Three-to-one internal resonance in the non linear oscillation of shallow arch

Physica Scripta

Volumn 72, Issue 6, 2005, Pages 439-450

  El Bassiouny, A F ^a,b  

^a BENHA UNIVERSITY   (Egypt)

^b NONE   (Saudi Arabia)

Author keywords

[No Author keywords available]

Indexed keywords

DAMPING; FREQUENCY RESPONSE; NONLINEAR SYSTEMS; ORDINARY DIFFERENTIAL EQUATIONS; OSCILLATIONS;

INTERNAL RESONANCE; NONLINEAR OSCILLATIONS; TRANSVERSE LOADS;

STRUCTURAL LOADS;

EID: 29744436115     PISSN: 00318949     EISSN: None     Source Type: Journal    
DOI: 10.1088/0031-8949/72/6/003     Document Type: Article

References (48)

1. Miles, J. W., Parametric excitation of an internally resonant double pendulum. Journal of Applied Mathematics and Physics (ZAMP) 36, 337-345 (1985).
2. Nayfeh, A. H., The response of multidegree-of-freedom systems with quadratic non-linearities to a harmonic parametric resonance. Journal of Sound and Vibration 90, 237-244 (1983).
3. Nayfeh, A. H., The response of two-degree-of-freedom systems with quadratic non-linearities to a parametric excitation. Journal of Sound and Vibration 88, 547-557 (1983).
4. Nayfeh. A. H., Parametric excitation of two internally resonant oscillators. Journal of Sound and Vibration 119, 95-109 (1987).
5. Nayteh, A. H. and Zavodny, L. D., The response of two-degree-of-freedom systems with quadratic nonlinearities to a combination parametric resonance. Journal of Sound and Vibration 107, 329-350 (1986).
6. Streit, D. A., Bajaj, A. K., Krousgrill, C. M., Combination parametric resonance leading to periodic and chaotic response in two-degree-of-freedom systems with quadratic non-linearities. Journal of Sound and Vibration 124, 297-314 (1988).
7. Asrar, W., Two-degree-of-freedom systems with quadratic nonlinearities subjected to parametric and self excitation. Journal of Sound and Vibration 150, 447-456 (1991).
8. Ibrahim, R. A. and Barr, A. D. S., Autoparametric resonance in a structure containing a liquid. Part I: Two mode interaction. Journal of Sound and Vibration 42, 159-179 (1975).
9. Holmes, P. J., Chaotic motions in a weakly nonlinear model for surface waves. Journal of Fluid Mechanics 162, 365-388 (1986).
10. Nayfeh, A. H., Surface waves in closed basins under parametric and internal resonances, Phys. Fluids 30, 2976-2982 (1987).
11. Gu, M. X. and Sethna, P. R., Resonant surface waves and chaotic phenomena. Journal of Fluid Mechanics 183, 543-565 (1987).
12. Nayfeh, A. H. and Nayfeh, J. F., Surface waves in closed basins under principal and autoparametric resonances. Physics of Fluids A2, 1635-1648 (1990).
13. Elnaggar, A. M. and El-Bassiouny, A. F., Principal parametric resonances of two-degree-of-freedom system with quadratic and cubic non-linearities. Bull. Sci. Assuit Univ. 24(2), 25-40 (1995).
14. El-Bassiouny, A. F., Parametrically excited non-linear systems: a comparison of two methods, Mechanics and Mechanical Engineering International Journal 5(1), 21-40 (2001).
15. Maccari. A., Modulated motion and infinite-period bifurcation for two nonlinearly coupled and parametrically excited van der Pol oscillators. International Journal of Non-Linear Mechanics 36, 335-347 (2001).
16. Bi. Q., Dynamical analysis of two coupled parametrically excited Van der Pol oscillators. International Journal of Non-Linear Mechanics 39(1), 33-54 (2004).
17. Tso. W. K. and Asmis, K. G., Multiple parametric resonance in a nonlinear two degree of freedom system. International Journal of Nonlinear Mechanics 9, 269-277 (1974).
18. Tezak, E. G., Mook, D. T. and Nayfeh, A. H., Nonlinear analysis of the lateral response of columns to periodic loads. Journal of Mechanical Design 100, 651-659 (1978).
19. Asmis, K. G. and Tso, W. K., Combination resonance in a nonlinear two-degree-of-freedom system. Journal of Applied Mechanics 39, 832-834 (1972).
20. Ciliberto, S. and Gollub. J. P., Chaotic mode competition in parametrically forced surface waves. Journal of Fluid Mechanics 158, 381-398 (1985).
21. Meron, E. and Procaccia, I., Low-dimensional chaos in surface waves: Theoretical analysis of an experiment. Physical Review A34, 3221-3237 (1986).
