The fundamental problem solving of interior ballistics using the physical law of powder burning

8 Pages·0.079 MB·English

by  Titica Vasile, Doru Safta, Cristian Barbu.

AARMS MECHANICS
Vol.3,No.3(2004)407–414
The fundamental problem solving of interior ballistics
using the physical law of powder burning
TITICA VASILE,DORU SAFTA,CRISTIAN BARBU
Military Technical Academy, Bucharest, Romania
In this paper a mathematical model of firing phenomenon is presented, using the
physical law of powder burning. On the base of the mathematical model an interior
ballistic software was elaborated, which allows to study the variation of the powder gas
pressure and the projectile velocity versus its displacement inside of grooved barrel and
versus time. For an extant weapon, the theoretical results, obtained with the aid of this
software and the experimental data are compared.
Introduction
The fundamental problem of interior ballistics is solved, in many cases, accepting a
series of simplified hypotheses, due to complexity of the firing phenomenon. Thus, in
many works,1–13 among others, the geometrical law for powder burning, proposed by
French researcher Vieille as well as a constant index of burning rate law are assumed.
In the proposed mathematical model, for the powder burning was utilized the
physical law which reflects better the reality of firing phenomenon, using the
experimental results obtained during the powder burning in the manometric bomb at
densities much small than the densities from guns. The index from the burning rate law is
variable and is determined as a function of the pressure using the results from the
manometric bomb, too.
The elaborated mathematical model is consisting of differential and algebraic
equations. By solving the differential and algebraic equations numerically with the aid
of the computer, one can obtain the variation of gas pressure and projectile velocity
versus its displacement within the barrel and versus time since the beginning of powder
burning, which defines the content of fundamental problem of interior ballistics.
Mathematical model
The elaborated mathematical model is based on the main differential and algebraic
equations of interior ballistics,3,6,8,10,12–16 thus:
– the fundamental equation of interior ballistics:
Received: April 19, 2004
Address for correspondence:
TITICA VASILE
Military Technical Academy
81–83 George Cosbuc Avenue, Bucharest, Romania
E-mail: [email protected]
T.VASILE: Interior ballistics
( )
splψ+l = fω(cid:4)(cid:6)ψ+ v2 (cid:1)(cid:3), (1)
(cid:5) vl2im (cid:2)
where: p– the average pressure of powder gas in gun; v– the velocity of projectile; l –
the displacement of projectile inside of barrel; ψ– the fraction of burned powder;v –
lim
the limit velocity of projectile; ω– the mass of powder charge; f– the force of powder;
s – the cross section of barrel; l(cid:1) – the fictive length of free volume of charge
chamber;
– the equation of projectile translation motion:
ϕqdv =sp (2)
dt
or
ϕqvdv =sp, (2’)
dl
where: ϕ – the coefficient of projectile fictitious mass; t – the time since the beginning
of powder burning in gun;
– the burning rate law:
de = APν (3)
dτ
or
de = Apν, (3’)
dt
where: 2e– the powder burned web; τ– the time since the beginning of powder burning
in the manometric bomb; P – the average pressure of powder gas in the manometric
bomb; A– the coefficient of burning rate law; ν– the index of burning rate law;
– the rate of gas forming:
dψ= χσAPν (4)
dτ e
1
or
dψ= χσApν, (5)
dt e
1
where: χ– the shape coefficient of powder grain; 2e – the powder web.
1
These equations have been adequately transformed, in order to allow the utilizing of
experimental data, obtained during the powder burning in manometric bomb, at the
solving of fundamental problem of interior ballistics.
408 AARMS 3(3)(2004)
T.VASILE: Interior ballistics
The differential equation of displacement is obtained with the aid of the fundamental
equation of interior ballistics and of the equation of projectile translation motion, thus:
dl = ϕq vdv , (6)
lψ+l fωψ+ v2
vl2im
( ) ( ) ( )
where: lψ=l0(a−bψ); a=1−δ∆; b=∆α−δ1 ;ψ= ∆1b − δ1 / Pf +α− δ1 ; δ – the
density of powder; α – the covolume; ∆ – the charge density in gun; ∆ – the charge
b
density in manometric bomb.
