Computational atmospheric trajectory simulation analysis of spin-stabilised projectiles and small bullets

Transcription

1 Int. J. Computing Science and Mathematics, Vol. x, No. x, xxxx 1 Computational atmospheric trajectory simulation analysis of spin-stabilised projectiles and small bullets.n. Gkritzapis* Laboratory of Firearms and Tool Marks Section, Criminal Investigation ivision, Captain of Hellenic Police, Hellenic Police, Athens, Greece gritzap@yahoo.gr *Corresponding author E.E. Panagiotopoulos,.P. Margaris and.g. Papanikas Fluid Mechanics Laboratory, University of Patras, Patras, Greece hpanagio@mech.upatras.gr margaris@mech.upatras.gr papanikas@mech.upatras.gr Abstract: A mathematical model is based on the full equations of motion set up in the no-roll body reference frame and is integrated numerically from given initial conditions at the firing site. The computational flight analysis takes into consideration the Mach number and total angle of attack effects by means of the variable aerodynamic coefficients. For the purposes of the present work, linear interpolation has been applied for aerodynamic coefficients from the official tabulated database. Static stability is examined. The aerodynamic jump has the most important effect and is examined more closely for projectile and bullet flight trajectories. Keywords: aerodynamic jump; constant and variable aerodynamic coefficients; Coriolis effect; Magnus effect; static or gyroscopic stability. Reference to this paper should be made as follows: Gkritzapis,.N., Panagiotopoulos, E.E., Margaris,.P. and Papanikas,.G. (xxxx) Computational atmospheric trajectory simulation analysis of spin-stabilised projectiles and small bullets, Int. J. Computing Science and Mathematics, Vol. x, No. x, pp.xxx xxx. Biographical notes:.n. Gkritzapis is Captain of Hellenic Police in Laboratory of Firearms and Tool Marks Section. Also, is post-graduate student (Ph) in Mechanical Engineering and Aeronautics epartment, at University of Patras, for thesis of Ballistics. He took part in AFM Conference, AIAA, in South Carolina on August He was in the conference of WASET at Copyright 200x Inderscience Enterprises Ltd.

2 2.N. Gkritzapis et al. Venice and also he took part in other two International conferences in Greece. His papers were acceptance of the Open Mechanics Journal, Journal of Pyrotechnics, Journal of Battlefield Technology and International Journal of Mathematical, Physical and Engineering Sciences for publication. E.E. Panagiotopoulos is a post-graduate student in Mechanical Engineering and Aeronautics epartment at University of Patras. In his Ph thesis is dealing with the prediction of nominal entry descent flight path trajectories including real gas flow phenomena and modelling the particular conditions of the most basic planetary atmospheres. Furthermore, is interested in the scientific field of exterior ballistics atmospheric motions in addition to the complicated non-linear aerodynamic phenomena for big and small projectile types. He is participating in over 15 international conferences on the above scientific areas and recently has a publication on the Open Mechanics Journal..P. Margaris is Assistant Professor in department of Mechanical and Aeronautics Engineering at the University of Patras, Greece. His research activities/fields: 1. Multiphase flows of gas-liquid-solid particles. 2. Gas-liquid two-phase flow air-lift pumps performance. 3. Centrifugal and T-junction separation modelling in gas-liquid two-phase flow. 4. Experimental and theoretical investigation of hot air dehydration of agricultural products. 5. Experimental and theoretical investigation of capillary pumped loops. 6. Steady and transient flows in pipes and network. 7. Numerical simulation of centrifugal pump performance. 8. Fluid dynamics analysis of wind turbines and aerodynamic installations. 9. Aero-acoustic analysis and environmental impacts of wind turbines. Participating in several research projects supported by HAI, GSRT, CEC-THERMIE. He is member of AIAA, and ASME..G. Papanikas is ex-professor in department of Mechanical and Aeronautics Engineering at the University of Patras, Greece. His research activities/fields: 1. Exterior Ballistics Programs with collaboration of Hellenic efense Systems. 2. Aerodynamic hypervelocity vehicles. 3. Multiphase flows of gas-liquid-solid particles. 4. Gas-liquid two-phase flow air-lift pumps performance. 5. Centrifugal and T-junction separation modelling in gas-liquid two-phase flow. 6. Experimental and theoretical investigation of capillary pumped loops. 7. Steady and transient flows in pipes and network. 8. Numerical simulation of centrifugal pump performance. 9. Fluid dynamics analysis of wind turbines and aerodynamic installations. 10. Aero-acoustic analysis and environmental impacts of wind turbines. Participating in several research projects supported by HAI, GSRT, CEC-THERMIE. He is senior member of AIAA, and ASME. 1 Introduction The first stone hurled by prehistoric man was probably the earliest example of external ballistics. The advantages of being able to throw farther and with more power led to devices such as slings and spears. Next came the bow, and an extension of it called the ballista from which ballistics derives its name. In turn, the word ballista (McCoy, 1999) owes its origin to a Greek word ballein, meaning to throw. The ballista was a complicated device used for propelling large arrows.

