This paper describes a nonlinear phenomenon in the dynamical behavior of a nonlinear system under two non-ideal excitations: the self-synchronization of unbalanced direct current motors. The considered model is taken as a Duffing system that is excited by two unbalanced direct current motors with limited power supplies. The results obtained by using numerical simulations are discussed in details

We apply the Bogoliubov Averaging Method to the study of the vibrations of an elastic foundation, forced by a Non-ideal energy source. The considered model consists of a portal plane frame with quadratic nonlinearities, with internal resonance 1:2, supporting a direct current motor with limited power. The non-ideal excitation is in primary resonance in the order of one-half with the second mode frequency. The results of the averaging method, plotted in time evolution curve and phase diagrams are compared to those obtained by numerically integrating of the original differential equations. The presence of the saturation phenomenon is verified by analytical procedures.