Analysis of axial vibration of non-uniform nanorod using boundary characteristic orthogonal polynomials

In this paper, the longitudinal vibration of nanorod based on Eringen’s nonlocal elasticity theory was studied using Rayleigh-Ritz method. A non-uniform nano-rod with variable cross-sectional area, density and Young’s modulus were considered. In the present work, boundary polynomials with orthogonal polynomials were used as shape functions in the Rayleigh-Ritz method which causes the vibrational analysis to be computationally efficient and imposing of boundary conditions to be easier. Using the mentioned polynomials the convergence rate of the obtained results was increased. All of the equations used in this study were made to have no dimensional to reduce the number of effective parameters in the solution. The influence of the nonlocal and in-homogeneity parameters on the vibrational behavior of nanorod was investigated. The results were compared to available results in the literature and a good agreement has been achieved. The results showed that nanorod frequencies were depended to the small scale effect, non-uniformity, and boundary conditions. For instance, an increase in frequency ratio causes the scale coefficient in all vibration modes to be increased, especially in higher modes. In addition, the frequencies were increased by increasing in the length of the nanorod.