Numerical solving of ballistic flight equations for big bore air rifle

6 Pages·0.223 MB·English

by  Cumin Josip, Grizelj Branko, Scitovski Rudolf.

J . Cumin, B . Grizelj, R . Scitovski Numeričko rješavanje balističkih jednadžbi za zračnu pušku velikog kalibra
ISSN 1330-3651
UDC/UDK 531.555 : 623.442
NUMERICALSOLVING OF BALLISTIC FLIGHT EQUATIONS FOR BIG BOREAIR RIFLE
Josip Cumin, Branko Grizelj, Rudolf Scitovski
Subject review
Thispaperisaboutsolvingballisticequationsbymeansofnumericalmathematics.Ballisticflightequationsareappliedformodernbigboreairrifles,operated
withhighpressurecarbon-dioxide(CO )gas.Ballisticequationsuseairdragfunctionforobtainingresults.Todaytherearemanycomplexcommercialballistic
2
programsonmarket,basedonmodifiedmasspointmodel–"MPMM"or"6DOF"model.Bigboreairriflecommonlyusesballprojectiles,andvelocitiesofthe
projectilesarelowerthanthespeedofsound.Thereforesimplifiedmodelsforquickcalculationsofballistictrajectoriesandprojectilevelocitiescanbeused.
Key words:ballistic equations, big bore air rifle, air drag
Numeričko rješavanje balističkih jednadžbi za zračnu pušku velikog kalibra
Pregledničlanak
Temaovogradajerješavanjebalističkihjednadžbiuzpomoćnumeričkematematike.Balističkejednadžbesuprimijenjenezamodernezračnepuškevelikog
kalibra,punjenesugljik-dioksid(CO )plinom.Balističkejednadžbekoristeotporzrakomispunjenogprostorapriizračunurezultata.Danaspostojeraznovrsni
2
kompliciranikomercijalnibalističkiprograminatržištu,kojisubaziranina"modifiedpointmass–MPMM"modelu,ilipakna"6DegreesofFreedom"
modelu. Zračna puška velikog kalibra koristi kuglu kao projektil, i brzine projektila su uglavnom manje od brzine zvuka. Stoga se mogu koristiti
pojednostavnjenimodelizabrzoizračunavanjebalističkihputanjaibrzinaprojektila.
Ključne riječi:balističke jednadžbe, zračna puška velikog kalibra, otpor zraka
1 T–thermodynamic temperature, K
Introduction
R 8314
Uvod R (cid:2) m (cid:2) (cid:2)286,987 J/(kg (cid:4)K) (3)
air M 28,97
Today on the world market one can find various R –gas constant(cid:2)R =8314J/(kmol(cid:4)K)
m m
commercialballisticsoftware,whichcancalculateballistic M–molecularmassof air(cid:2)M=28,97 kg/kmol
trajectories,andprojectilespeedlossduringaflight.Allof Forthe givendata,and temperature(cid:5)=20 °C,
these programs use database with significant projectile
T=293,15 K
characteristics, needed for ballistic calculations. Main
characteristics of the projectiles are given by v (cid:2) (cid:3)(cid:4)R (cid:4)T (cid:2) 1,4 (cid:4) 286,987 (cid:4) 293,15 (cid:2) 343,2 m/s
manufacturers,andthisdataisobtainedbymeasurements s air
or numeric simulations. These programs are complex
Forbigboreairriflemodel,velocityofaprojectileis
because they also take into account a change of drag
highest when projectile is leaving bore, and it is getting
coefficientduringflightofprojectileasavariable.Thedrag
smallerwitheveryshotbecauseCO tankpressurereduces.
coefficient C is a function of projectile velocity among 2
d Even with the first shot, maximal velocity of projectile is
other variables. It is also measured or obtained by
smallerthanthespeedofsound.
