zbMATH — the first resource for mathematics

\(2:1\) and \(1:1\) frequency-locking in fast excited van der Pol-Mathieu-Duffing oscillator. (English) Zbl 1170.70344
Summary: The frequency-locking area of \(2:1\) and \(1:1\) resonances in a fast harmonically excited van der Pol-Mathieu-Duffing oscillator is studied. An averaging technique over the fast excitation is used to derive an equation governing the slow dynamic of the oscillator. A perturbation technique is then performed on the slow dynamic near the \(2:1\) and \(1:1\) resonances, respectively, to obtain reduced autonomous slow flow equations governing the modulation of amplitude and phase of the corresponding slow dynamics. These equations are used to determine the steady state responses, bifurcations and frequency-response curves. Analysis of quasi-periodic vibrations is carried out by performing multiple scales expansion for each of the dependent variables of the slow flows. Results show that in the vicinity of both considered resonances, fast harmonic excitation can change the nonlinear characteristic spring behavior from softening to hardening and causes the entrainment regions to shift. It was also shown that entrained vibrations with moderate amplitude can be obtained in a small region near the \(1:1\) resonance. Numerical simulations are performed to confirm the analytical results.

70K30 Nonlinear resonances for nonlinear problems in mechanics
70K65 Averaging of perturbations for nonlinear problems in mechanics
70K43 Quasi-periodic motions and invariant tori for nonlinear problems in mechanics
Full Text: DOI
[1] Tondl, A.: On the interaction between self-excited and parametric vibrations. National Research Institute for Machine Design, Monographs and Memoranda No. 25, Prague (1978)
[2] Schmidt, G.: Interaction of self-excited forced and parametrically excited vibrations. In: The 9th International Conference on Nonlinear Oscillations. Application of The Theory of Nonlinear Oscillations, vol. 3. Naukowa Dumka, Kiev (1984)
[3] Szabelski, K., Warminski, J.: Self excited system vibrations with parametric and external excitations. J. Sound. Vib. 187(4), 595–607 (1995) · doi:10.1006/jsvi.1995.0547
[4] Szabelski, K., Warminski, J.: The nonlinear vibrations of parametrically self-excited system with two degrees of freedom under external excitation. Nonlinear Dyn. 14, 23–36 (1997) · Zbl 0896.70016 · doi:10.1023/A:1008227315259
[5] Belhaq, M., Clerc, R.L., Hartman, C.: Etude numérique d’une 4-résonance d’une équation de Liénard forcée. C.R. Acad. Sci. Paris 303(II-10), 873–876 (1986)
[6] Belhaq, M.: Numerical study for parametric excitation of differential equation near a 4-resonance. Mech. Res. Commun. 17(4), 199–206 (1990) · Zbl 0702.70025 · doi:10.1016/0093-6413(90)90079-R
[7] Belhaq, M., Fahsi, A.: Higher-order approximation of subharmonics close to strong resonances in the forced oscillators. Comput. Math. Appl. 33(8), 133–144 (1997) · doi:10.1016/S0898-1221(97)00061-8
[8] Yano, S.: Analytic research on dynamic phenomena of parametrically and self-exited mechanical systems. Ing. Arch. 57, 51–60 (1987) · Zbl 0609.73062 · doi:10.1007/BF00536811
[9] Yano, S.: Considerations on self- and parametrically excited vibrational systems. Ing. Arch. 59, 285–295 (1989) · Zbl 0689.70011 · doi:10.1007/BF00534368
[10] Abouhazim, N., Belhaq, M., Lakrad, F.: Three-period quasi-periodic solutions in self-excited quasi-periodic Mathieu oscillator. Nonlinear Dyn. 39, 395–409 (2005) · Zbl 1098.70018 · doi:10.1007/s11071-005-3399-2
[11] Pandey, M., Rand, R.H., Zehnder, A.: Perturbation analysis of entrainment in a micromechanical limit cycle oscillator. Commun. Nonlinear. Sci. Numer. Simul. 12, 1291–1301 (2007) · Zbl 1124.34026 · doi:10.1016/j.cnsns.2006.01.017
[12] Pandey, M., Rand, R.H., Zehnder, A.: Frequency locking in a forced Mathieu–van der Pol–Duffing system. Nonlinear Dyn. (2007), doi: 10.1007/s11071-007-9238-x · Zbl 1279.34042
[13] Chelomei, V.N.: Mechanical paradoxes caused by vibrations. Sov. Phys. Dokl. 28, 387–390 (1983)
[14] Tcherniak, D.: The influence of fast excitation on a continuous system. J. Sound. Vib. 227(2), 343–360 (1999) · doi:10.1006/jsvi.1999.2349
[15] Thomsen, J.J.: Some general effects of strong high-frequency excitation: stiffening, biasing, and smoothening. J. Sound. Vib. 253(4), 807–831 (2002) · doi:10.1006/jsvi.2001.4036
[16] 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(2), 397–402 (2000) · Zbl 1110.74497 · doi:10.1115/1.1304824
[17] Hansen, M.H.: Effect of high-frequency excitation on natural frequencies of spinning discs. J. Sound. Vib. 234(4), 577–589 (2000) · doi:10.1006/jsvi.1999.2796
[18] Chatterjee, S., Singha, T.K., Karmakar, S.K.: Non-trivial effect of fast vibration on the dynamics of a class of nonlinearly damped mechanical systems. J. Sound. Vib. 260(4), 711–730 (2003) · doi:10.1016/S0022-460X(02)00993-8
[19] Thomsen, J.J.: Using fast vibrations to quench friction-induced oscillations. J. Sound. Vib. 228(5), 1079–1102 (1999) · doi:10.1006/jsvi.1999.2460
[20] Thomsen, J.J., Fidlin, A.: Analytical approximations for stick-slip vibration amplitudes. Nonlinear Mech. 38, 389–403 (2003) · Zbl 1346.74139 · doi:10.1016/S0020-7462(01)00073-7
[21] Mann, B.P., Koplow, M.A.: Symmetry breaking bifurcations of a parametrically excited pendulum. Nonlinear Dyn. 46, 427–437 (2006) · Zbl 1170.70359 · doi:10.1007/s11071-006-9033-0
[22] 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 (2006), doi: 10.1016/j.chaos.2006.10.040 · Zbl 1142.34332
[23] Belhaq, M., Sah, S.M.: Horizontal fast excitation in delayed van der Pol oscillator. Commun Nonlinear. Sci. Numer. Simul. (2007), doi: 10.1016/j.cnsns.2007.02.007 · Zbl 1142.34332
[24] Thomsen, J.J.: Slow high-frequency effects in mechanics: problems, solutions, potentials. Int. J. Bif. Chaos 15(9), 2799–2818 (2005) · Zbl 1093.70508 · doi:10.1142/S0218127405013721
[25] Blekhman, I.I.: Vibrational Mechanics–Nonlinear Dynamic Effects, General Approach, Application. World Scientific, Singapore (2000)
[26] Belhaq, M., Houssni, M.: Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations. Nonlinear Dyn. 18, 1–24 (1999) · Zbl 0969.70017 · doi:10.1023/A:1008315706651
[27] Rand, R.H., Guennoun, K., Belhaq, M.: 2:2:1 Resonance in the quasi-periodic Mathieu equation. Nonlinear Dyn. 31, 187–193 (2003) · Zbl 1062.70596 · doi:10.1023/A:1023216817293
[28] Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)
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.