Assuming that in the manometric bomb and in the gun the same thickness of powder
grain has been burned3,10,13 and taking into account the equation of projectile
translation motion, it is obtained:
ν−1
dv = s (cid:4)(cid:6)P(cid:1)(cid:3) . (7)
dI ϕq(cid:5) p(cid:2)
The differential equation for time has got the following form:
dt = dl . (8)
v
It is adopted as an independent variable ζ, defined as ratio between impulse value of
powder gas pressure I and impulse value of powder gas pressure at the fragmentation
moment I , in the case of multiperforated powders. In the case of simple shape powders,
s
impulse I is changed with impulse I , i.e. the impulse at the end of powder burning.
s k
In order to increase level of generality of these equations following nondimensional
variables are utilized:
η= v ; λ= l ; ε= t ,
vlim l0 tlim
where: vlim = 2θϕfωq ; tlim =ϕsq vlPim ; θ=γ−1; l0 – the fictive length of charge
chamber volume; γ – the ratio of specific heats of powder gas at constant pressure and
at constant volume.
Utilizing the nondimensional variables, the equation (6) becomes
dλ=2Eηdη, (9)
dζ θ dζ
AARMS 3(3)(2004) 409
T.VASILE: Interior ballistics
where E = a−bψ+λ.
ψ−η2
Taking into account that dI =I dζ and v=ηv , the equation (7) gets the form
s lim
dη= Bθ(cid:4)(cid:6)P(cid:1)(cid:3)ν−1, (10)
dζ 2 (cid:5) p(cid:2)
where B= s2Ik2 .
fωϕq
The differential equation for time (8), after the utilizing of the nondimensional
variables and the changing of dλ/dζ with its expression from equation (9), becomes
( )
dε = 2 l0 E Bθ P ν−1. (11)
dζ t v θ 2 p
lim lim
After introducing the nondimensional variables in the equation of projectile
translation motion, this gets the form
p= f∆ . (12)
E
By eliminating the pressure from equations (9) for displacement, from (10) for
velocity, from (11) for time, finally, it is obtained:
ν−1
dλ= 2BηE(cid:4)(cid:6)EP(cid:1)(cid:3) (13a)
dζ θ (cid:5) f∆(cid:2)
dη= Bθ(cid:4)(cid:6)EP(cid:1)(cid:3)ν−1 (13b)
dζ 2 (cid:5) f∆(cid:2)
ν
dε= 1 Is (cid:4)(cid:6)EP(cid:1)(cid:3) (13c)
dζ t P (cid:5) f∆(cid:2)
lim
p= f∆ . (13d)
E
For the index v from burning rate law a variable value was taken that is calculated
on the base of experimental data from manometric bomb, only for two densities of
charge in a first approximation, with the aid of relation10,13,16
logdI2 logdτ1
ν=1− dI1 = dτ2 , (14)
logP2 logP2
P1 P1
∆ ∆
where the magnitudes with index 1 and 2 correspond to densities ,respectively ,
1 2
from manometric bomb.
410 AARMS 3(3)(2004)
T.VASILE: Interior ballistics
The system of differential and algebraic equations (13) can be used in the first
period as well as in the second period of the firing phenomenon.
The differential and algebraic equations of projectile motion within grooved barrel
(13) represents a new mathematical model which allows to study the variation of gas
pressure and projectile velocity versus its displacement and versus time in conditions
closer to the reality of the firing phenomenon.
Integration of the system of differential and algebraic equations
The system of differential and algebraic equations (13) was integrated using the Runge-
Kutta IV method, with the aid of following initial conditions:
ζ=ζ ≠0; η=η =0; λ=λ =0;ε=ε ≠0,
0 0 0 0
τ
(cid:7) 0Pdτ
in which: ζ = I0 = 0 ; ε = t0 = 2.303 Ik(1−α∆)log(p0); p – the gas pressure
0 Is (cid:7)τsPdτ 0 tlim tlim f∆ pa 0
0
at the motion beginning of projectile; τ – the time when in manometric bomb burned
0
the fraction of powder which corresponds projectile motion beginning; τ – the time
s
when the grains of powder was broken.
The first period, in the case of multiperforated powder grains has two phases. In the
first phase, from the moment of the inflammation of powder until the moment of
breaking of powder grains, when ψ≤ψ and v≤1, it is integrated the equations system
s
(13). In the second phase, from the moment of powder grains breaking until the moment
of the end of powder charge burning, when ψ ≤ψ≤1 and v=1, the system (13) becomes:
s
dλ= 2BηE (15a)
dζ θ
dη= Bθ (15b)
dζ 2
dε= 1 IsE (15c)
dζ t f∆
lim
p= f∆ . (15d)
E
In the equations of this system the pressure, P is not included. However, the solution
of this system is done with the aid of the curve P(τ), because the values for ψ from E
relation are calculated on the base of experimental data.
AARMS 3(3)(2004) 411
T.VASILE: Interior ballistics
In the second period, from the moment of the end of powder charge burning until
the moment when projectile leaves the barrel, ψ=1 and v=1, the expression for E
becomes
E= a−b+λ.
1−η2
The initial conditions for the integration of the system of differential and algebraic
equations (15) are just values of ballistic magnitudes at the moment of end of powder
charge burning.
