Free vibration of micro/nano homogeneous beams coated by functionally graded porous layer using analytical and numerical methods

In this paper, the free vibration of homogeneous micro / nano beams coated by a functionally graded layer is investigated based on the Eringen's nonlocal elasticity theory. In order to extract the equations of motion, the first-order shear deformation beam theory (Timoshenko's beam theory) has been used. The obtained equations of motion are solved using two different analytical and numerical methods for different boundary conditions. In the analytical method, the equations of motion are first solved and the general functions for the displacements are obtained. The functions include a number of unknown parameters and constants. Then, by considering the boundary condition equations at both ends of the beam, a set of algebraic equations is extracted. Finally, the natural frequencies are obtained form the nonzero solution of the algebraic equations. In the numerical solution, the generalized differential quadrature method is used to solve the equations of motion. In the results section, first, the validity of present methods should be confirmed. Hence, the results obtained from this article are compared with the corresponding results presented in the litterature. Then, the results of the two analytical and numerical methods are compared, which confirms the consistency of the results of both two methods. The effects of thickness of porous layer and also the percentage of porosity of porous layer on the natural frequencies of beams are studied.