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By Farzad Ebrahimi

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20] G. Adomian, A review of the decomposition method and some recent results for nonlinear equation, Math. Comput. , 13(7), 1992, 17-43. H. He, Variational iteration method: a kind of nonlinear analytical technique, Int. J. Nonlin. , 34, 1999, 699-708. H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Nonlin. , 35, 2000, 37-43. H. He, An elemantary introduction to the homotopy perturbation method, Computers and Mathematics with Applications, 57(2009), 410-412.

29] B. B. Coskun, “The homotopy perturbation method for free vibration analysis of beam on elastic foundation”, Structural Engineering and Mechanics, An Int. Journal, 37(4), (2010). [30] B. Öztürk, “Free vibration analysis of beam on elastic foundation by variational iteration method“, International journal of Nonlinear Science and Numerical Simulation, 10 (10) 2009, 1255-1262. 22 Advances in Vibration Analysis Research [31] B. B. Z. T. Atay, Homotopy perturbation method for free vibration analysis of beams on elastic foundations, IOP Conf.

3 The crack model Numerous crack models for a cracked beam can be found in the literature. , 1991; Pandy & Biswas, 1994). , 1990; Chaudhari & Maiti, 2000). Christides and Barr (1984) developed the one-dimensional vibration theory for the lateral vibration of a cracked Euler-Bernoulli beam with one or more pairs of symmetric cracks. According to Christides and Barr(1984), the variation of bending stiffness EI d ( x ) along the cracked beam length takes up the form of EI d ( x ) = EI 0 1 + (c − 1)exp( −2α x − xc / d ) (16) where E is the Young’s modulus of the beam, I 0 = wd 3 / 12 is the second moment of area of the intact beam, c = 1 /(1 − C r )3 , C r = dc / d is the crack depth ratio and dc and d are the depth of crack and the beam, respectively, xc is the location of the crack.

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