The dynamic of a coupled three degree of freedom mechanical system

In this paper, a nonlinear coupled three degree-of-freedom autoparametric vibration system with elastic pendulum attached to the main mass is investigated numerically. Solutions for the system response are presented for specific values of the uncoupled normal frequency ratios and the energy transfer between modes of vibrations is observed. Curves of internal resonances for free vibrations and external resonances for vertical exciting force are shown. In this type system one mode of vibration may excite or damp another one, and except different kinds of periodic vibration there may also appear chaotic vibration. Various techniques, including chaos techniques such as bifurcation diagrams and: time histories, phase plane portraits, power spectral densities, Poincarè maps and exponents of Lyapunov, are used in the identification of the responses. These bifurcation diagrams show many sudden qualitative changes, that is, many bifurcations in the chaotic attractor as well as in the periodic orbits. The results show that the system can exhibit various types of motion, from periodic to quasi-periodic and to chaotic, and is sensitive to small changes of the system parameters.