simulations. If velocity of projectile is smaller than the
Drag function can be calculated by using:
speedofsoundintheair,thedragcoefficientisassumedto
beconstant.Thereforeforvelocitiesairriflecanproduce,C
d 1
is constant. By raising projectile velocity over speed of F (cid:2) (cid:4)(cid:6)(cid:4)A(cid:4)v2(cid:4)C , N (4)
sound,dragcoefficientchanges.GenerallyMachnumberis d 2 d
ratiobetweenprojectilevelocity(v)atsomemomentduring
where :
flight,andthespeedofsoundintheair(v),[1]:
s (cid:6)–air density, kg/m3
v A–cross section of projectile, m2
M (cid:2) (1)
a v v–projectile velocity, m/s
s
C –drag coefficient.
d
IfM <1, it is subsonic region
a C isthevaluethatisdeterminedforeveryshapeofthe
ifM >1, it is supersonic region d
a projectile. If C is small, then the drag force on
ifM = 1, it is transitional region. d
a body/projectile is small and vice-versa. C depends on
d
dimension of projectile, shape of projectile, surface
Speed of sound in the air can be calculated by [2]:
roughness and Reynold’s number. For this model
approximatevalueforsmallsphereC =0,45willbeused
v (cid:2) (cid:3)(cid:4)R(cid:4)T m/s, (2) [3]. d
s
Also for calculating trajectories commercial software
Where: usesotherfactorslikeprojectile/bulletspin,whichaffects
(cid:2)- ratio of specific heat capacitiesc/c, for air (cid:2)=1,4 dynamical stabilization. So if projectile has a high
p v
R -individual gas constant, J/(kg(cid:4) K) frequency of rotation it is over stabilized and vice-versa.
air
Technical Gazette 16,1(2009), 41-46 41
Numerical solving of ballistic flight equations for big bore air rifle J . Cumin, B . Grizelj, R . Scitovski
Figure 1 illustrates: a) perfectly stabilized projectile, b) Coordinatesystemforthisproblemiscalled–carriedlocal
(cid:2)
understabilizedprojectile,c)overstabilizedprojectile.The coordinatesystem,andvelocityismarkedas v.
variablewhichdescribesthisphenomenoniscalledyawof
angle and it represents the angle between projectile 2.1
principalaxisandtangentonballistictrajectory.
Euler model without wind
Inthecaseofairrifle,aballisusedasaprojectile,and
Eulerov model putanje zrna -bez bočnog vjetra
for its symmetry this phenomenon of stabilized projectile
can be ignored. In order to determine the yaw of angle in
AccordingtoFigure2,onecanseeforcesactingupon
everymomentoftime,complexmathematicalsolutionslike
projectile. Force of air drag is trying to slow down the
MPMM(modifiedmasspointmodel),or6DOF(6degrees
projectile,andtheforceofgravityispullingtheprojectile
offreedom)aretobeused.
down.Vectorequationcanbewrittenas[4]:
dv (cid:3)(cid:4) (cid:6)(cid:4)v2 v
m(cid:4) (cid:2)m(cid:4)g(cid:7) (cid:4)A(cid:4)C (cid:4) . (5)
dt 2 d v
Figure 2Drag force, and Earth gravity on the projectile
Slika 2Sila otpora zrakom ispunjenog prostora i ubrzanje Zemljine
sile teže koje djeluju na projektil
The term
v v (cid:4) v (cid:4) v (cid:4) (cid:4) (cid:4) (cid:4)
(cid:2) x i (cid:11) y j(cid:11) z k (cid:2)(cos(cid:8) ) i (cid:11) (cos (cid:9) ) j (cid:11) (cos (cid:10) )k
Figure 1Various cases of projectile stabilization v v v v
Slika 1Različiti slučajevi stabilizacije projektila
representsdirectioncosinesoftrajectorytangent.
