Dynamic Longitudinal Stability Equations for the Re-Entry Ballistic Missile

The limitations of the linearized equations predicting the dynamic stability during rapid deceleration of a nonrolling missile are discussed for the usual ballistic missile which has a longitudinal plane of symmetry and is trimmed to fly at a nearly zero angle of attack trajectory. In order to demonstrate that only the first-order terms can be rigorously justified, the ordinary second-order linear differential equation predicting the oscillations in angle of attack is derived by two entirely different approaches. The first method follows the usual procedure of small perturbations in the flight trajectory equations, while the second method utilizes Euler's dynamic equations, for axes that are rigidly fixed in the moving body. I t is found that acceleration increases the damping while deceleration decreases the damping of any stable oscillation. For Eqs. (22) to (24), when a « w/u -> 0 and q = dB/dt • 0,