Dynamic Analysis of Micro Functionally Graded Beam under Moving Mass based on Modified Couple Stress Theory

In this paper, numerical solution procedures are proposed for dynamic response of a functionally graded (FG) micro beam under action of a moving micro mass based on the modified couple stress theory (MCST) within the framework of Timoshenko beam theory for various boundary conditions. The governing equation and the related boundary conditions are derived by using Hamilton’s principle. Then the proposed solution for obtaining the natural frequencies and mode shapes of free vibration are presented by expressing the micro beam displacement fields in terms of the series of Legendre polynomials to find the numerical solution regardless of boundary conditions. Likewise, the computed eigenvalue of the system is developed in the modal expansion method to obtain the dynamic response. For validation purposes, the free vibration frequencies of the micro beam and the dynamic responses using the Timoshenko beam theory are compared with previously published studies and very good agreements have been observed. Furthermore, more numerical results for natural frequencies and dynamic deflection of the beam are presented and the effects of some parameters, such as material length scale parameter, thickness to length ratio of micro beam, boundary condition, the velocity of the moving load and the FG material power index are examined.