The prediction of instabilities of linear differential equations with periodic coefficients

Abstract

Summary The stability of the solutions of a system of differential equations with periodic coefficients has been examined using Floquet\\'s theorem and a general method of solution has been programmed in ICL 1900 Fortran. The application of the method is illustrated by the solution of two dynamical systems both of which are unsymmetrical rigid rotors in unsymmetrical bearings and the program has been used to obtain solutions for up to six simultaneous second-order differential equations with periodic coefficients. The relevance of the solutions for an associated system of equations having constant coefficients, to the solutions of the periodic system is discussed and used to propound a method for predicting the unstable regions of periodic systems of particular form. For Lagrangian systems without viscous damping a method is presented for calculating the limits of an unstable region.