Skip to main content
SpringerLink
Log in

The swinging spring — Approximate analyses for low and very high energy, II

  • Published:
Celestial mechanics Aims and scope Submit manuscript

Abstract

The two families of periodic solutions emanating from the lower equilibrium of a planar spring-pendulum system are analyzed both at and near resonance. Hamiltonian perturbation theory is used to obtain approximate formulas for the characteristic and the period of the motion. As the energy is increased to very high levels, a circulatory and an almost rectilinear periodic motion persist. Expressions for the trace of a stability matrix are determined for both these solutions. Comparisons are made throughout with accurate results from numerical integration.

Résumé

Les deux familles de solutions périodiques qui proviennent de l'équilibre inférieur d'un pendule élastique plan sont analysées pour le cas de résonnance et dans le voisinage de résonnance. On emploie la théorie perturbatrice Hamiltonienne pour obtenir des formules d'approximation pour la caractéristique et pour la période du mouvement. Quand l'énergie est augmentée à des niveaux très élevés, un mouvement périodique circulatoire et un mouvement périodique presque rectiligne sont préservés. Des expressions pour la trace d'une matrice de stabilité sont déterminées pour les deux familles de solutions périodiques. Des comparaisons sont faites entre les formules d'approximation et des résultats précis obtenus par l'integration numérique.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Breakwell, J. V.: 1967 and 1973,Lecture Notes — Advanced Space Mechanics, Stanford University.

  • Breakwell, J. V.: February 1974, private communication.

  • Broucke, R. and Baxa, P. A.: 1973a,Celes. Mech. 8, 261–267.

    Google Scholar 

  • Broucke, R. and Baxa, P. A.: 1973b, ‘Large Amplitude Periodic Oscillations of a Spring-Pendulum System’, to be published.

  • Byrd, P. F. and Friedman, M. D.: 1954,Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag.

  • Heinbockel, J. H. and Struble, R. A.: 1963,ZAMP,14, 262–269.

    Google Scholar 

  • Hénon, M.: 1965,Ann. Astrophys. 28, 992–1007.

    Google Scholar 

  • Hénon, M.: 1969,Astron. Astrophys. 1, 223–238.

    Google Scholar 

  • Henrard, J.: 1973,J. Diff. Eqts. 14, 431–441.

    Google Scholar 

  • Hitzl, D. L.: 1972,Celes. Mech. 5, 433–450.

    Google Scholar 

  • Hitzl, D. L.: 1974, ‘The Swinging Spring — Families of Periodic Solutions and Their Stability, I’, to be published inAstron. Astrophys.

  • Kane, T. R. and Kahn, M. E.: 1968,J. Appl. Mech. 547–552.

Download references

Author information

Authors and Affiliations

  1. Mathematics and Systems Analysis Dept., Lockheed Research Laboratory, 3251 Hanover Street, 94304, Palo Alto, Calif., USA

    Donald L. Hitzl

Authors
  1. Donald L. Hitzl
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hitzl, D.L. The swinging spring — Approximate analyses for low and very high energy, II. Celestial Mechanics 12, 359–382 (1975). https://doi.org/10.1007/BF01228569

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01228569

Keywords

Search

Navigation