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6 The Open Mechanics Journal, 2008, 2, 6-11 Open Access Modified Linear Theory for Spinning or Non-Spinning Projectiles D.N. Gkritzapis*,1, E.E. Panagiotopoulos2, D.P. Margaris3 and D.G. Papanikas4 1Laboratory of Firearms and Tool Marks Section, Criminal Investigation Division, Hellenic Police, and Post graduate Student, Mechanical Engineering and Aeronautics Department, University of Patras, Greece 2Post-graduate Student, Mechanical Engineering and Aeronautics Department, University of Patras, Greece 3Professor, Mechanical Engineering and Aeronautics Department, University of Patras, Greece 4Ex-Professor, Mechanical Engineering and Aeronautics Department, University of Patras, Greece Abstract: Static and dynamic stability are the most important phenomena for stable flight atmospheric motion of spin and fin stabilized projectiles. If the aerodynamic forces and moments and the initial conditions are accurately known, an es- sentially exact simulation of the projectile’s synthesized pitching and yawing motion can be readily obtained by numerical methods. A modified trajectory linear theory of the same problem implies an approximate solution. INTRODUCTION etc.) where the rifling of the barrel provides the required axial spin to projectile. In describing this condition, a gyro- More than 80 years ago, English ballisticians [1] con- scopic stability factor can be applied, which is obtained from structed the first rigid six-degree-of-freedom projectile exte- the roots of the modified linear theory in the equations of rior ballistic model. Their model contained a reasonably projectile motion. complete aerodynamic force and moment expansion for a spinning shell and included aerodynamic damping along Also, dynamic stability is defined as the condition where with Magnus force and moment. Guided by an extensive set a system is perturbed and the ensuing oscillatory has a ten- of yaw card firings, these researchers also created the first dency to either decrease or increase. Note that this definition approximate analytic solution of the six-degree-of-freedom assumes that the static stability is present, otherwise the os- projectile equations of motion by introducing a set of simpli- cillatory motion would not occur. fications based on clever linearization by artificially separat- PROJECTILE MODEL ing the dynamic equations into uncoupled groups. The re- sulting theory is commonly called projectile linear theory. The present analysis considers two different types of rep- Kent [2], Neilson and Synge [3], Kelley and McShane [4], resentative projectiles: a spin-stabilized of 105mm and a and Kelley et al. [5] made refinements and improvements to mortar fin-stabilized of 120 mm. projectile linear theory. Basic physical and geometrical characteristics data of the Projectile linear theory has proved an invaluable tool in above-mentioned 105 mm HE M1 and the non-rolling, understanding basic dynamic characteristics of projectiles in finned 120 mm HE mortar projectiles are illustrated briefly atmospheric flight, for establishing stability criteria for fin- in Table 1. and spin-stabilized projectiles, and for extracting projectile Table 1. Physical and Geometrical Data of 105 mm and aerodynamic loads from spark range data. 120mm Projectiles Types In the present work, the full six degrees of freedom (6- DOF) projectile flight dynamics atmospheric model is considered for the accurate prediction of short and long