Dynamic instability of finned missiles caused by unbalanced fin forces

Abstract

A dynamic instability is described for rolling, finned missiles with canted fins. The instability occurs as an undamping of the angle of attack and is caused by differential lift and drag from unequal effectiveness of the windward and leeward fins. The instability is similar to a Magnus instability in that a yawing moment is produced that unequally damps the positive and negative precession motions. The equivalence of the Magnus coefficient and the differential fin forces is derived, and the instability is demonstrated with computer simulations of the equations of motion for the angle-of-attack convergence of a re-entry vehicle. The computer results are compared with a closed-form solution for the angle-of-attack convergence envelope. NomenclatureA -slope of fin lift effectiveness vs angle of attack B -slope of fin drag effectiveness vs angle of attack Cio = aerodynamic fin-induced roll moment coefficient Ci s = aerodynamic fin lift derivative C mq = aerodynamic pitch damping derivative Cnpa -aerodynamic Magnus moment derivative C Na == aerodynamic normal force derivative C' Na = 2C Na I/md* d -aerodynamic reference diameter D = fin drag force / = pitch or yaw moment of inertia I x -roll moment of inertia L = fin lift force m = vehicle mass M{ -aerodynamic yaw moment Mq -aerodynamic pitch moment M% = aerodynamic roll moment p = roll rate p r = reduced roll rate, pp/2 q = dynamic pressure S = aerodynamic reference area u -vehicle velocity Xf -distance of fins aft of center of mass x s t = static margin (distance of center of pressure aft of center of mass) x ,y,z = body-fixed coordinates X,Y,Z = inertial coordinates 5 == effective fin cant angle 5 0 = fixed fin cant angle relative to body axis of symmetry f = yaw axis t\ -pitch axis 0 -total angle of attack (Euler angle) K = roll moment coefficient, Ci 0 qSd/I x \ = fin drag coefficient, C.o 5 <?Sd/2/ /it = inertia ratio, I X /I v = pitch or yaw damping coefficient, (qSd*/2Iu)(-C mq v m = (nC Na qS/mu) + (8 0 rA -£ = roll axis fin lift coefficient, Ci$qSx//I roll orientation relative to wind (Euler angle) precession angle (Euler angle) natural pitch frequency Paper presented at the AIAA 8th Aerospace