Analytical periodic solution and stability assessment of 1 DOF parametric systems with time varying stiffness

The presented paper deals with an approach to analytical periodic solution and to stability assessment of one-degree-of-freedom linear vibrating systems. It is supposed that these systems are excited by the time periodic force and contain time periodic stiffness. The periodic Green's function determined as a response to a Dirac chain of unit impulses repeating with period of excitation is used to transform the equation of motion into the Fredholm integral equation with degenerated kernel. If the Dirac chain is expressed as a Fourier series and a limited number of terms is taken into account, the solution of the integral equation can also be obtained in a series form. It has been found that the real eigenvalues of the system matrix determine the critical values of the fluctuation stiffness parameter. The values of this real parameter correspond to the borders of (in)stability in the plane given by the variation of the angle frequency and of the fluctuation stiffness parameter. Moreover, very interesting property of the system matrix was observed. The positive sign of the real valued determinant of the system matrix means the existence of periodic solution (system is stable). In the opposite case, the periodic solution does not exist (system is unstable). The verification of obtained results was performed on two case studies. The Floquet method was used to validate the stability assessment. Presented analytical periodic solution was compared with steady state obtained by the Runge-Kutta continuation. A very good agreement was achieved in both cases.