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Abstract
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In this study, strongly nonlinear free vibration behaviour 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’s method, the governing nonlinear partial differential equation is reduced to a nonlinear ODE which is related to the size effect of the beam. Because of the large coefficient of the 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 first order approximate solution of the presented method and numerical technique demonstrates the high accuracy of the new method for a wide range of vibration amplitudes. 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.
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