Nonintegrability of an Infinite-Degree-of-Freedom Model for Unforced and Undamped, Straight Beams

[+] Author and Article Information
K. Yagasaki

Department of Mechanical and Systems Engineering, Gifu University, Gifu, Gifu 501-1193, Japan

J. Appl. Mech 70(5), 732-738 (Oct 10, 2003) (7 pages) doi:10.1115/1.1602483 History: Received July 04, 2002; Revised February 20, 2003; Online October 10, 2003
Copyright © 2003 by ASME
Your Session has timed out. Please sign back in to continue.


Woinowsky-Krieger,  S., 1950, “The Effect of an Axial Force on the Vibration of Hinged Bars,” ASME J. Appl. Mech., 17, pp. 35–36.
Burgreen,  D., 1951, “Free Vibrations of a Pin-Ended Column With Constant Distance Between Pin Ends,” ASME J. Appl. Mech., 18, pp. 135–139.
Eisley,  J. G., 1964, “Nonlinear Vibration of Beams and Rectangular Plates,” Z. Angew. Math. Phys., 15, pp. 167–175.
Srinivasan,  A. V., 1966, “Nonlinear Vibrations of Beams and Plates,” Int. J. Non-Linear Mech., 1, pp. 179–191.
Ray,  J. D., and Bert,  C. W., 1969, “Nonlinear Vibrations of a Beam With Pinned Ends,” ASME J. Eng. Ind., 91, pp. 997–1004.
Bennet,  J. A., and Eisley,  J. G., 1970, “A Multiple Degree-of-Freedom Approach to Nonlinear Beam Vibrations,” AIAA J., 8, pp. 734–739.
Tseng,  W. Y., and Dugundji,  J., 1970, “Nonlinear Vibrations of a Beam Under Harmonic Excitation,” ASME J. Appl. Mech., 37, pp. 292–297.
Bennett,  J. A., and Rinkel,  R. L., 1972, “Ultraharmonic Vibrations of Nonlinear Beams,” AIAA J., 10, pp. 715–716.
Bennett,  J. A., 1973, “Ultraharmonic Motion of a Viscously Damped Nonlinear Beams,” AIAA J., 11, pp. 710–715.
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, John Wiley and Sons, New York.
Guckenheimer, J., and Holmes, P. J., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.
Wiggins, S., 1990, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York.
Holmes,  P. J., and Marsden,  J. E., 1981, “A Partial Differential Equation With Infinitely Many Periodic Orbits: Chaotic Oscillations of a Forced Beam,” Arch. Ration. Mech. Anal., 76, pp. 135–165.
Tseng,  W. Y., and Dugundji,  J., 1971, “Nonlinear Vibrations of a Buckled Beam Under Harmonic Excitation,” ASME J. Appl. Mech., 38, pp. 467–476.
Moon,  F. C., and Holmes,  P. J., 1979, “A Magnetoelastic Strange Attractor,” J. Sound Vib., 65, pp. 275–296.
Yagasaki,  K., 2000, “Horseshoes in Two-Degree-of-Freedom Hamiltonian Systems With Saddle-Centers,” Arch. Ration. Mech. Anal., 154(2000), pp. 275–296.
Yagasaki, K., 2003, “Homoclinic and Heteroclinic Orbits to Invariant Tori in Multi-Degree-of-Freedom Hamiltonian Systems With Saddle-Centers,” submitted for publication.
Yagasaki,  K., 2001, “Homoclinic and Heteroclinic Behavior in an Infinite-Degree-of-Freedom Hamiltonian System: Chaotic Free Vibrations of an Undamped, Buckled Beam,” Phys. Lett. A, 285, pp. 55–62.
Arnold,  V. I., 1964, “Instability of Dynamical Systems With Many Degrees of Freedom,” Sov. Math. Dokl., 5, pp. 581–585.
Lochak, P., 1999, “Arnold Diffusion: A Compendium of Remarks and Questions,” Hamiltonian Systems with Three or More Degrees of Freedom, edited by C. Simó, Kluwer, Dordrecht, pp. 168–183.
Yagasaki,  K., 2002, “Numerical Evidence of Fast Diffusion in a Three-Degree-of-Freedom Hamiltonian System With a Saddle-Center,” Phys. Lett. A, 301, pp. 45–52.
Yagasaki,  K., 1992, “Chaotic Dynamics of a Quasiperiodically Forced Beam,” ASME J. Appl. Mech., 59, pp. 161–167.
Yagasaki,  K., 1995, “Bifurcations and Chaos in a Quasi-Periodically Forced Beam: Theory, Simulation, and Experiment,” J. Sound Vib., 183, pp. 1–31.
