Far Eastern Mathematical Journal

To content of the issue


Stability of coupled oscillators


Guzev M.A., Dmitriev A.A.

2015, issue 2, Đ. 166-191


Abstract
We study a system of two coupled oscillators and a modified system of these oscillators whose rods intersect and slide without friction relative to each other. The oscillators posed vertically in a uniform gravity field and its interaction is described by a potential depending on distance. We demonstrate that both systems have symmetrical and asymmetrical equilibrium states. Stability of the states depend on the interaction energy and distance between the oscillators' suspension centers. Stability regions for Hooke and Coulomb potentials are calculated in the parameter plane.

Keywords:
coupled oscillators, equilibrium, stability

Download the article (PDF-file)

References

[1] Zommerfel'd A., Mehanika, NIC źReguljarnaja i haoticheskaja dinamika╗, Izhevsk, 2001.
[2] Maianti M., Pagliara S., Galimberti G., "Mechanics of two pendulums coupled by a stressed spring", Am. J. Phys., 77:9 (2009), 834-838.
[3] Ramachandran P., Krishna S.G., Ram Y.M., "Instability of a constrained pendulum system", Am. J. Phys., 79:4 (2011), 395-400.
[4] Koluda P., Perlikowski P., Czolczynski K., Kapitaniak T., "Synchronization configurations of two coupled double pendula", Communications in Nonlinear Science and Numerical Simulation, 19:8 (2014), 977-990.
[5] Huynh H.N., Chew L.Y., "Two-coupled pendulum system: bifurction, chaos and the potential landscape approach", Int. J. Bifurcation Chaos, 20:8 (2010), 2427-2442.
[6] Huynh H.N., Nguyen T.P.T., Chew L.Y., "Numerical simulation and geometrical analysis on the onset of chaos in a system of two coupled pendulums", Communications in Nonlinear Science and Numerical Simulation, 18:2 (2013), 291-307.
[7] A. Stephenson, "On an induced stability", Phil. Mag., 15 (1908), 233-236.
[8] Kapica P.L., "Majatnik s vibrirujushhim podvesom", Uspehi fizicheskih nauk, 44:5 (1951), 7-20.
[9] I.I. Blehman, Vibracionnaja mehanika, źNauka╗, M., 1994.
[10] Butikov E.I., "An improved criterion for Kapitza's pendulum stability", J. Phys. A: Math. Theor., 44:29 (2011), 7-20.
[11] Markeev A.P., "O dvizhenii svjazannyh majatnikov", Nelinejnaja dinamika, 9:1 (2013), 27-38.
[12] Tarasov B.G., Guzev M.A., "Mathematical Model of Fan-head Shear Rupture Mechanism", Key Engineering Materials, 592:11 (2013), 121-124.
[13] Baboly M.G., Su M.F., Reinke S.M., Alaie S., Goettler D.F., El-Kady I., Leseman Z.C, "The effect of stiffness and mass on coupled oscillations in a phononic crystal", AIP Advances, 3 (2013), 112121, 1-7.
[14] Lipfert J., Hao X., Dekker N.H., "Quantitative Modeling and Optimization of Magnetic Tweezers", Biophysical Journal, 96:12 (2009), 5040-5049.

To content of the issue