3 Computational atmospheric trajectory simulation analysis 3 Pioneering English ballisticians Fowler et al. (1920) constructed the first rigid six-egree-of-freedom (6-OF) projectile exterior ballistic model. Various authors has extended this projectile model for lateral force impulses (Cooper, 2001; Guidos and Cooper, 2000), linear theory in atmospheric flight for dual-spin projectiles (Costello and Peterson, 2000; Burchett et al., 2002), aerodynamic jump extending analysis due to lateral impulsives (Cooper, 2004) and aerodynamic asymmetry (Cooper, 2003), instability of controlled projectiles in ascending or descending flight (Murphy, 1981). Costello s modified linear theory (Hainz and Costello, 2005) has also applied recently for rapid trajectory projectile prediction. The present work address a full 6-OF projectile flight dynamics analysis for accurate prediction of short and long range trajectories of high spin-stabilised projectiles and small bullets. The proposed flight dynamic model takes into consideration the influence of the most significant force and moment variations, in addition to gravity, Magnus and Coriolis effects. The applied aerodynamic coefficient analysis takes into consideration the variations depending on the Mach number flight and total angle of attack. Gyroscopic stability and deflection from the initial line of departure due to the aerodynamic jump phenomenon are also very important and examined more closely for the presented exterior atmospheric flight estimation of the examined projectile types. The program of the present work gives more accurate results than the programs with constant aerodynamic coefficients and run very fast until linear interpolation with the aerodynamic coefficients are used. The efficiency of the developed method gives satisfactory results compared with published data of verified experiments and computational codes on dynamics model analysis of short and long-range trajectories of spin-stabilised projectiles and small bullets. 2 Projectile model The present analysis considers two different types of representative projectiles. A typical formation of the cartridge 105 mm HE M1 projectile is presented in Figure 1, and is used with various 105 mm howitzers such as M49 with M52, M52A1 cannons, M2A1 and M2A2 with M101, M101A1 cannons, M103 with M108 cannon, M137 with M102 cannon as well as NATO L14 MO56 and L5. Cartridge 105 mm HE M1 is of semi-fixed type ammunition, using adjustable propelling charges in order to achieve desirable ranges. The projectile producing both fragmentation and blast effects can be use against personnel and materials targets. Also a 0.30 caliber (0.308 diameter), 168 grain (10.9 gr) Sierra International bullet used by National Match M14 rifle is loaded into 7.62 mm M852 match ammunition for high power rifle competition shooting, as shown in Figure 2. The cartridge is intended and specifically prepared for used in those weapons designed as competitive rifles and for marksmanship training. This bullet is not for combat use. The cartridge case head stamping of MATCH identify the cartridge. It also has a knurl at the base of the cartridge case and a hollow point boat-tail bullet.

4 4.N. Gkritzapis et al. Figure mm HE M1 high explosive projectile artillery ammunition for howitzers (see online version for colours) Figure mm match ammunition with a diameter of 0.30 caliber representative small bullet types (see online version for colours) Basic physical and geometrical characteristics data of the above-mentioned 105 mm HE M1 projectile and 7.62 mm bullet illustrated briefly in Table 1. Table 1 Physical and geometrical data of 105 mm big projectile and 7.62 mm small bullet types Characteristics 105 mm HE M1 projectile 7.62 mm M852 bullet Reference diameter (mm) Total length (mm) Total mass (kg) Axial moment of inertia (kg m 2 ) Transverse moment of inertia (kg m 2 ) Centre of gravity from the base (mm) Trajectory flight simulation model A 6-OF rigid-projectile model (Etkin, 1972; Joseph et al., 2006; Amoruso, 1996; Costello and Anderson, 1996) has been employed in order to predict the free atmospheric trajectory to final target area without any control practices. The mathematical model analysis comprises the three translation components (x, y, z) describing the position of the projectile s centre of mass and three Euler angles (,, ) describing the orientation of the projectile body with respect to Figure 3.

