A New Method of Determining Regions of Instability of a System with Parametric Excitations

Abstract

Vibrations of a system with periodically variable coefficients are treated and a new method of determining regions of instability is presented. A solution of such a system is introduced according to Lyapunov's theorem. Substituting the solution in the equation of motion, two kinds of characteristic equations are obtained. Eigenvalues of those kinds of characteristic equations are obtained. Eigenvalues of those equations are determined by approximating them by a characteristics equation with a finite order. All eigenvalues can be described by only two representative eigenvalues which are classified into three cases : (i) two imaginary numbers, (ii) one imaginary and one real and (iii) a pair of complex conjugate numbers. Cases (ii) and (iii) correspond to the unstable solutions.