R-693, Jan 1949. Love, A. E. H., and Pidduck, Phil. Trans. Roy. Soc. 222: F. B. The Lagrange Ballistic Problem. 167, London, 1923.

4.

5.

6.

7. 8.

Kent, 11.H. Some Special Solutions for the Motion of the Powder Gas. Physics, VII, No. 9: 319, 1936. Frankle, Jerome M., and Hudson, James R. Propellant Surface Area Calculations for Interior Ballistic Systems. Aberdeen Proving Ground: BRL M-1187, Jan 1959. Frankle, Jerome M. Special Interior Ballistic Tests of llgm XM378 Slug Rounds. Aberdeen Proving Ground: BRL M-1266, May 1960 (Confidential). Baer, Paul G., and. Frankle, Jerome M. Reduction of Interior Ballistic Data from Artillery Weapons by High-Speed Digital Computer. Aberdeen Proving Ground: BEL M-ll48, Jun 1958. Gill, S. Runge-Kutta-Gill Numerical Procedure for Solving Systems Proceedings of thp. of First Order Ordinary Differential Equations. 96, Jan 1951. Cambridge Philosophical Society, 7, Part I: Hitchcock, Henry P. Tables for Interior Ballistics. Aberdeen Proving Ground: BRL R-993, Sep 1956 and BRL TN-]298, Feb 1960. 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.

9.

10.

11.

12.

13. 14.

31

15.

Strittmater, R. C. A Single Chart System of Interior Ballistics. Aberdeen Proving Ground: BRL R-1126, Mar 1961. Scarborough, James B. Numerical Mathematical Analysis. The Johns Hopkins Press, 2nd ed., 1950. Baltimore:

16.

17.

Tables of Computed Thermodynamic Baer, Paul G., and Bryson, Kenneth R. Aberdeen Proving Ground: Properties of Military Gun Propellants.

BEL M-1338, Mar 1961.18. Numerical Calculus. Milne, William Edmund. Princeton University Press, 1949. Princeton, New Jersey:

32

APPENDICES

A. B. C.

FORM FUNCTION EQUATIONS COMPUTATION ROUTINE INPUT AND OUTPUT DATA

D.

COMPARISON OF EXPERIMENTAL AND PREDICTEDPERFORMANCE FOR TYPICAL 105MM HOWITZER FIRING

33

APPENDJJ

A

Form Function Equati~ons

FORM FCTION EQUATIONSGeometrical Equations 1. Initial Volume of a Propellant Grain V (D 2

-

id

2

)L

(A-i)

where:

V = volume of an unburned propellant grain, in.

3

Di = outside diameter of grain, in. Ni -- number of perforations, dimensionless d, - diameter of perforation, in. Li = length of grain, in. 2. Volume of a Partially Burned Propellant Grain

Vg(l - zi)

[(Di-

ui

)2-

Ni (d

+ u)2]

(Li - ui)

(A-2)

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

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

3.

Initial Surface Area of a Propellant Grain8

g gi '

(Dn +Nidi)

1i

Dr(Li) +

22

12 NIdii]

(A-3)

where: 1.

S

= surface area of an unburned propellant grain, In.2

Surface Area of a Partially Burned Propellant Grain2 (D -u )

N (dei)

2

where:

S

surface area of partially b~rued I th propellant grain at a given time, in.

37

Equations for Newton-Raphson Method* for Finding Approxiate Values ofthe Real Roots of a Numerical Equation1. Rearrange Equation (A-2) to set f(u 1 0:

i2 f(U)1 (N:Ll) u13-

[L(N,-l)

-

2(D+Nidi)] u ud1

2

-

2L D2

+

Nid)2

+ (D12

-

+ Li (Dn

- Nd

)}

-

Vg

(lzi)

(A-5)

2.

Differentiate Equation (A-5) with respect to u

f, (u ) =

=

7

{3(N -)u 2

-2

[L

(cN-1 - 2 (Di.+ Nid )] u,

[2L(Di+N:Ld [

+

(Dj 1-

Nidi 2)]}1

(A-6)then:

3.

The value of the root of Equation (A-2) is f(ui) U.i+ 1-"

fu(uui+1 = the improved value of the root, where the first estimate of the root is ui.

(A-7)

where:

Procedure 1. For each propellant, determine z i by integration of Equation (29). 0). In the

initial calculation of each zi, Equation (A-3) Si (Si= Sgi when u=

is used to compute each

For subsequent calculations of each zi, Equation

(A-4) is used with ui determined as described below.

See Reference (16) for a discussion of this method.

