Through developing constructive nonlocal models, both in- and out-of-plane free dynamic analyses of single-layered of double-walled carbon nanotubes (DWCNTs) are going to be investigated carefully. By exploiting the nonlocal Rayleigh and Timoshenko beams, the rigorous governing equations of the nanosystem that explicitly express its transverse vibrations are extracted by continuum-based simulating the van der Waals forces. The abilities of the nonlocal continuous-based models in estimating the results of the nonlocal discrete-based models are proved. Afterward, the nonlocal transverse frequencies are evaluated and the roles of influential factors plus to the shear effect on the free dynamical response are elucidated and discussed methodically. The obtained results from this scrutiny provide a reasonably smooth path for inspecting mechanical behavior of more complex systems composed of vertically aligned DWCNTs.

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