Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 2, pp. 375-385, Warsaw 2013 NUMERICAL TESTING OF FLIGHT STABILITY OF SPIN-STABILIZED ARTILLERY PROJECTILES Leszek Baranowski Military University of Technology, Faculty of Mechatronics and Aerospace, Warsaw, Poland e-mail: leszek.baranowski@wat.edu.pl The paper presents results of numerical research on the effect of the twist rate, muzzle velocity, Magnus moment and firing disturbances (cross and range wind) on the stability of flight of a Denel 155mm artillery projectile (Assegai M2000 Series) for flat and steep trajectories. Key words: spin-stabilized projectile, flight stability, Magnus moment 1. Introduction Analytical solutions of equations ofmotion of spin-stabilized artillery projectiles obtained under certain simplifying assumptions (small total angle of attack αt and fixed factors in linearized differential equations of motion) identify essential conditions for stabilizing projectiles on the whole trajectory (Dmitrievskij, 1979; Gacek, 1998; Shapiro, 1956) — for the initial (near-straight-line) part of the trajectory, it is required that σ=1− b a2 > 0 (1.1) where a is the precession velocity a= Ixp0 2Iy b= MAα Iy = CAmαρV 2Sd 2Iy — then, for the curvilinear section of the trajectory, it is required that the angle δr contained between the vector of velocity of the projectile and the dynamic balance axis is small enough not to cause overturning (tumbling) of the projectile near the trajectory apex δr = 2a b gcosγ V = 2Ixp0gcosγ ρV 3SdCAmα (1.2) English-language literature (McCoy, 1999; PRODAS v3Usermanual; Textbook of Ballistics and Gunnery, 1987) proposes the following equation for “gyroscopic stability factor” based on linearized equations of motion of the projectile as a rigid body Sg = 2I2xp 2 πρIyd 3V 2CMα (1.3) The condition for stable projectile flight in the initial section of the trajectory is as follows Sg > 1 (1.4) Thepaper attempts to take a look at theproblemof stability of flight of artillery projectile on thewhole flight trajectory based on a non-linearmathematical model ofmotion of the projectile as a rigid body presented in the earlier Author’s work (2011). 376 L. Baranowski For this purpose, a computer application was developed for simulating the firing of the test projectile fromHowitzer-gun “Krab” using the aforementionedmathematical model. Denel 155mmartillery projectile (AssegaiM2000 Series) was used as the test projectile. The computer application was used to carry out all-inclusive tests of flight stability of the projectile fired with different muzzle velocities from a 52 caliber barrel to determine the twist rate and quadrant elevation range optimal from the point of view of projectile flight stability. The paper presents selected results of analysis of firingwith extreme charges (minimum velocity VK0 =319m/s for the 1st charge and themaximum velocity VK0 =935m/s for the 6th charge) designed to reveal the following: • the effect of the twist rate in the end of the barrel expressed in calibers η on the stability of flight of the projectile fired at a small and at the maximum quadrant elevation QE; • the effect of muzzle velocity on dynamic properties of the projectile fired at a small QE (flat trajectory) and the maximum QE (steep trajectory); • the effect of firing disturbances on the behavior of the projectile on the trajectory. 