S3 8

2.

The zi's obtained from Equation (29) are used to compute the ui's from Equation (A-7) and then the new Sits are computed from Equation (A-4). In the initial calculation of ui, the first estimate of its value is zero. With u

Equation (A-7) is used to calculate the improved value, u

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- inch.

39

APPENDIX B Computation Routine Generalized Flow Diagram VMST Listing

1. 2.

'I1]

UUC2 ON a WW

~~JAj CL L=air CLC E

r-j2 1I' CciCLLJcip

C.),a

=3 c

Interior Ba11hatice Program for Guns 1"ORAST LISTING A I2Y,7l~ -Tsijn-uj5)St 1)-5

8LOCUoi-o7)Cl-CS)F1-F 5GAt-CAiACOVI.-Co)V0ToI-TO5)RHqU1-RHO5)ETI-BET5) ('ON1AL.P1-ALP5)1)Ii-US (DPi DP5)iLI L5)NPI-~,)C CP)CIH5 INTF.R(A %4F.ADNANI )1.i)% READ-FURmAT(Ui)"(WP)XM)VU)AP)PFi Dflj)PPmAX)% RFEhD-Fut)MAT(O2

O000 00061

Al

8!

Bill. B2 82.1-

-( DT)NI.)I(V)KX)D)EVP)% SET(J=O0 91 PrAD-FURMAT(Ot)-(XCi,,J)PRlJ)% COliNTr90oHN(Jh3OTOc9l.l)% -in FNITEP(!NTE(GEk)N!)N)%SETI *J=L)% INTCNC.Pr-2*N)% 001i kEAD-F'-MAT(Ot)-(Ct,J)F.,dJIQAI,J)CflV1.J)TO1.J)PHOIJ2)1 IP v X)-4mt4 j;xi.j 2-(RETI J)ALIJI,J)DI ,!)r'Pl J)Ll ,J)NF'1,J)% 10121 COIIN-T( ,NJ INC.j)(TO(hR2lFT(J=On 0013 EN7ER(A.PlJNCH) AN1)l )% EN1FR(A.PLUN(H)AN89)i2% 141~FNrFh(A.PuJNCH)AN9)j)%I

PilNC,..VO0rMAIIt))-(j )CWF)XM)VUJAP)D)FILPF)PPmAO)% PuICj'A-lACI:8Cl/TGIlxCrT=F1.CT/TOl% FVM=EVP.12% RCI.,JFI ,j*C I I AI, J-uAI AC I, JaACI I/Tflt ,j % CCI #J2F I, J C I.J COmT71"r,j%HCI ,JU?'I.1,j(Dn j4.NP1,.Jsi1Pi,,J)+Cp1,J..?-NPl,J*flP1 .U*2)s

25 26 27 0035 0J66 00370039

____GC,lJwIlI..JNP1,J-1.)-C[.I,J+NP1,JiflP1.j)%

00400042 004,1 04

JCI .I3 # 416 *IC I-J/ 4F(,'UN Tc ,N ) IN (J ) Gf) T0 4. 1 TISEf (j 0 U) 85CT:0%TP1fI ____ __ 'rP cc j, * A1,j+T F I% CT =U I,J+C T % C 01iINTC.N ICJ0TC9-.1

GAPnIP1 /C'%5T(,J:Ol)%('F:QAP/(CQAP-1 FP:CT/FP EPI =1+E.PDELIfTPI./(CfA-1)%TIF2:i/Cc 2*TP1 .3j/nEL.C2,TPi*2),EpI%MCDW*TDLT'=UAWA*B./P

00451 00 4 6j0047

kEPffFXP(GCAF*LOGCI-TP2)% TP~uEXP(1.5#1.oG.(Dfl%____C0NT(n,))% ____-

TFP2uEXPC2.1?b*LOG

004S0049

-P3 3fF-PPT-8 -T5 *L-OG CT-)) TP4*CIt.J*TO1,J.TP4%_COuNJT(,N)jNUJ)GOTn(B5.2)% SETC~jxo)x HCI (.3P 1-2 0 P ( A+ U/ A FF V (TP-4/-C7129 8,)(( -n6* /T 3 P P CONTFEVm* *2)% ppLEAIRC7)NGS,AT(KI)% CrEAHC(7-NOS,ATcYJi% CLFAR(7)NOSATCQi2N 56 7

