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COMPUTER SIMULATION OF THE 7.62mm TT PISTOL EXTERNAL BALLISTICS USING TWO DIFFERENT AIR RESISTANCE LAWS
Vadim L. Khaikov
independent researcher, Krasnodar, Russian Federation, e-mail: [email protected],
ORCIDiD: ©http://orcid.org/0000-0003-1433-3562
DOI: 10.5937/vojtehg66-16534; https://doi.org/10.5937/vojtehg66-16534
FIELD: Mechanics - Ballistics ARTICLE TYPE: Original Scientific Paper ARTICLE LANGUAGE: English
Abstract:
A description of a pistol (rifle) cartridge often involves two ballistic coefficients that characterize its ballistic qualities with respect to various air resistance laws (ARLs). How close are the obtained ballistic trajectories with varied ARL specifications and what are the differences between them? How to evaluate ballistics if the ARLs are to be expressed in various mathematical forms? In this paper, the evaluation of external ballistics trajectories is given for two ARLs (the law brought in 1943 and the Siacci law). All the obtained results relate to the TT pistol with 7.62 x 25mm Tokarev cartridge. The paper also presents the answer to the question: how to calculate the ballistic trajectory if the ARL is expressed as a rational function, piecewise function or spline. For the 1943 ARL, a graphical interpretation of the function Cd (i, v) in the form of a surface is shown. This paper shows that, due to the selection of ballistic coefficients, it is possible to obtain sufficiently similar form of ballistic trajectories. A method of graphical comparison of external ballistic parameters is presented as well as the mathematical tools for quantitative analysis of a shape of ballistic curves.The difference between the two trajectories is proposed to be estimated using a relative error in regard to а selected ballistic parameter. Computer simulation considered for the 1943 and Siacci ARLs for the 7.62*25mm Tokarev cartridge indicates that the profiles of the function of instantaneous projectile velocity vs time of flight (TOF) had the greatest non-coincidence in relation to other ballistic parameters (e.g. horizontal range, height of the trajectory, etc.) The obtained maximum of the relative error was 0.8%. Its magnitude localizes at the point of impact.
Key words: computer simulation, external ballistics, TT pistol, air resistance law, drag function, the 1943 year law, bullet trajectory, spline, Mathcad.
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
There are no dangerous weapons; there are only dangerous men.
Robert A. Heinlein
Introduction
For one projectil type (bullet) that has equal initial conditions (x0, yo, в0, v0), but its motion is characterized by two different ARLs, it is possible to calculate so-called «twins-trajectories». These are two trajectories with a practically identical form, but due to differences in ARLs descriptions, they have different values of the ballistic coefficients C. Errors related to inequality of such trajectories are usually not reported and it’s completely unclear which of the external ballistic parameters x, y, в, v for each of «twins-trajectories» has the greatest inconsistency.
In Europe and in the countries of North and South America, external ballistics of small arms projectiles is generally based on the use of well-known G1/G7 air drag models; however, ARLs like the 1943 year law and the Siacci law are often used in the Commonwealth of Independent States or in countries - former members of the Warsaw Pact (or in countries that had in the past a military-technical cooperation with that defense treaty). One of the objectives of this article is to show how to carry out a ballistic simulation by using the 1943 and Siacci ARLs with various forms of their mathematical expressions. The second task is to present equality or inequality of the ballistic curves obtained as a result of the estimation process. For reducing the computer simulation (calculation) time, we will use the Mathcad 15 computer algebra system.
From the point of view of external ballistics, it is interesting to estimate the ballistics of one of well-known pistols, for instance, of the 7.62mm Tokarev-TT1 pistol using two previously mentioned ARLs. It is known that pistols based on the TT construction were produced in many countries and the 7.62^25 cartridge is widespread.
In the scientific article (Bogdanovich, 2012, p.42) one can find «...one of the best pistols based on the 7.62mm TT design was certainly the M57. This gun was constructed in Yugoslavia, at the «Zastava» plant and produced by Serbian «Zastava Arms» for export to various countries, including Europe and America». The arms plant «Crvena Zastava» (Kragujevac) began to produce the pistol-predecessor of the M57, namely the M54, in 1954 and at the same time the ammunition factory «Prvi Partizan» (Uzice) launched a serial production of the
1 7.62mm Tokarev self-loading pistol model TT 1930 (TT-30) / TT 1933 (TT-33). The abbreviation «TT» means Tula-Tokarev.
7.62x25mm Tokarev cartridges. In addition, it should be said that the TT pistol and its upgrades were manufactured in PRC (Type 51 & Type 54), in Hungary (M48, Tokagypt 58 with cartridge 9x19mm Para), in Romania (TTC), in DPR Korea (Type 68) and in other countries.
Ballistic and technical data. Table 1 shows the necessary technical specifications of TT-33, M54, and M57 pistols, important for evaluating their ballistics.
Table 1 - Technical specifications of TT-33, M54, and M57 pistols Таблица 1 - Технические характеристики пистолетов ТТ-33, М54, M57 Табела 1 - Техничке карактеристике пишто^а ТТ-33, М54 и М57
№/№ Specifications Units Model of pistols
TT-33, M54 M57
1 Chambering mm 7.62x25 TT 7.62x25 TT
2 Fire modes - Semi-Auto, Single Action Semi-Auto, Single Action
3 Bullet weight g 5.49-5.52 5.49-5.52
4 Bullet length mm 142 -
5 Bore length mm 116 116
6 Rifling length mm 100 -
7 Number of grooves - 4 RH 4 RH
8 Number of lands - 4 4
9 Twist rate mm per turn 240 240
inch per turn 9.45 9.45
clb3 per turn 31.496 31.496
° . .. 541'45'' 5°41'45''
10 Initial velocity mps 420 440
11 Bullet muzzle energy Joule 485.54 532.88
12 Bullet spin rate rps 1750 1750-1896
13 Effective firing range m 50 50
14 Bullet flight range m 800-1000 1640
15 Sight radius mm 156 158
16 Max mean pressure kg/cm2 2234 -
17 Practical rate of fire rpm 30 -
18 Precision (range: 50 m) m 0.25 -
Sources: (Bogdanovich, 2012, p.49) and author's estimations
2
for ordinary bullet Р-type (cyrillic: пуля «П» - простая)
clb - bullet caliber
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
Based on (http://popgun.ru,nd), the curves of pressure and bullet velocity vs rifling length (time) for the 7.62mm bullet of the TT-33 pistol in a logarithmic scale were built (Figure 1). The main advantage of the logarithmic scale is that it allows «to stretch the graph» in the direction to the origin (to the point «0 mm» by using the argument bore length and to the point «0 seconds» by using the argument time). The fragment a of Figure 1 shows the change of the projectile velocity in the barrel; the fragment b indicates the internal ballistic curve of the mean pressure in the barrel.
