Computational Prediction of Complicated Atmospheric Motion for Spinning or non- Spinning Projectiles

Transcription

1 ol:1, No:7, 007 omputational Prediction of omplicated Atmospheric Motion for Spinning or non- Spinning Projectiles imitrios N. Gkritzapis, Elias E. Panagiotopoulos, ionissios P. Margaris, and imitrios G. Papanikas International Science Index, Aerospace and Mechanical Engineering ol:1, No:7, 007 aset.org/publication/875 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 spin and fin-stabilized projectiles ia atmospheric flight to final impact point. The projectiles is assumed to be both rigid (non-flexible), and rotationally symmetric about its spin axis launched at lo and high pitch angles. The mathematical model is based on the full equations of motion set up in the no-roll body reference frame and is integrated numerically from gien initial conditions at the firing site. The projectiles maneuering motion depends on the most significant force and moment ariations, in addition to ind and graity. The computational flight analysis takes into consideration the Mach number and total angle of attack effects by means of the ariable aerodynamic coefficients. For the purposes of the present ork, linear interpolation has been applied from the tabulated database of Mcoy s book. The deeloped computational method gies satisfactory agreement ith published data of erified experiments and computational codes on atmospheric projectile trajectory analysis for arious initial firing flight conditions. Keyords onstant-ariable aerodynamic coefficients, lo and high pitch angles, ind. I. INTROUTION ALLISTIS is the science that deals ith the motion of Bprojectiles. The ord ballistics as deried from the Latin ballista, hich as an ancient machine designed to hurl a jaelin. The modern science of exterior ballistics [1] has eoled as a specialized branch of the dynamics of rigid bodies, moing under the influence of graitational and aerodynamic forces and moments. Exterior ballistics existed for centuries as an art before its first beginnings as a science. Although a number of sixteenth and seenteenth century European inestigators contributed to the groing body of This ork as supported in part by the Hellenic Police and Mechanical Engineering and Aeronautics epartment at Uniersity of Patras. imitrios N. Gkritzapis Laboratory of Firearms and Tool Marks Section, riminal Inestigation iision, First Lieutenant of Hellenic Police, Hellenic Police, 115 Athens, Greece (corresponding author to proide phone: ; gritzap@yahoo.gr). Elias. P. Panagiotopoulos is Postgraduate Student, Mechanical Engineering and Aeronautics epartment, Uniersity of Patras ( hpanagio@mech..upatras.gr). ionissios.e. Margaris is assistant professor Mechanical Engineering and Aeronautics epartment, Uniersity of Patras ( margaris@mech..upatras.gr). imitrios G. Papanikas is professor Mechanical Engineering and Aeronautics epartment, Uniersity of Patras ( papanikas@mech..upatras.gr). renaissance knoledge, Isaac Neton of England ( ) as probably the greatest of the modern founders of exterior ballistics. Neton s las of motion established, ithout hich ballistics could not hae adanced from an art to a science. Pioneering English ballisticians Foler, Gallop, Lock and Richmond [] constructed the first rigid six-degree-of-freedom projectile exterior ballistic model. arious authors has extended this projectile model for lateral force impulses [3]-[4], linear theory in atmospheric flight for dual-spin projectiles [5]-[6], aerodynamic jump extending analysis due to lateral impulsies [7] and aerodynamic asymmetry [8], instability of controlled projectiles in ascending or descending flight [9]. ostello s modified linear theory [10] has also applied recently for rapid trajectory projectile prediction. The present ork 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 and fin-stabilized projectiles. The proposed flight dynamic model takes into consideration the influence of the most significant force and moment ariations, in addition to graity. The applied ariable aerodynamic coefficients analysis takes into consideration the ariations depending on the Mach number