On the Global Geometric Structure of the Dynamics of the Elastic Pendulum

Nonlinear Dynamics

Volumn 18, Issue 1, 1999, Pages 51-68

  Georgiou, Ioannis T ^a,b  

^a SCIENCE APPLICATIONS INTERNATIONAL CORPORATION   (United States)

^b NAVAL RESEARCH LABORATORY   (United States)

Author keywords

Energy transfer; Fast and slow chaos; Lyapunov exponents; Separatrix; Slow and fast invariant manifolds

Indexed keywords

BIFURCATION (MATHEMATICS); DEGREES OF FREEDOM (MECHANICS); ELASTICITY; ENERGY TRANSFER; LYAPUNOV METHODS; OSCILLATIONS;

LYAPUNOV EXPONENTS; PLANAR ELASTIC PENDULUMS;

PENDULUMS;

EID: 0032665041     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1008356204490     Document Type: Article

References (27)

1. Fujii, A. H. and Inchiki, W., 'Nonlinear dynamics of the tethered subsatellite system in the station keeping phase', Journal of Guidance and Control 20(2), 1997, 403-406.
2. Rupp, C. C. and Laue, J. H., 'Shuttle/tethered satellite system', Journal of the Astronautical Sciences XXVI(1), 1978, 1-17.
3. Breitenberger, E. and Mueller, R. D., 'The elastic pendulum: A nonlinear paradigm', Journal of Mathematical Physics 22(6), 1981, 1196-1210.
4. Numez-Yepez, H. N., Salas-Brito, A. L., and Vargas, C. A., 'Onset of chaos in an extensible pendulum', Physics Letters A 145(2/3), 1990, 101-105.
5. Contopoulos, G., Grousouzakou, E., and Polymidis, C., 'Distribution of periodic orbits and the homoclinic tangle', Celestial Mechanics and Dynamical Astronomy 64, 1996, 363-381.
6. Hitzl, D. L., 'The swinging spring-families of periodic solutions and their stability. I.', Astronautics & Astrophysics 40, 1975, 147-159.
7. Hitzl, D. L., 'The swinging spring-invariant curves formed by quasi-periodic solutions. III', Astronautics & Astrophysics 41, 1975, 187-198.
8. Rigney, D. R. and Goldberger, A. L., 'Nonlinear mechanics of the heart's swinging during pericardial effusion', American Physiological Society 257, 1989, H1292-H1305.
9. Bayly, P. V. and Virgin, L. N., 'An empirical study of the stability of periodic motion in the forced spring-pendulum', Proceedings of the Royal Society of London A 443, 1993, 391-408.
10. Kane, T. J. and Kahn, M. E., 'On a class of two-degree-of-freedom oscillations', ASME Journal of Applied Mechanics 35, 1968, 547-552.
11. Ryland II, G. and Meirovitch, L., 'Stability boundaries of a swinging spring with oscillating support', Journal of Sound and Vibration 51(4), 1977, 547-560.
12. Narkis, Y., 'On the stability of a spring-pendulum', Journal of Applied Mathematics and Physics (ZAMP) 28, 1977, 343-348.
13. Heinbockel J. H. and Struble, R. A., 'Resonant oscillations of an extensible pendulum', Journal of Applied Mathematics and Physics (ZAMP) 14(3), 1963, 262-269.
14. Van Der Burgh, A. H. P., 'On the higher order asymptotic approximations for the solutions of the equations of motion of an elastic pendulum', Journal of Sound and Vibration 42(4), 463-475.
15. Fenichel, N., 'Geometric singular perturbation theory for ordinary differential equations', Journal of Differential Equations 31, 1979, 53-98.
16. Nipp, N., 'Invariant manifolds of singularly perturbed ordinary differential equations', Journal of Applied Mathematics and Physics (ZAMP) 36, 1985, 311-320.
17. Georgiou, I. T., 'Nonlinear dynamics and chaotic motions of a singularly perturbed nonlinear viscoelastic beam', Ph.D. Dissertation, Purdue University, West Lafayette, IN, 1993.
18. Georgiou, I. T., Bajaj, A. K., and Corless, M., 'Slow and fast invariant manifolds, and normal modes in a two degree of freedom structural dynamical system with multiple equilibrium states', International Journal of Non-Linear Mechanics 33(2), 1998, 275-300.
19. Knobloch, H. W. and Aullback, B., 'Singular perturbations and integral manifolds', Journal of Mathematical and Physical Sciences 18(5), 1984, 415-423.
20. Kokotovic̀, P. V., Khalil, H. K., and O'Reilly J., Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London, 1986.
21. Georgiou, I. T. and Schwartz, I. B., 'The slow invariant manifold of a conservative pendulum-oscillator system', International Journal of Bifurcation and Chaos 6(4), 1996, 673-692.
22. Georgiou, I. T. and Schwartz, I. B., 'Slaving the in-plane motions of a nonlinear plate to its flexural motions: An invariant manifold approach', ASME Journal of Applied Mechanics 64, 1997, 175-182.
23. Kauderer, Nichtlineare Mechanic, 1958, pp. 593-612.
24. Lichtenberg, A. J. and Lieberman, M. A., Regular and Chaotic Dynamics, Springer-Verlag, New York, 1992.
25. Vakakis, A. F. and Rand, R. H., 'Normal modes and global dynamics of a two-degree-of-freedom non-linear system. I. Low energies', International Journal of Non-Linear Mechanics 27(5), 1992, 861-873.
26. Vakakis, A. F. and Rand, R. H., 'Normal modes and global dynamics of a two-degree-of-freedom non-linear system. II. High energies', International Journal of Non-Linear Mechanics 27(5), 1992, 875-888.
27. Georgiou, I. T., 'Fast and slow chaos and chaotic burstings in structural dynamics' (in preparation).