Nonlocal Timoshenko representation and analysis of multi-layered functionally graded nanobeams

The vibration properties of functionally graded multiple nanobeams is studied using Eringen's nonlocal elasticity theory. Beam layers are considered to be continuously connected by layer of linear springs and the nonlocal Timoshenko beam theory is used to model each layer of beam which applies the size dependent effects in FG beam. The material behaviors of FG nanobeams are assumed to vary over the thickness based to the power law. The Hamilton's principle is used to derive the governing differential equations of motion according to Eringen nonlocal theory and a Chebyshev spectral collocation method is employed to convert the coupled equations of motion into algebraic equations. The discretized boundary conditions are applied to adjust the Chebyshev differentiation matrices, and the system of equations is then expressed in the matrix-vector form. Next, the coupled natural frequencies and corresponding mode shapes are obtained by solving the standard eigenvalue problem. The model is confirmed by comparing the obtained results with benchmark results existing in the literature. Next, a parametric study is carried out to determine the influences of the material gradation, length scale, and stiffness parameters on the vibration properties of multiple functionally graded nanobeams. It is demonstrated that these parameters are vital in examination of the free vibration of a multi-layered FG nanobeam.