Free Vibration Analysis of Non-Uniform Euler Beams on Elastic Foundation via Homotopy Perturbation Method
In this study Homotopy Perturbation Method (HPM) is employed to investigate free vibration of an Euler beam with variable stiffness resting on an elastic foundation. HPM is an easy-to-use and very efficient technique for the solution of linear or nonlinear problems. HPM produces analytical approximate expression which is continuous in the solution domain. This work shows that HPM is a promising method for free vibration analysis of nonuniform Euler beams on elastic foundation. Several case problems have been solved by using the technique and solutions have been compared with those available in the literature.
<p>[1] M. Balkaya, M.O. Kaya, and A. Sağlamer, “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.
[2] B. Ozturk, S.B. Coskun, “The Homotopy Perturbation Method for free
vibration analysis of beam on elastic foundation”, Structural Engineering
and Mechanics, vol.37, no.4, pp.415-425, 2011.
[3] I.E. Avramidis, K. Morfidis, “Bending of beams on three-parameter
elastic foundation”, International Journal of Solids and Structures,
vol.43, pp.357–375, 2006.
[4] M.A. De Rosa, “Free vibration of Timoshenko beams on two-parameter
elastic foundation”, Computers and Structures, vol.57, no.1, pp.151-156,
1995.
[5] H. Matsunaga, “Vibration and buckling of deep beam-columns on twoparameter
elastic foundatins”, Journal of Sound and Vibration, vol.228,
no.2, pp.359-376, 1999.
[6] M. El-Mously, “Fundamental frequencies of Timoshenko beams
mounted on Pasternak foundation”, Journal of Sound and Vibration,
vol.228, no.2, pp. 452-457, 1999.
[7] C.N. Chen, “Vibration of prismatic beam on an elastic foundation by the
differential quadrature element method”, Computers and Structures,
vol.77, pp.1–9. 2000.
[8] C.N. Chen, “DQEM vibration analyses of non-prismatic shear
deformable beams resting on elastic foundations”, Journal of Sound and
Vibration, vol.255, no.5, pp. 989-999, 2002.
[9] I. Coskun, “The response of a finite beam on a tensionless Pasternak
foundation subjected to a harmonic load”, European Journal of
Mechanics A/Solids, vol.22, pp.151–161, 2003.
[10] W.Q. Chen, C.F. Lu, and Z.G. Bian. “A mixed method for bending and
free vibration of beams resting on a Pasternak elastic foundation”,
Applied Mathematical Modelling, vol.28, pp. 877–890, 2004.
[11] P. Maheshwari, S. Chandra, and P.K. Basudhar, “Response of beams on
a tensionless extensible geosynthetic-reinforced earth bed subjected to
moving loads”, Computers and Geotechnics, vol.31, pp.537–548, 2004.
[12] N.M. Auciello, M.A. De Rosa, “Two approaches to the dynamic
analysis of foundation beams subjected to subtangential forces”,
Computers and Structures, vol.82, pp.519–524, 2004.
[13] U. Mutman, “Free Vibration Analysis of an Euler Beam of Variable
Width on the Winkler Foundation Using Homotopy Perturbation
Method”, Mathematical Problems in Engineering, Vol.2013, Article ID
721294, 2013.
[14] J.H. He, “A coupling method of a homotopy technique and a
perturbation technique for non-linear problems”, International Journal of
Non-Linear Mechanics, vol.35, no.1, pp.37-43, 2000.
[15] J.H. He, “The homotopy perturbation method for non-linear oscillators
with discontinuities”, Applied Mathematics and Computations, vol.151,
no.1, pp. 287-292, 2004.
[16] J.H. He, “Application of homotopy perturbation method to non-linear
wave equation”, Chaos, Solitons and Fractals, vol.26, no.3, pp.695-700,
2005.
[17] J.H. He, “Asymptotology by homotopy perturbation method”, Applied
Mathematics and Computations, vol.156, no.3, pp.591-596, 2004.
[18] J.H. He, “The homotopy perturbation method for solving boundary
problems”, Physics Letter A, vol.350, no.1, pp.87-88, 2006.
[19] J.H. He, “Limit cycle and bifurcation of nonlinear problems”, Chaos,
Solitons and Fractals, vol.26, no.3, pp.827-833, 2005.</p>
<p>[1] M. Balkaya, M.O. Kaya, and A. Sağlamer, “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.