range Characteristics 105 mm HE M1 120 mm HE mortar trajectories of high spin and fin-stabilized projectiles. It takes projectile projectile into consideration the influence of the most significant forces and moments, in addition to gravity. Reference diameter, mm 114.1 119.56 Projectiles, which are inherently aerodynamically unsta- Total Length, mm 494.7 704.98 ble, can be stabilized with spin. For this condition, the spin Total mass, kg 15.00 13.585 rate must be high enough to develop a gyroscopic moment, which overcomes the aerodynamic instability, and the pro- Axial moment of inertia, 2326x10-2 2335x10-2 jectile is said to be gyroscopically stable. This is the case for kg-m2 the most gun launched projectiles (handguns, rifles, cannons, Transverse moment of 23118x10-2 23187x10-2 inertia, kg-m2 *Address correspondence to this author at the Laboratory of Firearms and Centre of gravity from 113.4 422.9 Tool Marks Section, Criminal Investigation Division, Hellenic Police, and the base, mm Post graduate Student, Mechanical Engineering and Aeronautics Depart- ment, University of Patras, Greece; E-mail: [email protected] 1874-1584/08 2008 Bentham Science Publishers Ltd. Modified Linear Theory for Spinning The Open Mechanics Journal, 2008, Volume 2 7 TRAJECTORY FLIGHT SIMULATION MODEL and for the orientation of the flight body with the classical Euler angles (cid:1), (cid:2), (cid:3). From the two laws of Newton’s motion Flight mechanics of most projectile configurations can be the following equations (5) and (6) are derived, respectively: captured using a rigid body six degrees of freedom dynamic model. The six degrees of freedom flight analysis comprise (cid:1)u(cid:1) (cid:5) (cid:1)F(cid:1)x /m(cid:5) NRF TOTAL the three translation components (x, y, z) describing the posi- (cid:4) (cid:4)= (cid:4) (cid:4) + (cid:2)v(cid:1) (cid:6) (cid:2)F(cid:1)y /m(cid:6) tion of the projectile’s center of mass and three Euler angles NRF TOTAL ((cid:1), (cid:2), (cid:3)) describing the orientation of the projectile body (cid:3)(cid:4)w(cid:1)NRF(cid:7)(cid:4) (cid:3)(cid:4)F(cid:1)zTOTAL /m(cid:7)(cid:4) with respect to (Fig. 1). (cid:3) 0 r(cid:1) (cid:1)q(cid:1) (cid:6) (cid:1)u(cid:1) (cid:5) +(cid:4) NRF NRF (cid:7) (cid:4) NRF(cid:4) (5) (cid:1)r(cid:1) 0 (cid:1)r(cid:1) t (cid:2)v(cid:1) (cid:6) (cid:4) NRF NRF (cid:2)(cid:7) NRF (cid:4) (cid:4) (cid:5)(cid:4)q(cid:1)NRF r(cid:1)NRFt(cid:2) 0 (cid:8)(cid:7) (cid:3)w(cid:1)NRF(cid:7) (cid:3) (cid:9)L(cid:1) (cid:19) (cid:6) 0 (cid:1)r(cid:1) q(cid:1) (cid:16)(cid:13) TOTAL NRF NRF (cid:1)p(cid:1) (cid:5) (cid:4)(cid:4) (cid:10)(cid:12)M(cid:1)TOTAL(cid:20)(cid:12)(cid:1)(cid:7)(cid:7) r(cid:1)NRF 0 r(cid:1)NRFt(cid:2)(cid:17)(cid:17)(cid:14)(cid:14) (cid:2)(cid:4)q(cid:1)NRF(cid:6)(cid:4)= (cid:6)I(cid:1)1(cid:16)(cid:4)(cid:4) (cid:11)(cid:12)N(cid:1)TOTAL(cid:21)(cid:12) (cid:8)(cid:7)(cid:1)q(cid:1)NRF (cid:1)r(cid:1)NRFt(cid:2) 0 (cid:18)(cid:17)(cid:14)(cid:14) (6) NRF (cid:8) (cid:18) (cid:3)(cid:4)r(cid:1)NRF (cid:7)(cid:4) (cid:4)(cid:4) (cid:6)(cid:7)IXX IXY IXZ(cid:16)(cid:17)(cid:9)(cid:12)p(cid:1)NRF(cid:19)(cid:12) (cid:14)(cid:14) (cid:4) (cid:7)IYX IYY IYZ(cid:17)(cid:10)q(cid:1)NRF(cid:20) (cid:14) (cid:5)(cid:4) (cid:8)(cid:7)IZX IZY IZZ(cid:18)(cid:17)(cid:11)(cid:12)r(cid:1)NRF (cid:21)(cid:12) (cid:15)(cid:14) The total force acting on the projectile in equation (5) comprises the weight W , the aerodynamic forceA and Fig. (1). No-roll (moving) and fixed (inertial) coordinate systems