Stability of a flexible missile described by asymptotics of the eigenvalues of fourth order boundary value problems

TL;DR

This paper analyzes the eigenvalue asymptotics of fourth order boundary value problems relevant to elastic rods and flexible missile stability, providing explicit formulas for the eigenvalues' behavior.

Contribution

It derives explicit asymptotic formulas for eigenvalues of non self-adjoint fourth order problems with eigenvalue-dependent boundary conditions, applied to missile stability analysis.

Findings

Explicit eigenvalue asymptotics for elastic rod problems.

Asymptotic formulas for missile stability eigenvalues.

Analysis of non self-adjoint operator eigenvalues.

Abstract

Fourth order problems, with the differential equation $y^{(4)}-(gy')'=\lambda^2y$, where $g\in C^1[0,a]$ and $a>0$, occur in engineering on stability of elastic rods. They occur as well in aeronautics to describe the stability of a flexible missile. Fourth order Birkhoff regular problems with the differential equation $y^{(4)}-(gy')'=\lambda^2y$ and eigenvalue dependent boundary conditions are considered. These problems have quadratic operator representations with non self-adjoint operators. The first four terms of the asymptotics of the eigenvalues of the problems as well as those of the eigenvalues of the problem describing the stability of a flexible missile are evaluated explicitly.