flight and total angle of attack. The efficiency of the deeloped method gies satisfactory results compared ith published data of erified experiments and computational codes on dynamics model analysis of short and long-range trajectories of spin and fin-stabilized projectiles. II. PROJETILE MOEL The present analysis considers to different types of representatie projectiles. A typical formation of the cartridge 105mm HE M1 projectile is presented in Fig. 1, and is used ith arious 105mm hoitzers such as M49 ith M5, M5A1 cannons, MA1 & MA ith M101, M101A1 cannons, M103 ith M108 cannon, M137 ith M10 cannon as ell as TO L14 MO56 and L5. artridge 105 mm HE M1 is of semifixed type ammunition, using adjustable propelling charges in order to achiee desirable ranges. The projectile producing both fragmentation and blast effects can be use against personnel and materials targets. The 10 mm (Fig. ) Mortar System proides an organic indirect-fire support capability to the manoeure unit commander. It is a conentional smoothbore, muzzle-loaded mortar system that proides increased range, lethality and safety compared to the World War II-intage 4.-inch heay International Scholarly and Scientific Research & Innoation 1(7) scholar.aset.org/ /875

2 ol:1, No:7, 007 International Science Index, Aerospace and Mechanical Engineering ol:1, No:7, 007 aset.org/publication/875 mortar system it replaces in mechanized infantry, motorized, armored and caalry units. A complete family of 10 mm Enhanced Mortar Ammunition is being produced by seeral goernment and commercial sources. The M933/934 high explosie round also receied full materiel release and is in production. The M99 hite phosphorus/smoke receied full materiel release in the second quarter of 1999 and is in production. Basic physical and geometrical characteristics data of the aboe-mentioned 105 mm HE M1 projectile and the nonrolling, finned 10 mm HE mortar projectile illustrated briefly in Table I. Fig mm HE M1 high explosie projectile Fig. The non-rolling finned 10 mm HE mortar projectile TABLE I PHYSIAL AN GEOMETRIAL ATA OF 105 MM AN 10MM PROJETILES TYPES III. TRAJETORY FLIGHT SIMULATION MOEL A six degree of freedom rigid-projectile model [11-14] has been employed in order to predict the "free" nominal atmospheric trajectory to final target area ithout any control practices. The six degrees of freedom flight analysis comprise the three translation components (x, y, z) describing the position of the projectile s center of mass and three Euler angles (φ, θ, ψ) describing the orientation of the projectile body ith respect to Fig. 3. Fig. 3 No-roll (moing) and fixed (inertial) coordinate systems for the projectile trajectory analysis To main coordinate systems are used for the computational approach of the atmospheric flight motion. The one is a plane fixed (inertial frame, IF) at the firing site. The other is a no-roll rotating coordinate system on the projectile body (no-rollframe, NRF, φ = 0) ith 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. The tele state ariables x, y, z, φ, θ, ψ, u,,, p, q and r are necessary to describe position, flight direction and elocity at eery 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 in Eqs (1, ) ith the dimensionless arc length s as an independent ariable, the folloing full equations of motion for six-dimensional flight are deried: x = cosψ cosθ sinψ cosψ sinθ (1) y = cosθ sinψ cosψ sinθ sinψ () z = sinθ cosθ (3) p tan φ = θ r q θ = ψ = r (4, 5, 6) cosθ u = g sinθ Κ 1 X Κ1 X α Κ1 X β q r (7) ( Κ = Κ ) p 1 MaF r tan 1 α θ r (8) q = Κ Κ Κ ( ) = g cosθ Κ 1 Κ p q tan r 1 MaF β θ P q r = Κ P (9) 5 π p = p 16I ρ (10) R XX K GP MaM L GM I Κ r p r OM tan θ ( ) L p ( ) I XX YY Κ ( ) L p ( ) GP r Κ OM p q I I XX YY MaM q r tanθ (11) L GM (1) The projectile dynamics trajectory model consists of tele highly first order ordinary differential equations, hich are soled simultaneously by resorting to numerical integration using a 4th order Runge-Kutta method. In these equations, the folloing sets of simplifications are employed: elocity International Scholarly and Scientific Research & Innoation 1(7) scholar.aset.org/ /875

3 ol:1, No:7, 007 International Science Index, Aerospace and Mechanical Engineering ol:1, No:7, 007 aset.org/publication/875 u replaced by the total elocity because the side elocities and are small. The aerodynamic angles of attack α and sideslip β are small for the main part of the atmospheric trajectory α /, β /, the projectile is geometrically symmetrical I XY = I YZ = I XZ = 0, I YY = I ZZ and aerodynamically symmetric. With the afore-mentioned assumptions, the expressions of the distance from the center of mass to the standard aerodynamic and Magnus centers of pressure are simplified I. INITIAL SPIN RATE ESTIMATION In order to hae a statically stable flight for spin-stabilized projectile trajectory motion, the initial spin rate p o prediction at the gun muzzle in the firing site us important. According to Mcoy definitions 1, the folloing form is used: p = π / η ( rad / s ) (13) o o here o is the initial firing elocity (m/s), η the rifling tist rate at the gun muzzle (calibers per turn), and the reference diameter of the projectile type (m). Typical alues of rifling tist η are 1/18 calibers per turn for 105mm projectile. The 10 mm mortar projectile has uncanted fins, and do not roll or spin at any point along the trajectory.. OMPUTATIOL SIMULATION The flight dynamic models of 105 mm HE M1 and 10 mm HE mortar projectile types inoles the solution of the set of the tele first order ordinary differentials, Eqs (1-1), hich are soled simultaneously by resorting to numerical integration using a 4th order Runge-Kutta method, and regard to the 6- nominal atmospheric projectile flight. The results gie the computational simulation of the 6- non-thrusting and non-constrained flight trajectory path for some specific spin and fin-projectiles types. Initial flight conditions for both dynamic flight simulation models ith constant and ariable aerodynamic coefficients are illustrated in Table II for the examined test cases. TABLE II INITIAL FLIGHT PARAMETERS OF THE PROJETILE EXAMINE TEST ASES 3.64 rad/s at 45 o and 70 o, respectiely, are indicated in Fig. 4. The calculated impact points of the aboe no-ind trajectories ith the proposed constant aerodynamic coefficients compared ith accurately estimations of Mcoy s flight trajectory analysis [1] proide basic differences for the main part of the atmospheric flight motion for the same initial flight conditions. The mortar projectile of 10 mm diameter is also examined for its atmospheric constant flight trajectories predictions in at pitch angles of 45 o, 65 o, 85 o, ith initial firing elocity of 318 m/s, initial ya angle 3 and pitch rate rad/s. The impact points of the aboe trajectories are compared ith an accurately flight path prediction ith Mcoy s trajectory data [1] as presented in Fig. 5. At 45 the Mcoy model for 105 mm M1 projectile, fired at sea-leel neglecting ind conditions, gies a predicted range to impact of approximately 11,500 m and a maximum height at almost 3,490 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 shon in Fig. 4. Also at 70, the predicted leel-ground range of Mcoy s model is 7,310 m ith maximum height at about 6 km hile the proposed trajectory simulation gies 7,550 m and 6,100 m, respectiely. At 45 o, the 10 mm mortar projectile, fired at no ind conditions, gies a range to impact at 7,000 m ith a maximum height at almost 050 m. At 65 o, the predicted leel-ground range is approximately 5,30 m and the height is 3,80 m and at 85 o gies 1,35m and 3,950m respectiely. For the same initial pitch angles, the 10 mm mortar projectile of Mcoy s data has longer range to impact points. Fig. 4 Impact points and flight path trajectories ith constant aerodynamic coefficients for 105 mm projectile compared ith Mcoy s trajectory data I. RESULTS AN ISUSSION The flight path trajectory motion ith constant aerodynamic coefficients of the 105 mm projectile ith initial firing elocity of 494 m/sec, initial ya angle 3 degrees, rifling tist rate 1 turn in 18 calibers (1/18) and initial ya rates 3.61 rad/s and Fig. 5 Impact points and flight path trajectories ith constant aerodynamic coefficients for 10 mm at quadrant eleation angles of 45 o, 65 o and 85 o compared ith Mcoy s model data International Scholarly and Scientific Research & Innoation 1(7) scholar.aset.org/ /875

4 ol:1, No:7, 007 In Fig. 6, the present study of the 105 mm HE M1 projectile trajectory motion ith ariable aerodynamic coefficients compared ith Mcoy s flight atmospheric model at pitch angles of 45 and 70, proide satisfactory agreement for the same conditions. The diagram shos that the 105 mm HE M1 projectile, fired at sea-leel ith an angle of 45 (cyan solid line) and no ind, the predicted range to impact is 11,500 m and the maximum height is 3,490 m. At 70 (green solid line), the predicted impact point is 7,310 m, and the maximum height is slight oer 6,000 m. The flight path trajectories ith initial pitch angles of 15, 30 and 60 are also shon in the same figure in comparison ith the 45 and 70 flight motions. It can be stated that the maximum impact range is at 45 initial firing angle hile the minimum presents at 15. Fig. 8 Wind model path trajectories at eleation angles of 45 and 70 degrees, for 105 mm projectile International Science Index, Aerospace and Mechanical Engineering ol:1, No:7, 007 aset.org/publication/875 Fig. 6 Impact points and flight path trajectories ith ariable aerodynamic coefficients for 105 mm projectile at lo and high quadrant eleation angles of 15 o, 30 o, 45 o, 60 o and 70 o compared ith Mcoy s trajectory data Fig. 7 Impact points and flight path trajectories ith ariable aerodynamic coefficients for 10 mm bullet at quadrant eleation angles of 45 o, 65 o and 85 o compared ith Mcoy s data trajectory The mortar projectile of 10 mm diameter is examined for its atmospheric ariable flight trajectories predictions in Fig. 7 at lo and high pitch angles of 45 o, 65 o, 85 o, ith initial firing elocity of 318 m/s, initial ya angle 3 and pitch rate rad/s. The impact points of the aboe trajectories are compared ith an accurately flight path prediction ith Mcoy s trajectory data. At 45 o, the 10 mm projectile, fired at no ind conditions gies a range to impact at 7,300 m ith a maximum height at almost,100 m. At 65 o, the predicted leel-ground range is approximately 5,570 m and the height is 3,380 m and at 85 o gies 1,75 and 4,070 respectiely. For the same initial pitch angle, the 10 mm projectile of Mcoy s data proides satisfactory agreement as 105 mm projectile. Fig. 9 Wind models of 10 mm, at pitch angles of 45 and 65 degrees omparatie computed trajectories of the 105 mm projectile at pitch angles of 45 and 70 ith a 10.0 m/s mean crossind bloing are indicated in Fig. 8. From the computational results of the applied method at 45, the predicted range to impact (black solid line) is almost 11,400 m and the maximum height is slightly oer 3,400 m. In addition, the predicted leel-ground range (red solid line) at 70 for the crossind trajectory estimation gies the corresponding alues 7,100 m and 5,800 m, respectiely. Furthermore, the computational results for 10 mm projectile at eleation angles of 45 and 65 ith a 10.0 m/s mean crossind bloing are illustrated in Fig. 9. At 45 flat-fire trajectory, the range ith the ind simulation model is almost 7,50 m (black solid line) and the maximum height,100 m. At 65 pitch angle, the ind predicted range is 5,500 m (red solid line) and the height is almost 3,370 m. Figs. 10 and 11 sho the deflection of the flight trajectory at sea leel ith no-ind for the 105 mm projectile at pitch angles of 15 o, 30 o, 45 o, 60 o, 65 o and for 10 mm projectile at eleation angles 45 o, 65 o, 85 o respectiely. The present analysis trajectory of the 105 mm HE M1 projectile ith initial positie ya rate 3.61 rad/s at pitch angle of 15 o gies positie (right) deflection at about 146 m. On the other hand, as the initial pitch angle increases, the cross range prediction becomes negatie (to the left) in comparison ith the dashed reference line and increases continually at almost 700 m. Moreoer, the mortar projectile of 10 mm diameter ith initial firing elocity of 318 m/s, initial ya angle 3 degrees and pitch rate rad/s, gies alays positie alues of cross range 380 m, 90 m and 65 m at arious lo and high pitch angles, respectiely. International Scholarly and Scientific Research & Innoation 1(7) scholar.aset.org/ /875