Figure 1 - Internal ballistic curves of the TT pistol (argument - bore length)
Рис. 1 - Внутрибаллистические кривые пистолета ТТ (аргумент - длина
ствола пистолета)
Слика 1 - Унутрашше балистичке криве пишто^а ТТ (аргумент - дужина цеви)
Figure 2 - Internal ballistic curves of the TT pistol (argument time): a - the projectile velocity in the barrel; b - internal ballistic curve of the mean pressure in the barrel. Рис. 2 - Внутрибаллистические кривые пистолета ТТ (аргумент - время): а - скорость снаряда в стволе; b - внутрибаллистическая кривая среднего
давления в стволе.
Слика 2 - Унутрашше балистичке криве питона ТТ (аргумент - време): а) брзина про]ектила унутар цеви, б) унутрашъа балистичка крива средне вредности притиска унутар цеви.
The dependences of the bullet velocity and the mean pressure as a function of time are also obtained (Figure 2). The graphs show that the bullet initial velocity is 420 mps, and the mean pressure maximum is 2234 kg/cm2. The duration of the intraballistic cycle for the TT-33 pistol is approximately 2.5 milliseconds.
The mathematical model. Longitudinal motion of a pistol bullet in the Earth's atmosphere can be described by the system of ODEs with an independent argument TOF (t) (Regodic, 2006). This type of mathematical expression belongs to the Point-mass Trajectory Model type:
dv
dt
g sin(0)--^AC
2m
dd _ g cos(d)
dt v
dx
— _ v cos dt
(1)
— _ v sin(d) dt
where v - the instantaneous bullet velocity, m/s; t - the time of flight, s; g - the acceleration of gravity at the point of departure, m/s2; в - the angle of the velocity vector relative to the base of a trajectory, radian; p - the air density, kg/m3; m - the mass of projectile, kg; A - the cross section of the projectile, m2; Cd - a drag function, dimensionless; x - the abscissa (horizontal range) of the trajectory, m; y - the ordinate of the trajectory, m.
The density of the airp as a function of the projectile flying altitude y:
p=porH(y),
where p0 is the density of the air at the ground-level; H(y) is the function which indicates a relative variation of the air density with respect to the altitude y.
Using the standard ARLs Cdst and the coefficient i, it is possible to transform the first ODE of system (1):
dv
dt
-g sin(d)
Pv
2m
AiC
dst
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
where i - coefficient taking into account a shape of (launched) bullet (so-called form coefficient4), dimensionless; Cdst - the standard air drag function, dimensionless.
Coefficients i and ARL models. According to collected sources, the i coefficients for the 7.62mm pistol bullet and 9mm bullets are shown in Table 2.
Table 2 - Values of the coefficient i for pistol bullets Таблица 2 - Значение коэффициента i для пистолетных пуль Табела 2 - Вредности коефицц'ента i за зрна пишто^а
Type of cartridge and bullet Initial velocity v0, mps Bullet weight, g ARL Source
1943 Siacci
7.62*25mm Tokarev 420 5.505 1.35 0.75 (Kirillov, 1963, p.68)
9x18mm Makarov5 315 6.1 - 0.98 (Vodorezov, 2017, p.166)
9x19mm Luger (FMJ) 376 7.4 1.526 0.77 (https:// forum.guns.ru, nd)
9x19mm Luger (HP) 308 9.4 1.509 0.755 (https:// forum.guns.ru, nd)
A comparison of the values of the coefficients i for 7.62*25mm Tokarev indicates that its value for the ARL of Siacci is 1.8 times lower than i for the ARL of the 1943 year.
The 1943 law drag model is often used to describe ballistics of pistol bullets, for example, for 9mm Para cartridge (Jankovych, 2012, p.29) or for 9mm Luger (https://guns.ru, nd).
The coefficient i can be calculated by the following formula (Faraponov et al, 2017, p.35)
m
i =-------г C
1000d2
4 In the book of Semikolenov, Bondarenko & Krasner «Principles of small unit weapons firing» (Semikolenov et al, 1971) on the p.67 i is named as "the coefficient for the projectile shape".
5 The files for software Exterior Ballistics 2.5 (http://ballistics.eu/index.html, nd) contain data for the cartridge 9x18 Makarov with FMJ bullet (6.22 g.) and F0=346 mps: BC=0.1; drag model G1.
where i - the form coefficient, dimensionless; m - the mass of the projectile/bullet, kg; d - the caliber of the projectile/bullet, m; C - the ballistic coefficient of the projectile, m2/kg.
«Although the coefficient i is usually regarded as a constant value, as it can be seen from the expression
i = Cd
Cd
it, strictly speaking, depends on the instantaneous projectile velocity.
Therefore, using a projectile (bullet) of the same shape in different ranges of velocity, we can get some discrepancies in the numerical values of the coefficient i. For the same reason, the value of the coefficient i for the same projectile and for the same initial velocity depends on the angle of departure (AOD). This is explained by the fact that changing of AOD gives a change in the velocity range along the trajectory» (Shapiro, 1946, p.58).
For example, the relationship between the 1943 year ARL and the Siacci ARL in the range of up to 5 M is shown in Figure 3 (Khaikov, 2017, p.83). The coefficient i as a function of the Mach number (M) is complicated, i.e. it is not monotonous. The graph of the i(M) function has two local minimums and one local maximum (see the right graphics window). However, i(M) can be characterized by some average value, which is equal to half of the area under the i(M) function graph.
