Dynamic Stability of a Missile in Rolling Flight

Abstract

The paper sets down the equations of motion for a symmetric rolling missile with respect to axes attached to the missile. The missile may be jet (or rocket) propelled or coasting under accelerating or decelerating conditions, respectively, wherein the variable rolling velocity is derived from intentionally or unintentionally fins and/or wings. The equations contain a force and moment system that includes, in addition to the usual forces and moments, those due to magnus effects, misaligned surfaces, canted surfaces, jet misalignment, and the linear accelerations in the plane normal to the missile axis. The results present general stability criteria for a rolling missile which are summarized in the Discussion of Stability. LIST OP SYMBOLS (I) English Letters A — moment of inertia in roll about x axis, pd, lb0 sec. A = A/pd\itr B = moment of inertia about y axis, lb. sec. B = B/pd*. hr C ~ moment of inertia about z axis, lb. sec. CD = drag coefficient, dimensionless c = wing or tail chord, d — diameter of missile, ft, G = linear momentum vector H = angular momentum vector i, j = unit vectors along y and z axes, respectively L, M, N = moments about x, y, and z axes, respectively m = mass of missile, lb. sec. ft. m — m/pd, dimensionless P = rolling velocity of missile about x axis, rad. per sec. pz = P / F , r a d . p e r f t . q, r — angular velocities in pitch (about y axis) and in yaw (about z axis) 5 = area, sq.ft. T — thrust force, lbs. T = T/Pd\4 ft./sec. 2 t — time, sec. 11, v, w = components of linear velocity of the missile along x, y, z axes, respectively, per sec. u, v,w = derivatives with respect to time V = scalar velocity of missile along flight trajectory per sec. X, Y, Z — forces acting along x, y, z axes, respectively x, y, z = designations of coordinate axes z = distance along trajectory (II) Definition of Coefficients K = ?/V 1/ft. Kz , Km = dimensionless coefficients of force and moment, a a respectively, due to angle of attack or yaw Kz , Km = Kz ,/d and Km /d, respectively, 1/ft. Q'