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In this study, Strongly nonlinear free vibration behavior of a microbeam considering the structural damping effect is investigated analytically on the basis of modified couple stress theory. Employing Von Karman’s strain-displacement relations and implementing the Galerkin method, the governing nonlinear partial differential equation is reduced to a nonlinear ordinary differential equation which is related to the size effect of the beam. Because of large coefficient of nonlinear term and due to existence of the damping effect, none of the traditional perturbation methods leads to a valid solution. Also, there are many difficulties encountered in applying homotopy techniques when the damping effect is taken in to account in the Strongly nonlinear damped system. To overcome these limitations, here, a new analytical method is presented which is based on classical perturbation methods and fundamentals of Fourier expansion with an embedding nondimensional parameter. To solve the equation, the nonlinear frequency is assumed to be time dependent. The comparison between time responses of the system obtained by the presented approach and numerical method indicates the high accuracy of the new method. To validate the results of the presented method with those available in the literatures which are obtained for a special case of an undamped system, the damping coefficient is set to zero. The comparison shows a good agreement between the results for a wide range of vibration amplitudes.

Purpose: This paper proposes the use of different analytical methods in obtaining approximate solutions for nonlinear differential equations with oscillations.Methods: Three methods are considered in this paper: Lindstedt-Poincare method, the Krylov-Bogoliubov first approximate method, and the differential transform method.Results: Figures that are given in this paper give a strong evidence that the proposed methods are effective in handling nonlinear differential equations with oscillations.Conclusions: This study reveals that the differential transform method provides a remarkable precision compared with other perturbation methods.

Analyzing the nonlinear vibration of beams is one of the important issues in structural engineering. According to this, an impressive analytical method which is called modified iteration perturbation method (MIPM) is used to obtain the behavior and frequency of a cantilever beam with geometric nonlinear. This new method is combined by the Mickens and iteration methods. Moreover, this method does not require small parameter in the equation which is difficult to be found for nonlinear oscillation. The accuracy of the solution that is obtained by the use of MIPM has been shown graphically and compared with exact solution. Comparison shows that good adaptation is obtained and MIPM as a powerful method for solving the vibrational behavior of structures analytically.