Asymptotic solutions to a stability problem

This paper is concerned with the lateral stability of a free flying column subjected to an axial thrust with directional control. The stability curve (i.e., eignevauevs. thrust, in the neighborhood of zero eigenvalues) and the associated eignefunctions of this problem have not been fully understood. Here, asymptotic expansions are used to examine closely, for all values of the thrust directional control parameter, both the intersection of the eigenvalue curves with the zero branch and the associated eigenfunctions of zero and nearly zero eigenvalues. Several analytical proofs are provided substantiating previous numerical findings.