2. Characteristics of the physical model of the test projectile The development of the flight simulation computer program of artillery projectiles requires de- termination of the so-called physicalmodel (Dziopa et al., 2010;Koruba et al., 2010; Kowaleczko and Żyluk, 2009; Ładyżyńska-Kozdraś, 2012), which includes the following characteristics: a) Structure characteristics: – geometries – mass and inertia – elasticity b) Aerodynamic characteristics. c) Surrounding environment: – density, viscosity, temperature, pressure, velocity and wind direction depending on weather, flight altitude, etc. The simulation adopted the International Standard At- mosphere (ISO 2533, 1975) as the reference. Because of compact design and high rigidity of artillery projectiles, the frequency of proper vibration of elastic components of the deliberated test projectile is many times higher than the frequency of its oscillation around the center of mass, which enables treating the test projectile as a non-deformable solid body with 6degrees of freedom. Fig. 1. Overview: test projectile solid model In linewith the prevailing trend, the theoretical calculation of themass and inertia characte- ristics of the test projectile used the SolidWorks software package from SolidWorks Corporation (one of popular CAD/CAM suites). The computation of the characteristics assumes that the projectile is an axial-symmetric solidwith symmetricmass and inertia. SeeFig. 1 for an overview Numerical testing of flight stability of spin-stabilized artillery projectiles 377 Fig. 2. Main dimensions of the test projectile of the test projectile (solid model) and Fig. 2 for themain dimensions used for the computation of geometric and aerodynamic characteristics. The mass and inertia characteristics of the test projectile computed using the SolidWorks software are as follows: — mass: m=43.7kg — coordinate of position of the center of projectile mass relative to the nose (Fig. 2): xCG =0.563m — moments of inertia of the projectile Ix, Iy, Iz in the body-fixed system 0xyz: Ix =0.1444kgm2, Iy =1.7323kgm2, Iz =1.7323kgm2. Aerodynamic characteristics of the test projectilewere determined using an off-shelf software application: Arrow Tech PRODAS 3.5.3 dedicated to computer-aided designing weaponry. See Tables 1 and 2 for the results of computation of aerodynamic characteristics as a function of the Mach number M for: p∗ = pd/(2V ), q∗ = qd/(2V ), r∗ = rd/(2V ) and S=πd2/4. Table 1.Aerodynamic characteristics of the test projectile as a function of theMach number M CAX0 C A X α2 CAZα C A Ypα C A lp C A mα C A mq [–] [–] [rad−2] [rad−1] [rad−1] [–] [rad−1] [–] 0.010 0.144 1.90 1.624 −0.85 −0.0308 3.755 −9.5 0.400 0.144 1.90 1.623 −0.85 −0.0308 3.784 −9.2 0.600 0.144 1.91 1.629 −0.85 −0.0308 3.774 −9.5 0.700 0.144 2.10 1.633 −0.86 −0.0308 3.763 −9.8 0.800 0.146 2.21 1.638 −0.87 −0.0308 3.785 −10.3 0.900 0.160 2.30 1.655 −0.88 −0.0308 3.843 −11.0 0.950 0.202 2.50 1.661 −0.91 −0.0309 3.825 −11.9 0.975 0.240 2.64 1.694 −0.93 −0.0308 3.736 −12.8 1.000 0.284 2.74 1.746 −0.95 −0.0306 3.577 −14.0 1.025 0.313 2.89 1.823 −1.06 −0.0301 3.570 −15.2 1.050 0.332 3.09 1.902 −1.20 −0.0296 3.558 −16.8 1.100 0.337 3.30 1.962 −1.07 −0.0293 3.601 −18.8 1.200 0.340 3.51 2.006 −0.99 −0.0290 3.675 −20.8 1.500 0.321 3.87 2.128 −0.92 −0.0291 4.014 −22.6 2.000 0.276 4.36 2.209 −0.86 −0.0297 3.774 −22.8 2.500 0.240 4.86 2.299 −0.78 −0.0302 3.583 −24.3 3.000 0.214 4.37 2.359 −0.70 −0.0299 3.460 −25.6 378 L. Baranowski Table 2. Derivative of the Magnus moment coefficient for the test projectile as a function of the total angle of attack αt and theMach number CAnp [–] M αt [deg] [–] 0 1 2 3 5 10 20 0.010 0 −0.0069 −0.0002 0.0286 0.1316 0.3313 0.6349 0.400 0 −0.0069 −0.0002 0.0286 0.1316 0.3313 0.6349 0.600 0 −0.0069 −0.0002 0.0286 0.1316 0.3313 0.6349 0.700 0 −0.0091 −0.0047 0.0227 0.1348 0.3353 0.6425 0.800 0 −0.0144 −0.0137 0.0120 0.1229 0.2971 0.5670 0.900 0 −0.0054 −0.0002 0.0223 0.1028 0.2519 0.4764 0.950 0 0.0033 0.0144 0.0393 0.1340 0.3120 0.5892 0.975 0 0.0068 0.0194 0.0420 0.1174 0.2879 0.5447 1.000 0 0.0099 0.0243 0.0463 0.1124 0.2181 0.4089 1.025 0 0.0136 0.0304 0.0523 0.1084 0.2104 0.3945 1.050 0 0.0148 0.0317 0.0525 0.1062 0.2059 0.3865 1.100 0 0.0140 0.0296 0.0480 0.0909 0.1838 0.3443 1.200 0 0.0121 0.0257 0.0413 0.0723 0.1377 0.2553 1.500 0 0.0126 0.0251 0.0377 0.0640 0.1232 0.2289 2.000 0 0.0124 