A Generalized Hill’s Method for the Stability Analysis of Parametrically Excited Dynamic Systems

Abstract

A method is described for investigating the stability of the null solution for a general system of linear second-order differential equations with periodic coefficients. The method is based on a generalization of Hill’s analysis and leads to a generalized Hill’s infinite determinant. Following a proof of its absolute convergence, a closed-form expression for the characteristic infinite determinant is obtained. Methods for the stability analysis utilizing different forms of the characteristic determinant are discussed. For cases where the instabilities are of the simple parametric type, a truncated form of the determinant may be used directly to locate the boundaries of the resonance regions in terms of appropriate system parameters. The present generalized Hill’s method is applied to a multidegree-of-freedom discretized system describing pipes conveying pulsating fluid. It is demonstrated that the method is a flexible and efficient computational tool for the stability analysis of general periodic systems.