Vectorofvelocityisbydefinition,derivationofradii-
For comparison of cases, in literature [8, 9], one can
vectorintime.Radii-vectordescribespositionofprojectile
find expressions for ballistic equations in vacuum ("in
fromtheoriginofthecoordinatesystem.
vacuo")model.
(cid:4)
dr
2 (cid:2)v (6)
Ballistic equations including wind drag dt
Balističke jednadžbe s obzirom na otpor zrakom
First vector equation (5) is projected on tangent and
ispunjenog prostora
normal of trajectory. Equation (6) is projected on axes of
localcoordinatesystem[4].
As already mentioned(4), drag function can be
expressedas:
dv (cid:6)(cid:4)v2
(cid:2)(cid:7)g(cid:4)sin(cid:10)(cid:7) (cid:4)A(cid:4)C ,
1 dt 2m d
F (cid:2) (cid:4)(cid:6)(cid:4)A(cid:4)v2(cid:4)C , N.
d 2 d d(cid:10) (cid:2) (cid:7)g(cid:4)cos(cid:10),
dt v
(7)
Itisassumedthatprojectileisflyinginx-ycoordinate dx
(cid:2)v(cid:4)cos(cid:10) ,
system, with x coordinate representing the range, and y
dt
coordinaterepresentingtheheightofflight.Thedragforce
dy
canbeprojectedonaxes–forcedividedtocomponents.In (cid:2)v(cid:4)sin(cid:10) ,
thecaseofsidewind,theprojectiledriftssideways,andthis dt
isprojectedonzaxis.Therecanalsoberangewind(blowing
inpositiveornegativedirectionofxaxis),andverticalwind, where(cid:3)isanglebetweenvelocityvectorv,andpositivex
butitisveryrare.Verticalwindhasalmostnoeffectonthe axis.
projectile height, so it is neglected in equations. While Solvingtheseequationsinclassicalwaywouldbevery
calculating with side and range wind, components of difficult, but they can be approximated by means of
velocity change, and velocity vector composed of those numericalmathematics.
components also changes. This motion is complex. Withtheinitialconditionssetas:v =183m/s,massof
0
42 Tehnički vjesnik16,1(2009), 41-46
J . Cumin, B . Grizelj, R . Scitovski Numeričko rješavanje balističkih jednadžbi za zračnu pušku velikog kalibra
bulletm=14,6grams,angleofdeparturemeasuredfromx
axis(cid:3)=3°,bulletdiameterd=12,7mm(0,50inch),ambient
pressurep =101325Pa,thermodynamictemperatureT=
0
293,15K,densityofaircanbecalculatedasfollows:
p 101325
(cid:6)(cid:2) (cid:2) (cid:2)1,2044 kg/m3 (8)
R (cid:4)T 286,987 (cid:4) 293,15
air
Withdt=0,02s,andapplyingRunge-Kuttamethodon
equations (7), discrete values for the variables are
calculated.FourthorderRunge-Kuttamethodisappliedon
equations (7) in such a way that first the velocity v is
calculated. Next step is calculation of dv, d(cid:3), dx and dy.