St
Рис. 3 - Отношение двух законов сопротивления воздуха Слика 3 - Однос измену два закона отпора ваздуха
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
If i(M) and the coefficient i are considered as a constant, then the function Cd(i,v) can be represented as a surface (Figure 4). For i=1, we
have a standard function Cd [ — I, which is a section of the surface
* ^ a )
(orange line). For if-1, the individual function Cd, such as a purple line (i>1).
Using the value of the coefficient i, the caliber of the bullet, its mass, it is possible to calculate the ballistic coefficient C (Germershausen, 1982, p.159)
C = 1000 • i • d2 • q 1
In this case, we obtain the function Cd(C, v).
However, there is an alternative formula for calculating the ballistic
coefficient (BC) C = m • (d2 • i)1. In order to avoid misunderstanding, it is necessary to indicate a type of a calculation formula for ballistic coefficient determination.
o striking velocity (velocity of collision) о initial bullet velocity
Figure 4 - The function Cd(i,v) in the form of a surface Рис. 4 - Функция Cd(i,v) в форме поверхности Слика 4 - Функцц'а Cd(i,v) у облику површине
So, if i(v) = const, then the surface (Figure 4) shows all possible individual drag-functions which depend on i. When the coefficient i is
multiplied by Cdt j^—j , a linear transformation of the function takes place:
for i >1, the graph is stretched from the abscissa axis i times; for 0< i <1 this is the compression of the graph to the x-axis by 1/i times.
Therefore, the standard function Cd I — | is the boundary between
a
the «compression» and «stretching» zones of the Cd(i,v) surface. Since any ballistic trajectory has the initial and striking velocity of the projectile, due to the surfaces Cd(i,v) or Cd(C,v) we can show the range of the coefficient Cd that is necessary for flight path calculations.
Different forms of ARL expressions. ARLs or function can be described as: a classical analytical function; a piecewise function and a spline function. The spline function can be regarded as a special kind of the piecewise function.
Analytical forms for the ARL of the 1943 year. In view of the fact that the summit of the bullet trajectory in air for pistol external ballistics is not a large value, the speed of sound can be considered as a constant value. That is in the formula
M=v/a,
where v - instantaneous bullet velocity; а - local speed of sound (constant).
The ARL of the 1943 year is a table-valued function that can be found in (Konovalov & Nikolayev, 1979, p. 191) and approximated using a rational function:
C
d 43 R
v
a
Г
The conducted investigation of possible approximation forms for the ARL of the 1943 year led to the following rational function (Khaikov, 2017, p.85) (0.1 < v/a < 4):
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
C
d 43 R
1.378212 - 7.13796051 -1 +15.4986811 -
v I l a 1 l a
a 1 8.7777403 - 45.498974f - I + 99.290858f -
l a 1 l a j
- 17.77837бГ+ 10.605229(- 1.7807148Г
a
a
a
( ' N \3 ( ( v V '
-17.7783761 + 74.911081 vI4 v I - 21.3318141 v I
l a l a j l aj
/ v I6 ( v Y ( •v Л8
-1.6876336 I +1.1643621 v I - 0.28739041 v I
l va. 1 l aj l aj
( v i 6 ( v Л7 ( v V
- 3.02221381 | + 4.0786158 (v I - -1.0962723 v I
l a, l aj 1 l aj
+ 0.0259858441 -
l a j
+ 0.1012291
a
2
9
9
v
(2)
Another form of mathematical expression for the ARL of the 1943 year is a sum of rational and exponential functions (R&EF) (Kozlitin & Omelyanov, 2016, p.29):
C
d 43 R
a
p\-
a
+-
^0
Q a j 1+exp
r
^] + *2 l la j j
T + d0
v
(3)
It should be noted that the rational function (i.e. first summand) uses only even powers (from 0 to 12, namely 0, 2, 4,..10, 12). R&EF is expressed by the following formula (0.1 < v / a < 4.0) with eighteen empirical coefficients
C
d 43 R
- 1.9382 + 4.298of+ 0.320?f
a
a
. 2
a J 296.9213 - 853.9492[-] + 985.5873| -
l a J l a j
- 9.461o|+ 8.9342f- 0.947б|
a
a
a
- 580.864з!+ 178.669o|- 15.407l|
aJ
\ 12
a
a
(4)
-
4
+ 0.0525| -a
-
- + ■
f
+ 1.0000|-| 1 + exp
aJ
V
12
0.0531
90.5063| -
Va
---------^ + 0.1639
+ 85.5194
J
The research carried out in (Khaikov, 2017, p.88) showed that the matrixes P, Q, B and D in formula (3) may have different coefficients. For example, the matrices Pi, Qi, B1 and D1 with alternative coefficients are presented below:
Pi =
40.189924313 ^ '- 0.9050248435''
- 32.2497749054 2.8653174742
42.0499139169 - 3.7411757325
- 28.8388279297 Qi = 2.5742663163
9.989953385 - 0.8961295841
- 0.6976279168 0.0627439736
V0.0403773785 J v- 0.0036275618y
'0.06274397' \
Bi = 16.399062 Di = (11.416713 )
V57.358636 J
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
The ARL of the 1943 year can be expressed as a piecewise function (5) (Khaikov, 2017, p.80) consisting of 9 unequal intervals. This function is a modification of the formula from (Konovalov& Nikolayev, 1979, p.84) in which one more interval is added
C
d 43 .
a
0.157
0.0331 -J + 0.133
1.5| - I-1.176 a
a
0.384 sin
a
( (v^
1.851 -
V
У
0.29|-I + 0.172 a
- 0.011| - I + 0.301
Va
0.259
0.1 <V aa j / 4 < 0.73
0.73 < [ . a, < 0.82
0.82 < ( vN V a. < 0.91
0.91 < ( v V a, j< 1.00
1.00 <| ( vл V a, J< 1.18
1.18 <| ( vл V a, J< 1.62
1.62 <[ : aa: / 4 |< 3.06
3.06 < ( v V a, j< 3.53 4
3.53 < : a |< 4.0
(5)
2
v 1 ______.1 v
a
a
2
vv
v
a
-1
The «PWF» subscript denotes a piecewisefunction.
