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NASA NTRS1974

Effect of thrust on the longitudinal oscillations of missiles

Exact linearized solutions are provided for the longitudinal oscillations of missiles when the trajectory is nearly horizontal and the thrust is either zero, a specified constant value, or increasing with the square of the increasing flight speed. The results indicate that in all cases any increase in the thrust, or any acceleration along the nearly horizontal trajectory, will directly increase the damping of the angle-of-attack oscillations, whose increase in frequency is at first negligible, and then grows very slightly with time.

Laitone, E. V.↗

NASA NTRS1972

Stability criteria for accelerating and decelerating aircraft.

Derivation of the linearized differential equation for the longitudinal oscillations of an aircraft for the general case when the atmospheric density, thrust force, and velocity are varying. Exact solutions and stability criteria are presented for several different conditions. For the zero thrust coasting aircraft that is being decelerated by its drag force, new stability criteria are derived for both ascending and descending flight. A critical altitude is found for ascending aircraft that are attempting to coast out of the atmosphere. Above this critical altitude the oscillations cease, and there is a monotonic increase in the angle of attack. Exact solutions for the oscillations of coasting aircraft are presented in terms of the confluent hypergeometric functions. It is shown that previous attempts to predict a critical altitude for ascending vehicles were in error because they did not include the second type, or logarithmic solution, of the confluent hypergeometric function.

Laitone, E. V.↗