Analytical and Numerical Study on the Free Vibration of DWCNT Purposing the Nonlocal Effect between Its Walls

In this research, free vibration analysis of single and double walled carbon nanotubes (DWCNT) are considered, Analytical and numerically. In FEM simulation, two node beam elements are used and spring elements added as Van Der Waals (VDW) interaction forces between various walls of nanotube. For the first time in this research, the effect of nonlocal forces are prescribed as inclined nonlinear spring elements between nanotube walls. For better understanding of nonlocal force effects, free vibration analysis of DWCNT is performed analytically and numerically without nonlocal effect, too. In the analytical section, parabolic shear deformation theory is used. The nonlocal mode of FEM shows higher non-dimensional natural frequencies as compared to the analytical and linear FEM solutions. The later result is because of the increment in the flexural stiffness of simulated nanotube, by addition of nonlocal springs. However in the higher mode frequencies, the effects of nonlocal forces are reduced. Thus, nonlocal effect is higher in low frequency mode numbers. By comparing the results of analytics and FEM results, there shown more agreement between them by raising up the frequency mode number.