Analytical forms for ARL of Siacci. The F-curve for the Siacci law is written (Mori, 2013, p.41)
F (v ) = 0.2002v
48.05^ (0.1648v - 47.95)2 + 9.6 + Q-Q442v(v 3Qq)
v
Due to division by 4.74 • 10 4v2 we can transform the F-curve into the Cd - type function (Shapiro, 1946, p.37)
C (v ) = Fs (”)
d^V) лпл 1r.-4 2
4.74 • 10~4 v
(7)
The ARL of Siacci as a table-valued function can be found in (Konovalov & Nikolayev, 1979, p.191).
The technology of using spline functions is demonstrated in the appendix to this articleas well as in (Khaikov, 2018).
Different forms of the mathematical notations for ARLs of the 1943 year and Siacci law are combined in Table 3.
Table 3 - Forms of the mathematical notations for the ARLs of the 1943 year and Siacci
law
Таблица 3 - Формы математических обозначений законов сопротивления воздуха 1943 года и Сиаччи
Табела 3 - Форме математичких поjмова за законе отпора ваздуха из 1943. године и Siacci-jевог закона
ARL Analytic functions Table-valued function
Classical analytic form Piecewise form
1943 Formulas 2 - 4 Formula 5 * 6
Siacci Formulas 6, 7 - *
Mathcad programming code. The commented Mathcad-code is presented below. The characteristics for a pistol bullet are determined: caliber (0.00762m = 7.62mm), weight (0.0055kg) and a value of the i coefficient (i_43) (according to the the chosen law):
d:=0.00762 q:=0.0055 /_43: = 1.35
6 *
- source corresponds (Konovalov et al, 1979, p.191).
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
An angle of departure (in radians) is calculated as a set of angular degrees, minutes and seconds:
Gradus:=0 Min: = 10 Sec:=0
в
ж
180
„ , Min Sec
Gradus H----1-----
60 3600
2.909 • 10 3
At the point of departure, the value of the acceleration coefficient of gravity is determined as 9.18 m/s2. Further, it is necessary to determine the time interval of integration: its boundaries and the total number of integration points:
tbeg- 0 tend- 1.1 npoints- 1000
The initial conditions (for formula (1)) are determined as a matrix-column y, which will contain their known numerical values:
f У0 = v(0fi .= y = e(°)
У ' У2 = X(0)
v Уз = У (0)y
In view of the fact that the initial velocity of the 7.62mm TT bullet is 420 m/s, the matrix-column y will look like:
У •
' 420 ^
2.051 x 10~3 0
v 0 ,
The matrix D(t, y) has the form
D{t, y):
g • sin (yx )-Py°L. a . i _43. Cd (y0 2 • m ^ a
- g •(cos(y1 ))2
Уо
Уо•cos(yj)
Уо • sin(yj)
(8)
The matrix-column D(t,y) (8) is the right-hand part of the system of ODEs (1). It includes the following variables: the instantaneous projectile velocity v -y0, the angle of inclination of the tangent в —yh the abscissa of the trajectory x -y2; the ordinate of the trajectory y - y3. In connection with the fact that Cd in formulas (2), (3) or other presenta long and cumbersome expression, it is given in (8) only as a «short» notation.
If a calculation in the Mathcad system is implemented, then Cd must be replaced by the complete mathematical expression. In this formula, the sign of v (velocity) is replaced by y0. i _43 - the form coefficient of the 1943 year law in the Mathcad program.
For example, the matrix-column D(t, y) will have the form (for Cd only
y0
three initial terms are given; the powers of — are from 0 to 2). A Mathcad
a
script for D(t,y )is given below:
D( y ): =
- g • sin(y1)- P'(>0) • A • i 43-
1.378212 - 7.1379605
y о
+...
2 • m
8.7777403 - 45.4989741 ^ 1 + 99.2908581 ^ I +.
- g • (c°s(y1))2
y0
y0 • cos(y1) y 0 • sin(y1)
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
The complete Mathcad script of
Cd *■' (a)and
Cd43 I — I expressions is given in formulas (2) and (4). In order to use
a
the piecewise function (5) for the calculation in Mathcad, we transform it to the form:
D{t, У ):=
■ ( ) P'iy 0)2 -g • sinCyi)—2—
2 • m
• A • i 43
0.157г/| 0.1 < ^ < 0.73
0.033У^ + 0.133If1 0.73 < y^ < 0.082 a ) V a
3.91 I - 6.4194^ + 2.8025831
a ) a
1.5y^ - 1.176 L/f 0.91 < y^ < 1.0
- 1.6|^°| + 3.7632 y^ - 1.8287616 a
if| 0.082 < ^ < 0.91
if | 1.0 < y^- < 1.18
( ( M
1.85
0.384 sin
y0 V a ))
iff 1.18 < ^ < 1.62
\
iff 1.62 < y^ < 3.06
0.29— + 0.172
У0
- 0.011У^ + 0.301 If(3.06 < У^ < 3.53 a ) V a
a
a
2
a
a
0.259f| 3.53 < y^ < 5.0
- g •(cos(y1 ))2
У0
s(y1)
У 0 • sin(y1)
a
Below we give an example of the matrix D(t,y) for the description of the ARL and the system of ODEs (1) by using the cubic spline. The import of the table-valued function of ARL is carried out from an external file. Its can be the text «.txt» file or the Microsoft Excel «.xls» file7.
d( У):
g • sin(yi)
P (y°) • A • i _ 43 • int erp(cspline(vel, Cd), vel, Cd, y0) 2 • m
- g • (c°s(yi))2
y0
y°•cos(yi)
У° • sin(yi)
Alternatively, the data for the ARL may not be imported from the file, but be part of the D(t, y) matrix. In this case, data are written in the form of row-matrices (separated for velocities and separately for Cd data). Next, as in the previous example, we use cubic spline interpolation.
vel 0 200 400 • •!
