Abstract
<title>Abstract</title> <p>This study analyzes the longitudinal vibrational dynamics of a harmonic mass-spring chain under fixed-free boundary conditions and its application to the interpretation of vibrations in carbon nanotubes (CNTs). The analytical analysis reveals a nearly linear increase in displacement variance from the fixed end toward the free end, while confirming energy conservation in the harmonic regime. Molecular dynamics simulations (\textsc{LAMMPS}) of an armchair (10,10) carbon nanotube described by the AIREBO potential reproduce the same spatial evolution of displacement variance and show that, for excitation amplitudes between 0.01 and 0.10~\AA, this variance remains practically independent of the amplitude. In contrast to the displacement variance, which stays independent of amplitude in the studied linear regime, the axial force variance exhibits a significant increase with excitation. Spectral analysis shows that this dependence is accompanied by the appearance of harmonics in the force, whereas the displacement maintains a response dominated by the fundamental frequency. These results indicate that the first signatures of nonlinearity appear in the internal stresses before significantly altering the displacement dynamics. The agreement between the harmonic model and the atomistic simulations demonstrates that the joint analysis of displacements and forces provides an effective approach for characterizing the early nonlinear effects in CNTs and offers a relevant framework for the development of nano-electromechanical devices (NEMS).</p>