0.0247 0.0366 0.0609 0.1156 0.2154 2.500 0 0.0122 0.0241 0.0356 0.0605 0.1144 0.2131 3.000 0 0.0121 0.0240 0.0356 0.0605 0.1145 0.2133 3. Effect of the twist rate on stability of the projectile fired at small and large quadrant elevation The effect of the twist rate in the end of the gun barrel η (expressed in calibers per revolution) and effect of theMagnusmoment on stability of the projectile on the trajectory was checked by stimulating the firing with the minimum (VK0 = 319m/s) and maximum (VK0 = 935m/s) initial (muzzle) velocities and a flat trajectory (QE = 10deg) and steep trajectory (QE =70deg) for four values of the twist rate η=15, 20, 25, 30 calibers. See Table 3 for the basic inputs to the simulation and corresponding factors that were used for evaluating stability in the initial section of the trajectory using equations (1.1) and (1.3). Table 3.Main initial inputs to the firing simulation for testing of the twist rate η Twist Charge 1 Charge 2 rate η VK0 =319m/s VK0 =935m/s [caliber] p0 [rad/s] Sg [–] σ [–] √ σ [–] p0 [rad/s] Sg [–] σ [–] √ σ [–] 15 862.0 3.2 0.69 0.83 2526.8 3.5 0.71 0.84 20 646.6 1.8 0.45 0.67 1895.1 1.96 0.49 0.7 25 517.3 1.15 0.13 0.36 1516.1 1.25 0.2 0.45 30 431.0 0.8 −0.25 1263.4 0.8 −0.15 The initial (muzzle) value of the spin rate of the projectile was computed using the known dependence p0 = 2πVK0 ηd (3.1) Numerical testing of flight stability of spin-stabilized artillery projectiles 379 In addition, it was assumed at themuzzle that the total angle of attack αt =0 and angular velocities of the projectile (in the plane perpendicular to the longitudinal axis of the projectile) dependon the quadrant elevation QE and initial velocity VK0 (seeTable 4) but are independent of the twist rate η. Table 4.Typical initial values of projectile angular velocities in the plane perpendicular to the longitudinal axis of the projectile (McCoy, 1999) Quadrant Charge 1 Charge 2 elevation VK0 =319m/s VK0 =935m/s QE [deg] q0 [rad/s] r0 [rad/s] q0 [rad/s] r0 [rad/s] 10 1.235 0 4.000 0 70 1.270 0 4.015 0 Figures 3 and 4 show line diagrams of the total angle of attack αt(t) (for different η=15, 20, 25 and 30) in the simulation of a flat trajectory (QE = 10deg) with the minimum and maximum initial velocities. In addition, the left side diagrams illustrate the effect of omitting theMagnus moment from the computation, while the right side one takes it into account. Fig. 3. Total angle of attack αt versus time for different twist rates η, in the case of flat trajectories and the minimum initial velocity VK0 =319m/s; (a) without theMagnusmoment, (b) with theMagnus moment Fig. 4. Total angle of attack αt versus time for different twist rates η, in the case of flat trajectories and the maximum initial velocity VK0 =935m/s; (a) without theMagnusmoment, (b) with theMagnus moment Likewise, Figs. 5 and 6 show line diagrams of the total angle of attack αt(t) for a steep trajectory (QE =70deg) with theminimum andmaximum initial velocities, with andwithout theMagnus moment. 380 L. Baranowski Fig. 5. Total angle of attack αt versus time for different twist rates η, in the case of steep trajectories and the minimum initial velocity VK0 =319m/s; (a) without theMagnus moment, (b) with the Magnusmoment Fig. 6. Total angle of attack αt versus time for different twist rates η, in the case of steep trajectories and the maximum initial velocity VK0 =935m/s; (a) without theMagnus moment, (b) with the Magnusmoment Initial numerical tests performed using the non-linear model of projectile motion confirmed, following the existing literature, a huge effect of the twist rate η on the details of projectile motion around the center of its mass both for small and the maximum quadrant elevation (see Figs. 3-4 and Figs. 5-6, respectively). For η = 30 calibers, the projectile is clearly unstable (σ< 0, Sg < 1), which is confirmed by αt(t) lines