Nonlinear Vibration Analysis of Microscale Functionally Graded Timoshenko Beams using the Most General form of Strain Gradient Elasticity

Abstract

Based on the Timoshenko beam model, the nonlinear vibration of microbeams made of functionally graded (FG) materials is investigated under different boundary conditions. To consider small scale effects, the model is developed based on the most general form of strain gradient elasticity. The nonlinear governing equations and boundary conditions are derived via Hamilton's principle and then discretized using the generalized differential quadrature technique. A pseudo-Galerkin approach is used to reduce the set of discretized governing equations into a time-varying set of ordinary differential equations of Duffing-type. The harmonic balance method in conjunction with the Newton-Raphson method is also applied so as to solve the problem in time domain. The effects of boundary conditions, length scale parameters, material gradient index and geometrical parameters are studied. It is found that the importance of the small length scale is affected by the type of boundary conditions and vibration mode. Also, it is revealed that the classical theory tends to underestimate the vibration amplitude and linear frequency of FG microbeams.

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