This paper presents approximate series solutions for nonlinear free vibration of suspended cables via the Lindstedt-Poincare method and homotopy analysis method, respectively. Firstly, taking into account the geometric nonlinearity of the suspended cable as well as the quasi-static assumption, a mathematical model is presented. Secondly, two analytical methods are introduced to obtain the approximate series solutions in the case of nonlinear free vibration. Moreover, small and large sag-to-span ratios and initial conditions are chosen to study the nonlinear dynamic responses by these two analytical methods. The numerical results indicate that frequency amplitude relationships obtained with different analytical approaches exhibit some quantitative and qualitative differences in the cases of motions, mode shapes, and particular sag-to-span ratios. Finally, a detailed comparison of the differences in the displacement fields and cable axial total tensions is made. 1. Introduction As a basic and significant structural element, the suspended cable has been widely applied in many mechanical systems and engineering fields [1, 2], such as civil, ocean, and aerospace engineering. Generally speaking, the nonlinear dynamics of the suspended cable is very complicated and attracts more and more attention in recent years, for example, referring to the literature reviews by Rega [3, 4]. The suspended cable is a typical weakly nonlinear continuous system which contains the quadratic and cubic nonlinearity terms, and the nonlinear free vibration of the system has been studied through many analytical methods in the previous researches. Recently, Hagedorn and Sch？fer  investigated the nonlinear free vibrations of suspended cables in the case of small sag via the Lindstedt method. Luongo et al. [6, 7] applied multiple scales method and Lindstedt-Poincare method to study the nonlinear planar free vibrations of an elastic cable, respectively. Rega et al.  examined the nonlinear phenomenon in a large range of the cable sag-to-span ratios by the numerical investigation. Benedettini et al.  applied an order-three perturbation expansion to obtain the solutions of the free nonplanar coupled equations. Srinil et al.  presented a model to analyze large amplitude free vibrations of the suspended cable in three dimensions. In these studies, the perturbation method is the most significant analytical way to study the nonlinear vibration of the suspended cable. However, it is noted that the perturbation method should be based on the small parameter assumption. Because of the
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