Stability of a spinning body containing an elastic membrane via Liapunov's direct method.

Abstract

A general method of analysis is presented for studying the dynamic stability of a spinning rigid body containing a continuous elastic member. The method utilizes Liapunov's direct method applied to the nonlinear equations of motion with a modal expansion used for the spatial description of the elastic media. A model consisting of a rigid carrier vehicle with an attached, square elastic membrane is used to demonstrate the method which is applicable to a broad class of hybrid combined elastic-rigid body systems. The stability of the nominal spin about a principal axis of the system is investigated by constructing an appropriate Liapunov function and forming the Hessian matrix in such a way that additional modes may be readily included by simply adding new rows and columns to the matrix without affecting the previous principal minor. In this way, questions relating to the effect of truncation errors on the stability of the system can be readily studied. For the problem presented, a general set of stability criteria in terms of the system parameters is formulated which functionally shows the effect of including additional modes on the stability of the elastic system. Quantitatively the analysis shows that highly elastic systems at low spin rates require more normal mode terms for accurately establishing the boundaries to the stability regions than do relatively stiff systems.