in, basically, all diagrams, particularly if the Magnus moment is included in the computation (Figs. 3b, 4b, 5b and 6b). The case of η = 25 calibers is peculiar because based on the criteria of stability (σ > 0, Sg > 1) the projectile should be stable while computations taking into account the Magnus moment for theminimuminitial velocity (Figs. 3band5b) showthat theprojectile is dynamically unstable. The cause is a negative Magnus moment occurring at small total angle of attack and subsonic flight velocities (see Table 2). Based on the analysis of flight stability of the test projectile, it was assumed for the purposes of further simulation tests that the twist rate in the end of the gun barrel η would be equal to 20 calibers. 4. Effect of muzzle velocity on the dynamic properties of the projectile fired at a small and large quadrant elevation This test simulated the following parameters: twist rate η = 20calibers, minimum muzzle ve- locity VK0 = 319m/s, maximum muzzle velocity VK0 = 935m/s, small quadrant elevation QE=10deg, large quadrant elevation QE=70deg, remaining initial conditions as in Tables 3 Numerical testing of flight stability of spin-stabilized artillery projectiles 381 and 4. In order to determine the composite effect of theMagnusmoment on the stability of flight of the projectile, the computations were made for two variants: with and without considering theMagnus moment. Flight stabilitywas evaluated based on the variation of the total angle of attack in time αt(t) and the tracemarked by the vertex of the projectile on the plane perpendicular to the vector of flight velocity in form of the function αz = f(αy). Illustrations of the angle αy (projectile deviation from the vector of velocity in the horizontal plane)andangle αz (projectile deviation fromthevector of velocity in thevertical plane), defined as symmetrical, are shown in Fig. 7. The values of the angles were computed using the following equations αy =arctan ( − vK −vW uK −uW ) αz =arctan (wK −wW uK −uW ) (4.1) Fig. 7. Illustration of spatial positions of angles αz and αy (Baranowski, 2006) Fig. 8. Total angle of attack αt versus time in the case of flat trajectories QE=10deg and the minimum initial velocity VK0 =319m/s; (a) without theMagnus moment, (b) with theMagnus moment Fig. 9. Pitching and yawingmotion in the case of flat trajectories QE=10deg and the minimum initial velocity VK0 =319m/s; (a) without theMagnus moment, (b) with theMagnusmoment 382 L. Baranowski Figures 8-11 showmotion of the projectile around the center of its mass on a flat trajectory (QE =10deg) for 11 initial seconds of flight. Tomake the diagrams clearer, three flight phases with related projectile motion parameters (total angle of attack αt versus time and the trace marked by the vertex of the projectile illustrating pitching and yawing motion in form of the function αz = f(αy)) were distinguished with gray scale. In addition, the left side diagrams omit, and the right side one take into account, theMagnus moment. Fig. 10. Total angle of attack αt versus time in the case of flat trajectories QE=10deg and the maximum initial velocity VK0 =935m/s; (a) without theMagnus moment, (b) with theMagnus moment Fig. 11. Pitching and yawingmotion in the case of flat trajectories QE=10deg and the maximum initial velocity VK0 =935m/s; (a) without theMagnus moment, (b) with theMagnusmoment Likewise, as for the flat trajectory, Figs. 12-15 show the results of testing the effect ofmuzzle velocities (min. and max.) on the dynamic properties of the projectile flying along the steep trajectory, fired at a large quadrant elevation QE=70deg. Fig. 12. Total angle of attack αt versus time in the case of steep trajectories QE=70deg and the minimum initial velocity VK0 =319m/s; (a) without theMagnus moment, (b) with theMagnus moment Numerical testing of flight stability of spin-stabilized