If the solution is effectuated for the powder that burns without breaking and in the
conditions of the burning rate law u=u P, in which v=1, than it is utilized the system
1
(15), taking into consideration the particularities of the second phase of the powders that
burn with breaking of the grains.
Results and conclusions
In Figures 1 and 2 the variation of gas pressure and projectile velocity is presented
versus its displacement and time, obtained with the aid of the interior ballistics software
that was elaborated on the basic of proposed mathematical model, for an extant gun.
Velocity
Pressure
Figure 1. The variation of pressure and velocity versus time
412 AARMS 3(3)(2004)
T.VASILE: Interior ballistics
Velocity
Pressure
Displacement
Figure 2. The variation of pressure and velocity versus displacement
The magnitudes from both graphics were obtained as ratio between current values of
magnitudes and their maximum values.
From the analysis of the data, one of them are presented in the diagrams from
Figures 1 and 2, can be concluded that this way of the fundamental problem solving of
interior ballistics it is closer of firing phenomenon intimacy with guns that use
multiperforated powders.
It is necessary to remark that the finish of the powder burning takes place after the
projectile leaves the barrel of gun.
For an extant gun, the experimental and theoretical data are presented in Table 1.
Table 1
Way of determinations Experimental data Theoretical data
Magnitudes
Maximum pressure 2400 2566.1
[daN/cm2]
Muzzle velocity 840 847.1
[m/s]
From the comparison of the experimental data with the theoretical values obtained
in the case of the proposed mathematical model, the difference between the muzzle
velocity values is approximately 0.5% and between the maximum pressure values is
approximately 6%.
AARMS 3(3)(2004) 413
T.VASILE: Interior ballistics
The advantages of this manner of the fundamental problem solving as compared to
the manner based on geometric law of the powder burning are as follows:
– it utilizes the whole curve of the pressure P(τ) that is obtained at the combustion
of the powder in manometric bomb and it is taken into account all particularities
of powder burning;
– it uses variable values for the index v from the burning rate law, in the case of
the powders that burn with the breaking of the grains, which allow to present all
aspects of the powder burning.
References
1.CHARBONNIER, P., Balistique interieure, Octave Doin Editeur, Paris, 1908.
2.CORNER, J., Theory of Interior Ballistics of Guns, New York – London, 1950.
3.CIURBANOV, E. V., Vnutrenniaia Ballistika, Izdatelstvo VAOLKA, Leningrad, 1975.
4.CRANZ, C., Interior Ballistics, IDT Publishing House, Bucharest, 1954.
5.JACHINO, C., Teoria dei canoni. Balistica interna ed esterna, Ulrica Hoelpi Milano, 1935.
6.KADANKA, V., Vnitrni balistika hlavnovych zbrani, Nase Vojsko, Praha 1985.
7.KOVÁTS, Z., Népszer(cid:1) ballisztika, Zrínyi Katonai Kiadó, Budapest, 1966.
8.CVETKOVI(cid:2), M., Unustrasnja balistika, Vol. I, II, TSC, Zagreb, 1981.
9.SUGOT, G., Balistique interieure, Gauthier-Villare et Co. Editeurs, Paris, 1928.
10.SEREBRIAKOV, M. E., Internal Ballistics of Guns and Solid Propellant Rocket Engines, Vol. 1 and 2,
Military Academy Publishing House, Bucharest, 1970.
11.WOLFF, V., Innre Ballistik, Deutscher Militar Verlag, Berlin, 1961.
12.VENTZEL, D. A., Interior Ballistics, IDT Publishing House, Bucharest, 1973.
13.VASILE, T., Internal Ballistics of Guns, Vol. I, II, Military Academy Publishing House, Bucharest, 1993, 1996.
14.VASILE, T., Contributions to the fundamental problem solving of interior ballistics, Proceedings of the
2nd International Conference “Artillery Barrel Systems, Ammunition, Means of Artillery
Reconnaissance and Fire Control”, October 27–29, 1998, Kiev, pp. 440–445.
15.VASILE, T., NISTOR, I., Mathematical modeling of projectile motion in grooved barrel, The XXIIIth
National Conference of Solides Mechanics, University Bulettin, “Petrol-Gaze” Ploiesti, Vol. LI, No. 1,
May, 28–29, 1999, Ploiesti, pp. 61–66.
16.VASILE, T., Studies and researches regarding the fundamental problem solving of interior ballistics for
weapon with grooved barrel, Proceedings of 3rd International Armament Conference on Scientific
Aspect of Armament Technology, October, 11–13, 2000, Waplewo, pp. 157–165.
414 AARMS 3(3)(2004)