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Mathematical Problems in Engineering
Volume 2006, Article ID 80705, 19 pages

Autoparametric vibrations of a nonlinear system with pendulum

Department of Applied Mechanics, Lublin University of Technology, Nadbystrzycka 36, Lublin 20-618 , Poland

Received 31 December 2004; Revised 18 May 2005; Accepted 11 July 2005

Copyright © 2006 J. Warminski and K. Kecik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Vibrations of a nonlinear oscillator with an attached pendulum, excited by movement of its point of suspension, have been analysed in the paper. The derived differential equations of motion show that the system is strongly nonlinear and the motions of both subsystems, the pendulum and the oscillator, are strongly coupled by inertial terms, leading to the so-called autoparametric vibrations. It has been found that the motion of the oscillator, forced by an external harmonic force, has been dynamically eliminated by the pendulum oscillations. Influence of a nonlinear spring on the vibration absorption near the main parametric resonance region has been carried out analytically, whereas the transition from regular to chaotic vibrations has been presented by using numerical methods. A transmission force on the foundation for regular and chaotic vibrations is presented as well.