A method for converting nonautonomous systems with periodic coefficients to autonomous systems

Abstract

A new method that eliminates time dependent periodic coefficients of nonautonomous equations to time independent autonomous systems and computes stability boundaries is developed. This simplifies the solution of complex nonautonomous rotodynamic equations with periodic coefficients encountered on rotating machines. The method collects all of the periodic coefficients in the equations into a unitary matrix, U, and writes the equations as x + Uy = 0, where x and y are vectors with dimension n. In this form, the matrix of periodic coefficients is readily eliminated by transposition to yield x*x – y*y = 0. This method is applicable in the rotating and stationary frames of reference. The method combines with algorithms may provide the first step towards selecting actuators for remotely piloted helicopter's air unit rotors and other autonomous rotating applications.