{"title":"Free Vibration Analysis of Non-Uniform Euler Beams on Elastic Foundation via Homotopy Perturbation Method","authors":"U. Mutman, S. B. Coskun","volume":79,"journal":"International Journal of Mechanical and Mechatronics Engineering","pagesStart":1353,"pagesEnd":1359,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/16353","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.","references":"<p>[1] M. Balkaya, M.O. Kaya, and A. Sa\u011flamer, “Analysis of the vibration of\r\nan elastic beam supported on elastic soil using the differential transform\r\nmethod”, Archive of Applied Mechanics, vol.79, no.2, pp.135-146,\r\n2009.\r\n[2] B. Ozturk, S.B. Coskun, “The Homotopy Perturbation Method for free\r\nvibration analysis of beam on elastic foundation”, Structural Engineering\r\nand Mechanics, vol.37, no.4, pp.415-425, 2011.\r\n[3] I.E. Avramidis, K. Morfidis, “Bending of beams on three-parameter\r\nelastic foundation”, International Journal of Solids and Structures,\r\nvol.43, pp.357–375, 2006.\r\n[4] M.A. De Rosa, “Free vibration of Timoshenko beams on two-parameter\r\nelastic foundation”, Computers and Structures, vol.57, no.1, pp.151-156,\r\n1995.\r\n[5] H. Matsunaga, “Vibration and buckling of deep beam-columns on twoparameter\r\nelastic foundatins”, Journal of Sound and Vibration, vol.228,\r\nno.2, pp.359-376, 1999.\r\n[6] M. El-Mously, “Fundamental frequencies of Timoshenko beams\r\nmounted on Pasternak foundation”, Journal of Sound and Vibration,\r\nvol.228, no.2, pp. 452-457, 1999.\r\n[7] C.N. Chen, “Vibration of prismatic beam on an elastic foundation by the\r\ndifferential quadrature element method”, Computers and Structures,\r\nvol.77, pp.1–9. 2000.\r\n[8] C.N. Chen, “DQEM vibration analyses of non-prismatic shear\r\ndeformable beams resting on elastic foundations”, Journal of Sound and\r\nVibration, vol.255, no.5, pp. 989-999, 2002.\r\n[9] I. Coskun, “The response of a finite beam on a tensionless Pasternak\r\nfoundation subjected to a harmonic load”, European Journal of\r\nMechanics A\/Solids, vol.22, pp.151–161, 2003.\r\n[10] W.Q. Chen, C.F. Lu, and Z.G. Bian. “A mixed method for bending and\r\nfree vibration of beams resting on a Pasternak elastic foundation”,\r\nApplied Mathematical Modelling, vol.28, pp. 877–890, 2004.\r\n[11] P. Maheshwari, S. Chandra, and P.K. Basudhar, “Response of beams on\r\na tensionless extensible geosynthetic-reinforced earth bed subjected to\r\nmoving loads”, Computers and Geotechnics, vol.31, pp.537–548, 2004.\r\n[12] N.M. Auciello, M.A. De Rosa, “Two approaches to the dynamic\r\nanalysis of foundation beams subjected to subtangential forces”,\r\nComputers and Structures, vol.82, pp.519–524, 2004.\r\n[13] U. Mutman, “Free Vibration Analysis of an Euler Beam of Variable\r\nWidth on the Winkler Foundation Using Homotopy Perturbation\r\nMethod”, Mathematical Problems in Engineering, Vol.2013, Article ID\r\n721294, 2013.\r\n[14] J.H. He, “A coupling method of a homotopy technique and a\r\nperturbation technique for non-linear problems”, International Journal of\r\nNon-Linear Mechanics, vol.35, no.1, pp.37-43, 2000.\r\n[15] J.H. He, “The homotopy perturbation method for non-linear oscillators\r\nwith discontinuities”, Applied Mathematics and Computations, vol.151,\r\nno.1, pp. 287-292, 2004.\r\n[16] J.H. He, “Application of homotopy perturbation method to non-linear\r\nwave equation”, Chaos, Solitons and Fractals, vol.26, no.3, pp.695-700,\r\n2005.\r\n[17] J.H. He, “Asymptotology by homotopy perturbation method”, Applied\r\nMathematics and Computations, vol.156, no.3, pp.591-596, 2004.\r\n[18] J.H. He, “The homotopy perturbation method for solving boundary\r\nproblems”, Physics Letter A, vol.350, no.1, pp.87-88, 2006.\r\n[19] J.H. He, “Limit cycle and bifurcation of nonlinear problems”, Chaos,\r\nSolitons and Fractals, vol.26, no.3, pp.827-833, 2005.<\/p>\r\n","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 79, 2013"}

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