5 Computational atmospheric trajectory simulation analysis 5 Figure 3 No-roll (moving) and fixed (inertial) coordinate systems for the projectile trajectory analysis (see online version for colours) Two main coordinate systems are used for the computational approach of the atmospheric flight motion. The one is a plane fixed (inertial frame) at the firing site. The other is a no-roll rotating coordinate system on the projectile body (no-roll-frame, NRF, = 0) with the X NRF axis along the projectile axis of symmetry and Y NRF, Z NRF axes oriented so as to complete a right hand orthogonal system. Therefore, the twelve state variables x, y, z,,,, u, v, w, p, q and r are necessary to describe position, flight direction and velocity at every point of the projectile s atmospheric flight trajectory. Introducing the components of the acting forces and moments expressed in the no-roll-frame (~) rotating coordinate system, with the dimensionless arc length s as an independent variable, the following full equations of motion for six-dimensional flight are derived: x cos cos sin v wcos sin V (1) V y cos sin vcos w sin sin V (2) V z sin w cos (3) V p tan r (4) V V q (5) V r (6) V cos

6 6.N. Gkritzapis et al u g sin V CX VCX V 8m 8m VCX v rq w 8m V V (7) 3 4 v CNA( v vw ) pcnpa pw tan r 8m 16m V (8) 3 4 w gcos CNA( w ww ) pcnpa qtan pv (9) V 8m 16m V (10) 5 p p CLP 16I XX v v q C ( w w ) LE C p LE 3 4 w NA w MCP YPA MCM 8IYY 16IYY V I C q C r p r tan (11) 5 4 XX 2 MQ MA 16IYY 8IYY V IYY V w w r C ( v v ) LE pc LE 3 4 w NA w MCP YPA MCM 8IYY 16IYY V I C r C pq qr tan. (12) I I V I V 5 4 XX MQ MA 16 YY 8 YY YY The projectile dynamics trajectory model consists of 12 highly first order ordinary differential equations, which are solved simultaneously by resorting to numerical integration using a 4th order Runge-Kutta method. In these equations, the following sets of simplifications are employed: velocity u replaced by the total velocity V because the side velocities v and w are small. The aerodynamic angles of attack and sideslip are small for the main part of the atmospheric trajectory wv /, vv /, the projectile is geometrically symmetrical I XY = I YZ = I XZ = 0, I YY = I ZZ and aerodynamically symmetric. With the aforementioned assumptions, the expressions of the distance from the centre of mass to the standard aerodynamic and Magnus centres of pressure are simplified. 4 Coriolis effect The Coriolis effect is the apparent deflection of objects from a straight path if the objects are viewed from a rotating frame or reference. The effect is named after Gaspard-Gustave Coriolis, a French scientist, who described it in 1835, though the mathematics appeared in the tidal equations of Laplace in The acceleration produced by the Coriolis Effect is the quantity that is independent of the projectile weight, but varies with the projectile velocity, latitude of the firing site (lat) and the azimuth of fire (azim), relative to North, by the following equation:

7 Computational atmospheric trajectory simulation analysis 7 ( v cos(lat) sin(azim) w sin(lat)) FCor 2 ( u cos(lat) sin(azim) w cos(lat) cos(azim)). ( u sin(lat) v cos(lat) cos(azim)) For the present exterior computational atmospheric projectile trajectory analysis, the firing site latitude (lat) assumes to be at 45 Northern with 90 and 270 azimuth angle (azim), respectively. (13) 5 Atmospheric model Atmospheric properties of air are being calculated based on a standard atmosphere from the International Civil Aviation Organization (ICAO). The decrease in the air temperature with increasing altitude is accurately described, for moderate altitudes, by the following equation: T(Y) = [T 0 (F) ] e GY (14) T 0 ( F): Air temperature at the firing site ( F) Y: Altitude above firing site (ft) T(Y): Air temperature at altitude Y ( F) G: Temperature-altitude decay factor (1/ft). The standard sea-level air temperature is 59F. Appropriate values of the temperature-altitude decay factor G, are given by: G = ( ) Y (1/ft) (15) The speed of sound in air is given by the following equation: a T( Y) (ft/s). (16) The decrease in air density with increasing altitude is accurately described, for moderate altitudes, by the following equation: (Y ) = o e hy (17) o : Air density at the firing site (pounds/ft 3 ) (Y ): Air density at altitude Y (pounds/ft 3 ) h: Air density-altitude decay factor (1/ft). At sea level, the standard value of air density is given by: 0 = (lbs./ft 3 ) (18) h = ( ) Y (1/ft). (19) For altitudes up to 20,000 ft above the sea level, the above equations give essentially exact results for the variation of air temperature and air density with increasing altitude. The air density ratio is obtained from the equation of an ideal gas. The humidity has a small effect on both the air density and the speed of sound in air. The humidity correction to the air density ratio at sea level is given by:

8 8.N. Gkritzapis et al. F RH = (RH P WV /29.92) (20) F RH : Humidity correction factor to the air density ratio RH: Relative humidity (%) P WV : Water vapour pressure at saturation local temperature taking the typical value of 0.18 in Hg at 32 degrees of Fahrenheit (0 o C). In addition to the above equation (20), the humidity correction to the speed of sound in air at sea level is given in the following manner: F RHa = (RH P WV /29.92). (21) In general, increasing humidity causes a slight decrease in air density, because the air density of water vapour is less than that of dry air. On the other hand, increasing humidity causes a slight increase in the speed of sound. For air temperature below 70F (21 o C), the changes in both air density and speed of sound for 100% change in humidity, are less than 1%, and can be neglected for all practical reasons. In addition, for temperature above 70F, the humidity correction to the air density is small, but not negligible. The small correction for the humidity effect on the speed of sound should also be made at temperature above 70 o F, but it actually important only when the projectile flight velocity is near the speed of sound (M 1), were a small change in the Mach number causes a relatively large change in the drag coefficient. 6 Nonlinear geometric and aerodynamic model The flight of projectiles with large amplitude pitching and yawing motion requires a proper treatment of both geometric and aerodynamic nonlinearities. Geometric nonlinearity arises from the large size of the pitching and yawing motion itself, and it becomes important whenever the angle of attack a or the sideslip angle are large enough that either cos() or cos() is significantly less than unity. The effect of geometric nonlinearity is therefore insignificant if the yaw is small everywhere along the trajectory. On the other hand, many projectiles show significant nonlinear behaviour, even at relatively small amplitude pitching and yawing motion. In our present analysis, we will assume that the yaw level is small enough to neglect geometric nonlinearity, but will retain all significant nonlinear aerodynamic forces and moments. The constant dynamic flight model (Gkritzapis et al., 2007) uses mean values of the experimental average aerodynamic coefficients variations (McCoy, 1999) and gives a first estimation of the dynamic flight path trajectory. Moreover, during the atmospheric flight of the projectile or bullet there is an important change of the angle of attack and the Mach number. Therefore, it is more convenient to use variable aerodynamic coefficients for the accurate computational simulation of the atmospheric flight motion of projectiles and bullets to final impact area. For this purpose, linear interpolation for variable aerodynamic coefficients has been applied taking from official tabulated exterior ballistics database (McCoy, 1999).

9 Computational atmospheric trajectory simulation analysis 9 7 Initial spin rate estimation In order to have a statically stable flight projectile trajectory motion, the initial spin rate p 0 prediction at the gun muzzle in the firing site us important. According to McCoy (1999) definitions, the following form is used: p 2 V / (rad/s) (22) 0 0 where V 0 is the initial firing velocity (m/s), the rifling twist rate at the gun muzzle (calibers per turn), and the reference diameter of the projectile type (m). Typical values of rifling twist are 1/18 calibers per turn for big projectile and 12 inches per turn for small bullet, respectively. 8 Static or gyroscopic stability Any spinning object will have gyroscopic properties. In spin-stabilised projectile, the centre of pressure, the point at which the resultant air force is applied, is located in front of the centre of gravity. Hence, as the projectile leaves the muzzle it experiences an overturning movement caused by air force acting about the centre of mass. It must be kept in mind that the forces are attempting to raise the projectile s axis of rotation. In Figure 4, two cases of static stability are demonstrated: in the top figure, CP lies behind the CG so that a clockwise (restoring) moment is produced. This case tends to reduce the yaw angle and return the body to its trajectory, therefore statically stable. Conversely, the lower figure, with CP ahead of CG, produces an anti-clockwise (overturning) moment, which increases a further and is therefore statically unstable. It also possible to have a neutral case in which CP and CG are coincident whereby no moment is produced. Figure 4 Static stability/instability conditions There is clearly an important relation between the distance between the centre of pressure and the centre of gravity and the centre of the round. This distance is called the static margin. By definition, it is positive for positive static stability, zero for neutral stability and negative for negative stability. Classical exterior ballistics defines the gyroscopic stability factor S g in the following generalised form:

10 10.N. Gkritzapis et al. S g I p 2I S V C 2 2 XX 2 YY ref OM. (23) This may be rearranged into: S g 2 2 2I XX p YY V COM I The aforementioned equation (24) shows that the static factor is proportional to four terms product, depending on the geometric technical characteristics of the projectile shape model, the square axial spin to velocity ratio, the aerodynamic overturning moment coefficient and the proposed atmospheric density model. (24) 9 Aerodynamic jump deflection The epicyclic pitching and yawing motion produces three separate and distinct effects on a flat-fire trajectory: aerodynamic jump, epicyclic swerve and drift. For modern high-velocity, small-yaw, ground-launched flat-fire trajectories, the epicyclic swerve and the drift are generally small and insignificant compared with the aerodynamic jump. According to McCoy (1999) analysis (with no mass symmetry), the following form is used: C I p 2 L X JA Ky i 0 0. COM IY V We will assume that the initial yaw 0 is zero and with the help of the equation (26), the initial complex yaw rate is calculated as: 0 i. (26) V 0 For a statically stable projectile or missile, the aerodynamic jump takes the same direction as the initial complex yaw rate. Conversely, the direction of aerodynamic jump for statically unstable, spin-stabilised projectile is reversed. If the initial yawing motion of a spin-stabilised shell starts out to the right, the aerodynamic jump will deflect it 180 degrees out of phase, or to the left of the initial line of departure. Equation (25) also shows that for spinning projectiles, a second component of aerodynamic jump can exist, proportional to the product of the non-dimensional axial spin, and an initial complex yaw (as opposed to an initial yaw rate). This part of aerodynamic jump acts in a direction perpendicular to the initial complex yaw, and it gives rise to such phenomena, when firing into a crosswind. (25) 10 Computational simulation The flight dynamic models of 105 mm HE M1 and 7.62 mm projectile types involves the solution of the set of the twelve nonlinear first order ordinary differentials,

11 Computational atmospheric trajectory simulation analysis 11 equations (1) (12), which are solved simultaneously by resorting to numerical integration using a 4th order Runge-Kutta method, and regard to the 6- nominal atmospheric projectile flight. The results give the computational simulation of the 6- non-thrusting and non-constrained flight trajectory path for some specific big projectiles and small bullets types. Initial flight conditions for both dynamic flight simulation models with constant and variable aerodynamic coefficients are illustrated in Table 2 for the examined test cases. Table 2 Initial flight parameters of the projectile examined test cases Initial flight data 105 mm HE M1 projectile 7.62 mm bullet x, m y, m z, m , deg , deg 15, 30, 45, 60 and , 10, 20 and 32, deg u, m/s v, m/s w, m/s p, rad/s ,335.0 q, rad/s r, rad/s 3.61 and Results and discussion The flight path trajectory motion with constant aerodynamic coefficients of the big 105 mm projectile with initial firing velocity of 494 m/sec, initial yaw angle 3 degrees, rifling twist rate 1 turn in 18 calibers (1/18) and initial yaw rates 3.61 rad/s and 3.64 rad/s at 45 and 70 o, respectively, are indicated in Figure 5. The calculated impact points of the above no-wind trajectories with the proposed constant aerodynamic coefficients compared with accurately estimations of McCoy s (1999) flight trajectory analysis provide basic differences for the main part of the atmospheric flight motion for the same initial flight conditions. The flight dynamic trajectory McCoy s prediction at pitch angles of 45 and 70 are in good agreement with the present flight analysis that takes into account variable aerodynamic coefficients at 25 C and humidity of 80% are in good agreement, as presented in Figure 6. At 45 the McCoy model for 105 mm M1 projectile, fired at sea-level neglecting wind conditions, gives a predicted range to impact of approximately 11,500 m and a maximum height at almost 3490 m. From the results of the presented applied method, the maximum range is 11,600 m and the maximum height is almost 3,490 m, as shown in Figure 6. Also at 70, the predicted level-ground range of McCoy s model is 7310 m with maximum height at about 6 km while the proposed trajectory simulation gives 7550 m and 6100 m, respectively.

12 12.N. Gkritzapis et al. Figure 5 Impact points and flight path trajectories with constant aerodynamic coefficients for 105 mm projectile compared with McCoy s trajectory data (see online version for colours) Figure 6 Present trajectory simulations with atmospheric firing conditions 25C and 80% humidity at pitch angles of 45 and 70 compared with McCoy s flight path motions (see online version for colours) In Figure 7, the present study of the 105 mm HE M1 projectile trajectory motion with variable aerodynamic coefficients compared with McCoy s flight atmospheric model at pitch angles of 45 and 70, provide satisfactory agreement for the same conditions. The diagram shows that the 105 mm HE M1 projectile, fired at sea-level with an angle of 45 (cyan solid line) and no wind, the predicted range to impact is 11,500 m and the maximum height is 3490 m. At 70 (green solid line), the predicted impact point is 7310 m, and the maximum height is slight over 6000 m. The flight path trajectories with initial pitch angles of 15, 30 and 60 are also shown in the same figure in comparison with the 45 and 70 flight motions. It can be stated that the maximum impact range is at 45 initial firing angle while the minimum presents at 15.

