Ваш браузер устарел.

Для того, чтобы использовать все возможности сайта, загрузите и установите один из этих браузеров.

скрыть

Article

  • Title

    Forced harmonic oscillations of the Euler-Bernoulli beam with resistance forces

  • Authors

    Krutiy Yuriy S.

  • Subject

    MACHINE BUILDING

  • Year 2015
    Issue 3(47)
    UDC 624.072.2
    DOI 10.15276/opu.3.47.2015.03
    Pages 9-16
  • Abstract

    The important issue in the oscillation theory is the study of resistance impact on oscillatory processes. Unlike the calculations of free oscilla-tions, that reside in determination of natural frequencies and waveshapes and unlike the calculations of forced oscillations far away from resonance, that are performing without reference to friction, the oscillations researches in vicinity of resonance need accounting of friction forces. Special attention is paid to forced transverse fluctuations in beams as an important technical problem for engineering and building. Aim: The aim of the work is constructing of analytical solution of the problem of forced transverse vibrations of a straight rod with constant cross-section, which is under the influence of the harmonic load taking into account external and internal resistances. Materials and Methods: The internal resistance is taken into account using the corrected hypothesis of Kelvin-Voigt which reflects the empirically proven fact about the frequency-independent internal friction in the material. The external friction is also considered as frequency-independent. Results: An analytical solution is built for the differential equation of forced transverse oscillations of a straight rod with constant cross-section which is under the influence of the harmonic load taking into account external and internal resistances. As a result, analytically de-rived formulae are presented which describe the forced dynamic oscillations and the dynamic internal forces due to the harmonic load applied to the rod thus reducing the problem with any possible fixed ends to the search of unknown integration constants represented in a form of initial parameters.

  • Keywords Euler-Bernoulli beam, forced oscillations, differential equation, initial parameters, exact solution
  • Viewed: 2215 Dowloaded: 29
  • Download Article
  • References

    Література
    1.    Василенко, М.В. Теорія коливань і стійкості руху / М.В. Василенко, О.М. Алексейчук. — К.: Вища школа, 2004. — 525 с.
    2.    Бабаков, И.М. Теория колебаний / И.М. Бабаков. — 4-е изд., испр. — М.: Дрофа, 2004. — 591 с.
    3.    Rao, S.S. Vibration of Continuous Systems / S.S. Rao. — Hoboken: John Wiley & Sons, 2007. — 720 p.
    4.    Wang, C.Y. Structural Vibration: Exact Solutions for Strings, Membranes, Beams, and Plates / C.Y. Wang, C.M. Wang. — Boca Raton: CRC Press, 2014. — 293 p.
    5.    Exact frequency equations of free vibration of exponentially non-uniform functionally graded Timoshenko beams / A.-Y. Tang, J.-X. Wu, X.-F. Li, K.Y. Lee // International Journal of Mechanical Sciences. — 2014. — Vol. 89. — PP. 1—11. doi:10.1016/j.ijmecsci.2014.08.017
    6.    Advances in Vibration Analysis Research / ed. by F. Ebrahimi. — Rijeka: In Tech, 2011. — 468 p. DOI:10.5772/639
    7.    Recent Advances in Vibrations Analysis / ed. by N. Baddour. — Rijeka: In Tech, 2011. — 248 p. DOI:10.5772/861
    8.    Rao, S.S. Mechanical Vibrations / S.S. Rao. — 5th Edition. — Upper Saddle River: Prentice Hall, 2011. — 1084 p.
    9.    Sinha, A.K. Vibration of Mechanical Systems / A.K. Sinha. – New York: Cambridge University Press, 2010. – 308 p.
    10.    Balachandran, B. Vibrations / B. Balachandran, E.B. Magrab. — 2nd Edition. — London: Cengage Learning, 2008. — 716 p.
    11.    Yavari, A. On nonuniform Euler–Bernoulli and Timoshenko beams with jump discontinuities: application of distribution theory / A. Yavari, S. Sarkani, J.N. Reddy // International Journal of Solids and Structures. — 2001. — Vol. 38, Issues 46–47. — PP. 8389—8406. doi:10.1016/S0020-7683(01)00095-6
    12.    Fryba, L. Vibration of Solids and Structures under Moving Loads / L. Fryba. — 3rd Edition. — Prague: Academia, 1999. — 494 p.
    13.    Inman, D.J. Engineering Vibration / D.J. Inman. — 4th Edition. — Boston: Pearson, 2014. — 707 p.
    14.    Critical load for buckling of non-prismatic columns under self-weight and tip force / D.J. Wei, S.X. Yan, Z.P. Zhang, X.-F. Li // Mechanics Research Communications. — 2010. — Vol. 37, Issue 6. — PP. 554—558. doi:10.1016/j.mechrescom.2010.07.024
    15.    Abu-Hilal, M. Forced vibration of Euler–Bernoulli beams by means of dynamic Green functions / M. Abu-Hilal // Journal of Sound and Vibration. — 2003. — Vol. 267, Issue 2. — PP. 191—207. doi:10.1016/S0022-460X(03)00178-0

