Torsional vibrations of restrained nanotubes using modified couple stress theory

In the current study, torsional vibration analysis of carbon nano tubes with general elastic boundary conditions is presented via modified couple stress theory. The model developed based on modified couple stress theory gives us opportunity to interpret small size effect. Two torsional springs are attached to a single-walled carbon nanotube at both ends. The idea of the proposed work is to obtain a coefficient matrix for eigen-value analysis involving the torsional spring coefficients. Stoke transformation is employed to work out the Fourier sine series for the carbon nanotube with general elastic boundary conditions. The direct expressions of the vibrational responses with torsional spring coefficients are obtained by using the non classical boundary conditions. In order to demonstrate the validity of the proposed method, results obtained for rigid boundary cases are presented for a comparison with those given in the literature and the results agree with each other exactly. The influences of torsional spring coefficients and small scale parameter on torsional frequencies are investigated in terms of the numerical results for both rigid and restrained boundary conditions.