Morales-Ruiz,  J. J., and Ramis,  J. P., 2001, “Galoisian Obstructions to Integrability of Hamiltonian Systems,” Methods Appl. Anal., 8, pp. 33–96.
Morales-Ruiz, J. J., 1999, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhäuser, Basel.
Morales-Ruiz,  J. J., 2001, “Meromorphic Nonintegrability of Hamiltonian Systems,” Rep. Math. Phys., 48, pp. 183–194.
Morales-Ruiz,  J. J., and Simó,  C., 1996, “Non-Integrability Criteria for Hamiltonians in the Case of Lamé Normal Variational Equations,” J. Diff. Eqns., 129, pp. 111–135; Corrigendum: 1998, 144, pp. 477–478.
Morales-Ruiz,  J. J., and Peris,  J. M., 1999, “On a Galoisian Approach to the Splitting of Separatrices,” Ann. Fac. Sci. Toulouse VI. Sér., Math., 8, pp. 125–141.
Yagasaki, K., 2003, “Galoisian Obstructions to Integrability and Melnikov Criteria for Chaos in Two-Degree-of-Freedom Hamiltonian Systems With Saddle-Centers,” Nonlinearity, to appear.
Moser, J., 1973, Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, NJ.
Arnold, V. I., 1989, Mathematical Methods of Classical Mechanics, 2nd ed., Springer-Verlag, New York.
Kaplansky, I., 1976, An Introduction to Differential Algebra, 2nd ed., Hermann, Paris.
Singer, M. F., 1989, “An Outline of Differential Galois Theory,” Computer Algebra and Differential Equations, edited by E. Tournier, Academic Press, London, pp. 3–57.
Van der Put, M., and Singer, M. F., 2003, Galois Theory of Linear Differential Equations, Springer-Verlag, New York.
Ziglin,  S. L., 1982, “Branching of Solutions and Non-Existence of First Integrals in Hamiltonian Mechanics, I,” Funct. Anal. Appl., 16, pp. 181–189.
Morales-Ruiz,  J. J., and Simó,  C., 1994, “Picard-Vessiot Theory and Ziglin’s Theorem,” J. Diff. Eqns., 104, pp. 140–162.
Whittaker, E. T., and Watson, G. N., 1927, A Course of Modern Analysis, Cambridge University Press, Cambridge.
Hairer, E., Nørsett, S. P., and Wanner, G., 1993, Solving Ordinary Differential Equations I, 2nd ed., Springer-Verlag, Berlin.
Dormand,  J. R., and Prince,  P. J., 1989, “Practical Runge-Kutta Processes,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput., 10, pp. 977–989.
Nusse, E. H., and Yorke, J. A., 1997, Dynamics: Numerical Explorations, 2nd ed., Springer-Verlag, New York.
Yagasaki, K., 2003, “Numerical Analysis for Local and Global Bifurcations of Periodic Orbits in Autonomous Differential Equations,” in preparation.
Yagasaki, K., 2003, HomMap: A Package of AUTO and Dynamics Drivers for Homoclinic Bifurcation Analysis for Periodic Orbits of Maps and ODEs, Version 2.0, Gifu University, Gifu, in preparation.
Abraham, R., and Marsden, J. E., 1978, Foundations of Mechanics, 2nd ed., Addison-Wesley, Redwood City, CA.
Marsden, J. E., and Ratiu, T., 1999, Introduction to Mechanics and Symmetry, 2nd ed., Springer-Verlag, New York.
Yagasaki, K., 2003, “Homoclinic and Heteroclinic Motions for Resonant Periodic Orbits in Forced, Two-Degree-of-Freedom Systems,” in preparation.
Lang, S., 1990, Undergraduate Algebra, 2nd ed., Springer-Verlag, New York.


Grahic Jump Location
Transverse deformation of an initially straight beam
Grahic Jump Location
Orbits of the Poincaré map of Eq. (8) with N=2 for j1=1,j2=2,ω1=1,ω2=3.2, and H=11. (b) is an enlargement of (a) near the origin.
Grahic Jump Location
Numerically computed stable and unstable manifolds of a periodic orbit for the Poincaré map of Eq. (8) with N=2. The same parameter values as in Fig. 2 are taken.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In