f f Magnus force M : for the projectile trajectory analysis. f tanbi ooopd-nlrTyaaonl wl (elan o rpof opim-xtrraeoaotdliialnn c-( gfhicr n oacoemoofror teditr,hadi lnNei nafRartaateFtmm e,s oyse(cid:1)sy)s=p stae0httem)e m trwhsi cemia t frhfolei irv tgihuinhnesgte g X dm sw iotfeiott.ihr o Ta nttxhhh.i eeesT pocahorlteohom neojeprgn cu eittst iha lieaes- (cid:1)(cid:2)(cid:4)(cid:3)(cid:4)FFF(cid:1)(cid:1)(cid:1)xyzTTTOOOTTTAAALLL(cid:5)(cid:6)(cid:4)(cid:7)(cid:4)= (cid:4)(cid:3)(cid:4)(cid:2)(cid:1)YZX~~~wwwfff(cid:4)(cid:7)(cid:4)(cid:6)(cid:5) + (cid:4)(cid:4)(cid:3)(cid:4)(cid:4)(cid:2)(cid:1)XZY~~~AAAfff(cid:4)(cid:4)(cid:7)(cid:4)(cid:4)(cid:6)(cid:5) + (cid:4)(cid:4)(cid:3)(cid:4)(cid:4)(cid:2)(cid:1)XZY~~~MMMfff(cid:4)(cid:4)(cid:7)(cid:4)(cid:4)(cid:6)(cid:5) (7) NRF projectile axis of symmetry and YNRF, ZNRF axes oriented so The total moment acting on the projectile in equation (6) as to complete a right hand orthogonal system. comprises the moment due to the standard aerodynamic Newton’s laws of the motion state that the rate of change forceA , due to Magnus aerodynamic force M and the m m of linear momentum must equal the sum of all the externally unsteady aerodynamic momentUA : applied forces (1) and the rate of change of angular momen- m tum must equal the sum of the externally applied moments (cid:1)L~ (cid:5) (cid:1)L~ (cid:5) (cid:1)L~ (cid:5) (cid:1)L~ (cid:5) (2), respectively. (cid:4)~TOTAL(cid:4) = (cid:4)~Am(cid:4) + (cid:4)~Mm(cid:4) + (cid:4)~UAm(cid:4) (8) mdV(cid:1) =(cid:1)F(cid:1)+mg(cid:1) (1) (cid:4)(cid:3)(cid:2)MN~TTOOTTAALL(cid:4)(cid:7)(cid:6) (cid:4)(cid:3)(cid:2)MN~AAmm(cid:4)(cid:7)(cid:6) (cid:4)(cid:3)(cid:2)MN~MMmm(cid:4)(cid:7)(cid:6) (cid:4)(cid:3)(cid:2)MN~UUAAmm(cid:4)(cid:7)(cid:6) dt All aerodynamic coefficients are based on Mach number (cid:1) dH (cid:1) and the aerodynamic angles of attack and sideslip: =(cid:1)M (2) dt (cid:2)=tan(cid:1)1(cid:3)w(cid:1)NRF(cid:6) (9) The total force acting on the projectile comprises the (cid:5)(cid:4)u(cid:1) (cid:8)(cid:7) NRF weight, the aerodynamic force and the Magnus force. The total moment acting on the projectile comprises the moment (cid:2)=tan(cid:1)1(cid:3)v(cid:1)NRF(cid:6) (10) due to the standard aerodynamic force, the Magnus aerody- (cid:4) (cid:7) (cid:5)u(cid:1) (cid:8) namic moment and the unsteady aerodynamic moment. The NRF dynamic equations of motion [6-9] are derived in the non- The total aerodynamic velocity given in equation: rolling frame and provided in equations (3) up to (6): V = u(cid:1) 2+v(cid:1) 2+w(cid:1) 2 (11) T NRF NRF NRF (cid:1)x (cid:5) (cid:4)cos(cid:2)cos(cid:3) (cid:1)sin(cid:3) sin(cid:2)cos(cid:3)(cid:7) (cid:1)u(cid:1) (cid:5) if NRF (cid:4) (cid:4)=(cid:5) (cid:8) (cid:4) (cid:4) (3) The weight force in no-roll system is: (cid:2)yif(cid:6) (cid:5)cos(cid:2)sin(cid:3) cos(cid:3) sin(cid:2)sin(cid:3)(cid:8) (cid:2)v(cid:1)NRF(cid:6) (cid:3)(cid:4)zif (cid:7)(cid:4) (cid:6)(cid:5) (cid:1)sin(cid:2) 0 cos(cid:2) (cid:9)(cid:8) (cid:3)(cid:4)w(cid:1)NRF(cid:7)(cid:4) (cid:1)(cid:4)X(cid:1)wf(cid:5)(cid:4)= (cid:3)(cid:6)(cid:1)sin(cid:2)(cid:7)(cid:6) (12) (cid:2)Y(cid:1)w (cid:6) mg(cid:4) 0 (cid:8) for the position of projectile’s center of mass and f (cid:3)(cid:4)Z(cid:1)wf (cid:7)(cid:4) (cid:5)(cid:6)cos(cid:2)(cid:9)(cid:6) (cid:4)(cid:1)(cid:8) (cid:2)1 0 t (cid:5) (cid:1)p(cid:1) (cid:5) (cid:7) (cid:7)=(cid:3) (cid:1) (cid:6) (cid:4) NRF(cid:4) (4) The aerodynamic force, which acts on the projectile at (cid:5)(cid:2)(cid:9) 0 1 0 (cid:2)q(cid:1) (cid:6) (cid:3) (cid:6) NRF aerodynamic center of pressure, is: (cid:6)(cid:7)(cid:3)(cid:10)(cid:7) (cid:4)(cid:3)0 0 1/cos(cid:1)(cid:7)(cid:6) (cid:3)(cid:4)r(cid:1)NRF (cid:7)(cid:4) 8 The Open Mechanics Journal, 2008, Volume 2 Gkritzapis et al. (cid:4)(cid:4)(cid:3)(cid:4)(cid:4)(cid:2)(cid:1)XZY~~~AAAfff(cid:4)(cid:4)(cid:7)(cid:4)(cid:4)(cid:6)(cid:5)=12(cid:3)VT2Sref(cid:1)CX0(cid:1)CX(cid:1)(cid:1)2CCw~NNVNAATRw2F~v~VN2NTRR(cid:1)FFCX2v~VNTRF22 (13) ppsrl=eodjTd1e hpc(cid:1)0ttiiisVtlc edthe,t icn w hgnh iaiqcnuhde ytcuaarwunsisne sgo tum hte o tteoiqo unba eitn i odvneespr yed nedcteoern nmtv ieonnfi nietghn ett h sie in z c (e1ot h8uoe)-f V T analysis of free-flight range data. The Magnus, which acts on projectile at the Magnus DIFFERENTIAL EQUATION OF MOTION force center of pressure, is: The differential equation governing the angular oscillatory motion for the complete linearized pitching and yawing ~ (cid:4)(cid:4)(cid:4)(cid:2)(cid:1)XY~~MMff(cid:4)(cid:4)(cid:4)(cid:6)(cid:5)=12 (cid:3) VT2Sref ~pNRFD2CV0(cid:3)(cid:4)2(cid:2)w~NRF (14) mdisottaionnce o tfr asvlieglhedtl ys sisy mshmowetnri bc eplorowje: ctiles [6] as a function of (cid:4)(cid:3)ZMf(cid:4)(cid:7) (cid:1)~p DCT v~ (cid:2)(cid:3)(cid:3)+(H(cid:1)iP)(cid:2)(cid:3)(cid:1)(M+iPT)(cid:2)=iPG (19) NRF (cid:3)(cid:4)(cid:2)NRF 2VT2 where (cid:9)(cid:6)L~~ATmh(cid:9)(cid:13)e m(cid:4)(cid:3)o0ment0 due to a0erody(cid:11)(cid:10)n(cid:4)(cid:3)Xa~~mAifc(cid:11)(cid:10) f orce is: (15) H = (cid:3)2SmdCL(cid:1)(cid:3)2SmdCD(cid:1)(cid:2)y(cid:1)2(cid:4)(cid:6)(cid:5)(cid:3)2SmdCMQ(cid:7)(cid:9)(cid:8) (20) (cid:9)(cid:8)(cid:7)MN~AAmm(cid:9)(cid:15)(cid:14)=(cid:4)(cid:4)(cid:5)00 R(cid:2)0MAC (cid:1)R(cid:2)0MAC(cid:11)(cid:11)(cid:12)(cid:4)(cid:4)(cid:4)(cid:5)ZY~AAff(cid:11)(cid:11)(cid:11)(cid:12) P=(cid:1)(cid:3)(cid:2)IIYX(cid:4)(cid:6)(cid:5)(cid:1)(cid:3)(cid:2) pVd(cid:4)(cid:6)(cid:5) (21) (cid:4)(cid:1)L~~MTmhe(cid:4)(cid:5) mo(cid:4)(cid:3)0ment 0due to M0agnu(cid:7)(cid:6)s (cid:4)(cid:3)fX~o~rMcef (cid:7)(cid:6)is : (16) M =(cid:2)y(cid:1)2(cid:4)(cid:6)(cid:5)(cid:3)2SmdCMA(cid:7)(cid:9)(cid:8) (22) (cid:4)(cid:3)(cid:2)MN~MMmm(cid:4)(cid:7)(cid:6)= (cid:4)(cid:4)(cid:5)00 R(cid:2)0MAX (cid:1)R(cid:2)0MAX(cid:7)(cid:7)(cid:8)(cid:4)(cid:4)(cid:4)(cid:5)ZY~MMff(cid:7)(cid:7)(cid:7)(cid:8) T =(cid:4)(cid:6)(cid:5)(cid:3)2SmdCL(cid:7)(cid:9)(cid:8)+(cid:2)x(cid:1)2(cid:4)(cid:6)(cid:5)(cid:3)2SmdCNPA(cid:7)(cid:9)(cid:8) (23) In addition, for the unsteady moment UAm is: G= gdcos(cid:1) (24) ~ V2 p DC NRF LP (cid:5)(cid:4)(cid:5)(cid:3)(cid:2)MNL~~~UUUAAAmmm(cid:5)(cid:8)(cid:5)(cid:7)(cid:6)=21(cid:1)VT2D Sref q~~rNNRRFF2DDVCC2MMVQQ(cid:1)+CCVMMAA (17) a(cid:3)e=roT(cid:1)dhy+insi a(cid:2)md i ifcf efroerncte isa la nedq umaot imone ntcso tnht aati nasf feacltl ththee p istcighni ni f g i( ca2an5nd)t 2V V yawing motion of a spinning or non-spinning symmetric projectile body. The author’s definition (cid:3)=(cid:1)+i(cid:2)was first The dynamic equations of motion (3-6) are highly non- chosen by Fowler et al. and was adopted by R. H. Kent. linear. Thus, numerical integration is commonly used to ob- Gunners and engineers usually prefer this definition. The tain solutions to this initial value problem. gunner observer looks downrange from a position located just behind the gun. Upward and to the right are always con- MODIFIED PROJECTILE LINEAR THEORY sidered as positive directions. It is nature for the gunner to To