Vibration amplitude control for a van der Pol–Duffing oscillator with time delay

Abstract Periodic solutions for the fundamental resonance response of a van der Pol–Duffing system under time-delayed position and velocity feedbacks are investigated. Using the asymptotic perturbation method, two slow-flow equations for the amplitude and phase of the fundamental resonance response are derived. Their fixed points correspond to limit cycles (phase-locked periodic solutions) for the original system. In the uncontrolled system, periodic solutions exist only for fixed values of amplitude and phase and depend on the system parameters and excitation amplitude. In many cases, the amplitudes of periodic solutions do not correspond to the technical requirements. It is demonstrated that, if the vibration control terms are added, stable periodic solutions with arbitrarily chosen amplitude can be accomplished. Therefore, the results obtained show that an effective vibration amplitude control is possible if appropriate time delay and feedback gains are chosen.

[1]  Earl H. Dowell,et al.  Resonances of a Harmonically Forced Duffing Oscillator with Time Delay State Feedback , 1998 .

[2]  Colin H. Hansen,et al.  The response of a Duffing-van der Pol oscillator under delayed feedback control , 2006 .

[3]  Clément Tchawoua,et al.  Strange attractors and chaos control in a Duffing–Van der Pol oscillator with two external periodic forces , 2004 .

[4]  Yoshiyuki Suzuki,et al.  Response and stability of strongly non-linear oscillators under wide-band random excitation , 2001 .

[5]  Errata: Periodic Solutions of Volterra’s Population Equation with Hereditary Effects , 1976 .

[6]  Andrew Y. T. Leung,et al.  Bifurcation Control of a Parametrically Excited Duffing System , 2002 .

[7]  Wischert,et al.  Delay-induced instabilities in nonlinear feedback systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  Attilio Maccari.  Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback , 2003 .

[9]  Neville J. Ford,et al.  Numerical Hopf bifurcation for a class of delay differential equations , 2000 .

[10]  A. Maccari.  Vibration Control for the Primary Resonance of a Cantilever Beam by a Time Delay State Feedback , 2003 .

[11]  Ali H. Nayfeh,et al.  Perturbation Methods in Nonlinear Dynamics—Applications to Machining Dynamics , 1997 .

[12]  Tass,et al.  Delay-induced transitions in visually guided movements. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  A. Maccari,et al.  The Response of a Parametrically Excited van der Pol Oscillator to a Time Delay State Feedback , 2001 .

[14]  Andrew Y. T. Leung,et al.  Resonances of a Non-Linear s.d.o.f. System with Two Time-Delays in Linear Feedback Control , 2002 .

[15]  A. Maccari.  Vibration Control for Two Nonlinearly Coupled and Parametrically Excited van der Pol Oscillators , 2005 .

[16]  Jinde Cao.  Periodic oscillation and exponential stability of delayed CNNs , 2000 .

[17]  Fatihcan M. Atay,et al.  VAN DER POL'S OSCILLATOR UNDER DELAYED FEEDBACK , 1998 .

[18]  Kwok-wai Chung,et al.  Effects of time delayed position feedback on a van der Pol–Duffing oscillator , 2003 .

[19]  Guanrong Chen,et al.  Bifurcation Control: Theories, Methods, and Applications , 2000, Int. J. Bifurc. Chaos.