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An Efficient Analytical Method for Vibration Analysis of a Beam on Elastic Foundation with Elastically Restrained Ends

DOI: 10.1155/2014/159213

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Abstract:

An efficient analytical method for vibration analysis of a Euler-Bernoulli beam on elastic foundation with elastically restrained ends has been reported. A Fourier sine series with Stoke’s transformation is used to obtain the vibration response. The general frequency determinant is developed on the basis of the analytical solution of the governing differential equation for all potential solution cases with rigid or restrained boundary conditions. Numerical analyses are performed to investigate the effects of various parameters, such as the springs at the boundaries to examine how the elastic foundation parameters affect the vibration frequencies. 1. Introduction Beams resting on elastic foundations have wide application in engineering practice. The vibration analysis of beams is investigated using various elastic foundation models, such as, Vlasov, Pasternak, and Winkler models. A number of studies have been performed to predict the dynamic response of beams on elastic foundations with different boundary conditions. Numerous works have been performed to explore the static deflection and vibration response of the beams resting on various elastic foundations. Chun [1] has investigated free vibration of hinged beam. Maurizi et al. [2] have considered the vibration frequencies for a beam with different boundary conditions. Vibration of beams on partial elastic foundations has been studied by Doyle and Pavlovic [3]. Laura et al. [4] have investigated beams which carry concentrated masses subject to an axial force. Abbas [5] has investigated vibration of Timoshenko beams with elastically restrained ends. Free vibration and stability behavior of uniform beams and columns with nonlinear elastic end rotational restraints has been considered by Rao and Naidu [6]. Free vibration behaviour of an Euler-Bernoulli beam resting on a variable Winkler foundation has been considered by Kacar et al. [7]. Civalek [8] has implemented differential quadrature and harmonic differential quadrature methods for buckling analysis of thin isotropic plates and elastic columns. H. K. Kim and M. S. Kim [9] have considered vibration of beams with generally restrained boundary conditions. A number of studies have been reported investigating the free vibration of beams on elastic foundation [10–25]. Although vibration analysis of beams on elastic foundation is a widely studied topic, there are only few papers that exist in the literature pertaining to the analysis of beams with elastically restrained ends. In this study, an efficient method is introduced for the analysis of the free

References

[1]  K. R. Chun, “Free vibration of a beam with one end spring-hinged and the other free,” Journal of Applied Mechanics, Transactions ASME, vol. 39, no. 4, pp. 1154–1155, 1972.
[2]  M. J. Maurizi, R. E. Rossi, and J. A. Reyes, “Vibration frequencies for a uniform beam with one end spring-hinged and subjected to a translational restraint at the other end,” Journal of Sound and Vibration, vol. 48, no. 4, pp. 565–568, 1976.
[3]  P. F. Doyle and M. N. Pavlovic, “Vibration of beams on partial elastic foundations,” Earthquake Engineering Structural Dynamics, vol. 10, no. 5, pp. 663–674, 1982.
[4]  P. A. A. Laura, P. Verniere de Irassar, and G. M. Ficcadenti, “A note on transverse vibrations of continuous beams subject to an axial force and carrying concentrated masses,” Journal of Sound and Vibration, vol. 86, no. 2, pp. 279–284, 1983.
[5]  B. A. H. Abbas, “Vibrations of Timoshenko beams with elastically restrained ends,” Journal of Sound and Vibration, vol. 97, no. 4, pp. 541–548, 1984.
[6]  G. V. Rao and N. R. Naidu, “Free vibration and stability behaviour of uniform beams and columns with non-linear elastic end rotational restraints,” Journal of Sound and Vibration, vol. 176, no. 1, pp. 130–135, 1994.
[7]  A. Kacar, H. T. Tan, and M. O. Kaya, “A note free vibration analysis of beams on variable Winkler elastic foundation by using the differential transform method,” Mathematical and Computational Applications, vol. 16, pp. 773–783, 2001.
[8]  ?. Civalek, “Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns,” Engineering Structures, vol. 26, no. 2, pp. 171–186, 2004.
[9]  H. K. Kim and M. S. Kim, “Vibration of beams with generally restrained boundary conditions using fourier series,” Journal of Sound and Vibration, vol. 245, no. 5, pp. 771–784, 2001.
[10]  M. Hetenyi, “Beams and plates on elastic foundation,” Applied Mechanics Reviews, vol. 19, pp. 95–102, 1966.
[11]  C. Miranda and K. Nair, “Finite beams on elastic foundation,” Journal Structural Division, vol. 92, pp. 2131–2142, 1966.
[12]  D. Z. Yankelevsky and M. Eisenberger, “Analysis of a beam column on elastic foundation,” Computers and Structures, vol. 23, no. 3, pp. 351–356, 1986.
[13]  M. Eisenberger and J. Clastornik, “Vibrations and buckling of a beam on a variable winkler elastic foundation,” Journal of Sound and Vibration, vol. 115, no. 2, pp. 233–241, 1987.
[14]  M. A. De Rosa, “Stability and dynamics of beams on Winkler elastic foundations,” Earthquake Engineering & Structural Dynamics, vol. 18, no. 3, pp. 377–388, 1989.
[15]  J. Wang, “Vibration of stepped beams on elastic foundations,” Journal of Sound and Vibration, vol. 149, no. 2, pp. 315–322, 1991.
[16]  Y. C. Lai, B. Y. Ting, W.-S. Lee, and B. R. Becker, “Dynamic response of beams on elastic foundation,” Journal of Structural Engineering ACSE, vol. 118, no. 3, pp. 853–858, 1992.
[17]  D. Zhou, “A general solution to vibrations of beams on variable winkler elastic foundation,” Computers and Structures, vol. 47, no. 1, pp. 83–90, 1993.
[18]  D. Thambiratnam and Y. Zhuge, “Free vibration analysis of beams on elastic foundation,” Composite Structure, vol. 60, pp. 971–980, 1996.
[19]  D. N. Paliwal and R. K. Pandey, “The free vibration of a cylindrical shell on an elastic foundation,” Journal of Vibration and Acoustics, Transactions of the ASME, vol. 120, no. 1, pp. 63–71, 1998.
[20]  P. Gülkan and B. N. Alemdar, “Exact finite element for a beam on a two-parameter elastic foundation: a revisit,” Structural Engineering and Mechanics, vol. 7, no. 3, pp. 259–276, 1999.
[21]  K. Al-Hosani, S. Fadhil, and A. El-Zafrany, “Fundamental solution and boundary element analysis of thick plates on Winkler foundation,” Computers and Structures, vol. 70, no. 3, pp. 325–336, 1999.
[22]  J.-H. Yin, “Closed-form solution for reinforced Timoshenko beam on elastic foundation,” Journal of Engineering Mechanics, vol. 126, no. 8, pp. 868–874, 2000.
[23]  J.-H. Yin, “Comparative modeling study of reinforced beam on elastic foundation,” Journal of Geotechnical and Geoenvironmental Engineering, vol. 126, no. 3, pp. 265–271, 2000.
[24]  D. M. Santee and P. B. Gon?alves, “Oscillations of a beam on a non-linear elastic foundation under periodic loads,” Shock and Vibration, vol. 13, no. 4-5, pp. 273–284, 2006.
[25]  P. A. A. Laura and R. H. Gutierrez, “Analysis of vibrating Timoshenko beams using the method of differential quadrature,” Shock and Vibration, vol. 1, pp. 89–93, 1993.
[26]  A. Kacar, H. T. Tan, and M. O. Kaya, “Free vibration analysis of beams on variable winkler elastic foundation by using the differential transform method,” Mathematical Computational Applications, vol. 16, pp. 773–783, 2011.
[27]  M. Balkaya, M. O. Kaya, and A. Sa?lamer, “Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method,” Archive of Applied Mechanics, vol. 79, no. 2, pp. 135–146, 2009.

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