zbMATH — the first resource for mathematics

Effect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator. (English) Zbl 1142.34332
Summary: We investigate the interaction effect of fast vertical parametric excitation and time delay on self-oscillation in a van der Pol oscillator. We use the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic and then we apply the averaging technique on this slow dynamic to derive a slow flow. In particular we analyze the slow flow to analytically approximate regions where self-excited vibrations can be eliminated. Numerical integration is performed and compared to the analytical results showing a good agreement for small time delay. It was shown that vertical parametric excitation, in the presence of delay, can suppress self-excited vibrations. These vibrations, however, persist for all values of the excitation frequency in the case of a fast vertical parametric excitation without delay [R. Bourkha, M. Belhaq, Chaos Solitons Fractals 34, No. 2, 621–627 (2007; Zbl 1167.70012)].

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
70K99 Nonlinear dynamics in mechanics
Full Text: DOI
[1] Blekhman, I.I., Vibrational mechanics – nonlinear dynamic effects, general approach, application, (2000), World Scientific Singapore
[2] Tcherniak, D.; Thomsen, J.J., Slow effects of fast harmonic excitation for elastic structures, Nonlinear dyn, 17, 227-246, (1998) · Zbl 0933.74032
[3] Fidlin A, Thomsen JJ. Non-trivial effect of strong high-frequency excitation on a nonlinear controlled system. In: Proceeding of the XXI international congress of theoretical and applied mechanics. 15-21 August 2004, Warsaw.
[4] Thomsen, J.J., Some general effects of strong high-frequency excitation: stiffening, biasing, and smoothing, J sound vib, 253, 4, 807-831, (2002)
[5] Thomsen, J.J., Using fast vibrations to quenching friction-induced oscillations, J sound vib, 228, 5, 1079-1102, (1999)
[6] Bourkha, R.; Belhaq, M., Effect of fast harmonic excitation on a self-excited motion in van der Pol oscillator, Chaos, solitons & fractals, 34, 2, 621-627, (2007) · Zbl 1167.70012
[7] Insperger, T.; Stepan, G., Stability chart for the delayed Mathieu equation, Proc royal soc math phys eng sci, 458, 1989-1998, (2002) · Zbl 1056.34073
[8] Maccari, A., The response of a parametrically excited van der Pol oscilator to a time delay state feedback, Nonlinear dyn, 26, 105-119, (2001) · Zbl 1018.93015
[9] Nana Nbendjo, B.R.; Woafo, P., Active control with delay of horseshoes chaos using piezoelectric absorber on a buckled beam under parametric excitation, Chaos, solitons & fractals, 32, 1, 73-79, (2007)
[10] Ji, J.C.; Hansen, C.H., Stability and dynamics of a controlled van der Pol-Duffing oscillator, Chaos, solitons & fractals, 28, 2, 555-570, (2006) · Zbl 1084.34040
[11] Kalmar-Nagy, T.; Stepan, G.; Moon, F.C., Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations, Nonlinear dyn, 26, 121-142, (2001) · Zbl 1005.70019
[12] Rand RH. Lecture Notes on Nonlinear Vibrations (version52). <http://www.tam.cornell.edu/randdocs/nlvibe52.pdf>, 2005.
[13] Nayfeh, A.H.; Mook, D.T., Nonlinear oscillations, (1979), Wiley New York
[14] Wirkus, S.; Rand, R.H., Dynamics of two coupled van der Pol oscillators with delay coupling, Nonlinear dyn, 30, 205-221, (2002) · Zbl 1021.70010
[15] Shampine LF, Thompson S. Solving delay differential equations with dde23. PDF <http://www.radford.edu/ thompson/webddes/tutorial.pdf>, 2000.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.