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In this work, a numerical Galerkin approach which can handle different boundary conditions in a general manner is presented to study the free vibration of double-walled carbon nanotubes (DWCNTs) modeled as nonlocal Euler beams. The effects of small scale, different boundary conditions, and geometrical parameters on the vibration of DWCNTs are examined in detail. It is also observed from the numerical results that the higher modes of DWCNT vibration (where the inner and outer nanotubes vibrate out of phase) are dominated by van der Waals interaction between the inner and outer nanotubes, and small scale effect and boundary conditions are noted to have minimal consequence on the first higher natural frequency mode of a DWCNT.