On the Harmonic Oscillations for the Motion of a Dynamical System
DOI:
https://doi.org/10.24297/jap.v13i2.5754Keywords:
Sub-harmonic resonance, Ultra-harmonic resonance, Perturbation methods, MSC2010, 70B10, 70E20, 70K28Abstract
This work touches two important cases for the motion of a pendulum called Sub and Ultra-harmonic cases. The small parameter method is used to obtain the approximate analytic periodic solutions of the equation of motion when the pivot point of the pendulum moves in an elliptic path. Moreover, the fourth order Runge-Kutta method is used to investigate the numerical solutions of the considered model. The comparison between both the analytical solution and the numerical ones shows high consistency between them.Downloads
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of a Wii remote to measure spasticity with the pendulum test: Proof of concept, Gait and Posture 43
(2016) 70-75.
[2] H. Huang, M. Ju, C. Lin, Flexor and extensor muscle tone evaluated using the quantitative pendulum
test in stroke and parkinsonian patients, Journal of Clinical Neuroscience 27 (2016) 48-52.
[3] M.R. Turner, T. J. Bridges, H. Alemi Ardakani ,The pendulum-slosh problem: Simulation using a
time-dependent conformal mapping, Journal of Fluids and Structures 59 (2015) 202-223.
[4] X. Dai, An vibration energy harvester with broadband and frequency-doubling characteristics based on
rotary pendulums, Sensors and Actuators A 241 (2016) 161-168.
[5] Y. H. Kim, S. H. Kim, Y. K. Kwak, Dynamic analysis of anon-holonomic two-wheeled inverted pendulum
robot. J Intell Robot Syst 44 (2005) 25-46.
[6] N. D. Anh, H. Matsuhisa , L. D.Viet, M. Yasuda, Vibration control of an inverted pendulum type
structure by passive mass-spring-pendulum dynamic vibration absorber. J Sound Vib 307 (2007) 187-
201.
[7] A. H. Nayfeh, Perturbations methods, WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim (2004).
[8] F. A. El-Barki,A. I. Ismail, M. O. Shaker, T. S. Amer, On the motion of the pendulum on an ellipse,
ZAMM, 79, 1, (1999)65-72, .
[9] Xu Xu and M. Wiercigroch, Approximate analytical solutions for oscillatory and rotational motion of
a parametric pendulum, Nonlinear Dyn 47 (2007) 311-320.
[10] Y. Song, H. Sato, T. Komatsuzaki, The response of a dynamic vibration absorber system with a para-
metrically excited pendulum, J Sound Vib 259 (2003) 747-59.
[11] T. S. Amer, M. A. Bek, Chaotic Responses of a Harmonically Excited Spring Pendulum Moving in
Circular Path, Nonlinear Analysis: Real World Applications 10 (2009)3196-3202.
[12] B. P. Koch, R. W. Leven, Subharmonic and homoclinic bifurcations in a parametrically forced pendulum,
Physica D 16 (1984) 1-13.
[13] J. Guckenheimer, Holmes P.J., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector
Fields, Springer, Berlin (1983).
[14] A. F. El-Bassiouny, Parametrically excited nonlinear systems: a comparison of two methods, Int. J.
Math. Math. Sci. 32, 12 (2002) 739-761.
[15] J. Miles, Resonantly forced motion of two quadratically coupled oscillators, Physica D 13(1984) 247-260.
[16] S. Tousi, A. K. Bajaj, Period-doubling bifurcations and modulated motions in force mechanical systems,
ASME J. Appl. Mech. 52 (1985) 446-452.
[17] W. Szemplinska-Stupnicka, E. Tyrkiel, A. Zubrzycki, The global bifurcations that lead to transient
tumbling chaos in a parametrically driven pendulum, Int. J. Bifurcat. Chaos 10 (9) (2000) 2161-2175.
[18] T. A. Nayfeh,W. Asrar, A. H. Nayfeh, Three-mode interactions in harmonically excited systems with
quadratic nonlinearities, Nonlinear Dyn. 58 (1991) 1033-1041.
[19] K. Zaki, S. Noah, K. R. Rajagopal, A.R. Srinivasa, Eect of nonlinear stiness on the motion of a exible pendulum, Nonlinear Dyn. 27, 1 (2002) 1-18.
[20] W. K. Lee, H. D. Park, Chaotic dynamics of a harmonically excited spring-pendulum system with
internal resonance, Nonlinear Dyn. 14 (1997) 211-229.
[21] P. R. Sethna, Vibrations of dynamical systems with quadratic nonlinearities, J. Appl. Mech. 32 (1965)
576-582.
[22] W. K. Lee, C. S. Hsu, A global analysis of a harmonically excited spring-pendulum system with internal
resonance, J. Sound Vib. 171,3 (1994) 335-359.
[23] I. G. Malkin, Some problems in the theory of nonlinear oscillations, U. S. Atomic Energy Commission.
Technical Information Service, AEC-tr-3766, (1959).
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Published
2017-02-15
How to Cite
Amer, W. S. (2017). On the Harmonic Oscillations for the Motion of a Dynamical System. JOURNAL OF ADVANCES IN PHYSICS, 13(2), 4657–4670. https://doi.org/10.24297/jap.v13i2.5754
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