Nonlinear oscillations of a coupled autoparametrical system with ideal and nonideal sources of power

Danuta, Sado; Maciej, Kot

doi:https://doi.org/10.1155/MPE/2006/82691

Mathematical Problems in Engineering

On this page

Abstract References Copyright Related Articles

Open Access

Volume 2006 | Article ID 082691 | https://doi.org/10.1155/MPE/2006/82691

Nonlinear oscillations of a coupled autoparametrical system with ideal and nonideal sources of power

Sado Danuta¹and Kot Maciej¹

Received16 Oct 2005

Revised05 Jun 2006

Accepted16 Jul 2006

Published29 Nov 2006

Abstract

An ideal and nonideal autoparametrical system excited by DC motor with unbalanced mass is presented in this work. The system consists of the body of mass M which is hung on a nonlinear spring with a nonlinear damper, and a pendulum of the length l and mass m mounted to the body of mass M. It is assumed that the motion of the pendulum is damped by nonlinear resistive forces. Vibrations of both models (ideal and nonideal) are researched. Solutions for the system response are presented for specific values of the parameters of system and the energy transfer between modes of vibrations is studied. Next excited vibrations for both models have been examined analytically and numerically. Except different kinds of periodic vibrations, there may also appear chaotic vibrations.

References

A. K. Bajaj, S. I. Chang, and J. M. Johnson, “Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system,” Nonlinear Dynamics, vol. 5, no. 4, pp. 433–457, 1994.
View at: Publisher Site | Google Scholar
A. K. Bajaj and J. M. Johnson, “Asymptotic techniques and complex dynamics in weakly nonlinear forced mechanical systems,” International Journal of Non-Linear Mechanics, vol. 25, no. 2-3, pp. 211–226, 1990.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. K. Bajaj and S. Tousi, “Torus doublings and chaotic amplitude modulations in a two-degree-of-freedom resonantly forced mechanical system,” International Journal of Non-Linear Mechanics, vol. 25, no. 6, pp. 625–641, 1990.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. L. Baker and J. P. Gollub, Chaotic dynamics: An Introduction, Cambridge University Press, Cambridge, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. M. Balthazar, D. T. Mook, H. I. Weber, R. M. L. R. F. Brasil, A. Fenili, D. Belato, and J. L. P. Felix, “An overview on non-ideal vibrations,” Meccanica, vol. 38, no. 6, pp. 613–621, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
D. Belato, J. M. Balthazar, H. I. Weber, and D. T. Mook, “On dynamical characteristics of the electromotor-pendulum,” in Nonlinear Dynamics, Chaos, Control and Their Applications to Engineering Sciences. Vol. 2: Vibrations with Measurements and Control, J. M. Balthazar, P. B. Gonçalvez, and J. Clayssen, Eds., vol. 2, pp. 222–235, Cruzeiro Editora e Artes Gráficas, Rio Claro, 1999.
View at: Google Scholar
D. Belato, J. M. Balthazar, H. I. Weber, and D. T. Mook, “Chaotic vibrations of a nonideal electro-mechanical system,” International Journal of Solids and Structures, vol. 38, no. 10–13, pp. 1699–1706, 2001.
View at: Google Scholar | Zentralblatt MATH
K. L. Cavalca, I. L. E. Santos, and J. M. Balthazar, “Analysis of combined effects of dry friction and non-linear restoring forces in rotors,” in Nonlinear Dynamics, Chaos, Control and Their Applications to Engineering Sciences. Vol. 2: Vibrations with Measurements and Control, J. M. Balthazar, P. B. Gonçalvez, and J. Clayssen, Eds., vol. 2, pp. 206–221, Cruzeiro Editora e Artes Gráficas, Rio Claro, 1999.
View at: Google Scholar
R. M. Evan-Ivanowski, Resonance Oscillations in Mechanical Systems, Elsevier, Amsterdam, 1976.
View at: Google Scholar | Zentralblatt MATH