- g • sinl
(yi)-
p
•(У0 )2
2 • m
• A • i 43 •
Cd, ^|7.i05e -15 4.92i 5i.533
T
D(t, У)
in t erp(cspline(vel, Cd ), vel, Cd, y 0) - g -(cos(yi ))2
y 0
y0 •cos(yi) y0 • sin(yi)
Four black circles in two lines (vel and Cd) symbolize the remaining elements of the matrix-lines vel (tabulated instantaneous velocity) and Cd (drag coefficient). The sign of T denotes the matrix transposition.
The following Mathcad command-line is showed using the solver-function rkfixed for the numerical solution of (1) (Kir'yanov, 2012, p.259):
Num Result := rkfixed(y, tbeg, tend, npoint, D) .
Variables vel, Cd should be described as global variables. They are introduced to the matrix D(t, y) using the «READFILE» Mathcad function.
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
The solver-function rkfixed is implemented in the non-stiff fourth order Runge-Kutta numerical method with a fixed step. More information about the integration of ODEs in Mathcad can be found in (Khaikov, 2018, p.298).
Calculations. Calculations were performed for the ARL of the 1943 year for 4 C-types (RF, R&EF, piecewise, spline-function) and 2 types of the Ciacci ARL (analytical and spline-function). The variable Num_Result is the matrix that contains the results of the numerical solution of (1). In this case, the matrix has the dimension of 5 x 1001 elements and contains 5005 numbers. Five columns of the Num_Result matrix are: independent argument TOF (t); and elements of the matrix У (or D) namely v, в, x, y. 1001 rows are the sum of 1 (initial condition) and n points. The first row of the matrix Num_Result includes t(0), v(0), в(0), x(0), y(0). The first column of the matrix Num_Result contains 1001 discrete TOF values: from tbeg= t(0) to tend. The 5-by-1001 matrix from the second to fifth columns (in each of them) has 1001 values of the quantities v, в, x, y respectively. This means that we have 1001 values of instantaneous velocity, 1001values of в, 1001 values of x and so on.
It was shown previously that for obtaining two trajectories with the same horizontal range but characterized by different ARLs, it is necessary to make selection of ballistic coefficients. This procedure allows obtaining sufficiently close forms of both ballistic trajectories. However, due to the fact that the bullet movement for each flight path is determined by the intrinsic ARL, then the bullet retardation process will not coincide with «twins-trajectories». For a comparison of dependencies between the elements of ODEs (1), the method developed in (Khaikov, 2018, p.281303) will be used below. The solution of system (1) is represented as a five-dimensional space. Each element of this 5D-space is a function between the variables (x, y, в, v) and the argument (t), obtained as a result of numerical solution (1). The angle в is calculated in angular minutes (or minute of angle (MOA). Since the solution of system (1) depends on the ballistic coefficient, it becomes possible to compare the same-named dependencies (xj, yj, в1, vj, tj), (x2, y2, в2, v2, t2) obtained for different
values of the coefficients cj and c2. The entire set of relations between the variables and the independent argument of (1) is presented in Table 4.The order of the values (x, y, в, v, t) location in the 5-by-1001 matrix Num_Result and in Table 4 is different.
Table 4 - Structure of the relations for ODEs parameters (1) for two ARLs Таблица 4 - Структура соотношений параметров системы дифференциальных уравнений (1) при различных законах сопротивления Табела 4 - Структура односа параметара ОДJ (1) за два закона отпора ваздуха
X1, X2 Y1, Y2 01, 02 V1, V2 T1, T2
1 2 3 4 5
X1, X2 1 X1 vs X1 X2 vs X2 X1 vs Y1 X2 vs Y2 X1 vs 01 X2 vs 02 X1 vs V1 X2 vs V2 X1 vs T1 X2 vs T2
Y1, Y2 2 Y1 vs X1 Y2 vs X2 Y1 vs Y1 Y2 vs Y2 Y1 vs 01 Y2 vs 02 Y1 vs V1 Y2 vs V2 Y1 vs T1 Y2 vs T2
01, 02 3 01 vs X1 02 vs X2 01 vs Y1 02 vs Y2 01 vs 01 02 vs 02 01 vs V1 02 vs V2 01 vs T1 02 vs T2
V1, V2 4 V1 vs X1 V2 vs X2 V1vs Y1 V2vs Y2 V1 vs 01 V2 vs 02 V1 vs V1 V2 vs V2 V1 vs T1 V2 vs T2
T1, T 2 5 T1 vs X1 T2 vs X2 T1 vs Y1 T2 vs Y2 T1 vs 01 T2 vs 02 T1 vs V1 T2 vs V2 T1 vs T1 T2 vs T2
- relationship between the variables of ODEs (1);
- relationship between variables (1) and argument (t);
- diagonal cells.
The graphs lying inside the green backgroundare the functions between the variables of ODE (1) (x, y, в, v). The graphs located inside the blue background associate the variables with the argument TOF (t). The diagonal cells-graphs placed on a light yellow background of the graphics window show the functions depending on themselves, for example, «y is a function of y» and so on. A small red square on each of 25 graphs shows the starting point. If we plot a horizontal and vertical line through the starting points of any graph (see Figure 5), they will connect the starting points of the graphs along the vertical row and the horizontal line.
Results analysis. Determining the magnitude of the relative error (MRE) for pistol ballistics using two different ARLsis an important element of assessment. To do this, we find the MRE of the horizontal range, the height of the trajectory, the angle of inclination of the velocity vector and the instantaneous velocity as functions of TOF. The relative error is expressed in percent.
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
(blue dotted line) and 1943 year (red solid line) (in accordance with Table 4)
Рис. 5 - Соотношение между баллистическими параметрами системы (1) при законах сопротивления воздуха Сиаччи (синяя пунктирная линия) и 1943 года (красная сплошная линия) (согласно табл. 4)
Слика 5 - Однос измену балистичких параметара ОДJ (1) за Siacci-jев закон отпора ваздуха (плава испрекидана линца) и закона из 1943. године (пуна црвена
линца) (према табели 4)
Evaluation of the MRE for the horizontal range is
5,() = 100-X- X{‘У
X1
()43
each of 25 windows of Figure 5 contains 2 graphics, parameters in relation to the 1943 ARL and the Siacci ARL.
characterizing the ballistic
The MRE for the height of the trajectory is
Sy (t ) = 100
|y (t )43 - y 0 )s,
У ()43
The evaluation of the MRE for the instantaneous velocity of the projectile is
8 ()= 100
|Ч4з -v(t)ss
v()43
The MRE for the angle of velocity vector relative to the base of a trajectory
8,0 = 100|e(t]2()s|.
