Abstract
It is possible that self-excited vibrations in turbomachine blades synchronize due to elastic coupling through the shaft. The synchronization of four coupled van der Pol oscillators is presented here as a simplified model. For quasilinear oscillations, a stability condition is derived from an analysis based on linearizing the original equation around an unperturbed limit cycle and transforming it into Hill’s equation. For the nonlinear case, numerical simulations show the existence of two well-defined regions of phase relationships in parameter space in which a multiplicity of periodic attractors is embedded. The size of these regions strongly depends on the values of the oscillator and coupling constants. For the coupling constant below a critical value, there exists a region in which a diversity of phase-shift attractors is present, whereas for values above the critical value an in-phase attractor is predominant. It is observed that the presence of an anti-phase attractor in the subcritical region is associated with sudden changes in the period of the coupled oscillators. The convergence of the coupled system to a particular periodic attractor is explored using several initial conditions. The study is extended to non-identical oscillators, and it is found that there is synchronization even over a wide range of difference among the oscillator constants.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Winfree, A.T.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15–42 (1967)
Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization, a Universal Concept in Nonlinear Sciences. Cambridge University Press, London (2001)
Strogatz, S.H.: Sync: The Emerging Science of Spontaneous Order. Theia, New York (2003)
Kawahara, T.: Coupled van der Pol oscillators—a model of excitatory and inhibitory neural interactions. Biol. Cybern. 39(1), 37–43 (1980)
Fukuda, H., Tamari, N., Morimura, H., Kai, S.: Entrainment in a chemical oscillator chain with a pacemaker. J. Phys. Chem. A 109(49), 11250–11254 (2005)
Cai, W., Sen, M.: Synchronization of thermostatically controlled first-order systems. Int. J. Heat Mass Transf. 51(11–12), 3032–3043 (2008)
Woafo, P., Kadji, H.G.: Synchronized states in a ring of mutually coupled self-sustained electrical oscillators. Phys. Rev. E 69(046206), 1–9 (2004)
Huygens, C.: Letter to de Sluse. Letter No. 1333 of February 24, 1665, page 241. In: Oeuvres Complète de Christiaan Huygens. Correspondence, vol. 5, pp. 1664–1665. Société Hollandaise des Sciences, Martinus Nijhoff, La Haye (1893)
Bennett, M., Schatz, M.F., Rockwood, H., Wiesenfeld, K.: Huygens’s clocks. Proc. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 458(2019), 563–579 (2002)
Dimentberg, M., Cobb, E., Mensching, J.: Self-synchronization of transient rotations in multiple-shaft systems. J. Vib. Control 7(2), 221–232 (2001)
Balthazar, J.M., Felix, J.L.P., Brasil, R.M.: Some comments on the numerical simulation of self-synchronization of four non-ideal exciters. Appl. Math. Comput. 164(2), 615–625 (2005)
Katayama, N., Takata, G., Miyake, M., Nanahara, T.: Theoretical study on synchronization phenomena of wind turbines in a wind farm. Electr. Eng. Jpn. 155(1), 9–18 (2006)
Liew, K.M., Wang, W.Q., Zhang, L.W., He, X.Q.: A computational approach for predicting the hydroelasticity of flexible structures based on the pressure Poisson equation. Int. J. Numer. Methods Eng. 72(13), 1560–1583 (2007)
Gabbai, R.D., Benaroya, H.: An overview of modeling and experiments of vortex-induced vibration of circular cylinders. J. Sound Vib. 282(3-5), 575–616 (2005)
Facchinetti, M.L., de Langre, E., Biolley, F.: Vortex shedding modeling using diffusive van der Pol oscillators. C.R. Mec. 330(7), 451–456 (2002)
Facchinetti, M.L., de Langre, E., Biolley, F.: Coupling of structure and wake oscillators in vortex-induced vibrations. J. Fluids Struct. 19, 123–140 (2004)
Mathelin, L., de Langre, E.: Vortex-induced vibrations and waves under shear flow with a wake oscillator model. Eur. J. Mech. B: Fluids 24(4), 478–490 (2005)
Violette, R., de Langre, E., Szydlowski, J.: Computation of vortex-induced vibrations of long structures using a wake oscillator model: comparison with DNS and experiments. Comput. Struct. 85(11–14), 1134–1141 (2006)
van der Pol, B., van der Mark, J.: The heartbeat considered as a relaxation oscillation, and an electrical model of the heart. Philos. Mag. 6(38), 763–775 (1928)
Ablowitz, R.: The theory of emergence. Philos. Sci. 6(1), 1–16 (1939)
Aggarwal, J.K., Richie, C.G.: On coupled van der Pol oscillators. IEEE Trans. Circuit Theory CT13(4), 465–466 (1966)
Storti, D.W., Rand, R.H.: A simplified model of two coupled relaxation oscillators. Int. J. Non-Linear Mech. 22(4), 283–289 (1987)
Storti, D.W., Reinhall, P.G.: Phase-locked mode stability for coupled van der Pol oscillators. ASME J. Vib. Acoust. 122(3), 318–323 (2000)
Bakri, T., Nabergoj, R., Tondl, A.: Multi-frequency oscillations in self-excited systems. Nonlinear Dyn. 48(1–2), 115–127 (2007)
Ookawara, T., Endo, T.: Effects of the deviation of element values in a ring of three and four coupled van der Pol oscillators. IEEE Trans. Circuits Syst. 46, 827–840 (1999)
Hasegawa, A., Endo, T.: Multimode oscillations in a four fully-interconnected van der Pol oscillators. In: Proceedings of the 2001 International Symposium on Circuits and Systems, Sydney, Australia (2001)
Endo, T., Mori, S.: Mode analysis of a ring of a large number of mutually coupled van der Pol oscillators. IEEE Trans. Circuits Syst. CAS-25(1), 7–18 (1978)
Reinhall, P.G., Storti, D.W.: A numerical investigation of phase-locked and chaotic behavior of coupled van der Pol oscillators. In: Proceedings of the 1995 ASME Design Engineering Technical Conference, Boston, MA (1995)
Aronson, D.G., Ermentrout, G.B., Koplell, N.: Amplitude response of coupled oscillators. Physica D 41, 403–449 (1990)
Nana, B., Woafo, P.: Synchronization in a ring of four mutually coupled van der Pol oscillators: theory and experiment. Phys. Rev. E 74(4), 1–8 (2006)
Rand, R., Wong, J.: Dynamics of four coupled phase-only oscillators. Commun. Nonlinear Sci. Numer. Simul. 13(3), 501–507 (2008)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)
Thompson, J.M.T., Stewart, H.B.: Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists. Wiley, New York (1986)
Farkas, M.: Periodic Motions. Springer, New York (1994)
Magnus, W., Winkler, S.: Hill’s Equation. Interscience, New York (1966)
Moore, G.: Floquet theory as a computational tool. SIAM J. Numer. Anal. 42(6), 2522–2568 (2005)
Cai, W., Sen, M., Yang, K.T., McClain, R.L.: Synchronization of self-sustained thermostatic oscillations in a thermal-hydraulic network. Int. J. Heat Mass Transf. 49, 4444–4453 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barrón, M.A., Sen, M. Synchronization of four coupled van der Pol oscillators. Nonlinear Dyn 56, 357–367 (2009). https://doi.org/10.1007/s11071-008-9402-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-008-9402-y