Existence of analytical solution, stability assessment and periodic response of vibrating systems with time varying parameters

Abstract

The paper is focused on the solution of a vibrating system with one-degree-of-freedom with the objective to dealwith  the method for periodical response calculation (if exists)  reminding Harmonic Balance Method of linearsystems having time dependent parameters of mass, damping and stiffness under arbitrary periodical excitation.As a starting point of the investigation, a periodic Green’s function (PGF) construction of the stationary part ofthe original differential equation is used. The PGF then enables a transformation of the differential equation tothe integro-differential one whose analytical solution is given in this paper. Such solution exists only in the casethat the investigated system is  stable and  can be expressed in  exact form. The second goal of the paper is toassess the stability and solution existence. For this purpose, a methodologyof (in)stable parametric domain borderdetermination is developed.