The subscript «43» denotes that the calculations characterize the 1943 ARL and «Si» stands for Siacci.
The results of the calculations are shown in Figure 6. The MRE is found for a TOF interval of 0-1.1 s. The time of 1.1 seconds corresponds to the time of impact.
Figure 6a gives the function MRE of the horizontal range vs TOF8X(t) and the instantaneous velocity vs TOF 8v(t). The results of the
calculations and comparisons show that 8X (t) and 8V (t) are TOF-increasing functions. The maximum MRE for 8X(t) is 0.3% for TOF 1.1 seconds. A function characterizing the MRE for the instantaneous velocity 8v (t )has a similar character (Figure 6a). The maximum value of this function is 0.8% (for the same TOF point).
The result of dividing the function 8v (t) by the function 8X (t) is
shown in Figure 6b. Thus, the MRE for the instantaneous velocity is approximately 2.8-3 times larger than the MRE for the horizontal range.
In contrast to the functions mentioned above, the functions 8e(t),
8y (t) do not have an increasing character; moreover, they have discontinuities (Figures 6c, 6d). The discontinuity for the function 8e(t) corresponds to the vertex of the trajectory. At this point, the angle в is
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
zero (division by zero). The discontinuity for the function s, (t)
corresponds to the point of impact. At this point, the height of the trajectory y is zero too and we have division by zero.
Figure 6 - Results of the error analysis: Sx (t), Sv (t), Se(t), Sy (t)
Рис. 6 - Результаты анализа погрешностей: функции Sx(t), Sv(t), Se(t), Sy(t) Слика 6 - Резултати анализе грешака Sx(t), Sv (t), Se(t), Sy(t)
In contrast to the functions mentioned above, the functions Se(t) and Sy (t) do not have an increasing character as functions Sx (t) and Sv (t);
moreover, they have discontinuities. The discontinuity for the function 6(t) corresponds to the vertex of the trajectory (t = 0.084 s). At this point, the 6(t) = 0 (division by zero). The discontinuity for the function Sy, (t)
corresponds to the point of collision (t = 1.1 s). At this point, the height of the trajectory is zero (division by zero).
Summary and conclusions
The evaluations and external ballistics trajectories of the TT pistol with 7.62x25mm Tokarev cartridge are given for two ARLs (the 1943 year and Siacci). The characteristic of the internal-ballistic period for the TT is shown. The calculation feature is to use various forms of the ARL mathematical notation: the classical analytical formulas, the piecewise formula and the function-tables form.
For the ARL of the 1943 year, a graphical interpretation of Cd(i,v) function in the form of a surface and its main elements is visualized. Depending on the value of the form coefficient i, it is demonstrated how the standard drag-function Cd(v) is transformed.
A method of graphical comparison of the ballistic trajectory parameters is represented. This comparison takes place in a 5x5 matrix.
It is shown that due to the selection of the ballistic coefficients, it is possible to obtain sufficiently close «twins-trajectories». However, in connection with the fact that the movement of the bullet in each of them is determined by different ARLs, then the slowing down of the bullet on each of them will have its own independent nature and therefore will not coincide with the «twins-trajectories».
The computer simulation considered for the ARLs of the 1943 year and Siacci for the 7.62x25 Tokarev cartridge indicates that the profiles of the function of instantaneous projectile velocity vs time of flight had the greatest non-coincidence in relation to other ballistic parameters (e.g. horizontal range, height of the trajectory, etc.) The obtained maximum of the relative error was 0.8%. Its magnitude localizes at the point of impact. The simulation results showed that the MRE for the instantaneous velocity is approximately 2.8-3 times larger than the MRE for the horizontal range.
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
Appendix
Analytical form of the cubic spline function that expresses the air resistance law of the 1943 year in the Mach number range from 0.1 to 4.0
The spline function Cd 43s {M) is given on the interval divided into 39
segments (parts). A cubic spline is a function that: on each segment there is a cubic polynomial and it has continuous first and second derivatives on the whole interval 0.1-4.0 M.Table 5 serves to describe a cubic spline Cd 43>cs {M) which is defined on 39 segments. All 39 parts of
the spline function have the same length equal to 0.1 M.
' f„ M) LSB0 < M < RSB0 л
Cd 43cs {M )_ f,(M ) LSB1 < M < RSBt
4 f» M) LSB38 < M < RSB38,
where f {M) - the cubic polynomial assegments(parts) with the number i; LSBj; RSBj - the left and right segment borders; M- the Mach number as an argument of Cd 43s {M).
Each of the 39 cubic polynomials f {M) has 4 coefficients. The total number of coefficients for Cd43cs is 156. The numbers in columns four through seven (Table 5) describe a polynomial in the form:
f {m) = a3iM 3 + a2iM 2 + ahM + а0г , where a3, a2, a1t a0 - the calculated coefficients.
The spline function consisting of 39 segments and described by the coefficients from Table 5 is shown in Figure 7. All of the 39 cubic polynomials drawn in the same range of Mach numbers, i.e. in a range of 0.1- 4 M, give a large number of «branches». Figure 7 and Table 5 were obtained using software (Arndt Brunner, 2018).
