Vibration of single-walled carbon nanotubes with elastic boundary conditions

The vibrational behaviours of single-walled carbon nanotubes (SWCNTs) bridged on a silicon channel are investigated using a three-segment Timoshenko beam model and a one-segment Timoshenko beam model with elastic boundaries together with molecular dynamics (MD) simulation. A modified Fourier series method (MFSM) is proposed to analyse the free vibration of the Timoshenko beam models with elastic boundary conditions. Explicit formulas are derived for the van der Waals (vdW) interaction coefficients between the SWCNTs and silicon substrates. The boundary elastic constants of the SWCNTs bridged on the silicon channel are obtained by fitting the bending curve of SWCNTs subjected to a static uniformly distributed lateral load simulated via the MD method. The MD simulations show that both the three-segment Timoshenko beam model and the one-segment Timoshenko beam model with elastic boundaries have a relatively good ability to predict the vibrational behaviours of SWCNTs bridged on a silicon channel.