Nonlinear Oscillations of an Autoparametrical System With Two Coupled Pendulums

Abstract

The nonlinear dynamics of a three degree of freedom autoparametrical vibration system with two coupled pendulums in the neighborhood internal and external resonances is presented in this work. It was assumed that the main body is suspended by an element characterized by non-linear elasticity and non-linear damping force and is excited harmonicaly in the vertical direction. The two connected by spring pendulums characterized are mounted to the main body. It is assumed, that the motion of the pendulums are damped by nonlinear resistive forces. 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 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, Poincare` 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 to chaotic, and is sensitive to small changes of the system parameters.