Research Article. Physicomathematical Simulation Analysis for Small Bullets

Transcription

1 Jest Jounal of Engineeing Science and Technology Review 1 (2008) Reseach Aticle JOURNAL OF Engineeing Science and Technology Review Physicomathematical Simulation Analysis fo Small Bullets. N. Gkitzapis,*, 1 N. E. Tsiatis, 2 E E. Panagiotopoulos 3 and. P. Magais Physicist in Laboatoy of Fieams and Tool Maks Section, Ciminal nvestigation ivision, Captain of Hellenic Police, Hellenic Police, Athens, Lectue in Hellenic Militay Academy and postgaduate Student in Mechanical Engineeing and Aeonautics ept., Univesity of Patas, Hellas 2 Laboatoy of Fieams and Tool Maks Section, Ciminal nvestigation ivision, Captain of Hellenic Police, Hellenic Police, Athens, Hellas 3 postgaduate Student in Mechanical Engineeing and Aeonautics ept., 4 Pofesso in Mechanical Engineeing and Aeonautics ept., Fluid Mechanics Laboatoy, Univesity of Patas, Patas, Hellas Received 3 Apil 2008; Accepted 17 Septembe 2008 Abstact A full six degees of feedom (6-OF) flight dynamics model is poposed fo the accuate pediction of shot and long-ange tajectoies of small bullets via atmospheic flight to final impact point. The mathematical model is based on the full equations of motion set up in the no-oll body efeence fame and is integated numeically fom given initial conditions at the fiing site. The pojectile maneuveing motion depends on the most significant foce and moment vaiations, in addition to gavity and Magnus effect. The computational flight analysis takes into consideation the Mach numbe and total angle of attack effects by means of the vaiable aeodynamic coefficients. Fo the puposes of the pesent wok, linea intepolation has been applied fo aeodynamic coefficients fom the official tabulated database. The developed computational method gives satisfactoy ageement with published data of veified expeiments and computational codes on atmospheic pojectile tajectoy analysis fo vaious initial fiing flight conditions. Keywods: constant and vaiable aeodynamic coefficients, six degees of feedom Nomenclatue C = dag foce aeodynamic coefficient C L = lift foce aeodynamic coefficient C LP = oll damping moment aeodynamic coefficient C = pitch damping moment aeodynamic coefficient C = ovetuning moment aeodynamic coefficient C YPA = Magnus moment aeodynamic coefficient x, y, z = pojectile position coodinates in the inetial fame, m m = pojectile mass, kg = pojectile efeence diamete, m s = dimensionless ac length = total aeodynamic velocity, m/s u, v, w = pojectile velocity components expessed in the no-ollfame, m/s p, q, = pojectile oll, pitch and yaw ates in the moving fame, espectively, ad/s g = gavity acceleation, m/s 2 = pojectile inetia matix = pojectile axial moment of inetia, kg m 2 = pojectile tansvese moment of inetia about y-axis though the cente of mass, kg m 2 Ι ΧΧ, Ι ΥΥ, Ι ΖΖ = diagonal components of the inetia matix Ι ΧΥ, Ι ΥΖ, Ι ΧΖ = off-diagonal components of the inetia matix LE MCM = distance fom the cente of mass (CG) to the Magnus cente of pessue (CM) along the station line, m LE MCP = distance fom the cente of mass (CG) to the aeodynamic cente of pessue (CP) along the station line, m ρ = density of ai, kg/m 3 φ, θ, ψ = pojectile oll, pitch and yaw angles, espectively, deg α, β = aeodynamic angles of attack and sideslip, deg Subscipts o = initial values at the fiing site * addess: Hgitzap@yahoo.g SSN: Kavala nstitute of Technology. All ights eseved. 70

