A Six Degree of Freedom Trajectory Analysis of Spin- Stabilized Projectiles

nternational onference of omputational Methods in Sciences and Engineering 007,MSE 007, orfu, Greece, 5-30 Sep. 007 A Six egree of Freedom Trajectory Analysis of Spin- Stabilized Projectiles imitrios N. Gkritzapis **, Elias E. Panagiotopoulos 1, ionissios P. Margaris and imitrios G. Papanikas 3 ** Laboratory of Firearms and Tool Marks Section, riminal nvestigation ivision, First Lieutenant of Hellenic Police, Hellenic Police, 115 Athens Greece and Postgraduate student of Fluid Mechanics Laboratory, University of Patras, 6500 Patras, Greece 1 Postgraduate student, Assistant Professor, 3 Professor of Mechanical Engineering and Aeronautics ept, University of Patras, 6500 Patras, Greece Abstract. A full six degrees of freedom (6-OF) flight dynamics model is proposed for the accurate prediction of short and long-range trajectories of high and low spin-stabilized projectiles via atmospheric flight to final impact point. The projectile is assumed to be both rigid (non-flexible), and rotationally symmetric about its spin axis launched at low and high pitch angles. The projectile maneuvering motion depends on the most significant force and moment variations in addition to gravity and Magnus Effect. 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 from the tabulated database of Mcoy s book. The aforementioned variable flight model is compared with a trajectory atmospheric motion based on appropriate constant mean values of the aerodynamic projectile coefficients. Static stability, also called gyroscopic stability, is examined as a necessary condition for stable flight motion in order to locate the initial spinning projectile rotation. Static stability examination takes into account the overturning moment variations with Mach number flight motion. The developed method gives satisfactory results compared with published data of verified experiments and computational codes on atmospheric dynamics model analysis. Keywords: Trajectory ynamics Simulation, ariable Aerodynamic Analysis, High and Low Pitch Angles, Magnus Effects, Symmetric Projectiles, Static Stability riteria. PAS: 00, 0, 04 SYMBOLS = zero-yaw drag force aerodynamic coefficient = zero-yaw axial force aerodynamic coefficient X = square drag force aerodynamic coefficient = yaw angle square axial force aerodynamic coefficient X L = lift force aerodynamic coefficient = normal force aerodynamic coefficient NA LP = roll damping aerodynamic coefficient = pitch damping aerodynamic coefficient MQ = pitch moment coefficient YPA = Magnus moment coefficient = Magnus force aerodynamic coefficient NPA S = stability factor g x,y,z = projectile position, in inertial space u,v,w = projectile velocity components, expressed in no roll-frame p = projectile roll rate, expressed in no-roll-frame

nternational onference of omputational Methods in Sciences and Engineering 007,MSE 007, orfu, Greece, 5-30 Sep. 007 q,r = projectiles pitch and yaw rates expressed in no-roll-frame θ, ψ = projectiles pitch and yaw angles φ = projectile roll angle = projectile inertia matrix XX,, ZZ = diagonal components of the inertia matrix XY,YZ, XZ = off-diagonal components of the inertia matrix = total aerodynamic velocity ρ = atmospheric density S ref = projectile reference area (d /4) m = mass of projectile α, β = aerodynamic angles of attack and sideslip R = vector from the projectile center of mass to the center of pressure R = vector from the projectile center of mass to the Magnus center of pressure X R = distance from center of mass to the Magnus center of pressure along station line MM R = distance from center of mass to center of pressure along station line MP = projectile reference diameter g = gravity NTROUTON Ballistics is the science that deals with the motion of projectiles. The word ballistics was derived from the Latin ballista, which was an ancient machine designed to hurl a javelin. The modern science of exterior ballistics [1] has evolved as a specialized branch of the dynamics of rigid bodies, moving under the influence of gravitational and aerodynamic forces and moments. Pioneering English ballisticians Fowler, Gallop, Lock, and Richmond [] constructed the first rigid six-degree-offreedom projectile exterior ballistics model. arious authors have extended this projectile model for lateral force impulses [3, 4] dual-spin projectiles [5, 6] etc. The present work address a full six degrees of freedom (6-OF) projectile flight dynamics analysis for accurate prediction of short and long range trajectories of high spin-stabilized projectiles. The proposed flight dynamic model takes into consideration the influence of the most significant force and moment variations, in addition to wind and gravity forces. The variable aerodynamic coefficients are taken into account variations depending on the Mach number flight and total angle of attack. Additionally, this flight trajectory analysis is compared with another flight model based on appropriate constant mean values of the aerodynamic coefficients. The developed computational method gives satisfactory agreement with published data of verified experiments and computational codes on atmospheric projectile trajectory analysis for various initial firing flight conditions. Gyroscopic stability factor is very important for stable flight. A new proposal definition for static stability is applied. The present analysis considers the cartridge 105mm HE M1 with 105mm howitzer such as M103 with M108 cannon, as a representative projectile type. Projectile Model A typical formation of the artridge 105mm HE M1 projectile is presented (see figure 1), and is used with various 105mm Howitzers (see figure ) such as US M49 with M5, M5A1 cannons, 1 & with M101, M101A1 cannons, M103 with M108 cannon, M137 with M10 cannon as well as NATO L14 MO56 and L5 (see table 1). artridge 105mm 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 used against personnel and materials targets. Technical characteristics and operational data of 105mm HE M1 are illustrated (see table ).

nternational onference of omputational Methods in Sciences and Engineering 007,MSE 007, orfu, Greece, 5-30 Sep. 007 FGURE 1: 105mm HE M1 High Explosive Projectile Artillery Ammunition For Howitzers. FGURE : Howitzer of 105mm Projectile at firing site. annon harge M5 M5A1,M101,M101A1 M10 M103 annon harge 1 7 1 7 Muzzle elocity, m/s 198,1 47,4 05,0 494,0 Maximum Range, m 3510 1170 3700 11500 TABLE 1: Maximum Range mpact of 105mm HE M1 epending on annon harge and Muzzle elocity at Firing Site. Technical escription The cartridge 105mm HE M1 consists Projectile: Hollow steel forging High explosive charge: TNT (,09kg) Supplementary charge: TNT (0,14kg) artridge case:m14 Propelling charge: M67 (7 propelling charge increments) Percussion primer: M8B Technical ata Total length: 790mm Total weight: 18.15kg Temperature limits: Operation: -40 º to +5 º Storage: -6 º to 71 º TABLE : Technical haracteristics And Operational ata Limits of 105mm HE M1 Howitzers Projectile. Mathematical Flight Simulation Model And Simplifications A six degree of freedom rigid-projectile model [7,8,9,11] has three rotations and three translations. The three translations components (x,y,z) describing the position of the projectile s center of mass and the three Euler angles (φ,θ,ψ) describing the orientation of the projectile with respect to translation from the body frame (no-roll-frame, NRF, φ=0) to the plane fixed (inertial frame). For such flight bodies, X axis of the projectile in the no-rollframe coordinate system usually lies along the projectile axis of symmetry andy NRF, Z NRF axes are then oriented so as to complete a right-hand orthogonal system (Figure 3). NRF FGURE 3: No-roll (moving) and Fixed (inertial) coordinate systems for the projectile trajectory analysis Newton s laws of motion state that rate of change of linear momentum must equal the sum of all the externally applied forces and rate of change of angular momentum must equal the sum of all the externally applied moments, respectively. The force acting on the projectile comprises the weightw, the aerodynamic force A, and Magnus f f

nternational onference of omputational Methods in Sciences and Engineering 007,MSE 007, orfu, Greece, 5-30 Sep. 007 force M f. The moment acting on the projectile comprises the moment due to the standard aerodynamic force A m, the Magnus aerodynamic force M, and the unsteady aerodynamic momentua. All aerodynamic coefficients are m based on Mach number and the aerodynamic angles of attack and sideslip. The independent variable is changed from time to dimensionless arc length the equations that determine the pitching and yawing motion become independent of the size of the projectile, which turns out to be very convenient in the analysis of free-flight range data. The transformations to the no-roll-frame ( ) we have: x = cosψ cosθ sinψ v + w cosψ sinθ (1) m y = cosθ sinψ + v cosψ + w sinθ sinψ () z = sinθ+ w cosθ (3) p tan φ = + θ r q θ = ψ = r (4, 5, 6) cosθ u 3 3 3 = g sinθ ρ v X ρ X α ρ X β + r q w (7) 8m 8m 8m v 3 = ρ NA v 8m w θ 16m NPA ( v ) + 4 pρ α p w tan r (8) w 3 ( w w ) 4 = g cosθ ρ NA w pρ β + q + tanθ p v (9) 8m 16m NPA 5 p = p 16 ρ (10) LP q = 3 ρ w 8 NA w MP 16 + 5 ρ q + 4 ρ r XX 16 MQ 8 ( w ) LE + 4 ρ p( v ) YPA p r tan θ v w LE MM + (11) r = 3 8 + 5 16 ρ NA ( v ) LE + 4 pρ ( w ) v w ρ r 4 MQ 8 MP ρ + 16 p q XX YPA + q r tanθ w w LE MM + (1) The projectile dynamic model consists of 1 highly first order ordinary differential equations, which are solved simultaneously by resorting to numerical integration using 4th order Runge-Kutta method.

nternational onference of omputational Methods in Sciences and Engineering 007,MSE 007, orfu, Greece, 5-30 Sep. 007 n these equations, the following sets of simplifications are employed: the aerodynamic angles of attack α, β are small so to beα w /, β v /. Also the yaw and pitch angles ψ, θ are small. A change of variables is introduced velocity u replaced by total velocity because the side velocities v and w are small. Magnus force and moment is a calculable quantity in comparison with weight and aerodynamics forces so there are maintained in the forces and moments, respectively. The projectile is geometrically symmetrical so: XY = YZ = XZ = 0, = ZZ The projectile is aerodynamically symmetric so the expressions of the distances from the center of mass to both the standard aerodynamic and Magnus centers of pressure are simplified. Static Or Gyroscopic Stability Any spinning object will have gyroscopic properties. n a spin stabilized projectile, the center of pressure, the point at which the resultant air force is applied, is located in front of the center of gravity. Hence, as the projectile leaves the muzzle it experiences an overturning movement caused by air forces acting about the center of mass. t must be kept in mind that the forces are attempting to raise the projectile s axis of rotation. lassical exterior ballistics [10] defines the gyroscopic stability factor S, as: S g g = XX (13) ρ p Sd The static factor is proportional to four terms product, which are the geometric technical characteristics of projectile shape model, the square axial spin to velocity ratio, the aerodynamic Magnus effect variation and the proposed density atmospheric model. This may be rearranged into: S g = XX d 3 p 1 1 ρ (14) The estimation of the projectile stability factor is effected from the last three terms separately, due to the fact that the first term is a constant parameter for a given projectile body motion. omputational Simulation The flight dynamic model of 105mm HE, M1, projectile involves, the solution of the set of the twelve nonlinear first order ordinary differential Eqs. (1-1) which are solved simultaneously by resorting to numerical integration using a 4th order Runge-Kutta method. Physical characteristics of the projectile 105mm, HE, M1 are listed (see table 3). 105MM, HE, M1 projectile Reference diameter : 104.8mm Projectile total length: 49.47cm Projectile weight : 15.00Kg Axial moment of inertia: 0.036Kg-m Transverse moment of inertia: 0.3118 Kg-m enter of gravity: 18.34cm from the base TABLE.3: Physical and Geometrical haracteristics of the 105mm projectile. The constant dynamic flight model uses mean values of the experimental aerodynamic coefficients variations [1] (see table 4). 105mm, HE, M1 projectile with constant aerodynamic coefficients = 0.43, = 3.76, L = 1.76, LP = 0.381, = 0.0108 = 0.15 = 9.300, YPA NPA TABLE 4: Trajectory aerodynamic parameters of dynamic model. MQ

nternational onference of omputational Methods in Sciences and Engineering 007,MSE 007, orfu, Greece, 5-30 Sep. 007 For the trajectory parameters of the projectile with variable coefficients, linear interpolation has been applied taking from official tabulated database [1]. nitial data for both dynamic trajectory models with constant and variable aerodynamic coefficients are:x = 0.0 m, y = 0.0 m, z = 0.0 m, ϕ = 0.0, θ = 45.0 and 70.0, ψ = 0.0, u =494 m/sec, v = 0.0 m/sec, w = 0.0 m/sec, p = 1644 rad/sec, q = 0.0 rad/sec and r = 0.0rad/sec. Giving the initial rifling twist rate η at gun muzzle (calibers/turn) the axial spin rate is calculated from the expression: p = / η (rad/sec) (15) The density and pressure are calculated as function of altitude from the simple exponent model atmosphere, and gravity acceleration