Modelling vibrating nano-strings by lattice, finite difference and Eringen's nonlocal models

This paper focuses on the modelling of a tightly stretched nano-string with elastically supported ends by the central finite difference model (FDM), the lattice model and the Eringen's nonlocal model (ENM). The equivalence between the FDM and the lattice model is confirmed provided that the nodal spacing of FDM is equal to the segmental length of lattice model. The analytical vibration solutions for lattice model or FDM can be obtained by solving the linear second-order finite difference equation derived herein. By matching the vibration solutions for lattice model (or FDM) and ENM, the Eringen's small length scale coefficient e₀ is calibrated, which is a constant equal to 0.289, regardless of the vibration modes or boundary conditions. The influence of end lateral spring stiffnesses on the vibration solutions for a taut local string is also discussed. For a taut symmetrically restrained local string with sliding ends, the fundamental frequency always approaches to zero and the second frequency always approaches to π.