5 ol:1, No:7, 007 [11] Etkin, B., ynamics of Atmospheric Flight, John Wiley and Sons, Ne York, 197. [1] Joseph K., ostello, M., and Jubaraj S., Generating an Aerodynamic Model for Projectile Flight Simulation Using Unsteady Time Accurate omputational Fluid ynamic Results, Army Research Laboratory, ARL-R-577, 006. [13] Amoruso, M. J., Euler Angles and Quaternions in Six egree of Freedom Simulations of Projectiles, Technical Note, [14] ostello, M., and Anderson,., Effect of Internal Mass Unbalance on the Terminal Accuracy and Stability of a projectile, AIAA Paper, International Science Index, Aerospace and Mechanical Engineering ol:1, No:7, 007 aset.org/publication/875 Fig. 10 ross range ersus range ith ariable aerodynamic coefficients for 105 mm projectile Fig. 11 ross range computational predictions of 10 mm mortar projectile at eleation angles of 45 o, 65 o and 85 o II. ONLUSION The complicated 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 lo spin and finstabilized projectiles. It takes into consideration the Mach number and the total angle of attack ariation effects by means of the ariable and constant aerodynamic coefficients. The computational results of the proposed synthesized analysis are in good agreement compared ith other technical data and recognized exterior atmospheric projectile flight computational models. REFERENES [1] Mcoy, R., Modern Exterior Ballistics, Schiffer, Attlen, PA, [] Foler, 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., Influence of Ya ards on the Ya Groth 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, AIAA Paper, 000. [5] ostello, M., and Peterson, A., Linear Theory of a ual-spin Projectile in Atmospheric Flight, Journal of Guidance, ontrol, and ynamics, ol.3, No. 5, 000. [6] Burchett, B., Peterson, A., and ostello, M., Prediction of Sering Motion of a ual-spin Projectile ith Lateral Pulse Jets in Atmospheric Flight, Mathematical and omputer Modeling, ol. 35, No. 1-, 00. [7] ooper, G., Extending the Jump Analysis for Aerodynamic Asymmetry, Army Research Laboratory, ARL-TR-365, 004. [8] ooper, G., Projectile Aerodynamic Jump ue to Lateral Impulsies, Army Research Laboratory, ARL-TR-3087, 003. [9] Murphy,., Instability of ontrolled Projectiles in Ascending or escending Flight, Journal of Guidance, ontrol, and ynamics, ol.4, No. 1, [10] Hainz, L., and ostello, M., Modified Projectile Linear Theory for Rapid Trajectory Prediction, Journal of Guidance, ontrol, and ynamics, ol.8, No. 5, 005. NOMENLATURE X = axial force aerodynamic coefficient = normal force aerodynamic coefficient MaF = magnus force aerodynamic coefficient R = roll damping moment aerodynamic coefficient P = pitch damping moment aerodynamic coefficient OM = oerturning moment aerodynamic coefficient MaM = magnus moment aerodynamic coefficient MaF = magnus force aerodynamic coefficient x, y, z = projectile position coordinates in the inertial frame, m m = projectile mass, kg = projectile reference diameter, m S = dimensionless arc length = total aerodynamic elocity, m/s u,, = projectile elocity components expressed in the no-roll-frame, m/s u,, = ind elocity components in no-roll-bodyframe, m/s p, q, r = projectile roll, pitch and ya rates in the moing frame, respectiely, rad/s ρ = density of air, kg/m 3 φ, θ, ψ = projectile roll, pitch and ya angles, respectiely, deg α, β = aerodynamic angles of attack and sideslip,deg g = graity acceleration, m/s I = projectile inertia matrix I XX = projectile axial moment of inertia, kg m I YY = projectile transerse moment of inertia about y-axis through the center of mass, kg m Ι ΧΧ, Ι ΥΥ, Ι ΖΖ = diagonal components of the inertia matrix Ι ΧΥ, Ι ΥΖ, Ι ΧΖ = off-diagonal components of the inertia matrix L GM = distance from the center of mass (G) to the Magnus center of pressure (M) along the station line, m L GP = distance from the center of mass (G) to the aerodynamic center of pressure (P) along the station line, m Κ 1, Κ = dimensional coefficients, π ρ 3 /8m and π ρ 3 /16Ι ΥΥ, respectiely K y = non-dimensional transerse moment of inertia International Scholarly and Scientific Research & Innoation 1(7) scholar.aset.org/ /875