13 Computational atmospheric trajectory simulation analysis 13 Figure 7 Impact points and flight path trajectories with variable aerodynamic coefficients for 105 mm projectile at low and high quadrant elevation angles of 15, 30, 45, 60 and 70 compared with McCoy s trajectory data (see online version for colours) The small bullet of 7.62 mm diameter is also examined for its atmospheric variable flight trajectories predictions in Figure 8 at low and high pitch angles of 0.84, 10, 20, 32, with initial firing velocity of 793 m/s, initial yaw angle 2, yaw rate 25 rad/s and rifling twist 12 inches per turn. The impact points of the above trajectories are compared with an accurately flight path prediction with Nennstiel s (1996) trajectory analysis for cartridge 7.62 mm ball M80 bullet type with initial firing velocity of 838 m/s. At 0.84 o, the 7.62 mm M852 bullet, fired at no wind sea-level conditions gives a range to impact at 920 m with a maximum height at almost 5 m. At 32 o, the predicted level-ground range is approximately 4280 m and the height is 1140 m. For the same initial pitch angle, the 7.62 mm M80 ball-bullet of Nennstiel s flight path has a smaller range to impact and a maximum height at 1170 m. Figure 8 Impact points and flight path trajectories with variable aerodynamic coefficients for 7.62 mm bullet at quadrant elevation angles of 0.84, 10, 20 and 32 compared with Nennstiel s model trajectory (see online version for colours)

14 14.N. Gkritzapis et al. Figures 9 and 10 show the deflection of the flight trajectory at sea level with no-wind for the 105 mm projectile and 7.62 mm bullet. The present analysis trajectory of the 105 mm HE M1 projectile with initial firing velocity of 494 m/s, initial yaw angle 3 degrees, rifling twist rate 1 turn in 18 calibers (1/18) and initial positive yaw rate 3.61 rad/s, at pitch angles of 15, 30 and 45, gives the values of deflection as 146, 140 and 530 m, respectively. Moreover, for 60 and 65 the value of deflection is almost 700 m. Figure 9 Cross range vs. range with variable aerodynamic coefficients for 105 mm projectile (see online version for colours) Figure 10 Cross range computational predictions of 7.62 mm M852 bullet at initial elevation angles of 0.84, 10, 20 and 32 (see online version for colours) The small bullet of 7.62 mm diameter at low and high pitch angles of 0.84, 10, 20 and 32, with initial firing velocity of 793 m/s, initial yaw angle 2 degrees, yaw rate 25 rad/s and rifling twist 12 inches per turn, gives positive values of cross range 30, 90, 110 and 115 m, respectively. The presented flight simulation trajectory prediction model for 105 mm projectile at 45 degrees latitude, gives a deflection due to Coriolis effect of +13 m (relative to the nominal impact range estimation of 11,500 m) at 90 degrees and 13 m at 270 degrees azimuth angle. This deflection of 13 m is not an insignificant amount for high accuracy projectile motion analysis. In addition, for the flight trajectory of 7.62 mm bullet, the Coriolis effect is very small, so it can be treated as negligible. It is readily apparent that

15 Computational atmospheric trajectory simulation analysis 15 the Coriolis effect is an important consideration, when firing at very long ranges (i.e., Paris Gun, 210 mm, was used to bombard the city of Paris in 1918, from a range of approximately 120 km, the Coriolis effect was 400 m to hit the shell shortly). The strong characteristic alterations of the total angle of attack influence the distributions of the most basic projectile trajectory phenomena. Its effects are indicated in Figure 11 for the 105 mm M1 projectile type fired from a 1/18 twist cannon with a muzzle velocity of 494 m/s at quadrant elevation angles of 45 and 70 degrees, respectively. Figure 11 Comparative total angle of attack on the range of 105 mm projectile at 45 and 70 degrees After the damping of the initial transient motion, at apogee, the stability factor for 45 degrees has increased from 3.1 at muzzle to 23 and then decreased to 8 at final impact point area, as presented in Figure 13. The corresponding flight behaviour at 70 degrees initial pitch angle shows that the transient motion damps out quickly and the yaw of repose grows nearly 13 degrees at apogee, where the gyroscopic static stability factor has increased from 3.1 to 121 and then decreased at almost 6.6 at the impact point. It is very important to investigate more closely the two critical areas Z 1 and Z 2 (crooked lines). The static stability factor s variation is not continuous due to the aerodynamic overturning moment coefficient C OM, which affects static margin. Therefore, any variations in the positions of either CP or CG will directly affect the value of C OM. The position of CP will certainly vary during flight if the projectile passes through the transonic flow regime, due to complicated shock and expansion fan movements. In particular, it rises abruptly as the inverse C OM term is highly reduced. This therefore means that S g reduces under transonic conditions (M 1). Figure 12 shows the angle of attack variations with range for 7.62 mm bullet, fired from a 12 twist per turn with a muzzle velocity of 793 m/s at quadrant elevation angles of 10 and 32 degrees, respectively. The transient motion damps out very quickly for the two cases. The gyroscopic stability factor for 32 is 1.7 at muzzle, grows to 29 at the summit of the trajectory, and then decrease to value of 12 at the impact point, as indicated in Figure 14. In addition, the stability factor for 10 initial firing angle was 1.7 at muzzle and then grows to value of 17 at the impact point. The estimation of static stability factors, equation (24), for the examined projectile test cases is based only on square axial spin to velocity ratio due to the variable overturning aerodynamic moment. This magnitude can be state that if a projectile or bullet is statically stable at the muzzle, it will be statically stable for the rest of its

