Abstract
In this paper, the dynamics of an electromechanical system is investigated. In this model, the phase of the system is a dynamical variable. The existence of four periodic orbits is proved. Two of them are asymptotically stable and the others are unstable. In the absence of viscous damping, when the parameters associated with the dimensionless voltages are adequately changed, each orbit is subject to a change of stability of the kind stable \(\rightarrow \) unstable \(\rightarrow \) stable with repetition of this pattern. This phenomenon is known as Sommerfeld effect. Moreover, the stability of these orbits depends on the zeros of the Bessel function \(J_{1}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balthazar J, Mook D, Weber H, Brasil R, Felini A, Belato D, Felix J (2003) An overview on non-ideal vibrations. Meccanica 38:613–621
Blekhman II (2000) Vibrational mechanics: nonlinear dynamic effects, general approach, applications. World Scientific, Singapore
Dantas M, Balthazar J (2007) On the existence and stability of periodic orbits in non ideal problems: general results. Z Angew Math Phys (ZAMP) 58:940–958
Dantas M, Sampaio R, Lima R (2014) Asymptotically stable periodic orbits of a coupled electromechanical system. Nonlinear Dyn 78:29–35
Dantas M, Sampaio R, Lima R (2016a) Existence and asymptotic stability of periodic orbits for a class of electromechanical systems: a perturbation theory approach. Z Angew Math Phys (ZAMP) 67:1–14
Dantas M, Sampaio R, Lima R (2016b) Phase bifurcations in an electromechanical system. IUTAM Symp Anal Methods Nonlinear Dyn Procedia IUTAM 19:193–200
Eckert M (1996) The ’Sommerfeld-Effekt’: Theorie und Geschichte eines bemerkenswerten Resonanzphänomens. Eur J Phys 17:285–289
Eckert M (2013) Arnold sommerfeld: science, life and turbulent times 1868–1951. Springer, New York
Fidlin A (2006) Nonlinear oscillations in mechanical engineering. Springer, New York
Hale J (2009) Ordinary differential equations. Dover Publications, Mineola
Karnopp D, Margolis D, Rosenberg R (2006) System dynamics: modeling and simulation of mechatronic systems, 4th edn. Wiley, New York
Kononenko VO (1969) Vibrating systems with a limited power supply. London Iliffe Books LTD, London
Lima R, Sampaio R (2012) Analysis of an electromechanic coupled system with embarked mass. Mec Comput XXXI:2709–2733
Lima R, Sampaio R (2016a) Two parametric excited nonlinear systems due to electromechanical coupling. J Braz Soc Mech Sci Eng 38:931–943
Lima R, Sampaio R (2016b) Energy behavior of an electromechanical system with internal impacts and uncertainties. J Sound Vib 373:180–191
Lima R, Soize C, Sampaio R (2015) Robust design optimization with an uncertain model of a nonlinear vibro-impact electro-mechanical system. Commun Nonlinear Sci Numer Simul 23:263–273
Nayfeh AH, Mook DT (1979) Nonlinear oscillations. Wiley, New York
Sommerfeld A (1902) Beiträge zum dynamischen Ausbau der Festigkeitslehre. Zeitschrift des Vereines deutscher Ingenieure 46:391–394
Stoker J (1950) Nonlinear vibrations in mechanical and electrical systems. Interscience Publishers Inc, New York
Whittaker E, Watson G (1973) A course in modern analysis. Cambridge University Press, London
Acknowledgements
The first author acknowledges the support given by FAPEMIG and the second and third authors acknowledge the support given by FAPERJ, CNPq, and CAPES
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Luz de Teresa.
Rights and permissions
About this article
Cite this article
Dantas, M.J.H., Sampaio, R. & Lima, R. Sommerfeld effect in a constrained electromechanical system. Comp. Appl. Math. 37, 1894–1912 (2018). https://doi.org/10.1007/s40314-017-0428-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-017-0428-y