Abstract
We analyze the effect of a fast harmonic excitation on hysteresis and on entrainment area in a forced van der Pol–Duffing oscillator near the primary and the 3:1 subharmonic resonances. Analytical treatment based on perturbation techniques is performed to capture the entrainment zone, the quasiperiodic modulation domain and the hysteresis area in the vicinity of the two resonances. Specifically, it is shown that a fast harmonic excitation can suppress hysteresis for a certain range of the fast excitation leading to a smooth transition between the quasiperiodic and the frequency-locked responses near these resonances. Furthermore, the influence of different system parameters on the hysteresis area has been investigated. In particular, the results reveal that the amplitude of the fast excitation and the nonlinear damping significantly affect the domain of hysteresis suppression near the primary and the 3:1 subharmonic resonances.
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References
Nayfeh, A.H., Younis, M.I.: Dynamics of MEMS resonators under superharmonic and subharmonic excitations. Micromech. Microeng. 15, 1840–1847 (2005)
Nayfeh, A.H., Younis, M.I., Abdel-Rahman, E.M.: Dynamic pull-in phenomenon in MEMS resonators. Non-linear Dyn. 48, 153–163 (2007)
Pandey, M., Rand, R.H., Zehnder, A.T.: Perturbation analysis of entrainment in a micromechanical limit cycle oscillator. Commun. Non-linear Sci. Numer. Simul. 12, 1291–1301 (2007)
Rajesh, S., Nandakumaran, V.M.: Control of bistability in a directly modulated semiconductor laser using delayed optoelectronic feedback. Physica D 213, 113–120 (2006)
Fahsi, A., Belhaq, M., Lakrad, F.: Suppression of hysteresis in a forced van der Pol–Duffing oscillator. Commun. Non-linear Sci. Numer. Simul. (2008). doi:10.1016/j.cnsns.2008.03.003
Stephenson, A.: On induced stability. Philos. Mag. 15, 233–236 (1908)
Hirsch, P.: Das Pendel mit oszillierendem Aufhängepunkt. Z. Angew. Math. Mech. 10, 41–52 (1930)
Kapitza, P.L.: Dynamic stability of a pendulum with an oscillating point of suspension. Zurnal Eksp. Teor. Fiz. 21, 588–597 (1951). (In Russian)
Thomsen, J.J.: Some general effects of strong high-frequency excitation: stiffening, biasing, and smoothening. J. Sound Vib. 253(4), 807–831 (2002)
Jensen, J.S., Tcherniak, D.M., Thomsen, J.J.: Stiffening effects of high-frequency excitation: experiments for an axially loaded beam. J. Appl. Mech. 67, 397–402 (2000)
Hansen, M.H.: Effect of high-frequency excitation on natural frequencies of spinning discs. J. Sound Vib. 234(4), 577–589 (2000)
Tcherniak, D., Thomsen, J.J.: Slow effect of fast harmonic excitation for elastic structures. Non-linear Dyn. 17, 227–246 (1998)
Mann, B.P., Koplow, M.A.: Symmetry breaking bifurcations of a parametrically excited pendulum. Non-linear Dyn. 46, 427–437 (2006)
Sah, S.M., Belhaq, M.: Effect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator. Chaos Solitons Fractals 37(5), 1489–1496 (2008)
Belhaq, M., Sah, S.M.: Horizontal fast excitation in delayed van der Pol oscillator. Commun. Non-linear Sci. Numer. Simul. 13(8), 1706–1713 (2008)
Belhaq, M., Sah, S.M.: Fast parametrically excited van der Pol oscillator with time delay state feedback. Int. J. Non-linear Mech. 43(2), 124–130 (2008)
Belhaq, M., Fahsi, A.: 2:1 and 1:1 frequency-locking in fast excited van der Pol–Mathieu–Duffing oscillator. Non-linear Dyn. 53, 139–152 (2008)
Blekhman, I.I.: Vibrational Mechanics—Non-linear Dynamic Effects, General Approach, Application. World Scientific, Singapore (2000)
Belhaq, M., Houssni, M.: Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations. Non-linear Dyn. 18, 1–24 (1999)
Belhaq, M., Guennoun, K., Houssni, M.: Asymptotic solutions for a damped non-linear quasi-periodic Mathieu equation. Int. J. Non-linear Mech. 37, 445–460 (2000)
Rand, R.H., Guennoun, K., Belhaq, M.: 2:2:1 resonance in the quasi-periodic Mathieu equation. Non-linear Dyn. 31, 187–193 (2003)
Rand, R.H., Morrison, T.: 2:1:1 resonance in the quasi-periodic Mathieu equation. Non-linear Dyn. 40, 195–203 (2005)
Sah, S.M., Recktenwald, G., Rand, R.H., Belhaq, M.: Autoparametric quasiperiodic excitation. Int. J. Non-linear Mech. 43, 320–327 (2008)
Nayfeh, A.H., Mook, D.T.: Non-linear Oscillations. Wiley, New York (1979)
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Belhaq, M., Fahsi, A. Hysteresis suppression for primary and subharmonic 3:1 resonances using fast excitation. Nonlinear Dyn 57, 275–287 (2009). https://doi.org/10.1007/s11071-008-9438-z
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DOI: https://doi.org/10.1007/s11071-008-9438-z