Skip to main content
SpringerLink
Log in

Methods for lower bounds to frequencies of continuous elastic systems

  • Original Papers
  • Published:
Zeitschrift für angewandte Mathematik und Physik ZAMP Aims and scope Submit manuscript

Zusammenfassung

Es werden Verfahren zur Berechnung der unteren Grenzen des Frequenzgangs freier Schwingungen von kontinuierlichen linear-elastischen Systemen dargestellt. Diese Verfahren beruhen auf einer systematischen Verbesserung der durch explizite Lösung ähnlicher, jedoch einfacherer Schwingungsprobleme hergeleiteten unteren Grenzen. Die Allgemeinheit unserer Darstellung ermöglicht die Erfassung einer umfangreichen Klasse linearer Theorien. Die Anwendbarkeit dieser Verfahren wird an Beispielen aus der klassischen Plattentheorie erläutert, und es werden numerische Ergebnisse für vier Balkenprobleme angegeben.

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

References

  1. N. Bazley andD. W. Fox,Truncations in the Method of Intermediate Problems for Lower Bounds to Eigenvalues, J. Res. Nat. Bur. Stds.65B, 105–111 (1961).

    Google Scholar 

  2. N. Bazley andD. W. Fox,Lower Bounds to Eigenvalues Using Operator Decompositions of the Form B * B, Arch. Rat. Mech. Anal.10, 352–360 (1962).

    Google Scholar 

  3. N. Bazley andD. W. Fox,Lower Bounds for Eigenvalues of Schrödinger's Equation, Phys. Rev.124, 483–492 (1961).

    Google Scholar 

  4. N. Bazley andD. W. Fox,Lower Bounds for Energy Levels of Molecular Systems, J. Math. Phys.4, 1147–1153 (1963).

    Google Scholar 

  5. L. Collatz,Engenwertaufgaben mit technischen Anwendungen, Leipzig 1957, Beker und Erler.

    Google Scholar 

  6. A. Weinstein,Etudes des spectres des équations aux dérivées partielles, Mém. Sci. Math. fasc. 88, Paris 1937, Gauthier-Villars.

    Google Scholar 

  7. N. Aronszajn,Approximations Methods for Eigenvalues of Completely Continuous Symmetric Operators, Proc., Symp. Spectral Theory and Differential Problems, Stillwater, pp. 179–202 (1951).

  8. S. Gould,Variational Methods for Eigenvalue Problems, Toronto 1957.

  9. S. Michlin,Variationsmethoden der mathematischen Physik, Berlin 1962, Akademie-Verlag, pp. 306–308.

    Google Scholar 

  10. H. Weinberger,A Theory of Lower Bounds for Eigenvalues, Inst. for Fluid Dyn. and Appl. Math. Univ of Md. Tech. Note BN-183 (1959).

  11. H. Weyl,Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann.71, 1–11 (1912).

    Google Scholar 

  12. H. Lamb andR. Southwell,The Vibrations of a Spinning Disc, Proc. Roy. Soc.A99, 272–280 (1921).

    Google Scholar 

  13. W. Boyce, R. DiPrima, andG. Handelman,Vibrations of Rotating Beams of Constant Section, Proc. Second U.S. Natl. Congr. of App. Mech., Providence, 165–173 (1958).

  14. J. Hersch,Propriétés de convexité du type de Weyl pour des problèmes de vibration ou d'équilibre. ZAMP12, 298–322 (1961).

    Google Scholar 

  15. B. Zwahlen,Über die Eigenwerte der Summe zweier selbstadjungierter Operatoren, Battelle Mem. Tech. Note, Geneva (1964).

  16. I. Sokolnikoff,Mathematical Theory of Elasticity, New York 1956, McGraw-Hill.

    Google Scholar 

  17. K. Friedrichs,Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren, Math. Ann.109, 465–487, 685–713 (1934);110, 777–779 (1935).

    Google Scholar 

  18. M. Stone,Linear Transformations in Hilbert Space, New York 1932, Amer. Math. Soc. Colloquium Series, Vol. 15.

  19. T. Kato,Quadratic Forms in Hilbert Spaces and Asymptotic Perturbation Series, Univ. of Calif. math. dept. lect. notes, Berkeley 1955, 11–42.

  20. S. Sobolev,Applications of Functional Analysis in Mathematical Physics, Providence 1963, Vol. 7, Transl. of Mathematical Monographs, Amer. Math. Soc.

  21. H. Poincaré,Sur les équations aux dérivées partielles de la physique mathématique, Am. J. Math.12 (1890).

  22. N. Bazley,Lower Bounds for Eigenvalues, J. Math. Mech.10, 289–308 (1961).

    Google Scholar 

  23. J. Gay,Lower Bounds to the Eigenvalues of Hamiltonians by Intermediate Problems, Univ. of Florida dissertation, Gainesville (1963).

  24. J. Stadter,Bounds to Eigenvalues of Rhombical Membranes, App. Phys. Lab. Johns Hopkins Univ. Tech. Report CF-3084 (1964) (to appear in Jour. S.I.A.M.).

  25. N. Aronszajn,Application of Weinstein's Method with an Auxiliary Problem of Type II, Univ. of Kansas Tech. Report 4, Studies in eigenvalue problems.

  26. F. Tricomi,Sulle vibrazioni transversali di aste, specialmente di bielle, di sezione variabile. Ricerche di Ingegneria4, 47–53 (1936).

    Google Scholar 

  27. F. Tricomi,Integral Equations, New York 1957, Interscience, 141–143.

    Google Scholar 

  28. G. Fichera,Sul calcolo degli autovalori, Att. del Simp: Appl. dell'Anal. alla Fis. Mat., Cagliari (1964).

  29. B. Zwahlen,Untere Schranken für die Eigenwerte eines Differentialoperators, (to appear in ZAMM).

  30. R. Courant andD. Hilbert,Methods of Mathematical Physics, Vol. I, New York 1953, Interscience, 90, 321.

    Google Scholar 

  31. C. Colonna,High Precision Calculations of Parameters for Frequencies and Mode Shapes of Uniform Beams, App. Phys. Lab. Johns Hopkins Univ. Tech. Report CF-3093 (1964).

Download references

Author information

Authors and Affiliations

  1. Institut Battelle, Geneva, Switzerland

    Norman Bazley

  2. Applied Physics Laboratory, The Johns Hopkins University, Silver Spring, Maryland, USA

    David W. Fox

Authors
  1. Norman Bazley
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David W. Fox
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Sponsored in part by the Aeronautical Research Laboratories, OAR, through the European Office of Aerospace Research, United States Air Force.

Sponsored in part by the Department of the Navy under contract NOw-62-0604-c with the Bureau of Naval Weapons.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bazley, N., Fox, D.W. Methods for lower bounds to frequencies of continuous elastic systems. Journal of Applied Mathematics and Physics (ZAMP) 17, 1–37 (1966). https://doi.org/10.1007/BF01594084

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01594084

Keywords

Search

Navigation