Missile Stability Using Finite Elements—an Unconstrained Variational Approach

The stability behavior of a flexible missile idealized as a free-free beam is studied using the finite-element technique. The structure is assumed to be under a constant thrust which, in turn, is subjected to a directional control device. A concentrated mass is included to model a piece of heavy machinery. The solution formulation - finite element in conjunction with unconstrained variational principles -is shown here to be general, simple to use, and effective to overcome the difficulties arising from nonconservative forces, concentrated mass, and feedback control features. As a basis of this approach, an unconstrained variational statement and the associated adjoint problems are introduced. Numerical results from this study reveal that 1) for a free-free beam under a constant thrust, there exists a nonzero mode which seems to have escaped previous investigators; 2) since this newly realized mode is the lowest nonzero mode and is divergent in nature, a missile structure is unstable without feedback control; and 3) depending on the amount and location of a concentrated mass, it can improve the stability behavior of a missile.