Autoparametric vibrations of a nonlinear system with pendulum.

*(English)*Zbl 1100.70012Summary: Vibrations of a nonlinear oscillator with an attached pendulum, excited by the motion of its point of suspension, have been analysed. The derived differential equations of motion show that the system is strongly nonlinear and the motions of both subsystems, the pendulum and the oscillator, are strongly coupled by inertial terms, leading to the so-called autoparametric vibrations. It has been found that the motion of the oscillator, forced by an external harmonic force, has been dynamically eliminated by pendulum oscillations. The influence of a nonlinear spring on the vibration absorption near the main parametric resonance has been examined analytically, whereas the transition from regular to chaotic vibrations has been examined by numerical methods. A transmission force on the foundation for regular and chaotic vibrations is presented as well.

##### MSC:

70K28 | Parametric resonances for nonlinear problems in mechanics |

70K40 | Forced motions for nonlinear problems in mechanics |

70K55 | Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics |

##### Software:

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\textit{J. Warminski} and \textit{K. Kecik}, Math. Probl. Eng. 2006, Article ID 80705, 19 p. (2006; Zbl 1100.70012)

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##### References:

[1] | D. A. Acheson, “A pendulum theorem,” Proceedings of the Royal Society of London. Series A, vol. 443, no. 1917, pp. 239-245, 1993. · Zbl 0784.70019 |

[2] | 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, pp. 433-457, 1994. |

[3] | M. P. Cartmell and J. W. Roberts, “Simultaneous combination resonances in an autoparametrically resonant system,” Journal of Sound and Vibration, vol. 123, no. 1, pp. 81-101, 1988. · Zbl 1235.70067 |

[4] | W. K. Lee and C. S. Hsu, “A global analysis of an harmonically excited spring-pendulum system with internal resonance,” Journal of Sound and Vibration, vol. 171, no. 3, pp. 335-359, 1994. · Zbl 0925.70264 |

[5] | H. E. Nusse and J. A. Yorke, Dynamics: Numerical Explorations, vol. 101 of Applied Mathematical Sciences, Springer, New York, 1994. · Zbl 0805.58007 |

[6] | D. Sado, Energy Transfer in Nonlinearly Coupled Systems with Two Degrees of Freedom (Przenoszenie energii w nieliniowo sprz\ce\Dzonych układach o dwóch stopniach swobody), Prace Naukowe, Mechanika, z.166, Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa, 1997 (in Polish). |

[7] | Y. Song, H. Sato, Y. Iwata, and T. Komatsuzaki, “The response of a dynamic vibration absorber system with a parametrically excited pendulum,” Journal of Sound and Vibration, vol. 259, no. 4, pp. 747-759, 2003. |

[8] | A. Tondl, T. Ruijgrok, F. Verhulst, and R. Nabergoj, Autoparametric Resonance in Mechanical Systems, Cambridge University Press, Cambridge, 2000. · Zbl 0957.70001 |

[9] | F. Verhulst, “Autoparametric resonance, survey and new results,” in 2nd European Nonlinear Oscillation Conference (Prague, 1996), vol. 1, pp. 483-488, European Mechanics Society, 1996. |

[10] | A. Vyas and A. K. Bajaj, “Dynamics of autoparametric vibration absorbers using multiple pendulums,” Journal of Sound and Vibration, vol. 246, no. 1, pp. 115-135, 2001. · Zbl 1237.70118 |

[11] | J. Warminski, 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. · Zbl 1237.74094 |

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