Transverse Vibration and Wave Propagation of Functionally Graded Nanobeams with Axial Motion

Abstract

Based on the nonlocal elasticity theory, the vibration behavior and wave propagation of functionally graded Euler nanobeams with axial motion are investigated. Assuming that the axial velocity of nanobeams is a constant and the graded material parameters vary along the thickness direction in terms of power index, we apply the hypothesis of neutral plane to derive the axial displacement of geometry surface caused by non-uniformity of graded materials. Effects of graded index, axial velocity and nonlocal scale parameter on natural frequencies are analyzed through numerical examples. Also, the relationship between wave propagation frequency, wave velocity and wave number is revealed. The complex mode method is utilized to solve the governing equation, and the natural frequencies and wave velocity are obtained accordingly. To solve the derived transcendental equations, the Newton iteration method is used and a detailed solution flowchart is provided. For vibration, with the increase of non-dimensional axial velocity, non-dimensional nonlocal parameter and gradient index, natural frequencies decrease. For wave propagation, with the increase of wave number, frequency of wave propagation increases. However, with the increase of wave number, velocity of wave propagation decreases. Moreover, increasing the gradient index causes a decrease in frequency and velocity of wave propagation, increasing the axial velocity causes an increase in frequency and velocity of wave propagation, and increasing the nonlocal parameter causes a decrease in frequency and velocity of wave propagation.