A New Analytical Approach for Strongly Nonlinear Vibration of a Microbeam Considering Structural Damping Effect

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 straindisplacement 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.