Figure 5Velocity of projectile as a function of time t
Thesevaluesareaddedtovaluesofv,(cid:3),x,y,respectively. Slika 5Brzina projektila kao funkcija vremena t
Valuesv,(cid:3),x,yaresetatthebeginningofprogramfromthe
initialconditions.Ifconditionismet,programisinstructed 2.2
tostop,anddisplayresults.Ifconditionisnotmet,bythe Euler model with side wind
iterativeprocedureprogramcalculatesnewvaluesofdv,d(cid:3),
Eulerov model putanje zrna - sbočnim vjetrom
dx,dy,andaddsthemtothevaluesv,(cid:3),x,y,untilconditionis
met. In this way for each value of time t, values v, (cid:3), x, y
This model is actually just a minor modification of a
numericallyapproximateanalyticalsolution.Programming classicalEulermodel.Ituseswindvelocitycomponentsto
was done in Mathematica 5.2 for students. Programming change the projectile velocity components, and thus the
canbedoneinC++,Fortranorothersoftware.Following projectilevelocityvector.Accordingto[4]modifiedEuler's
diagramswereconstructedfromresults. equationsareasfollows:
dv (cid:3)(cid:4) (cid:6)(cid:4)v2 v
m (cid:2)m(cid:4)g(cid:7) (cid:4)A(cid:4)C (cid:4) (9)
dt 2 d v
Equation of radii vector:
(cid:4)
Figure 3Ballistic trajectory solved by Euler's equations dr (cid:4) (cid:4) (cid:3)(cid:3)(cid:4) (cid:3)(cid:3)(cid:4)
Slika 3Balistička putanja riješenaEulerovim jednadžbama (cid:2)v , v (cid:2) v (cid:11) v ,
dt k w
(cid:3)(cid:3)(cid:4) (cid:4) (cid:4) (cid:4)
v (cid:2)u i(cid:11)h j(cid:11)w k, (10)
(cid:3)(cid:3)k(cid:3)(cid:4) k (cid:4) k(cid:3)(cid:4) k (cid:4)
w (cid:2)u i(cid:11)0(cid:4)j(cid:11)w k,
w w w
wherev iswindvelocity,v projectilevelocityaccording
w k
toclassicalEuler'smodel.
Fromequations(9),(10)componentscanbewrittenas:
du (cid:12)u (cid:7)u (cid:13) dx
(cid:2)(cid:7)E(cid:4)C (cid:4) k w , (cid:2)u,
dt d v dt
dh h dy
(cid:2)(cid:7)E(cid:4)C (cid:4) k (cid:7)g, (cid:2)h, (11)
dt d v dt
SliFkaig 4urKeu 4tA(cid:3)nizgrlaež(cid:3)ena sk aao f ufunnckticoinja o vfr teimmeen ta t dw (cid:2)(cid:7)E(cid:4)C (cid:4)(cid:12)wk (cid:7)ww(cid:13), dz (cid:2)w,
dt d v dt
From Figure 3, one can see that trajectory is more
curvedattheend,andthisisclearlyvisibleeffectoftheair where:
drag. Figure 4 represents the angle between the velocity (cid:6)(cid:4)v2(cid:4)A
vectorandxaxis.Theangleofdeparture atthebeginning E (cid:2)
2(cid:4)m
was set as 3°, and in the moment when projectile hit the
ground, calculated angle was (cid:3)=(cid:4)4,4°. In vacuum model v(cid:2) (cid:12)u (cid:7)u (cid:13)2(cid:11)h2(cid:11)(cid:12)w (cid:7)w (cid:13)2 (seeFigure 6)
(withoutairdrag)thereisnodifferencebetweentheangleof k w k k w
departure and the angle (cid:3) when projectile hits the ground
(flat). Assuminginitialconditionsas:v =183m/s,massof
0
Figure 5 shows scalar value of velocity vector bulletm=14,6grams,theangleofdeparturemeasuredfrom
decreasingthroughtimebecauseofairdrag. x axis (cid:3) = 3°, bullet diameter d = 12,7 mm (0,50 inch),
Initial velocity was set as v = 183 m/s. Calculated densityofair (cid:5)=1,2044kg/m3 ,withambientpressurep =
0 0
velocityinthemomentwhenbullethitsthegroundisv= 101325 Pa, thermodynamic temperature T = 293,15 K,
105,3m/s. componentsofwindvelocityvectoru =5m/s,w =15m/s,
w w
dt=0,02s,andapplyingRunge-Kuttamethodonequations
Technical Gazette 16,1(2009), 41-46 43
Numerical solving of ballistic flight equations for big bore air rifle J . Cumin, B . Grizelj, R . Scitovski
(11), discrete values for variables are calculated, and
followingdiagramsconstructed(Figures7,8,9and10).