artillery projectiles 383 Fig. 13. Pitching and yawingmotion in the case of steep trajectories QE=70deg and the minimum initial velocity VK0 =319m/s; (a) without theMagnus moment, (b) with theMagnusmoment Fig. 14. Total angle of attack αt versus time in the case of steep trajectories QE=70deg and the maximum initial velocity VK0 =935m/s; (a) without theMagnus moment, (b) with theMagnus moment Fig. 15. Pitching and yawingmotion in the case of steep trajectories QE=70deg and the maximum initial velocity VK0 =935m/s; (a) without theMagnus moment, (b) with theMagnusmoment 5. Results of tests of projectile behavior on the trajectory during firing in non-standard conditions Thispointdiscusses the effect of perturbationofmeteorological conditions on theflightdynamics of the spinning projectile, including specifically the effect of cross and range wind. The following Figs. 16 and 17 show illustrative diagrams of selected parameters of flight of the test projectile flying flat trajectory (QE = 10deg) in calm conditions (vWg = 0m/s) and in conditions of strong cross wind blowing from: left side – vWg =20m/s and right side – vWg =−20m/s since second second of the projectile flight. 384 L. Baranowski Fig. 16. (a) Effect of cross wind on the trajectory in the horizontal plane, (b) disturbing effect of cross wind on the total angle of attack αt Fig. 17. Pitching and yawingmotion for: (a) vWg =−20m/s, (b) vWg =+20m/s 6. Summary and conclusions Results of testing flight stability in standard (undisturbed) conditions suggest the following: • In general, the Magnus moment accelerates the suppression of the total angle of attack but, because it is negative for subsonic flight velocities, for the minimum initial velocity VK0 = 319m/s the growth of the total angle of attack destabilizes the projectile in the final flight phase (Fig. 8b and Fig. 12b). • Forflat trajectories, themaximumtotal angle of attack occurs at the start of the trajectory, whereas for steep trajectories near the vertex. • For a large quadrant elevation, the details of projectile motion near the trajectory vertex (Fig. 13 and Fig. 15) are different than those provided by analytical solutions using the simplified model (Dmitrievskij, 1979; Gacek, 1998; Shapiro, 1956). • Similarly, verifying the classical criteria of stability is not sufficient for selecting the twist rate of rifling: additional simulations are required taking into account theMagnusmoment. The results of numerical simulations of non-standard conditions confirmed, among others, the known phenomenon of projectile trajectory deflection in the direction of wind (Fig. 16a), but showed also clear difference of left- and right-sidedwind on the projectilemotion around the center of mass (see Figs. 16b and 17). This interesting phenomenon requires further theoretical and empirical testing. Acknowledgement The research work was supported by the Polish Ministry of Science and Higher Education in the years 2010-2013within the framework of grant 423/B0/A. Numerical testing of flight stability of spin-stabilized artillery projectiles 385 References 1. BaranowskiL., 2006,Amathematicalmodelofflightdynamicsoffieldartilleryguidedprojectiles, 6th International Conference on Weaponry “Scientific Aspects of Weaponry”, Waplewo, 44-53 [in Polish] 2. 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The ISO Standard Atmosphere, 1975, ISO 2533 Badania numeryczne stabilności lotu klasycznego pocisku artyleryjskiego stabilizowanego obrotowo Streszczenie W pracy przedstawiono wyniki badań numerycznych wpływu długości skoku gwintu lufy, prędkości początkowejpocisku,momentuMagnusaorazzakłóceńwarunkówstrzelania (wiatrupodłużnego i boczne- go) na stabilność lotu 155mmpocisku artyleryjskiego firmyDenel (AssegaiM2000 Series) dla przypadku płaskiego i stromego toru lotu. Manuscript received May 8, 2012; accepted for print July 2, 2012