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Shock and Vibration
Volume 19 (2012), Issue 3, Pages 365-377

The Principal Parametric Resonance of Coupled van der Pol Oscillators under Feedback Control

Xinye Li,1 Huabiao Zhang,2 and Lijuan He3

1School of Mechanical Engineering, Hebei University of Technology, Tianjin, China
2School of Astronautics, Harbin Institute of Technology, Harbin, China
3School of Mechanical Engineering, Tianjin University of Science and Technology, Tianjin, China

Received 9 August 2010; Revised 15 December 2010

Copyright © 2012 Hindawi Publishing Corporation. 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.


The principal parametric resonance of two van der Pol oscillators under coupled position and velocity feedback control with time delay is investigated analytically and numerically on the assumption that only one of the two oscillators is parametrically excited and the feedback control is linear. The slow-flow equations are obtained by the averaging method and simplified by truncating the first term of Taylor expansions for those terms with time delay. It is found that nontrivial solutions corresponding to periodic motions exist only for one oscillator if no feedback control is applied although the two oscillators are nonlinearly coupled. Based on Levenberg-Marquardt method, the effects of excitation and control parameters on the amplitude of periodic solutions of the system are graphically given. It can be seen that both of the two oscillators can be excited in periodic vibration with proper feedback. However, the amplitudes of the periodic vibrations are independent of the sign of feedback gains. In addition, the influence of time delay on the response of the system is periodic. In terms of numerical simulations, it is shown that both of the two oscillators can also have quasi-periodic motions, periodic motions about a new equilibrium position and other complex motions such as relaxation oscillation when feedback control is considered.