Figure 10Projectile side drift as a function of range
Slika 10Bočno zanošenje projektila kao funkcija dometa
Difference from previous results (Euler's equations
withoutwind),canbeseeninFigure10.
Astheinitialspeedwassetrathersmall,theeffectofair
dragisnotquiteasvisibleinFigure3andFigure7.Results
obtainedbynumericallysolvingequations(7),(11)would
havebeendifferent,iftheinitialspeedwassethigher,likev
0
= 300 m/s. Limit is that the initial speed must not be set
higherthanthespeedofsound.
Airriflewithsmallercaliberprojectilethan12,7mm,
Figure 6Ballistic trajectory in space, with components
of velocity vector shown has higher initial velocity than 183 m/s, and the results
Slika 6Balistička putanja u prostoru, s prikazanim calculatedforthatcaseshowhigherexpressedappearance
komponentama vektora brzine oftheairdrag,inthemeansofballistictrajectories.
2.3
Flat fire model
Flat fire model
This is a bit different way of determining ballistic
Figure 7Ballistic trajectory in x-y coordinate system,
solved by Euler's equations including wind trajectories. It uses the same equations from the classic
Slika 7Balistička putanja u x-y koordinatnom sustavu, Euler'smodel.Maindifferenceinthiswayofsolvingisthat
riješen pomoću Eulerovih jednadžbi uzimajući u obzir i bočni vjetar velocity vector v is divided on v and v components.
x y
Becauseofthisseparation,variablesv andv havecertain
x y
limitations. This model was developed during World War
One,andinthattimeriflesweremadewithsmoothbores.
Projectileshadsmallexitvelocitiesatboreend,andsmall
range. The army still uses this model for calculating
ballisticsofhighcaliberguns,withsmallbulletvelocities.
Theseconditionsaresimilartotheproblemofbigboreair
rifle, therefore this model can be used. Assumptions are
following:therearenoCoriolisacceleration,Magnusforce
andforceofaerodynamiclift.
Equations are [5]:
(cid:4)
dv v i (cid:11) v j (cid:11) v k A(cid:4)(cid:6)(cid:4)C (cid:4) (cid:4)
(cid:2) x y z (cid:2)(cid:7) d (cid:4)v (cid:4) v (cid:7) g (cid:4) j (12)
dt dt 2m
Figure 8Total velocity of projectile as a function of time
Slika 8Ukupna brzina projektila kao funkcija vremena
Components can be written:
dv
v(cid:5) (cid:2) x (cid:2)(cid:7)C*(cid:4)v(cid:4)v ,
x dt d x
dv
v(cid:5) (cid:2) y (cid:2)(cid:7)C*(cid:4)v(cid:4)v (cid:7)g,
y dt d y (13)
dv
v(cid:5) (cid:2) z (cid:2)(cid:7)C*(cid:4)v(cid:4)v ,
z dt d z
v(cid:2) v2(cid:11)v2(cid:11)v2,
x y z
where:
Figure 9Ballistic trajectory in 3D, solved by Euler's A(cid:4)(cid:6)(cid:4)C
Slika 9Prostorni prikeaqz ubaatliiosntisč kinec kluridviunlgje w riinjedšene prema Eulerovim Cd* (cid:2) 2m d (cid:4)
jednadžbama uključujući i bočni vjetar
Assumptions:
v = 0(there is no side wind)
z
From initial velocity 183 m/s, because of wind drag, |v/v | =tan(cid:10)< 0,1(cid:2)(cid:10)<5,7°
velocity drops to v = 104 m/s, when projectile hits the y x
v/v< 0,5 %.
