Axial vibration analysis of nanorods with variable density based on nonlocal elastic theory and high-order finite difference method

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This paper proposes a novel high-order finite difference method (p-FDM) for predicting the axial vibration behaviour of elastic nanorods with variable density based on nonlocal elastic theory. For p-FDM, it first solves the nodal weighting coefficients for approximating the first-order derivatives of displacements at the discrete nodes by using moving least-squares approximation. Next, p-FDM recursively determines the nodal weighting coefficients for approximating the higher-order derivatives by using the obtained weighting coefficients for the first-order derivative. After verifying with exact solutions, results show that the proposed p-FDM yields accurate vibration solutions for a nanorod with either fixed–fixed ends or fixed–free ends. Only a small number of regularly or irregularly distributed grid nodes are required by a high order p-FDM to obtain a converged solution. p-FDM realises an exponential rate of convergence for frequency solution when setting p=n−2 where p is order of p-FDM and n is the number of grid nodes. New axial vibration solutions of nanorods with exponential density distribution are found by using p-FDM. It is shown that the axial frequencies are always larger for a nanorod with an exponential density distribution and no small-scale effect.