2 . N. Gkitzapsi, N. E. Tsiatis, E E. Panagiotopoulos and. P. Magais/ Jounal of Engineeing Science and Technology Review 1 (2008) ntoduction Ballistics is the science that deals with the motion of pojectiles. The wod ballistics was deived fom the Latin ballista, which was an ancient machine designed to hul a javelin. The moden science of exteio ballistics [1] has evolved as a specialized banch of the dynamics of igid bodies, moving unde the influence of gavitational and aeodynamic foces and moments. Exteio ballistics existed fo centuies as an at befoe its fist beginnings as a science. Although a numbe of sixteenth and seventeenth centuy Euopean investigatos contibuted to the gowing body of enaissance knowledge, saac Newton of England ( ) was pobably the geatest of the moden foundes of exteio ballistics. Newton s laws of motion established, without which ballistics could not have advanced fom an at to a science. Pioneeing English ballisticians Fowle, Gallop, Lock and Richmond [2] constucted the fist igid six-degee-of-feedom pojectile exteio ballistic model. The pesent wok addess a full six degees of feedom (6- OF) pojectile flight dynamics analysis fo accuate pediction of shot and long ange tajectoies of small bullets. The poposed flight dynamic model takes into consideation the influence of the most significant foce and moment vaiations, in addition to gavity and Magnus effect. The applied aeodynamic coefficient analysis takes into consideation the vaiations depending on the Mach numbe flight and total angle of attack. The efficiency of the developed method gives satisfactoy esults compaed with published data of veified expeiments and computational codes on dynamics model analysis of shot and long-ange tajectoies of spin-stabilized pojectiles and small bullets. 2. Pojectile Model The pesent analysis conside a 0.30 calibe (0.308 diamete), 168 gain ( 10.9 g) Siea ntenational bullet used by National Match M14 ifle is loaded into 7.62 mm M852 match ammunition fo high powe ifle competition shooting, as shown in Fig.1,2. The catidge is intended and specifically pepaed fo used in those weapons designed as competitive ifles and fo maksmanship taining. This bullet is not fo combat use. The catidge case head stamping of TCH identify the catidge. t also has a knul at the base of the catidge case and a hollow point boat-tail bullet. Basic physical and geometical chaacteistics data of the above-mentioned 7.62 mm bullet illustated biefly in Table 1. Fig mm match ammunition with a diamete of 0.30 calibe epesentative small bullet types. Table 1. Physical and geometical data of 7.62 mm small bullet type Chaacteistics 7.62 mm M852 bullet Refeence diamete, mm 7.62 Total length, mm Total mass, kg Axial moment of inetia, kg m Tansvese moment of inetia, kg m 2 Cente of gavity fom the base, mm 3. Tajectoy Flight Simulation Model A six degee of feedom igid-pojectile model [2], [3], [4], [5] has been employed in ode to pedict the "fee" atmospheic tajectoy to final taget aea without any contol pactices. The six degee of feedom flight analysis compises the thee tanslation components (x, y, z) descibing the position of the pojectile s cente of mass and thee Eule angles (φ, θ, ψ) descibing the oientation of the pojectile body with espect to Fig.3. Fig 3. No-oll (moving) and fixed (inetial) coodinate systems fo the pojectile tajectoy analysis. Fig mm bullet Two main coodinate systems ae used fo the computational appoach of the atmospheic flight motion. The one is a plane fixed (inetial fame) at the fiing site. The othe is a no-oll otating coodinate system on the pojectile body (no-oll-fame, NRF, φ = 0) with the X NRF axis along the pojectile axis of symmety and Y NRF, Z NRF axes oiented so as to complete a ight hand othogonal system. 71

3 . N. Gkitzapsi, N. E. Tsiatis, E E. Panagiotopoulos and. P. Magais/ Jounal of Engineeing Science and Technology Review 1 (2008) Newton s laws of the motion state that the ate of change of linea momentum must equal the sum of all the extenally applied foces and the ate of change of angula momentum must equal the sum of the extenally applied moments, as shown espectively in the following foms: d m = F tot (1) dt d H = M tot (2) dt whee the total foce acting on the pojectile compises the weight, the aeodynamic foce and the Magnus foce. Moeove, the total moment vecto compises the moment due to the standad aeodynamic foce, the Magnus aeodynamic moment and the unsteady aeodynamic moment. Theefoe, the twelve state vaiables x, y, z, φ, θ, ψ, u, v, w, p, q and ae necessay to descibe position, flight diection and velocity at evey point of the pojectile s atmospheic