00nn0cusp 00 00D4 0056 0057 00596

8s.2

86.1 81.2 67.1

CLEAkd5)NOS.ATtflJ)% XLST=OX PRLST~fl% Y3,Ju1.1% COUNTC5)IN(,J)(AOTO(86.13% SFT(Jnn2% Y3.jJOD COUNT(N~NCJ)QOGU96.2)% SFThJ~nfl005 lF~-I0TtNsl)8lOTn(B7,b)% SET(W3mPF?3 .1 !P'-lNT( Nx2flOTO(b7.6)% SET(SW42flA3.1.)% IF-lNT(Na3)QOTO(b7.7)% SET(SW~mDR3.i )% IF-!NT(N24-)3OTO187.8)I bE' (5W6KDR3,12% GOTO(81%

Cuot 0062

145

87. 7

SF(wz14ERAG.fTfltM)%

0flA

87,8 88 Ba*li

SET(SW6mB14)NGOT0(88)1 T P tou %00 SET( Jan1 X C;OuNTCPN)IN(4)GQTOC98.1)% TPiDC I J**rP1IPT=CI**1/(VO-TP1)% SF.T(Sw~c.ij5)SIW8=Rji.5)% PmA~uPT1 ENTE[R.RK.G.)DT),N)R9)y1)K1)01)% GnfTflC,Swl)% Y3:lKicAj151 OWRI.:fl%,j3:U1SFT(Sw1.1:81O)Jxfl)%GOTU(DRS. lI sI)G T ()I) IF 4 = r T~i 1)A F (q lI Y4tK=X~ORnjzXE(wl~lJlDnW) Y5'1S=i~r~IS~L.-I01SFT(SW11ufI21 2)Jr.PGOTO(n~) IF(Y7>=1)GQTr1(HI.1)'5 T(SW11=R41uSi~jI1GDOo(nR1)% I1x

0066 0068 0069 007n 7P7 007,4 74 Ws41 0079 b1 0081 00 Fq2 008x 06 0085 00861 00871 0o O8 009n 0092 0093 009 109Rj*K*+X*j U4 ~9_____________________

89.1 sin B10.1 * B.i 1.812_

Si.j B 814 *814.1 8 144.2 B1.4.3 *814.4 814.5 DR1

0R3* R3.j. DR4

Y7IKaS ~ %17DSTSW1=1MI4)J:4)%GflTOC ,Sw6% ~ SE7(,j~f.;TP~xO% TP 12D(1uI-,,j( -Y ,J FC I, J 0Y 3, J *T PIr)1f N T N IN Ja 0T0 (14 1 1 1 VC=V(,'AP#XI-TPI17E(J30)1TPIRSCI1 TP IvC I joy 3, j+TP 11C('N IC .N) IN ( J)GOT 0 ( 914 2) S cA)%ELCL_ TP2:ACI,J)*r3,j.TI1CUUNT(,IN)IN(~J)OTC(Rt4.3)% SFTc(,r=f %'TMPc(Tfl~-ALP)/TPPITP1 uCtJ% TPImC.CI J# Y3, J+7 P %C(JUN I ,tI N N ) GOT n (HI4 .4 SF~TCJmfQPTATFMP*TPj/VC% GOTO(#SW8u5 ENTER(P.lK.GD~t R =T EPIAPIjLOq(TIO~tl,)H=r ~ulii H3 =HUI , JX14 aI C1 , J1H5:JCI .i(IlY3 ,J)%H6 =I t, J1)I7 ml,J% tiu p .% 9 N iJ m (D I j M l~ l,-G Y (sM )I3SaR /DC1.pj% GTO(.SwltU1 P8=PT/FPI1PRH=HiB/EF'?% K~uAiW(P-PWK~t1%xty2%0097 1FcXT:1IAND(Y5)zijAND(Y6>u1) CONTAND(Y7>n1)GoTu(816.1)IGIJTO(117)% SET(STUCKsH22)% XF=X1/12%V=-Y1./i2%AF:KI/12%SETc(3 j Scfl DC?1,j:I{3TJTC1.J1CrWJT(, )!N(J)Q ?0(R17.1SI)NtT(Jeouo STSj$x~iT.)N()OO872%E(vI IFCP3RS-319 .11,i 10 . 4 )3 . :14+1

o 2. +I * 2-9 ' .10,1. ,20n{r),) . Vl I.

it'73,A S r'9 71 .

134

53

268341. 8108 ,vs

1-7,3 .I.14 17.3 6

1- 1+o 34

.7S;9

, U

.41.-

3 1.P~l+.7P5j , ,(1

0 .110

5 . 8 U10 5 . 8 ot1n

1 V * I]. I FI -, 15

tI.5.b"6; 1+ "(11 IIU.

R.