Affiliations: School of Mechanical Engineering, Hebei University of
Technology, Tianjin, P.R. China | School of Astronautics, Harbin Institute of
Technology, Harbin, P.R. China | School of Mechanical Engineering, Tianjin University
of Science and Technology, Tianjin, P.R. China
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.
Keywords: Principal parametric resonance, coupled van der Pol oscillators, feedback control, time delay, bifurcation