[2] B. Ozturk, S.B. Coskun, “The Homotopy Perturbation Method for free
vibration analysis of beam on elastic foundation”, Structural Engineering
and Mechanics, vol.37, no.4, pp.415-425, 2011.
[3] I.E. Avramidis, K. Morfidis, “Bending of beams on three-parameter
elastic foundation”, International Journal of Solids and Structures,
vol.43, pp.357–375, 2006.
[4] M.A. De Rosa, “Free vibration of Timoshenko beams on two-parameter
elastic foundation”, Computers and Structures, vol.57, no.1, pp.151-156,
1995.
[5] H. Matsunaga, “Vibration and buckling of deep beam-columns on twoparameter
elastic foundatins”, Journal of Sound and Vibration, vol.228,
no.2, pp.359-376, 1999.
[6] M. El-Mously, “Fundamental frequencies of Timoshenko beams
mounted on Pasternak foundation”, Journal of Sound and Vibration,
vol.228, no.2, pp. 452-457, 1999.
[7] C.N. Chen, “Vibration of prismatic beam on an elastic foundation by the
differential quadrature element method”, Computers and Structures,
vol.77, pp.1–9. 2000.
[8] C.N. Chen, “DQEM vibration analyses of non-prismatic shear
deformable beams resting on elastic foundations”, Journal of Sound and
Vibration, vol.255, no.5, pp. 989-999, 2002.
[9] I. Coskun, “The response of a finite beam on a tensionless Pasternak
foundation subjected to a harmonic load”, European Journal of
Mechanics A/Solids, vol.22, pp.151–161, 2003.
[10] W.Q. Chen, C.F. Lu, and Z.G. Bian. “A mixed method for bending and
free vibration of beams resting on a Pasternak elastic foundation”,
Applied Mathematical Modelling, vol.28, pp. 877–890, 2004.
[11] P. Maheshwari, S. Chandra, and P.K. Basudhar, “Response of beams on
a tensionless extensible geosynthetic-reinforced earth bed subjected to
moving loads”, Computers and Geotechnics, vol.31, pp.537–548, 2004.
[12] N.M. Auciello, M.A. De Rosa, “Two approaches to the dynamic
analysis of foundation beams subjected to subtangential forces”,
Computers and Structures, vol.82, pp.519–524, 2004.
[13] U. Mutman, “Free Vibration Analysis of an Euler Beam of Variable
Width on the Winkler Foundation Using Homotopy Perturbation
Method”, Mathematical Problems in Engineering, Vol.2013, Article ID
721294, 2013.
[14] J.H. He, “A coupling method of a homotopy technique and a
perturbation technique for non-linear problems”, International Journal of
Non-Linear Mechanics, vol.35, no.1, pp.37-43, 2000.
[15] J.H. He, “The homotopy perturbation method for non-linear oscillators
with discontinuities”, Applied Mathematics and Computations, vol.151,
no.1, pp. 287-292, 2004.
[16] J.H. He, “Application of homotopy perturbation method to non-linear
wave equation”, Chaos, Solitons and Fractals, vol.26, no.3, pp.695-700,
2005.
[17] J.H. He, “Asymptotology by homotopy perturbation method”, Applied
Mathematics and Computations, vol.156, no.3, pp.591-596, 2004.
[18] J.H. He, “The homotopy perturbation method for solving boundary
problems”, Physics Letter A, vol.350, no.1, pp.87-88, 2006.
[19] J.H. He, “Limit cycle and bifurcation of nonlinear problems”, Chaos,
Solitons and Fractals, vol.26, no.3, pp.827-833, 2005.</p>
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:57988", author = "U. Mutman and S. B. Coskun", title = "Free Vibration Analysis of Non-Uniform Euler Beams on Elastic Foundation via Homotopy Perturbation Method", abstract = "In this study Homotopy Perturbation Method (HPM) is employed to investigate free vibration of an Euler beam with variable stiffness resting on an elastic foundation. HPM is an easy-to-use and very efficient technique for the solution of linear or nonlinear problems. HPM produces analytical approximate expression which is continuous in the solution domain. This work shows that HPM is a promising method for free vibration analysis of nonuniform Euler beams on elastic foundation. Several case problems have been solved by using the technique and solutions have been compared with those available in the literature.", keywords = "Homotopy Perturbation Method, Elastic Foundation, Vibration, Beam", volume = "7", number = "7", pages = "1590-6", }
