Timoshenko beam model for vibrational analysis of double-walled carbon nanotubes bridged on substrate

Abstract

This paper proposes a continuum three-segment double Timoshenko beam (TSDTB) model to investigate free vibration of double-walled carbon nanotubes (DWCNTs) bridged on a silicon channel. Accurate explicit formulas of the van der Waals (vdW) interactions between each tube as well as between the tubes and a substrate are derived for a better prediction of the vibrational behaviors of the DWCNTs. An analytical modified Fourier series method (MFSM) is developed for the vibrational analysis of the TSDTB model. Numerical results show the good convergence characteristics and accuracy of the MSFM. The overlapped lengths between DWCNTs and the substrate at both ends have almost no effects on the lower natural frequencies of the TSDTB model when the overlapped lengths are long enough. Moreover, the natural frequencies of the TSDTB model are much less than of those of one-segment double Timoshenko beam (OSDTB) model with clamped supported boundary condition, whereas the TSDTB model can be equivalent to the simply supported OSDTB model under certain condition. In addition, the effects of vdW interaction coefficients on the vibrational behaviors of the DWCNTs are also revealed. The obtained results in this work should be greatly helpful in the design and application of CNTs-based nanomechanical resonators.