The extraordinary mechanical and physical properties of carbon nanotubes (CNTs) have been studied using a range of theories and methods. The mechanical properties of CNTs can be classified into statistic mechanics and dynamics properties. This study focuses on the dynamic mechanical properties of CNTs and single-walled carbon nanocones (SWCNCs). The dynamic mechanical behavior of CNTs and SWCNCs, including vibration and elastic waves, are studied by using molecular dynamics (MD) simulation and continuum mechanics models. The elastic wave propagation in CNTs is investigated with nonlocal and classical (local) continuum mechanics models and verified by MD simulation. The free vibrations of CNTs and SWCNCs with different boundary conditions are studied using continuum mechanics models and MD simulation. The applicability of the nonlocal and classical continuum models of CNTs is investigated through the problems of elastic waves and vibration. MD simulations of flexural wave, transverse wave, and torsional wave propagation in CNTs are generated. The second-generation reactive empirical bond order (REBO) potential is used in the MD to simulate the short-range interaction force between atoms. The propagation of elastic waves in CNTs is investigated by using nonlocal beam models and nonlocal shell models. The applicability of nonlocal continuum models for elastic waves in CNTs is verified by MD simulation. The scale parameter in nonlocal theory is estimated by matching the dispersion relation obtained from the continuum models with the MD results. It is found that the value of is mainly dependent on the different kinds of elastic waves investigated, rather than the structure of the CNT, such as the diameter, the chirality and the single walled or double walled CNTs. The nonlocal beam and shell models are found to offer a better prediction of the dispersion relationships than the classical models when the wavenumber is very large. The non-coaxial vibration of double-walled carbon nanotubes (DWCNTs) at high frequencies is also investigated with MD simulation. The van der Waals (vdW) interaction between the adjacent tubes is found to have little effect on the noncoaxial flexural vibration of CNTs. The nonlocal elastic Timoshenko beam theory is found to be inapplicable for modeling the noncoaxial wave propagation in CNTs. 0e0e The free transverse, longitudinal, and torsional vibrations of single-walled carbon nanotubes (SWCNTs) are investigated using a nonlocal beam model and a nonlocal rod model, and verified by MD simulation. The nonlocal Timoshenko beam model offers a better prediction of the fundamental frequencies of shorter SWCNTs than the local beam models. The nonlocal rod model is employed to study the longitudinal and torsional vibrations of SWCNTs, and is shown to give good predictions of the MD results for shorter SWCNTs. The nonlocal and classical continuum models are in good agreement with the MD simulation results for relatively longer SWCNTs. Finally, the free transverse and longitudinal vibrations of SWCNCs with different boundary conditions and apex angles are studied by MD simulation, and the fundamental frequencies are obtained. The fundamental frequencies of SWCNCs are also compared with those of SWCNTs with radii equal to top radii of SWCNC.