develop the modified projectile linear theory [7], the define the (cid:1) axis as positive upward, the (cid:1) axis as positive following sets of simplifications are employed: the axial ve- to right and the clockwise direction of all rotations as posi- locity u(cid:1) can be replaced by the total velocity V because tive for right hand twist rifling. NRF T the side velocities v(cid:1) and w(cid:1) are small. STATIC AND DYNAMIC STABILITY CRITERIA NRF NRF The aerodynamic angles of attack (cid:1) and sideslip (cid:2) are The solution of differential equation (19) tells us that the small for the main part of the atmospheric trajectory epicyclic frequencies depend only on the dimensionless roll rate P and overturning moment M, and are unaffected by any (cid:1)(cid:3)w(cid:1) /V ,(cid:2)(cid:3)v(cid:1) /V NRF T NRF T of other aerodynamic forces and moments. and the projectiles are geometrically symmetrical From the definition of an unstable motion, we are led I = I = I = 0, I = I naturally to the concept of static stability: XY YZ XZ YY ZZ Constant aerodynamic coefficients for the most important (P2(cid:1)4M)(cid:2)0 (26) forces and moments with respect to angle of attack and Mach number are taken into account. Flat-fire and small yaw tra- Classical exterior ballistics defines the static stability jectories are considered so the yaw angle (cid:1) is small: factor [5] Sg, as: sin((cid:1))(cid:1)(cid:1), cos((cid:1))(cid:1)1 S = IX2p2 (27) g 2(cid:1)I SdV2C The independent variable is changed from time t to di- Y MA mensionless arclength s, measured in calibers of travel: Modified Linear Theory for Spinning The Open Mechanics Journal, 2008, Volume 2 9 Eliminating P2 between equations (26-27), we have: x = 0.0 m, y = 0.0 m, z = 0.0 m, (cid:1) = 0.0°, (cid:1) = 45.0° and ~ ~ ~ 85.0°, (cid:1) = 8.0°, u =318 m/s, v = 0.0 m/s, w = 0.0 m/s, 4M(Sg(cid:1)1)(cid:2)0 (28) ~p = 0.0 rad/s, q~ = 1.795 rad/s and ~r = 0.0 rad/s. For statically unstable (spin-stabilized) projectile, M > 0 The density and pressure are calculated as function of and equation (28) reduces to the classical static stability cri- altitude from the simple exponent model atmosphere, and terion: gravity acceleration is taken into account with the constant value g = 9.80665 m/s2. S (cid:1)1 (29) g The flight dynamic models of 105 mm HE M1 and 120 Equation (26) is a more general result than the (29), be- mm HE mortar projectile types involves the solution of the cause it shows that a statically stable missile (M < 0), is set of the twelve first order ordinary differentials, equations statically stable without spin. (3)-(6), which are solved simultaneously by resorting to numerical integration using a 4th order Runge-Kutta method, Dynamic stability requires that both damping exponents and regard to the 6-D nominal atmospheric projectile flight. be negative throughout the projectile’s flight. For a non- spinning statically stable missile (M < 0) and P is either zero. RESULTS AND DISCUSSION For finned missiles, the pitch damping moment coefficient is The flight path trajectories motion with constant aerody- usually negative. The lift and drag force coefficients are both namic coefficients of the 105 mm projectile with initial fir- positive, therefore H is nearly always greater than zero, and ing velocity of 494 m/sec, initial yaw angle 3 deg and rifling the dynamic stability is assured. twist rate 1 turn in 18 calibers (1/18) at 45o and 70o are illus- The dynamic stability factor Sd is defined as: trated in (Fig. 2). S = 2T (30) d H and the expressions H>0 (31) 1 (cid:2)S (2(cid:1)S ) (32) S d d g are the generalized dynamic stability criteria for any spinning or non-spinning symmetric projectile flight body. COMPUTATIONAL SIMULATION The constant dynamic flight model [10] uses mean values Fig. (2). Impact points and flight path trajectories of 6-DOF and of the experimental aerodynamic coefficients variations (Ta- Modified Linear with constant aerodynamic coefficients for 105 ble 2). mm projectile. Table 2. Constant Aerodynamic Parameters for Dynamic At 45° the 6-DOF model for 105 mm M1 projectile, fired Trajectory Flight of the Two Projectile Types at sea-level neglecting wind conditions, gives a predicted range to impact of approximately 11,600 m and a maximum height at almost 3,600 m. From the results of the modified 105mm Projectile 120mm Projectile linear model, the maximum range and the maximum height CD = 0.243, CL = 1.76, CD = 0.14, CL = 2.76, are almost the same, as shown in (Fig. 2). Also at 70°, the CLP = -0.0108, CMQ = -9.300, CMQ = -22.300, predicted level-ground range of 6-DOF model is 7,500 m CMA = 3.76, CYPA = 0.381 CMA = -15.76 with maximum height at about 6,450 m and the modified CNPA = 0.215 linear trajectory simulation gives the same values. The mortar projectile of 120 mm diameter is also exam- Initial data for 105 mm dynamic trajectory model with ined for its atmospheric constant flight trajectories predic- constant aerodynamic coefficients are: tions at pitch angles of 45o and 85o, with initial firing velocity of 318 m/s, initial yaw angle 8° and pitch rate 1.795 rad/s, x = 0.0 m, y = 0.0 m, z = 0.0 m, (cid:1) = 0.0°, (cid:1) = 45.0° and ~ ~ ~ as shown in (Fig. 3). 70.0°, (cid:1) = 3.0°, u =494 m/s, v = 0.0 m/s, w = 0.0 m/s, ~p = 1644 rad/s, q~ = 0.0 rad/s and ~r = 0.0 rad/s. At 45o, the 120 mm mortar projectile, fired at no wind conditions, the 6-DOF trajectory gives a range to impact at The axial spin rate is calculated from 7,000 m with a maximum height at almost 2050 m. At 85o, p(cid:1) =2(cid:2)V /(cid:1)D(rad/s) (33) the predicted level-ground range is approximately 1,230 m T and the height is 3,950 m. For the same initial pitch angles, where V is the total firing velocity (m/s), (cid:1)the rifling twist T the modified linear model’s results have satisfactory agree- rate at the gun muzzle (calibers per turn), and D the refer- ment. (Fig. 3). ence diameter of the projectile type (m). In (Fig. 4), static stability factor for the 105 mm HE M1 In addition, the corresponding initial data for 120 mm projectile trajectory motion with constant aerodynamic coef- are: 10 The Open Mechanics Journal, 2008, Volume 2 Gkritzapis et al. ficients is calculated at pitch angles 45° and 70°, respectively. 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 value of 8 at final impact point. The corresponding flight behavior at 70 degrees initial pitch angle shows that the transient motion damps out quickly, where the gyroscopic static stability factor has increased from 3.1 to 121 and then decreased to value of almost 6.6 at the impact area. Fig. (5). Comparative stability factor versus Mach number with constant aerodynamic model at low and high quadrant angles for the 105 mm projectile. CONCLUSION A six degrees of freedom (6-DOF) simulation flight dynamics model is applied for the accurate prediction of short and long-range trajectories for spin and fin-stabilized projectiles. The modified projectile linear theory trajectory