Chaotic vibration of an autoparametrical system with a non-ideal source of power

This paper studies the dynamical coupling between energy sources and the response of a two degrees of freedom autoparametrical system, when the excitation comes from an electric motor (with unbalanced mass $ m_0$), which works with limited power supply. The investigated system consists of a pendulum of the length $ l$ and mass $ m$, and a body of mass $ M$ suspended on a flexible element. In this case, the excitation has to be expressed by an equation describing how the energy source supplies the energy to the system. The non-ideal source of power adds one degree of freedom, which makes the system have three degrees of freedom. The system has been searched for known characteristics of the energy source (DC motor). The equations of motion have been solved numerically. The influence of motor speed on the phenomenon of energy transfer has been studied. Near the internal and external resonance region, except for different kinds of periodic vibration, chaotic vibration has been observed. For characterizing an irregular chaotic response, bifurcation diagrams and time histories, power spectral densities, Poincaré maps and maximal exponents of Lyapunov have been developed.