taken with the constant value g = 9.80665 m/s. Results and iscussion The flight path trajectories of the present dynamic model with initial firing velocity of 494 m/sec and rifling twist rate 1 turn in 18 calibers (1/18) of the M103 Howitzer at initial pitch angles of 45 and 70 are indicated in Figure 4 for two cases: constant[1] and variable[13] aerodynamic coefficients. The impact points of the above trajectories with the present variable coefficients method compared with accurately estimations of Mcoy s flight atmospheric model [1] provide satisfactory agreement for the same conditions. The diagram shows that the 105mm M1 projectile, fired at sea-level with an angle of 45 (dark blue solid line) and no wind, the predicted range to impact is approximately 1150 m, the time of flight is slight greater than 53 sec, and the maximum height is 3465 m. At 70 (green solid line), the predicted level-ground range is 7340 m, the time of flight to impact is about 7,5 sec, and the maximum height is slightly over 6030 m. n the same diagram, the basic differences from Mcoy s flight data are remarked in trajectory models with constant aerodynamic coefficients. FGURE 4: mpact points and flight path trajectories with constant and variable aerodynamic coefficients for 105mm projectile compared with Mcoy s trajectory data. The three final bracketed terms of gyroscopic stability factor (eq.14), contain trajectory variables, which vary significantly during a projectile s flight, and each will be considered independently below: A) Square Axial Spin to elocity Ratio Effect The ratio of square axial to velocity (Figure 5) against range at 45 and 70 grows from value 11 at muzzle, to values of 49 and 04 at apogee, and then decreases to values of and 19 at impact, respectively. FGURE 5: Square axial to elocity ratio versus Range at quadrant elevation angles of 45 and 70 degrees.

nternational onference of omputational Methods in Sciences and Engineering 007,MSE 007, orfu, Greece, 5-30 Sep. 007 t can be stated that if a projectile is statically stable at the muzzle, it will be statically stable for the rest of its flight. This is a result of the static factor being proportional to the ratio of the projectile s rotational and transversal velocity. As the rotational velocity is much less damped than the transversal velocity (which is damped due to the action of the drag), the static factor increases, at least for the major part of the trajectory. B) Atmospheric ensity Effect and ) Magnus Aerodynamic Effect The inverse density against range at 45 and 70 (Figure 6) grows from 0,8 at muzzle, to values of 1,15 and 1,5 at apogee, and then decrease to a value 0,8 at impact point. Also S g will be at its lowest when density is at its highest. This will occur at low altitudes and in low temperature, high pressure atmospheric conditions. t is for this reason that firing trials for projectile impacts are often carried out in cold countries. FGURE 6: nverse ensity versus Range at quadrant elevation angles of 45 and 70 degrees for 105 mm projectile. FGURE 7: nverse Pitching moment versus Mach number flight at quadrant angles of 45 and 70 degrees. On the other hand the aerodynamic Magnus effect coefficient (Figure 7) clearly varies with anything which affects static margin. As a consequence, any variations in the positions of either P or G will directly affect the value of. The position of P will certainly vary during flight if the projectile passes through the transonic flow regime, due to complicated shock and expansion fan movements. n particular, it rises abruptly as the inverse highly reduced. This therefore means that S g reduces under transonic conditions. term is FGURE 8: omparative static stability on the Range with constant and variable coefficients at pitch angle of 45 degrees. FGURE 9: Gyroscopic stability factor variations on constant and variable coefficients at 70 degrees initial pitch angle. The gyroscopic stability factor with constant and variable aerodynamic coefficients at 45 (Figure 8), which was 3,1 at muzzle, grows to 3 and 19 at the summit of the trajectory and then decrease to values 8 and 5,8 at impact, respectively. Also S g estimations have been made for aerodynamic coefficients with constant and variable values at 70 (Figure 9). t begins with an initial 3,1 at muzzle (the same value as in 45 ), growing to 13 and 117 at the summit of the trajectory and then decrease to values 6,6 and 4,19 at impact point, respectively.