16 16.N. Gkritzapis et al. atmospheric flight motion. As the rotational velocity is much less damped than the transversal velocity (which is damped due to the action of the drag force), the static factor S g increases at least for the major part of the predicted projectile trajectory. Figure 12 Total angle of attack vs. range at initial pitch angles of 10 and 32 degrees for 7.62 mm bullet (see online version for colours) Figure 13 Comparative static stability variations at high and low quadrant angles for 105 mm projectile (see online version for colours) Figure 14 Static stability vs. range at firing pitch angles of 10 and 32 degrees, for 7.62 mm bullet (see online version for colours)

17 Computational atmospheric trajectory simulation analysis 17 Aerodynamic jump effect is also examined for the trajectory flight model of 105 mm projectile fired at a muzzle velocity of 494 m/s, from a barrel with 1 turn in 18 calibers (1/18) twist of rifling, with initial yaw rates of 3.61 and 3.64 rad/s, at pitch angles of 45 and 70 degrees, respectively. For the right-directed initial yaw rate, the aerodynamic jump at range of 200 m produces a deflection of almost 11 cm to the right, as indicated in Figure 15. The dashed curve seems to be the mean, or the average direction of the damped sinusoidal motion. Figure 15 eflection vs. range, showing the aerodynamic jump at elevation angles of 45 and 70 degrees for 105 mm projectile (see online version for colours) Also for the trajectory flight motion of 7.62 mm bullet, fired at a muzzle velocity 793 m/s at elevation angles 0.84 and 32, with initial yaw rate 25 rad/s, the aerodynamic jump deflection from the initial line of departure is estimated to be almost 2 cm to the right at 100 m range, as shown in Figure 16. The above diagrams are an excellent illustration of the fact that the aerodynamic jump phenomenon generally has a much more significant effect on the atmospheric complicated flight projectiles and small bullets. Figure 16 eflection from the initial line of departure for 7.62 mm bullet, at pitch angles 0.84 and 32 degrees (see online version for colours)

