Free vibration analysis of axially functionally graded nanobeam with radius varies along the length based on strain gradient theory

In this study the vibration of axially functionally graded material (AFGM) nanobeam is investigated by using strain gradient theory. In so doing, Euler-Bernoulli beam model is used, the nanobeam surroundings are modeled as visco-Pasternak foundation, and the beam has simply-supported boundary conditions. The governing equations and boundary conditions are derived by using Hamilton's principle, and differential quadrature method (DQM) is used to discretize equations of motion and solve the vibrational problem with simple-simple and clamped-clamped boundary conditions. The results demonstrate that the effect of the variation of Young's modulus, density, the diameter of the nanobeam, and size parameter along the length on the natural frequency of the nanobeam is significant. In addition, the effects of the stiffness and damping of the visco-Pasternak foundation on the natural frequency of the nanobeam are studied.