nternational onference of omputational Methods in Sciences and Engineering 007,MSE 007, orfu, Greece, 5-30 Sep. 007 onclusion The six degrees of freedom (6-OF) simulation flight dynamics model is applied for the accurate prediction of short and long range trajectories of high and low spin-stabilized projectiles. t takes into consideration the Mach number and total angle of attack variation effects by means of the variable and constant aerodynamic coefficients. riteria and analysis of gyroscopic or static stability are examined. The computational results are in good agreement compared with other technical data and recognized projectile flight dynamic models. REFERENES 1. Mcoy, R. L.: Modern Exterior Ballistics, Schiffer, Attlen, PA, 1999.. Fowler, R., Gallop, E., Lock,., and Richmond, H.: The aerodynamics of spinning shell, Philosophical Transactions of the Royal Society of London, Series A: Mathematical and Physical Sciences, ol.1, 190. 3. ooper, G., nfluence of yaw cards on the growth of spin stabilized projectiles, Journal of aircraft, ol.38, No., 001. 4. Guidos, B., and ooper, G., losed form solution of finned projectile motion subjected to a simple in-flight lateral impulse, AAA Paper 000-0767, 000. 5. ostello, M., and Peterson, A., Linear theory of a dual-spin projectile in atmospheric flight, Journal of Guidance, ontrol, and ynamics, ol.3, No5, 000, pp.789-797. 6. Burchett, B., Peterson, A., and ostello, M., Prediction of swerving motion of a dual-spin projectile with lateral pulse jets in atmospheric flight, Mathematical and omputer Modeling, ol. 35, No. 1-, 00, pp. 1-14 7. Etkin, B.: ynamics of Atmospheric Flight, John Wiley and Sons, New York, 197. 8. Hainz, L.., ostello, M.: Modified Projectile Linear Theory for Rapid Trajectory Prediction, Journal of Guidance, ontrol and ynamics, ol. 8, No. 5, Sept-Oct 005. 9. Amoruso, M. J.: Euler Angles and Quaternions in Six egree of Freedom Simulations of Projectiles, March 1996. 10. Nennstiel, R.: How do Bullets Fly? AFTE Journal, ol. 8, No., April 1996. 11. ostello, M. F., Anderson,. P.: Effect of nternal Mass Unbalance on the Terminal Accuracy and Stability of a Projectile, AAA Paper 96-3447, 1996. 1. Gkritzapis,. N., Panagiotopoulos, E. E., Margaris,. P., Papanikas,. G.: Atmospheric Flight ynamic Simulation Modelling of Spin-Stabilized Projectiles, Proceedings of the nd nternational onference on Experiments / Process / System Modelling / Simulation / Optimization, nd -EpsMsO, 4-7 July 007, Athens, Greece. 13. Gkritzapis,. N., Panagiotopoulos, E. E., Margaris,. P., Papanikas,. G.: omputational Atmospheric Trajectory Simulation Analysis of Spin-Stabilized Projectiles and Small Bullets, Atmospheric Flight Mechanics onference and Exhibit, AAA Paper 007-6584, 0-3 August 007, Hilton Head, South arolina.