A. Fenili, J. M. Balthazar, and D. T. Mook, “Some remarks on nonlinear and ideal or nonideal structure vibrating model,” in Proceedings of the 7th Conference on Dynamical Systems-Theory and Applications, J. Awrejcewicz, A. Owczarek, and J. Mrozowski, Eds., vol. 2, pp. 549–556, Łódź, 2003.
View at: Google Scholar
H. Hatwal, A. K. Mallik, and A. Ghosh, “Forced nonlinear oscillations of an autoparametric system—part 2: chaotic responses,” Transactions of the ASME: Journal of Applied Mechanics, vol. 50, pp. 663–668, 1983.
View at: Google Scholar
V. O. Kononenko, Vibrating Systems with Limited Power Supply, Illife Books, London, 1969.
View at: Google Scholar
T. S. Krasnopol'skaya and A. Yu. Shvets, “Chaotic interactions in a pendulum energy source system,” Prikladnaya Mekhanika, vol. 26, no. 5, pp. 90–96, 1990.
View at: Google Scholar | Zentralblatt MATH
F. C. Moon, Chaotic Vibrations, John Wiley & Sons, New York, 1987.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, John Wiley & Sons, New York, 1979.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
L. Půst and O. Szöllös, “Forced irregular oscillations of the two degrees of freedom nonlinear system,” in Proceedings of the 2nd European Nonlinear Oscillation Conference (EUROMECH '96), pp. 367–370, Prague, 1996.
View at: Google Scholar
D. Sado, “Periodic and chaotic oscillations of the autoparametric beam-pendulum system,” in Proceedings of the 3rd Biennal World Conference on Integrated Design and Process Technology (IDPT '98), vol. 6, pp. 206–213, Berlin, 1998.
View at: Google Scholar
D. Sado, “Chaos in autoparametric coupled mechanical systems,” in Proceedings of the 10th World Congress on the Theory of Machines and Mechanisms (IFToMM '99), vol. 4, pp. 1638–1643, Oulu, 1999.
View at: Google Scholar
D. Sado, “Chaotic behavior of beam-pendulum system with dry friction,” in Proceedings of the 5th Engineering Systems Design and Analysis Conference (ESDA '00), vol. 82, pp. 163–168, Montreux, 2000.
View at: Google Scholar
D. Sado, “The chaotic phenomenon of a system with inertial coupling,” Mechanics and Mechanical Engineering, vol. 6, no. 1, pp. 31–42, 2002.
View at: Google Scholar
D. Sado, “The dynamics of a coupled three degree of freedom mechanical system,” Mechanics and Mechanical Engineering, vol. 7, pp. 29–39, 2004.
View at: Google Scholar
D. Sado and K. Gajos, “Note on chaos in three degree of freedom dynamical system with double pendulum,” Meccanica, vol. 38, no. 6, pp. 719–729, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
D. Sado and M. Kot, “The chaotic vibration of an autoparametrical system with non ideal source of power,” in Proceedings of the 7th Conference on Dynamical Systems-Theory and Applications, J. Awrejcewicz, A. Owczarek, and J. Mrozowski, Eds., vol. 2, pp. 689–696, Łódź, 2003.
View at: Google Scholar
D. Sado and M. Kot, “The periodic and chaotic vibration of an autoparametrical system with non ideal source of power,” to appear in Journal of Computational Methods in Sciences and Engineering.
View at: Google Scholar
M. Tsuchida, K. de Lolo Guilherme, J. M. Balthazar, G. N. Silva, and B. I. Cheshankov, “On regular and irregular vibrations of a non-ideal system with two degrees of freedom. 1:1 resonance,” Journal of Sound and Vibration, vol. 260, no. 5, pp. 949–960, 2003.
View at: Publisher Site | Google Scholar
J. Warmiński, Regular and Chaotic Vibrations of Parametric and Self-Excited Systems with Ideal and Non-Ideal Energy Sources, Technical University of Lublin Press, Lublin, 2001.
View at: Google Scholar
J. Warmiński, J. M. Balthazar, and R. M. L. R. F. Brasil, “Vibrations of a non-ideal parametrically and self-excited model,” Journal of Sound and Vibration, vol. 245, no. 2, pp. 363–374, 2001.
View at: Google Scholar

Copyright

Copyright © 2006 Sado Danuta and Kot Maciej. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation Order printed copies

Views

311

Downloads

861

Citations