flight tajectoy. ntoducing the components of the acting foces and moments expessed in the no-oll-fame () otating coodinate system in Eqs (1, 2) with the dimensionless ac length s as an independent vaiable, the following full equations of motion fo six-dimensional flight ae deived: x = cosψ cosθ u sinψ v w cosψ sinθ (3) y = cosθ sinψ u v cosψ w sinθ sinψ (4) z = sinθ u w cosθ (5) φ = p tanθ (6) θ = q (7) ψ = (8) cosθ u = g sinθ ρ C ρ C α 8 m 8 m (9) ρ C β v q w 8 m ( C C ) v p w tan u v 3 = ρ θ 8m L ( C C ) w q u tan p v (10) w 3 = g cosθ ρ θ (11) 8m L 5 p = p ρ C 1 6 LP (12) The pojectile dynamics tajectoy model consists of twelve highly fist ode odinay diffeential equations, which ae solved simultaneously by esoting to numeical integation using a 4th ode Runge-Kutta method. n these equations, the following sets of simplifications ae employed: velocity u eplaced by the total velocity because the side velocities v and w ae small. The aeodynamic angles of attack α and sideslip β ae small fo the main pat of the atmospheic tajectoy α w /, β v /, the pojectile is geometically symmetical XY = YZ = XZ = 0, = ZZ and aeodynamically symmetic. With the afoementioned assumptions, the expessions of the distance fom the cente of mass to the standad aeodynamic and Magnus centes of pessue ae simplified. 4. nitial Spin Rate Estimation n ode to have a statically stable flight pojectile tajectoy motion, the initial spin ate p0 pediction at the gun muzzle in the fiing site us impotant. Accoding to McCoy definitions [1], the following fom is used: p = 2 / ( ad / s ) (15) 0 0 η whee ( C C ) 3 ρ 4 ρ v q = w L E C p LE 8 L MCP 16 YPA MCM 5 ρ 4 ρ C p 2 tanθ 16 q C 8 ( C C ) 3 ρ 4 w = v L E p ρ C LE 8 L MCP 16 YPA MCM 5 ρ 4 C ρ C p q 16 8 is the initial fiing velocity (m/s), o η the ifling twist ate at the gun muzzle (calibes pe tun), and the efeence diamete of the pojectile type (m). Typical values of ifling twist η ae 1/18 calibes pe tun fo big pojectile and 12 inches pe tun fo small bullet, espectively. 5. Computational Simulation q tan θ (13) (14) The flight dynamic model of 7.62 mm bullet involves the solution of the set of the twelve nonlinea fist ode odinay diffeentials, Eqs (3-14), which ae solved simultaneously by esoting to numeical integation using a 4th ode Runge- Kutta method, and egad to the 6- nominal atmospheic pojectile flight. 72

4 . N. Gkitzapsi, N. E. Tsiatis, E E. Panagiotopoulos and. P. Magais/ Jounal of Engineeing Science and Technology Review 1 (2008) The constant [7], [8] dynamic flight model uses mean values of the expeimental aeodynamic coefficients vaiations [1] (see table 2). Table 2 Tajectoy aeodynamic paametes of atmospheic flight dynamic model 7.62mm with constant aeodynamic coefficients C = 0.235,C = 2.205,C = 0.01 C = L,C LP = 2.92,C YPA = 1.26 The esults give the computational simulation of the 6- non-thusting and non-constained flight tajectoy path fo some specific big pojectiles and small bullets types. nitial flight conditions fo the dynamic flight simulation model with constant and vaiable aeodynamic coefficients ae illustated in Table 3 fo the examined test case. Table 3. nitial flight paametes of the pojectile examined test cases. nitial flight data 7.62 mm bullet x, m 0.0 y, m 0.0 z, m 0.0 φ, deg 0.0 θ, deg 1,7 and 15 ψ, deg 2.0 u, m/s v, m/s 0.0 w, m/s 0.0 p, ad/s 16,335.0 q, ad/s 0.0, ad/s Results and iscussion The flight path tajectoies of the pesent dynamic model with initial fiing velocity of 793 m/sec and ifling twist 12 inches pe tun at initial pitch angles of 1,7 and 15 ae indicated in Figue 4 fo two cases: constant 7 and vaiable 8 aeodynamic coefficients. The small bullet is examined fo its atmospheic constant flight tajectoies pedictions in Fig. 4, fied at no wind sea-level conditions at 1 o gives a ange to impact at 1,190 m with a maximum height at almost 6.7 m. At 7 o, the pedicted level-gound ange is appoximately 2,778 m and the height is 150 m, and at 15 degees the impact point and the height ae 3,410 m and 440 m, espectively. Fig 4. mpact points and flight