C
M
Figure 7 - Graphical construction of the Air Resistance Law of the 1943 year using 39 cubic polynomials in the MN range from 0.1 to 4 M Рис. 7 - Гоафическое конструирование закона сопротивления воздуха 1943 года с помощью 39 кубических полиномов, диапазон 0.1-4 М Слика 7 - Закон отпорa ваздуха из 1943. године, графички приказан помоПу 39 кубних полинома у опсегу од 0,1 до 4 маха
Table 5 - Interval borders and coefficients of the cubic polynomials that describe
the ARL of the 1943 year
Таблица 5 - Гоаницы интервалов и коэффициенты, описывающие закон сопротивления воздуха 1943 года
Табела 5 - Гоанице и коефици^енти интервала кубних полинома щи опису^у закон
отпорa ваздуха из 1943. године
Num- ber Borders Coefficients of the cubic polynomials with the argument M
Left Right Мл3 МЛ2 МЛ1 МЛ0
1 2 3 4 5 6 7
1 0.1 0.2 -0.00779 0.002337 -0.000156 0.157
2 0.2 0.3 0.038952 -0.025708 0.005453 0.156626
3 0.3 0.4 -0.148018 0.142565 -0.045029 0.161674
4 0.4 0.5 0.553119 -0.6988 0.291517 0.116801
5 0.5 0.6 -1.064459 1.727567 -0.921666 0.318999
6 0.6 0.7 1.704716 -3.256948 2.069043 -0.279143
7 0.7 0.8 -2.754406 6.10721 -4.485868 1.250336
8 0.8 0.9 35.312909 -85.254348 68.603379 -18.24013
9 0.9 1.0 -61.497231 176.13303 -166.645262 52.334463
10 1.0 1.1 23.676014 -79.386704 88.874473 -32.838782
11 1.1 1.2 2.793175 -10.473335 13.069766 -5.043723
12 1.2 1.3 0.151287 -0.962537 1.656809 -0.478541
13 1.3 1.4 1.601679 -6.619067 9.010297 -3.665052
14 1.4 1.5 -0.558002 2.451591 -3.688623 2.261111
15 1.5 1.6 0.630327 -2.89589 4.332598 -1.7495
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
Num- Borders Coefficients of the cubic polynomials with the argument M
ber Left Right МЛ3 МЛ2 МЛ1 МЛ0
16 1.6 1.7 -0.963308 4.753562 -7.906526 4.778033
17 1.7 1.8 1.222906 -6.396132 11.047955 -5.96284
18 1.8 1.9 -0.928316 5.22047 -9.861929 6.583091
19 1.9 2.0 0.49036 -2.865984 5.502333 -3.147608
20 2.0 2.1 -0.033122 0.274906 -0.779447 1.040245
21 2.1 2.2 -0.357871 2.320827 -5.075881 4.047749
22 2.2 2.3 0.464608 -3.107534 6.866514 -4.710007
23 2.3 2.4 -0.500559 3.552115 -8.450678 7.033173
24 2.4 2.5 0.537628 -3.922831 9.489192 -7.318723
25 2.5 2.6 -0.649953 4.984029 -12.777958 11.237235
26 2.6 2.7 1.062185 -8.370651 21.94421 -18.85531
27 2.7 2.8 -1.598787 13.183226 -36.251257 33.52061
28 2.8 2.9 1.332964 -11.443485 32.703532 -30.837193
29 2.9 3.0 -0.733069 6.530999 -19.42247 19.551276
30 3.0 3.1 0.59931 -5.460413 16.551765 -16.42296
31 3.1 3.2 -0.664173 6.289986 -19.874472 21.217485
32 3.2 3.3 1.057383 -10.236952 33.011728 -35.194462
33 3.3 3.4 -1.565357 15.728176 -52.673194 59.058953
34 3.4 3.5 1.204047 -12.519753 43.369764 -49.789733
35 3.5 3.6 -0.250832 2.756477 -10.097038 12.588203
36 3.6 3.7 -0.200721 2.215277 -8.148718 10.250219
37 3.7 3.8 0.053714 -0.608947 2.300908 -2.637653
38 3.8 3.9 -0.014135 0.164534 -0.63832 1.085369
39 3.9 4.0 0.002827 -0.033925 0.13567 0.079182
The Internet online service (Arndt Brunner, 2018) can be used for cubic spline calculation and spline function visualization.
References
-Arndt Brunner. 2018. Rechner fur kubische Splines. [Internet]. Available at: http://www.arndt-bruenner.de/mathe/scripts/kubspline1 .htm (in German). Accessed: 07.03.2018.
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Faraponov, V.V., Bimatov, V.I., Savkina, N.V., & Khristenko, Yu.F. 2017. Praktikum po aeroballistike. Tomsk: STT Publishing (in Russian). (In the original: Фарапонов, В.В., Биматов, В.И., Савкина, Н.В., Христенко, Ю.Ф. 2017. Практикум по аэробаллистике. Томск: STT Publishing).
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predmetu/0HZ/10%20Hlavhove%20zbrane%20Prechodova%20a%20vnejs[%20 balistika.pdf (in Czech). Accessed: 07.03.2018.
Khaikov, V. 2017. Review of mathematical formulas for the air resistance law of the 1943 year. Part 1. Electronic Information Systems, 4(15), pp.74-90 (in Russian). (In the original: Хайков, В. 2017. Обзор аналитических выражений закона сопротивления воздуха 1943 г. Часть 1. Электронные
информационные системы 4(15). C. 74-90).
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Kirillov, V.M. 1963. Osnovaniya ustroystva i proyektirovaniya strelkovogo oruzhiya. Penza: Penzenskoye Vyssheye Artilleriyskoye Inzhenernoye
Uchilishche (in Russian). (In the original: Кириллов, В.М. 1963. Основания устройства и проектирования стрелкового оружия. Пенза: Пензенское Высшее Артиллерийское Инженерное Училище).
Kir'yanov, D.V. 2012. Mathcad 15/Mathcad Prime 1.0. Sankt-Peterburg: BKhV (in Russian). (In the original: Кирьянов, Д.В. 2012. Mathcad 15/Mathcad Prime 1.0. Санкт-Петербург: БХВ).
Konovalov, A.A., & Nikolayev, Yu.V. 1979. Vneshnyaya ballistika. Moscow: Tsentral'nyy Nauchno-Issledovatel'skiy Institut Informatsii (in Russian). (In the original: Коновалов, А.А., Николаев, Ю.В. 1979. Внешняя баллистика. Mосква: Центральный Научно-Исследовательский Институт Информации).
Kozlitin, I.A., & Omelyanov, A.S. 2016. A method for smooth approximation of drag functions. Mathematical Models and Computer Simulations, 28(10), pp.23-32 (in Russian). (In the original: Козлитин, И.А., Омельянов, А.С. 2016. Метод построения гладкой аппроксимации законов сопротивления. Математическое моделирование, 28(10). С. 23-32).
