Longitudinal vibration characteristics analysis of nonlocal rod structure with arbitrary internal elastic supports

This manuscript obtained an exact solution for longitudinal vibrating characteristics of generally restrained nanorod structure with multiple internal elastic supports via an improved Fourier series method. Due to the nonlocal elasticity theory, the general boundary restraints have been formulated by introducing artificial springs. Then, classical boundary conditions can be easily realized by setting the restrained stiffness to zero or infinity accordingly. Energy formulation is derived for describing longitudinal vibration characteristics of generally restrained nanorod with multiple internal elastic supports. To improve the differential continuities at the elastically restrained boundary, nonlocal longitudinal vibrating displacement is expanded as the standard Fourier series combined with auxiliary polynomials, which makes the field function sufficiently smooth in the calculation region. By making use Rayleigh-Ritz procedure, the unknown coefficients can be obtained by solving the standard eigenvalue matrix. Then, numerical results are presented and compared with those in the literature to illustrate the reliability and effectiveness of the current model. The influence of parameters of the multiple internal elastic supports, nonlocal structural parameters, and boundary restraints on vibration characteristics are discussed and analyzed. This study provides an effective method for predicting longitudinal vibration characteristics of a generally restrained nanorod with multiple internal elastic supports, in which the nanorod has complex boundary and coupling conditions.