ground. x
44 Tehnički vjesnik16,1(2009), 41-46
J . Cumin, B . Grizelj, R . Scitovski Numeričko rješavanje balističkih jednadžbi za zračnu pušku velikog kalibra
According to [5] equations (13) can be transformed to (cid:22) (cid:16)v (cid:17)(cid:23)
use variable of rangex,instead of variable of timet. g (cid:4) t (cid:24)(cid:24) v x 0 (cid:4) (cid:29)t(cid:29)(cid:4)ln(cid:18)(cid:20) vx0(cid:19)(cid:21)(cid:25)(cid:25)
tan(cid:10)(cid:2)tan(cid:10) (cid:7) 1(cid:11) x (17)
0 (cid:24) (cid:25)
v 2x
dv dx x0 (cid:24) (cid:25)
dxx (cid:4) d t(cid:2)(cid:7)Cd*(cid:4)vx(cid:4)vx(cid:14)vx’(cid:2)(cid:7)Cd*(cid:4)vx, (14a) (cid:24)(cid:26) (cid:25)(cid:27)
dvy (cid:4)dx (cid:2)(cid:7)C*(cid:4)v (cid:4)v (cid:7)g(cid:14) dvy (cid:2)(cid:7)C*(cid:4)v (cid:7) g . (14b) Equation for describing projectile height y[5]:
dx dt d x y dx d y vx (cid:22) (cid:23)
projeIfcttihleesamiradlrlaegrtchoaenffsipceieendtoCfdsiosucnodn)s,ttahnatn(evqeuloactiiotyno(1f4thae) y(cid:2)y0(cid:11)x(cid:4)tan(cid:10)0(cid:7)g2 (cid:24)(cid:24)(cid:24)(cid:24) v xx0(cid:4)ln(cid:16)(cid:18)1vx0(cid:17)(cid:19)(cid:25)(cid:25)(cid:25)(cid:25)(cid:24)(cid:24)(cid:26)(cid:22)12(cid:18)(cid:20)(cid:16)vvxx0(cid:7)1(cid:19)(cid:21)(cid:17)2(cid:11)(cid:18)(cid:20)(cid:16)vvxx0(cid:7)1(cid:19)(cid:21)(cid:17)(cid:7)ln(cid:18)(cid:20)(cid:16)vvxx0(cid:19)(cid:21)(cid:17)(cid:25)(cid:25)(cid:27)(cid:23)
canbesolvedasfollows: (cid:24)(cid:26) (cid:20) vx (cid:21)(cid:25)(cid:27) (18)
x Assuminginitialconditionsas:v =183m/s,massof
(cid:7)Cd* (cid:15) d x 1 bulletm=14,6grams,angleofdepart0uremeasuredfromx
vx (cid:2)vx0(cid:4)e 0 axis (cid:3)=3°,bulletdiameterd=12,7mm(0,50inch),density
ofair(cid:5)=1,2044kg/m3 ,withambientpressurep =101325
and applying 0
Pa,thermodynamictemperatureT=293,15K,dt=0,02s,
k (cid:2)C*(cid:14)v (cid:2)v (cid:4)e(cid:7)k1 (cid:4) x andapplyingRunge-Kuttamethodonequations(14-18),
1 d x x0 discretevaluesforvariablesarecalculated,andfollowing
diagramsconstructed(Figure11,12,13,14and15).
