By Farzad Ebrahimi
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Additional resources for Advances in Vibration Analysis Research
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).  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  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.