Rotating solutions and stability of parametric pendulum by perturbation method

Journal of Sound and Vibration

Volumn 310, Issue 1-2, 2008, Pages 243-259

  Lenci, S ^a   Pavlovskaia, E ^b   Rega, G ^c   Wiercigroch, M ^b  

^a UNIVERSITÀ POLITECNICA DELLE MARCHE   (Italy)

^b UNIVERSITY OF ABERDEEN   (United Kingdom)

^c SAPIENZA UNIVERSITY OF ROME   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SIMULATION; DAMPING; HAMILTONIANS; NUMERICAL METHODS; PERTURBATION TECHNIQUES;

COMPREHENSIVE FORMULAS; ROTATING SOLUTIONS; SUBHARMONIC ROTATIONS;

PENDULUMS;

EID: 36248974220     PISSN: 0022460X     EISSN: 10958568     Source Type: Journal    
DOI: 10.1016/j.jsv.2007.07.069     Document Type: Article

References (29)

1. Salam F.M.A., and Sastry S.S. Dynamics of the forced Josephson junction circuit: the regions of chaos. IEEE Transaction on Circuits Systems CAS32 (1985) 784-796
2. 3. Physical Review Letters 46 (1981) 511-516
3. Blekhman I.I. Vibrational Mechanics (2000), World Scientific, Singapore
4. Guckenheimer J., and Holmes P. Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (1983), Springer, New York
5. Hagedorn P. Nonlinear Oscillations (1988), Oxford University Press, New York
6. Nayfeh A.H. Introduction to Perturbation Techniques (1981), Wiley-Interscience, New York
7. Butikov E. The rigid pendulum-an antique but evergreen physical model. European Journal of Physics 20 (1999) 429-441
8. van Dooren R. Chaos in a pendulum with forced horizontal support motion: a tutorial. Chaos, Solitons & Fractals 7 (1996) 77-90
9. Garira W., and Bishop S.R. Rotating solutions of the parametrically excited pendulum. Journal of Sound and Vibration 263 (2003) 233-239
10. Szemplinska-Stupnicka W., Tyrkiel E., and Zubrzycki A. The global bifurcations that lead to transient tumbling chaos in a parametrically driven pendulum. International Journal of Bifurcation and Chaos 10 9 (2000) 2161-2175
11. Bishop S.R., and Clifford M.J. The use of manifold tangencies to predict orbits, bifurcations and estimate escape in driven systems. Chaos, Solitons & Fractals 7 (1996) 1537-1553
12. Koch B.P., and Leven R.W. Subharmonic and homoclinic bifurcations in a parametrically forced pendulum. Physica D 16 (1985) 1-13
13. Bornemann P.B., Galvanetto U., and Crisfield M.A. Some remarks on the numerical time integration of non-linear dynamical systems. Journal of Sound and Vibration 252 (2002) 935-944
14. Abdel-Rahman E.M., Nayfeh A.H., and Masoud Z.N. Dynamics and control of cranes: a review. Journal of Vibrations and Control 9 (2003) 863-908
15. Chester W. The forced oscillations of a simple pendulum. Journal of the Institute of Maths Applications 15 (1975) 289-306
16. Struble R.A. On the subharmonic oscillations of a pendulum. ASME Journal on Applied Mechanics 30 (1963) 301-303
17. Szemplinska-Stupnicka W., and Tyrkiel E. The oscillation-rotation attractors in the forced pendulum and their peculiar properties. International Journal of Bifurcation and Chaos 12 (2002) 159-168
18. Blackburn J.A. Experimental study of chaos in driven pendulum. Physica D 26 (1987) 385-395
19. Kapitsa P.L. Dynamic stability of the pendulum when the point of suspension is oscillating. Journal of Experimental and Theoretical Physics 21 (1951) 588-597 (in Russian)
20. Starrett J., and Tagg R. Control of a chaotic parametrically driven pendulum. Physical Review Letters 74 (1995) 1974-1977
21. Leven R.W., Pompe B., Wilke C., and Koch B.P. Experiments on periodic and chaotic motions of a parametrically forced pendulum. Physica D 16 (1985) 371-384
22. Xu X., Pavlovskaia E., Wiercigroch M., Romeo F., and Lenci S. Dynamic interactions between parametric pendulum and electro-dynamical shaker. ZAMM-Zeitschrift fur Angewandte Mathematik und Mechanik 87 (2007) 172-186
23. Capecchi, D., Bishop, S., 1990. Periodic and non-periodic responses of a parametrically excited pendulum. Report 3/90 of the Dipartimento di Ingegneria delle Strutture, delle Acque e del Terreno, Università dell'Aquila, Italy.
24. Clifford M.J., and Bishop S.R. Rotating periodic orbits of the parametrically excited pendulum. Physics Letters A 201 (1995) 191-196
25. Xu X., Wiercigroch M., and Cartmell M.P. Rotating orbits of a parametrically-excited pendulum. Chaos, Solitons & Fractals 23 (2005) 1537-1548
26. Abramowitz M., and Stegun I. Handbook of Mathematical Functions (1970), Dover Publications, New York
27. Baker Jr. G.A. The theory and application of the Pade approximant method. In: Brueckner K.A. (Ed). Advances in Theoretical Physics Vol. 1 (1965), New York, Academic Press 1-58
28. Lenci S., and Rega G. Higher-order Melnikov functions for single-dof. oscillators: theoretical treatment and applications. Mathematical Problems in Engineering 2 2 (2004) 145-168
29. Xu X., and Wiercigroch M. Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum. Nonlinear Dynamics 47 (2007) 311-320