Nonlinear motion of a symmetric missile acted on by a double valued static moment.

The static moment of a symmetric missile is usually assumed to be single-valued although double-valued moments can occur. For example, lee side separation can occur at an angle of attack that is greater than that for reattachment and the moment for angles between these values can be multivalued. If a mathematically simple double-valued static moment consisting of two linear segments and two jump points is assumed (Fig. 1), an exact solution for planar motion and zero aerodynamic damping can be obtained. The effects of nonplanar motions and both linear and nonlinear damping moments are studied by the use of a quasilinear analysis. First regions in an amplitude plane are determined for which the multivalued moment is brought into play as well as regions for which the motion is only affected by the single-valued linear segments of the static moment curve. Next, quasilinear approximations are derived for frequencies and damping rates of the motion caused by this multivalued static moment coupled with one linear and two cubic damping moments. The accuracy of these approximations is determined by comparison with the exact solution for planar motion with no aerodynamic damping and is found to be better than 1% for most cases. The character of the planar singularity for linear damping is investigated and found to be a stable limit cycle.