Free vibration of nonlocal Timoshenko beams made of functionally graded materials by Symplectic method

The aim of this paper is to numerically investigate the size-dependent vibration behavior of the nano-beam made of functionally graded materials (FGMs). The material properties of the FGMs are considered to vary throughout the thickness direction of the beams. Based on the non-local theory and the material and dimensions of the beams, the total energy including the potential and kinetic energy of the FGMs beams is obtained. The first-order governing differential equations in the Hamilton systems of the beam are constructed by introducing the dual variables and with the help of the variational principle. A precise-constant method and extended Wittrick-Williams algorithm are applied to obtain the structural frequencies of the nano-beams with the clamped-free and clamped-clamped boundary conditions. The model is verified by comparing the results with the data available in the literature. In the following, a study is carried out to find the effects of the nonlocal parameter, power index, and aspect ratio on the vibration of the FGMs nano-beams with the clamped-free and clamped-clamped boundary conditions respectively. The results show the soft effect of the nonlocal parameter on the structural vibration and the increase of the nonlocal parameter leads to the decrease of the frequency. The power index and aspect ratio also have significant effects on the vibration of the beams. The increase of the power index can increase the ceramic volume fraction in the FGMs, which leads to the increase of structural frequencies. The vibration of the beams can be controlled by choosing proper values of the power index and aspect ratio.