A Semi-Analytical Nonlinear Approach for Size-Dependent Analysis of Longitudinal Vibration in Terms of Axially Functionally Graded Nanorods

Abstract

In this paper, the size dependency of the nonlinear free longitudinal vibration of axially functionally graded (AFG) nanorods is studied using the nonlocal elasticity theory. A power-law distribution is considered for the variations of the nanorod material properties through its length. The nonlinear equation of motion including the von-Karman geometric nonlinearity is derived using Hamilton’s principle. Then, a solution for the linear equation of motion is obtained using the harmonic differential quadrature method, and mode shapes and natural frequencies are extracted. In the next step, the nonlinear natural frequencies are calculated by solving the nonlinear equation of motion using the multiple scales method. Two types of boundary conditions are considered, i.e. fixed-fixed and fixed-free. The presented results include effects of various parameters like nanorod length and diameter, amplitude of vibration, small scale parameter, and frequency number, on the natural frequencies. In addition, a comparative study is conducted to evaluate the effects of the type of linear mode shape on the nonlinear natural frequencies.

Keywords

• nonlinear longitudinal vibration
• nanorod
• functionally graded materials
• nonlocal elasticity theory