Longitudinal Vibration Analysis of Single-Walled Carbon Nanobeams

In this study, the free longitudinal vibrations of single-walled carbon nanobeams are investigated using Eringen’s nonlocal elasticity theory. The governing differential equation for longitudinal motion under clamped-clamped boundary conditions is derived and solved analytically using Mathematica. To validate the developed analytical formulation, a fully three-dimensional finite element model is implemented in COMSOL Multiphysics. The fundamental natural frequency obtained from the analytical solution compared with numerical simulation indicates a relative error of less than 0.15%, which confirms the high accuracy of the modeling. Furthermore, the results of the parametric analyses demonstrated that increasing the nonlocal scale parameter leads to a reduction in natural frequencies across all modes, representing the stiffness softening effect. This effect is significantly more pronounced in higher vibrational modes, such that by increasing the nonlocal parameter, the frequency drop of the third mode reaches more than 42%. Finally, examining the aspect ratio showed that small-scale effects completely dominate in short nanobeams, and with an increasing length-to-diameter ratio, the system behavior approaches classical elasticity.