Expression for timetis calculated as:
x
dx (cid:7)(cid:15)Cd* d x 1
dt(cid:2) , v (cid:2)v (cid:4)e 0
v x x0
x
x2
(cid:15)Cd * d x 1
x dx xe0
t(cid:2)(cid:15) 2 (cid:2)(cid:15) dx
(cid:16) x2 (cid:17) v 2
0(cid:18)(cid:18) (cid:7)(cid:15)Cd * d x 1 (cid:19)(cid:19) 0 x0 (15)
(cid:18)v (cid:4)e 0 (cid:19)
(cid:18) x0 (cid:19)
(cid:20) (cid:21)
1 x
t(cid:2) (cid:15)ek1 x 2dx
v 2
x0 0 Figure 11Component vxof velocity as a function of time
Slika 11Komponenta v brzine kao funkcija vremena
t(cid:2) 1 (ek1(cid:4)x (cid:7)e0((cid:2) 1 (ek1(cid:4)x (cid:7)1( x
v (cid:4)k v (cid:4)k
x0 1 x0 1
Equation(14b):
dv g
y (cid:11)C * (cid:4) v (cid:2)(cid:7) ,
dx d y v
x
is linear differential equation of first order, and with the
initialconditions,v =v ,t=0andx=0,itissolvedas:
y y0
v (cid:2)v (cid:22)(cid:24)tan(cid:10) (cid:7) g (cid:4) t(cid:16)(cid:18)1(cid:11)v x 0 (cid:4) k 1 (cid:4) t(cid:17)(cid:19)(cid:23)(cid:25), Figure 12Component vyof velocity as a function of time
y x(cid:26) 0 vx0 (cid:20) 2 (cid:21)(cid:27) Slika 12Komponenta vybrzine kao funkcija vremena
v
y (cid:2)tan (cid:10) ,
v
x
From:
v (cid:2)v (cid:4)e(cid:7)k1 (cid:4) x (cid:28)k (cid:2) 1(cid:4)ln(cid:16)(cid:18) vx (cid:17)(cid:19),
x x0 1 x (cid:20)v (cid:21)
x0
follows:
(cid:22) (cid:16) 1 (cid:16)v (cid:17) (cid:17)(cid:23)
(cid:24) (cid:18) v (cid:4) (cid:4)ln(cid:18) x0 (cid:19)(cid:4)t(cid:19)(cid:25)
(cid:24) g (cid:4) t(cid:18) x0 x (cid:20) v (cid:21) (cid:19)(cid:25)
vy (cid:2)vx(cid:24)tan(cid:10)0(cid:7) (cid:18)1(cid:11) x (cid:19)(cid:25) (16)
v 2
(cid:24) x0 (cid:18) (cid:19)(cid:25)
(cid:24)(cid:26) (cid:18)(cid:20) (cid:19)(cid:21)(cid:25)(cid:27) Figure 13Total velocity v as a function of time
Slika 13Ukupna brzina v kao funkcija vremena
Technical Gazette 16,1(2009), 41-46 45
Numerical solving of ballistic flight equations for big bore air rifle J . Cumin, B . Grizelj, R . Scitovski
m. Converted it is 0,332 mm. Calculated velocity of
projectileattargetimpactv=144,2m/s.Kineticenergyis
calculatedasfollows:
m (cid:4) v2 14,6 (cid:4) 144,22
E (cid:2) (cid:2) (cid:2)151,8 J
k 2 2000
Calculated time of flightt= 0,62 s.
4
Figure 14Angle(cid:3)as a function of time
Slika 14Kut(cid:3)kao funkcija vremena Conclusion
Zaključak
This paper is about utilizing numerical mathematic
methodfornumericalapproximationofballistictrajectories
foraspecificproblem.Itisshownhowsimplifiedballistic
models can be applied for quick ballistic calculations.
Figure 15Ballistic trajectory in x-y coordinate system
Slika 15Balistička putanja u x-y koordinatnom sustavu Modelswereappliedformodernhighpressureairrifles.Itis
shown that new design of air rifles is quite advanced.
3 Today'smodernhighpressureairriflescanproducekinetic
powerofprojectileleavingboreexitequivalenttosomelow
Projectile kinetic energy calculations at different
power fire weapons. For some European countries legal
ranges
projectileenergylimitforairriflesislimitedto17J.
