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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.
Page intentionally blank
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
Page intentionally blank
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
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APPENDIX 'B
Computation Routine
1. Generalized Flow Diagram
2. FoRAST Listing
41
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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
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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
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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
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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.
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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
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, 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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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
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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