Theoretical study on synchronization of two exciters in a nonlinear vibrating system with multiple resonant types

Abstract

This paper aims at studying theoretically synchronization of two exciters in a nonlinear vibrating system (NVS), in which the behavior of NVS is mainly reflected at nonlinear restoring forces of springs with piecewise linear characters. The differential motion equations of two rigid frames in the horizontal direction are combined into their relative motion equation. Based on the asymptotic method, the nonlinear stiffness of springs is linearized equivalently as a function of the amplitude of the relative motion. Using Lagrange’s equations, the differential motion equations of the total system are deduced. The criterion of synchronization for two exciters in the analytical form is derived, by the average method. According to the principle that the stable solution of the synchronous states corresponds to a minimum point of Hamilton’s average action amplitude of the system, the criterion of stability of the synchronous states is achieved analytically. States of the system versus the operating frequency under consideration of multiple resonant types are presented. For the synchronous vibrating machines with two rigid frames used in engineering, the condition of the minimum of excitation to foundation is discussed, as well as the ideal working regions of the vibrating machines.