Računanje kinetičke energije hica na različitim
udaljenostima
5
References
Forsameinitialconditionsasbefore,valueofdeparture
Literatura
angle canbevaried,insuchwaythatprojectilehitstargetat
defined range. With small ranges angle of departure is
[1] NASA. Mach Number. URL:
small,andtrajectoryitselfisalmostflat.Withlongrangesit
http://www.grc.nasa.gov/WWW/K-12/airplane/mach.html,
isnecessarytoincreasetheangleofdepartureinorderfor
(26.11.2008).
projectiletohitthetargetatdesiredrange.Forthispurpose,
[2] NASA. Speed of sound. URL:
aclassicalEuler'smodelwasused(7). http://www.grc.nasa.gov/WWW/K-12/airplane/sound.htm,
Example 1: For distance x = 10 m, angle of departure (26.11.2008).
calculatedis(cid:3)=0,085°.Applyingvaluesofvariablesfrom [3] Dirac Delta science & engineering encyclopedia. Drag
the initial conditions, and Runge-Kutta method on coefficient.
URL:http://www.diracdelta.co.uk/science/source/d/r/drag%
equations (7), with range set as x = 10 m, as a result
20coefficient/source.html, (26.11.2008).
calculatedheightamountsy=0,0000368m.Transformedit
[4] Janković, S. Mehanika leta projektila. Sveučilište u
is 0,0368 mm deviation from gun's line of sight, and it is
Zagrebu,Fakultet strojarstva i brodogradnje, Zagreb, 1998.
satisfactory solution. As bullet diameter is 12,7 mm,
[5] Carlucci, D.; Jacobson, S.Theory of design of guns and
deviation from aimed target at range 10 m is 0,0368 mm, ammunition.Taylor and Francis group, 2007.
which is precise enough. Calculated velocity of projectile [6] Molitz, H.; Strobel, R. Äußsere Ballistik. Springer-Verlag
according to these initial conditions is 178,7 m/s. Now GmbH, Berlin, 1963.
projectile kinetic energy can be calculated according to [7] Matejiček, F. Primijenjena mehanika-KINEMATIKA.
expression: URL:http://www.sfsb.hr/ksk/statika/kinematika/3_sadržaj/i
ndex.htm. (02.12.2008).
14,6 [8] Matejiček, F. Primijenjena mehanika-KINETIKA.
m(cid:2) kg
URL:http://www.sfsb.hr/ks/statika/kinetika/3_sadržaj/index
1000
.htm. (02.12.2008).
m (cid:4) v2 14,6 (cid:4) 178,72 [9] Scitovski, R.Numerička matematika. Odjel za matematiku
E (cid:2) (cid:2) (cid:2)232,12 J
k 2 2000 Sveučilišta u Osijeku, Osijek, 2004.
Calculated time of flight ist= 0,055 s.
Authors' addresses
Example 2: For target at range x = 50 m, calculated
Adrese autora
angleofdepartureis(cid:3)=0,455°,andvalueofheightamounts
y=0,000278m.Converteditis0,278mm,andthisisalso Cumin Josip, dipl. ing. stroj.
small deviation from aimed point at target. Calculated Mechanical engineering faculty, University of Osijek
Trg I. Brlić-Mažuranić, 35000 Slavonski Brod, Croatia
velocityofprojectileatimpactv=162,4m/s.Kineticenergy
www.sfsb.hr, e-mail:[email protected]
iscalculatedasfollows:
Dr. sc. Grizelj Branko, full professor
m (cid:4) v2 14,6 (cid:4) 162,42 Mechanical engineering faculty, University of Osijek
E (cid:2) (cid:2) (cid:2)192,53 J Trg I. Brlić-Mažuranić, 35000 Slavonski Brod, Croatia
k 2 2000 www.sfsb.hr,e-mail:[email protected]
Calculated time of flightt= 0,291 s. Dr. sc Scitovski Rudolf, full professor
Department of Mathematics, University of Osijek
Example3:Fortargetatrange100m,calculatedangle
Trg Lj. Gaja 6, HR-31000 Osijek, Croatia.
ofdepartureis (cid:3)=0,988°,andvalueofheighty=0,000332
www.mathos.hr,e-mail:[email protected]
46 Tehnički vjesnik16,1(2009), 41-46