Survey of Blunt Body Dynamic Stability in Supersonic Flow <s> B. Governing Equations 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> B. Governing Equations 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 </s> Survey of Blunt Body Dynamic Stability in Supersonic Flow <s> B. Governing Equations of Motion <s> Report discussing various features of the stability of ballistic reentry shapes, including considerations for ballistic-missile and manned-satellite reentry capsules. Attainment of satisfactory stability of reentry bodies with subsonic terminal velocities was not found to be too difficult, but undesirable features that may cause marginal stability characteristics may be introduced in an effort to minimize weight. Reentry bodies with supersonic terminal velocities have fewer stability issues. <s> BIB003
It is informative to examine the equations of motion that describe the dynamic oscillation of a vehicle such that the subsequent discussions on the influence of various parameters in the system are grounded in the context of the governing physics. Thorough derivations of the differential equations governing the pitching motion of an entry vehicle as a function of time, typically starting from the planar equations of motion, can be found in the literature BIB001 , BIB002 ,  -  . Figure 1 describes the coordinate system of the planar formulation of the problem. A brief derivation presented here also begins with four planar equations of motion for an entry vehicle that describe rates of change of the altitude, velocity, flight path angle, and pitch angle with respect to time: Inherent to this description of the motion are several simplifying assumptions:  Motions are restricted to a plane  Aerodynamic derivatives are independent of Mach number and vary linearly with α  Small L/D  Small angle of attack (α < 30 o )   Constant acceleration due to gravity  Spherical, non-rotating planet  Constant mass vehicle  No contribution of atmospheric winds If we apply one further assumption to Eqs. BIB001 BIB002 BIB003 , namely that the contributions of the gravitational and centrifugal forces are negligible, the resulting differential description of the angle of attack can be obtained: