A Limiting Case for Missile Rolling Moments

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Missile rolling moments are studied by linearized theory for the limiting case in which infinitely many wings are symmetrically arranged around a circular cylinder. This case is physically interesting when interference effects prevent the rolling moments from getting too far out of the range of those practically obtainable with a finite number of wings. Mathematically, the limiting case is convenient because it can be analyzed through knowledge of the rate at which angular momentum is being added to the air, the rolling moment being proportional to this quantity. The analysis is independent of Mach Number, and, since there are few restrictions on the geometry of the system, it should furnish an upper limit for rolling moments for a variety of configurations and flight speeds. Many cases with a finite number of wings are difficult to analyze, and knowledge of a limiting value may help in checking intermediate cases. For a delta wing system with leading edges on the Mach lines, body of zero radius, and each wing acting as an aileron, the rolling moment coefficient for an infinite number of wings is -K times tha t for the planar wing and 1.85 times tha t for the cruciform wing. For narrow delta systems, the above values apply exactly for the planar case and approximately for the cruciform. Wing-tail interference effects on aileron rolling moments may also be studied for the limiting case. Here, all angular momentum is removed from the disc corresponding to the maximum cross section of the tail (unless opposite surfaces of the tail are designed to float relative to each other and are controlled only through the sum of their displacements). If the tail disc is as large as the wing disc, then no resultant rolling moment is produced by wing ailerons in the limiting case. This assumes the discs to be concentric. For a lifting wing, the effective displacement of the disc centers might introduce important nonlinear effects. Damping in roll can be studied by assuming the wing angle of attack proportional to the radius. Equating the damping moment to the aileron moment gives an equilibrium rate of roll which may be expressed as the ratio of tip helix angle to aileron displacement angle. For the delta wing system previously considered (with sonic leading edges), this ratio decreases as the number of wings increases. For an infinite number of wings, the ratio is two-thirds of the planar value or 0.73 times the cruciform value. For a given aileron displacement, the equilibrium rate of roll is therefore decreased by increasing the number of wings. Physically, this corresponds to the fact tha t interference effects on damping in roll are less than on aileron rolling moments. The above figures are approximately correct for narrow delta systems.