Abstract
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.
Highlights
The behavior of a pair of coupled limit cycle oscillators or a system with two degrees of freedom displays a significantly wider range of phenomenon than a single limit cycle oscillator [1,2]
The present work is aimed at investigating the effects of feedback control and parametric excitation on response of such a nonlinearly coupled van der Pol system when 1:1 internal resonance and principal parametric resonance exist
It can be seen that both of the two oscillators can be excited in periodic vibration with proper feedback
Summary
The behavior of a pair of coupled limit cycle oscillators or a system with two degrees of freedom displays a significantly wider range of phenomenon than a single limit cycle oscillator [1,2]. X. Li et al / The principal parametric resonance of coupled van der pol oscillators under feedback control oscillation of a parametrically excited viscoelastic moving belt with 1:3 internal resonance by the method of multiple scales and numerical method. Bi [11] discussed the dynamical behavior of two parametrically excited van der Pol oscillators with 1:1 internal resonance based on the averaging method. The present work is aimed at investigating the effects of feedback control and parametric excitation on response of such a nonlinearly coupled van der Pol system when 1:1 internal resonance and principal parametric resonance exist. The slow flow equations, which determine the principal parametric resonance response, are obtained with the aid of the averaging method Their fixed points correspond to different types of solutions of the system considered.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