re- ported here should prove useful to estimate flight trajectory’s phenomena as static and dynamic stability. Fig. (3). Flight path trajectories of 6-DOF and Modified Linear with constant aerodynamic coefficients for 120 mm at quadrant A LIST OF SYMBOLS elevation angles of 45o and 85o. C = zero-yaw drag aerodynamic coeffi- D0 cient C = yaw drag aerodynamic coefficient D2 C = lift aerodynamic coefficient L C = roll damping aerodynamic coefficient LP C = pitch damping aerodynamic coeffi- MQ cient C = overturning moment coefficient MA C = Magnus moment coefficient YPA C = Magnus force aerodynamic coefficient Fig. (4). Comparative static stability variation with constant aero- NPA dynamic model at low and high quadrant angles for the 105 mm S = static stability factor g projectile. S = dynamic stability factor d In (Fig. 5), at 45 and 70 degrees the Mach number was x ,y ,z = projectile position, m 1.45 at the muzzle, then decrease to 0.6 and 0.3 at the sum- if if if mit of the trajectory and then decrease to values of 0.8 and u(cid:1) ,v(cid:1) ,w(cid:1) = projectile velocity components ex- NRF NRF NRF 0.9 at the impact area, respectively. The constant value of pressed in no-roll-frame, m/s dynamic stability for 105 mm is 1.095 and lies within the p(cid:1) = projectile roll rate, rad/s interval (0 < Sd < 2). So the spin-stabilized 105 mm projec- NRF tile is gyroscopically and dynamically stable. q(cid:1) ,r(cid:1) = projectiles pitch and yaw rates ex- NRF NRF On the other hand the 120 mm mortar projectile has un- pressed in no-roll-frame, rad/s canted fins, and do not roll or spin at any point along the (cid:1),(cid:2) = projectiles pitch and yaw angles, deg trajectory. Because of that, the static stability is zero, and we examined only the dynamic stability for 120 mm projectile. (cid:1) = projectile roll angle, deg In the modified linear trajectory of the 120 mm mortar I = projectile inertia matrix projectile, the constant value of dynamic stability is 0.735 at I ,I ,I = diagonal components of the inertia pitch angles of 45 and 85 degrees. If the dynamic stability XX YY ZZ matrix factor S , lies within the interval (0<S <2), a statically stable d d projectile is always dynamically stable, regardless of spin. I ,I ,I = off-diagonal components of the inertia The mortar projectile of 120 mm belongs in the above inter- XY YZ XZ matrix val and H > 0, so is dynamically stable. Modified Linear Theory for Spinning The Open Mechanics Journal, 2008, Volume 2 11 V = total aerodynamic velocity, m/s REFERENCES T (cid:3) = atmospheric density, kg/m3 [1] Fowler R, Gallop E, Lock C, Richmond H. The Aerodynamics of Spinning Shell. London; 1920. D = projectile reference diameter, m [2] Kent R. An Elementary Treatment of the Motion of a Spinning Projectile About its center of Gravity. USA; 1937. S = projectile reference area ((cid:4)D2/4), m2 [3] Nielson K, Synge J. On the Motion of Spinning Shell. USA; 1943. ref [4] Kelley J, McShane E. On the Motion of a Projectile with Small or m = mass of projectile, kg Slowly Changing Yaw. USA; 1944. 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Atmospheric Flight Dynamic sure, m Simulation Modelling of Spin-Stabilized Projectiles; July 2007; g = gravity acceleration, m/s2 Athens, Greece; 2007. Received: December 06, 2007 Revised: December 31, 2007 Accepted: January 02, 2008 © Gkritzapis et al.; Licensee Bentham Open. This is an open access article licensed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.