    References
    1.    Vasilenko, M.V., & Aleksiichuk, O.M. (2004). Theory of Vibrations and Dynamic Stability. Kyiv: Vyshcha Shkola.
    2.    Babakov, I.M. (2004). Theory of Vibrations (4th Ed.). Moscow: Drofa.
    3.    Rao, S.S. (2007). Vibration of Continuous Systems. Hoboken: John Wiley & Sons. DOI:10.1002/9780470117866
    4.    Wang, C.Y., & Wang, C.M. (2014). Structural Vibration: Exact Solutions for Strings, Membranes, Beams, and Plates. Boca Raton: CRC Press.
    5.    Tang, A.-Y., Wu, J.-X., Li, X.-F., & Lee, K.Y. (2014). Exact frequency equations of free vibration of exponentially non-uniform functionally graded Timoshenko beams. International Journal of Mechanical Sciences, 89, 1—11. DOI:10.1016/j.ijmecsci.2014.08.017
    6.    Ebrahimi, F. (Ed.). (2011). Advances in Vibration Analysis Research. Rijeka: In Tech. DOI:10.5772/639
    7.    Baddour, N. (Ed.). (2011). Recent Advances in Vibrations Analysis. Rijeka: In Tech. DOI:10.5772/861
    8.    Rao, S.S. (2011). Mechanical Vibrations (5th Ed.). Upper Saddle River: Prentice Hall.
    9.    Sinha, A.K. (2010). Vibration of Mechanical Systems. New York: Cambridge University Press.
    10.    Balachandran, B., & Magrab, E.B. (2008). Vibrations (2nd Ed.). London: Cengage Learning.
    11.    Yavari, A., Sarkani, S., & Reddy, J.N. (2001). On nonuniform Euler–Bernoulli and Timoshenko beams with jump discontinuities: application of distribution theory. International Journal of Solids and Structures, 38(46–47), 8389—8406. DOI:10.1016/S0020-7683(01)00095-6
    12.    Fryba, L. (1999). Vibration of Solids and Structures under Moving Loads (3rd Ed.). Prague: Academia.
    13.    Inman, D.J. (2014). Engineering Vibration (4th Ed.). Boston: Pearson.
    14.    Wei, D.J., Yan, S.X., Zhang, Z.P., & Li, X.-F. (2010). Critical load for buckling of non-prismatic columns under self-weight and tip force. Mechanics Research Communications, 37(6), 554—558. DOI:10.1016/j.mechrescom.2010.07.024
    15.    Abu-Hilal, M. (2003). Forced vibration of Euler–Bernoulli beams by means of dynamic Green functions. Journal of Sound and Vibration, 267(2), 191—207. DOI:10.1016/S0022-460X(03)00178-0

  • Creative Commons License by Author(s)