22. Feng, Z. C. and Sethna, P. R., Symmetry-breaking bifurcation in resonant surface waves. Journal of Fluid Mechanics 199, 495-518 (1989).
23. Nayfeh, A. H. and Pai. P. F., Nonlinear non-planar parametric resonances of an inextensional beam. International Journal of Nonlinear Mechanics 24, 139-158 (1989).
24. Fu, F. C. L. and Nemat-Nasser, S., On the stability of steady-state response of certain nonlinear dynamic systems subjected to harmonic excitations, Ingenieur-Archiv 41, 407-120 (1972).
25. Fu, F. C. L. and Nemat-Nasser, S., Stability of solution of systems of linear differential equations with harmonic coefficients. AIAA J. 10.30-36 (1972).
26. Nayfeh, A. H. and Mook, D. T., Nonlinear oscillations. Wily-Interscience, New York (1979).
27. Nayfeh, A. H., Parametrically excited multi-degree-of-freedom systems with repeated frequencies. Journal Sound and Vibration 88, 145-150 (1983).
28. Tezak, E. G., Nayfeh, A. H. and Mook, D. T., Parametrically excited nonlinear multi-degree-of-freedom systems with repeated natural frequencies. Journal of Sound and Vibration 85, 459-172 (1982).
29. Namachchivaya. N. S. and Malhotra, N., Parametrically excited Hop bifurcation with nonsemisimple 1:1 resonance, Nonlinear Vibrations, ASME DE-Vol. 50/AMD-Vol. 144, 29-46 (1992).
30. Nayfeh, A. H., Method of Normal Forms, Wiley-lnterscience, New York (1993).
31. Nayfeh, A. H. and Chin, C.-M., Nonlinear interactions in a parametrically excited system with widely spaced frequencies. Nonlinear Dynamics 7, 195-216 (1995).
32. Nayfeh, A. H., Chin. C.-M. and Mook, D. T., Parametrically excited nonlinear two-degree-of-freedom system with repeated natural frequencies. Shock and Vibration 2(1), 43-57 (1995).
33. Arafat, H. N., Nayfeh. A. H. and Chin, C-M. Nonlinear nonplanar dynamics of parametrically excited cantilever beams. Nonlinear Dynamics 15. 31-61 (1998).
34. Wilson, J. F., Parametric spin resonance tor a spinner with orbiting pivot, International Journal of Non-Linear Mechanics 33(2), 189-200 (1998).
35. Pai, P. F., Wen, B., Naser, A. S. and Schulz, M. J., Structural vibration control using PTZ patches and non-linear phenomena, Journal of Sound and Vibration 215(2), 273-296 (1998).
36. Hyun, S. H. und Yoo, H. H., Dynamic modeling and stability analysis of axially oscillating cantilever beams, Journal of Sound and Vibration 228. 543-558 (1999).
37. Shen. H-C. and Chen, L-W., Lumped mass model for parameter instability analysis of cantilever shaft-disk systems, Journal of Sound and Vibration 234(2), 331-348 (2000).
38. Oueini, S. S., Chin, C-M. and Nayfeh, A.M., Response of two quadratically-coupled oscillators to a principal parametric excitation. Journal of Vibration and Control 6, 1115-1133 (2000).
39. El-Bassiouny, A. F., Kamel. M. M. and Abdel-Khalik, A., Two-to-one internal resonances in nonlinear two-degree-of-freedom system with parametric and external excitations, Mathematics and Computers in Simulations 63(1), 45-56 (2003).
40. Arafat, H. N. and Nayfeh, A. H., Non-linear responses of suspended cables to primary resonances, Journal of Sound and Vibration 266, 325-354 (2003).
41. Elnaggar, A. M. and El-Bassiouny, A. F., Periodic and non-periodic combination resonance in kinematically excited system of rods, Applied Mathematics and Computation 139, 179-199 (2003).
42. El-Bassiouny, A. F. and Eissa, M., Resonance of non-linear systems subjected to multi-parametric excited structures: (comparison between two methods, response and stability), Physica Scripta 70, 101-113 (2004).
43. El-Bassiouny, A. F., Effect of non-linearities in elastomeric material dampers on torsional oscillation control, Applied Mathematics and Computations 111 162, 835-854 (2004).
44. Rehfield, L. D. W., Nonlinear flexural oscillations of shallow arches, American Institute of Aeronautics and Astronautics Journal 12. 91-93 (1874).
45. Nayfeh, A. H., On the low-frequency drumming of bowed structures, Journal of Sound and Vibration 94(4), 551-562 (1984).
46. Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley-Interscience. New York (1981).
47. Nayfeh, A. H. and Balachandran, B., Applied Non-Linear Mechanics, Wiley-Interscience, New York (1994).
48. Nayfeh, A. H., Non-linear Interactions, Wiley-Interscience. New York (2000).