18 18.N. Gkritzapis et al. 12 Conclusions The complicated 6-OF simulation flight dynamics model is applied for the accurate prediction of short and long range trajectories results for high and low spin-stabilised projectiles and small bullets. It takes into consideration the Mach number and the total angle of attack variation effects by means of the variable and constant aerodynamic coefficients. The program of the present work runs so fast as with linear interpolation of the variable aerodynamic coefficients. Criteria and analysis of gyroscopic stability are also examined. The aerodynamic jump has the most important effect to the initial deflection caused by the pitching and yawing motion and investigated more closely. The computational results of the proposed synthesised analysis are in good agreement compared with other technical data and recognised exterior atmospheric projectile flight computational models. References Amoruso, M.J. (1996) Euler Angles and Quaternions in Six egree of Freedom Simulations of Projectiles, US Army Armament, Munitions and Chemical Command Picatinny Arsenal, New Jersey , Technical Note, pp Burchett, B., Peterson, A. and Costello, M. (2002) Prediction of swerving motion of a dual-spin projectile with lateral pulse jets in atmospheric flight, Mathematical and Computer Modeling, Vol. 35, Nos. 1 2, pp Cooper, G. (2001) Influence of yaw cards on the yaw growth of spin stabilized projectiles, Journal of Aircraft, Vol. 38, No. 2, pp Cooper, G. (2003) Projectile Aerodynamic Jump ue to Lateral Impulsives, Army Research Laboratory, Aberdeen Proving Ground, USA, ARL-TR-3087, pp.1 4. Cooper, G. (2004) Extending the Jump Analysis for Aerodynamic Asymmetry, Army Research Laboratory, Aberdeen Proving Ground, USA, ARL-TR-3265, pp.1 4. Costello, M. and Anderson,. (1996) Effect of Internal Mass Unbalance on the Terminal Accuracy and Stability of a Projectile, American Institute of Aeronautics and Astronautics. Costello, M. and Peterson, A. (2000) Linear theory of a dual-spin projectile in atmospheric flight, Journal of Guidance, Control, and ynamics, American Institute of Aeronautics and Astronautics, Vol. 23, No. 5, pp Etkin, B. (1972) ynamics of Atmospheric Flight, John Wiley and Sons, New York. Fowler, R., Gallop, E., Lock, C. and Richmond, H. (1920) The aerodynamics of spinning shell, Philosophical Transactions of the Royal Society of London, Series A: Mathematical and Physical Sciences, Vol. 221, pp Gkritzapis,.N., Panagiotopoulos, E.E., Margaris,.P. and Papanikas,.G. (2007) Atmospheric flight dynamic simulation modelling of spin-stabilized projectiles, Proceedings of the 2nd International Conference on Experiments/Process/System Modelling/Simulation/Optimization, 2nd IC-EpsMsO, 4 7 July, Athens, Greece, pp.1 8. Guidos, B. and Cooper, G. (2000) Closed Form Solution of Finned Projectile Motion Subjected to a Simple In-flight Lateral Impulse, American Institute of Aeronautics and Astronautics, Paper Hainz, L. and Costello, M. (2005) Modified projectile linear theory for rapid trajectory prediction, Journal of Guidance, Control, and ynamics, American Institute of Aeronautics and Astronautics, Vol. 28, No. 5, pp Joseph, K., Costello, M. and Jubaraj, S. (2006) Generating an Aerodynamic Model for Projectile Flight Simulation using Unsteady Time Accurate Computational Fluid ynamic Results, Army Research Laboratory, Aberdeen Proving Ground, USA, ARL-CR-577, pp.1 6.

19 Computational atmospheric trajectory simulation analysis 19 McCoy, R. (1999) Modern Exterior Ballistics, Schiffer, Attlen, PA, pp , 217, 218, 244, 248. Murphy, C. (1981) Instability of controlled projectiles in ascending or descending flight, Journal of Guidance, Control, and ynamics, Vol. 4, No. 1, pp Nennstiel, R. (1996) How do the bullets flight?, Journal of AFTE, Vol. 28, No. 2, pp Nomenclature C X Axial force aerodynamic coefficient CAN Normal force aerodynamic coefficient C NPA Magnus force aerodynamic coefficient C LP Roll damping moment aerodynamic coefficient C MQ Pitch damping moment aerodynamic coefficient C MA Overturning moment aerodynamic coefficient C YPA Magnus moment aerodynamic coefficient x, y, z Projectile position coordinates in the inertial frame, m m Projectile mass, kg Projectile reference diameter, m s imensionless arc length S g Stability factor S ref Projectile reference area ( 2 /4), m 2 V uvw,, Total aerodynamic velocity, m/s Projectile velocity components expressed in the no-roll-frame, m/s u, v, w Wind velocity components in no-roll-body-frame, m/s w w w pqr,, a t Projectile roll, pitch and yaw rates in the moving frame, respectively, rad/s 2 2 Total angle of attack, at a, deg g Gravity acceleration, m/s 2 I Projectile inertia matrix I XX Projectile axial moment of inertia, kg m 2 I YY Projectile transverse moment of inertia about y-axis through the center of mass, kg m 2,, iagonal components of the inertia matrix,, Off-diagonal components of the inertia matrix LE MCM istance from the centre of mass (CG) to the Magnus centre of pressure (CM) along the station line, m LE MCP F Cor lat azim istance from the centre of mass (CG) to the aerodynamic center of pressure (CP) along the station line, m Vector Coriolis force, N Latitude of the firing site Azimuth of the firing point

20 20.N. Gkritzapis et al. J A p / V 2 K y Aerodynamic jump Spin per caliber of travel, or in non-dimensional units, the ratio of axial spin to forward velocity Non-dimensional transverse moment of inertia IYY / m 2 C L Aerodynamic lift force coefficient, C C C cos( a ) L NA X t i Complex number ( i 1) ensity of air, kg/m 3,, Projectile roll, pitch and yaw angles, respectively, deg, 0 0 Subscripts o Aerodynamic angles of attack and sideslip, deg Initial complex yaw angle Initial complex yaw rate or tip off rate (rads/cal) Initial values at the firing site