An Equation Both More Consistent and Simpler Than the Bresse-Timoshenko Equation

Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 168)

Abstract

A simple equation is discussed which takes into account both shear deformation and rotary inertia in vibrating beams. This equation is both more consistent and simpler than the widely used one of Bresse-Timoshenko.

References

  1. 1.
    Abramovich H, Elishakoff I (1987) Application of the Krein’s method for determination of natural frequencies of periodically supported beam based on simplified Bresse-Timoshenko equations. Acta Mech 66:39–59CrossRefGoogle Scholar
  2. 2.
    Abramovich H, Elishakoff I (1990) Bolotin’s dynamic edge effect method incorporating shear deformation and rotary inertia. J Sound Vib 136:355–359CrossRefGoogle Scholar
  3. 3.
    Abramovich H, Elishakoff I (1990) Influence of shear deformation and rotary inertia on vibration frequencies based on Love’s equations. J Sound Vib 137:516–522CrossRefGoogle Scholar
  4. 4.
    Bresse M (1859) Cours de Mécanique Appliquée, Mallet-Bacheher, Paris (in French)Google Scholar
  5. 5.
    Egle DM (1969) An approximate theory for transverse shear deformation and rotary inertia effect in vibrating beams. NASA CR-1317Google Scholar
  6. 6.
    Elishakoff I, Livshits D (1989) Some closed-form solutions in random vibrations of Timoshenko beams, Prob Eng Mech 4:49–54CrossRefGoogle Scholar
  7. 7.
    Elishakoff I, Lubliner E (1984) Random vibration of a structure via classical and nonclassical theories. In: Eggwertz S, Lind NC (eds) Probabilistic methods in the mechanics of solids and structures. Springer, Berlin, pp 455–468Google Scholar
  8. 8.
    Elishakoff I, Abramovich H (1992) Note on dynamic response of large space structures. J Sound Vib 156:178–184CrossRefGoogle Scholar
  9. 9.
    Elishakoff I, Pentaras D (2009) Natural frequencies of carbon nanotubes based on simplified Bresse-Timoshenko theory. J Comput Theor Nanosci, accepted for publicationGoogle Scholar
  10. 10.
    Grigolyuk EI, Selezov IT (1973) Nonclassical theories of vibration of beams, plates and shells, “VINITI” Publishing House, Moscow (in Russian)Google Scholar
  11. 11.
    Laura PAA, Rossi RE, Maurizi MJ (1992) Vibration of Timoshenko beams. Institute of Applied Mechanics and Department of Engineering, Universidad Nacional del Sur, Bahia Blanca, ArgentinaGoogle Scholar
  12. 12.
    Lottati I, Elishakoff I (1987) Influence of shear deformation and rotary inertia on flutter of a cantilevered beam–extract and symbolic computerized solutions. In: Elishakoff I, Irretier H (eds) Refined dynamical theories in beams, plates and shells and their applications. Springer, Berlin, pp 261–273Google Scholar
  13. 13.
    Love MAA (1927) Treatise on the mathematical theory of elasticity, 4th edn. Dover, New York, pp 430–431Google Scholar
  14. 14.
    Nesterenko VV (1993) A theory for transverse vibrations of the Timoshenko beam. J Appl Math Mech 57(4):669–6777CrossRefGoogle Scholar
  15. 15.
    Novozhilov VV (1979) Mathematical models and accuracy of engineering analysis, Sudostroenie (Shipbuilding) 7:5–12 (in Russian)Google Scholar
  16. 16.
    Rayleigh L (JWS Strutt) (1877–11878), The Theory of Sound, Macmillan, London (see also Dover, New York, 1945)Google Scholar
  17. 17.
    Sayir M (1987) Theoretical and experimental results on the dynamic behavior of composite beams, plates and shells. In: Elishakoff I, Irretier H (eds) Refined dynamical theories of beams, plates and shells, and their applications. Springer, Berlin, pp 72–90Google Scholar
  18. 18.
    Sayir M (1992–1994) Personal communicationsGoogle Scholar
  19. 19.
    Stephen NG (1982) The second frequency spectrum of Timoshenko beams. J Sound Vib 80:578–582CrossRefGoogle Scholar
  20. 20.
    Stephen NG (2006) The second spectrum of Timoshenko beam theory. J Sound Vib 292:372–389CrossRefGoogle Scholar
  21. 21.
    Weaver W Jr, Timoshenko SP, Young DH (1990) Vibration problems in engineering. Wiley, New YorkGoogle Scholar
  22. 22.
    Timoshenko SP (1921) On the correction for shear of the differential equation for transverse vibration of prismatic bars. Phil Mag Ser 6, 41:744–746 (See also, Timoshenko SP, The Collected Papers, 288–290, McGraw-Hill, New York, 1953)Google Scholar
  23. 23.
    Timoshenko SP (1922) On the transverse vibration of bars of uniform cross sections. Phil Mag Ser 6, 43:125–131 (See also, Timoshenko SP, The Collected Papers, 288–290, McGraw-Hill, New York, 1953)Google Scholar
  24. 24.
    Tseitlin AI (1961) On the effect of shear deformation and rotary inertia in vibration of beams on elastic foundation. PMM-J Appl Math Mech 25:531–535CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringFlorida Atlantic UniversityBoca RatonUSA

Personalised recommendations