Mori, E. 2013. Balistica Pratica. Ilmiolibro Self Publishing (in Italian).
Regodic, D. 2006. Spoljna balistika. Belgrade: Vojna akademija (in Serbian).
Semikolenov, N.P., Bondarenko, F.G., & Krasner, N.Y. 1971. Principles of small unit weapons firing. Charlottesville, US: Army Foreign Science and Technology Center. Trans. from Russian.
Shapiro, Ya.M. 1946. Vneshnyaya ballistika. Moscow: Gosudarstvennoye izdatel'stvo oboronnoy promyshlennosti (in Russian). (In the original: Шапиро, Я.М. 1946. Внешняя баллистика. Москва: Государственное издательство оборонной промышленности).
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
Vodorezov, Yu.G. 2017. Teoriya i praktika strel'by iz nareznogo dlinnostvol'nogo strelkovogo oruzhiya. Chast' 1. Moscow: Moskovskiy
Gosudarstvennyy Tekhnicheskiy Universitet (in Russian). (In the original: Водорезов, Ю.Г. 2017. Теория и практика стрельбы из нарезного длинноствольного стрелкового оружия. Часть 1.).
КОМПЬЮТЕРНОЕ МОДЕЛИРОВАНИЕ ВНЕШНЕЙ БАЛЛИСТИКИ ПИСТОЛЕТА С ИСПОЛЬЗОВАНИЕМ ДВУХ РАЗЛИЧНЫХ ЗАКОНОВ СОПРОТИВЛЕНИЯ ВОЗДУХА (рассмотрение на примере пистолета 7.62 мм ТТ)
Вадим Л. Хайков,
независимый исследователь, г. Краснодар, Российская Федерация
ОБЛАСТЬ: механика - баллистика ВИД СТАТЬИ: оригинальная научная статья ЯЗЫК СТАТЬИ: английский
Резюме:
Для пистолетов М54, М57 (7.62*25 Токарев патрон) собраны баллистические параметры характеризующие их баллистику. В статье дан расчёт внешнебаллистических траекторий для двух законов сопротивления воздуха: 1943 года и Сиаччи, при этом использованы разные виды их математической записи (классические аналитические формулы, формулы кусочного вида, а также функции-таблицы). Для решения баллистической системы дифференциальных уравнений при табличном задании функции сопротивления воздуха используются сплайны. Для закона 1943 года показана графическая интерпретация функции Cd(i,v) в виде поверхности и её основные элементы. Показано, что такую поверхность можно построить для любого закона сопротивления воздуха. Показан способ графического сравнения баллистических траекторных параметров. Все вычисления выполнены в среде Mathcad 15, в статье приведён программный код расчёта. Показано, что за счёт подбора баллистических коэффициентов можно получить достаточно близкие по форме траектории. Однако в связи с тем, что движение пули по каждой из них определяется разными законами сопротивления воздуха, то замедление пули на каждой из них будет иметь свою собственную независимую форму и поэтому может не совпадать с «траекторией-двойником».
Ключевые слова: внешняя баллистика, пистолет M54, пистолет M57, патрон 7.62*25 ТТ, закон сопротивления воздуха, траектория пули, сплайн, Mathcad.
КОМШУТЕРСКА СИМУЛАЦИJА СПО^НЕ БАЛИСТИКЕ
ПИШТО^А ПРИМЕНОМ ДВА РАЗЛИЧИТА ЗАКОНА ОТПОРА ВАЗДУХА (на примеру пишто^а 7.62 мм ТТ)
Вадим Л. Ха]ков
независни истраживач, Краснодар, Руска Федерац^а
ОБЛАСТ: механика - балистика
ВРСТА ЧЛАНКА: оригинални научни чланак
ЗЕЗИК ЧЛАНКА: енглески
Сажетак:
За балистику метка пиштола (пушке) карактеристична су два балистичка коефици}ента ко}а се односе на различите законе отпора ваздуха. Колико су сличне балистичке путане доби}ене помогу различитих закона и какве су разлике измену лих? У овом раду процелу]е се сполна балистичка путала на основу два закона оптора ваздуха (закон из 1943. године и Б1асс1-]ев закон). Сви доби}ени резултати односе се на пиштол ТТ са метком „токарев” калибра 7.62*25 мм. У раду ]е, тако^е, приказан начин израчунавала балистичке путале ако }е закон отпора ваздуха изражен као рационална функциа, прекидна функциа или spline. Закон оптора ваздуха из 1943. године приказан }е као графичка интерпретаци]а функци]е Cd (i,v) у облику површине. Показано )е да )е могуче добити доволно сличан облик балистичких путала захвалу}уЬи избору балистичких коефици}ената. Представлен }е метод графичког поревела сполних балистичких параметара, као и
математички алати за квантитативну анализу облика балистичких кривих. Предложено )е да се разлика измену две путале одреди помогу релативне грешке у односу на изабрани балистички параметар. Комп]утерска симулаци}а два поменута закона отпора ваздуха за зрно „токарев” калибра 7.62*25 мм показу}е да су профили функци]е односа тренутне брзине про]ектила и времена лета имали на}веЬе неподударале у односу на остале балистичке параметре (нпр. хоризонтални домет, висину путале, итд.). На]веПа доб^ена вредност релативне грешке била )е 0,8%, локализована у тачки удара.
Клучне речи: комп}утерска симулаци}а, сполна балистика, поштол ТТ, закон отпора ваздуха, функциа чеоног отпора, закон из 1943. године, путала зрна, spline, Mathcad
Khaikov, V., Computer simulation of the 7.62mm TT pistol external ballistics using two different air resistance laws, pp.495-524
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2018, Vol. 66, Issue 3
Paper received on / Дата получения работы / Датум приема чланка: 12.02.2018. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 12.04.2018.
Paper accepted for publishingon / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 14.04.2018.
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© 2018 Автор. Опубликовано в «Военно-технический вестник / Vojnotehnicki glasnik / Military Technical Courier» (www.vtg.mod.gov.rs, втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией «Creative Commons» (http://creativecommons.org/licenses/by/3.0/rs/).
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