path tajectoies with constant and vaiable aeodynamic coefficients fo 7.62 mm bullet at low and high quadant elevation angles of 1 o, 7 o, and 15 o. The same figue shows the tajectoy flight path of 7.62 mm bullet with vaiable aeodynamic coefficients, and at 1 o pitch angle gives ange almost 1 km with height 6 m, at 7 degees gives impact point 2,600 m and height 125 m and at 15 degees the ange and the height ae 3,545 m and 388 m, espectively. Thee is big diffeence with the flight path tajectoy with constant aeodynamic coefficients. Fig 6. elocity vesus ange with constant aeodynamic coefficients fo 105 mm pojectile. Futhemoe, the computational esults fo 7.62 mm bullet flight path with constant and vaiable aeodynamic coefficients at elevation angles of 1, 7 and 15 ae illustated in Fig. 5 and 6. At pitch angle of 1 degee, the velocity fo the flight path with constant aeodynamic coefficients deceases to the values of almost 350 m/sec and 300 m/sec with vaiable aeodynamic coefficients. At pitch angle of 7 degees, the velocity deceases to the values of almost 124 and 169 m/sec. Moeove, at pitch angle of 15 degees the velocity deceases to values of 101 and 132 m/sec, espectively. 73

5 . N. Gkitzapsi, N. E. Tsiatis, E E. Panagiotopoulos and. P. Magais/ Jounal of Engineeing Science and Technology Review 1 (2008) Moeove, fom the esults of the pesented applied method with vaiable constant aeodynamic coefficients, at 1, 7 and 15 pitch angles, the oll ate deceases to the values 14,620 ad/s, 11,626 ad/s and 10,583 ad/s, espectively.( Fig. 9) Fig 7. elocity vesus ange of 7.62 mm M852 bullet at initial elevation angles of 1 o, 7 o and 15 o. Figue 8 shows the time of the flight tajectoy with constant and vaiable coefficients, at sea level with no-wind fo 7.62 mm bullet at elevation angles 1 o, 7 o and 15 o. The bullet with initial fiing velocity of 793 m/s, gives values of time 2,3 sec, 9,5 sec and 17 sec, espectively. Fig 9. Roll ate vesus ange at elevation angles of 1, 7 and 15 degees, fo 7.62mm bullet with vaiable and constant aeodynamic coefficients. 7. Conclusion The complicated six degees of feedom (6-OF) simulation flight dynamics model is applied fo the accuate pediction of shot and long-ange tajectoies esults fo small bullets. t takes into consideation the Mach numbe and the total angle of attack vaiation effects by means of the vaiable and constant aeodynamic coefficients. The computational esults of the poposed synthesized analysis ae in good ageement compaed with othe technical data and ecognized exteio atmospheic pojectile flight computational models. Fig 8. Time path tajectoies at elevation angles of 1, 7 and 15 degees, fo 7.62mm bullet with vaiable and constant aeodynamic coefficients. Refeences 1. McCoy, R., Moden Exteio Ballistics, Schiffe, Attlen, PA, 1999, pp , , 244, Fowle, R., Gallop, E., Lock, C., and Richmond H., The Aeodynamics of Spinning Shell, Philosophical Tansactions of the Royal Society of London, Seies A: Mathematical and Physical Sciences, ol. 221, Hainz, L., and Costello, M., Modified Pojectile Linea Theoy fo Rapid Tajectoy Pediction, Jounal of Guidance, Contol, and ynamics, ol.28, No. 5, 2005, pp Etkin, B., ynamics of Atmospheic Flight, John Wiley and Sons, New Yok, Amouso, M. J., Eule Angles and Quatenions in Six egee of Feedom Simulations of Pojectiles, Technical Note, Costello, M., and Andeson,., Effect of ntenal Mass Unbalance on the Teminal Accuacy and Stability of a pojectile, AAA Pape, Gkitzapis,., N., Panagiotopoulos, E. E., Magais,. P., Papanikas,. G.: Atmospheic Flight ynamic Simulation Modelling of Spin-Stabilized Pojectiles, Poceedings of the 2nd ntenational Confeence on Expeiments / Pocess / System Modelling / Simulation / Optimization, 2nd C-EpsMsO, 4-7 July 2007, Athens, Geece. 8. Gkitzapis,. N., Panagiotopoulos, E. E., Magais,. P., Papanikas,. G.: Computational Atmospheic Tajectoy Simulation Analysis of Spin-Stabilized Pojectiles and Small Bullets, Atmospheic Flight Mechanics Confeence and Exhibit, AAA Pape , August 2007, Hilton Head, South Caolina. 74