Survey of Blunt Body Dynamic Stability in Supersonic Flow <s> Analytical Applications of the Equation of Motion <s> From Summary: "An analysis is given of the oscillating motion of a ballistic missile which upon entering the atmosphere is angularly misaligned with respect to the flight path. The history of the motion for some example missiles is discussed from the point of view of the effect of the motion on the aerodynamic heating and loading. The miss distance at the target due to misalignment and to small accidental trim angles is treated. The stability problem is also discussed for the case where the missile is tumbling prior to atmospheric entry." <s> BIB001 </s> Survey of Blunt Body Dynamic Stability in Supersonic Flow <s> Analytical Applications of the Equation of Motion <s> From Summary: "An analysis is given of the oscillatory motions of vehicles which traverse ascending and descending paths through the atmosphere at high speed. The specific case of a skip path is examined in detail, and this leads to a form of solution for the oscillatory motion which should recur over any trajectory. The distinguishing feature of this form is the appearance of the Bessel rather than the trigonometric function as the characteristic mode of oscillation." <s> BIB002
In the literature, Eqs. (5) and (6) above have been manipulated to obtain analytical descriptions of key parameters or special cases which are useful in developing intuition and gaining insight into the expected dynamic response. By using an exponential atmosphere and other assumptions, Allen BIB001 was able to find an analytical expression for the velocity as function of altitude. By implementing this into the relations above, he then introduced a dynamic stability criterion: where ξ < 0 would indicate a dynamically stable configuration. Fig. 2 displays this parameter as a function of cone angle and location of the center of gravity for conical missile shapes. It can be seen that at the high cone angles typically used for entry vehicles due to heating concerns, the parameter becomes less negative which indicates the body is less dynamically stable. This criterion is similar to using a negative lift curve slope as an indicator that a vehicle is statically stable. While this parameter is useful for understanding the trends of dynamic stability, the numerous assumptions behind its derivation require that it be used with caution, as a negative ξ does not guarantee a dynamically stable vehicle. Often, in the literature and throughout this report, the pitch damping sum ( ̇) from the dynamic stability parameter is used independently as a measure of dynamic stability, with negative values indicating positive (favorable) damping. This term is effectively a measure of the damping experienced during small oscillation perturbations about a mean angle of attack  . Using the same closed form solution for the velocity as in BIB001 , multiple studies formulated expressions for bounding the maximum oscillation amplitude and frequency BIB001 , BIB002 ,  ,  . More recently, Schoenenberger et al  solved Eq. (6) with the Euler-Cauchy equation and used this solution to study the simplified oscillatory behavior of a body in which additional assumptions of a body in free flight ground testing apply. These analytical formulations are useful in understanding the fundamentals of the dynamic stability phenomenon. However, the numerous assumptions made do not accurately capture the complex nature of the flow field around a blunt body and limit their use as design and analysis tools.