Free vibration of a single-walled carbon nanotube containing a fluid flow using the Timoshenko beam model

The flexural vibration of the fluid-conveying single-walled carbon nanotube (SWCNT) is derived by the Timoshenko beam model, including rotary inertia and transverse shear deformation. The effects of the flow velocity and the aspect ratio of length to diameter on the vibration frequency and mode shape of the SWCNT are analyzed. Results show that the effects of rotary inertia and transverse shear deformation result in a reduction of the vibration frequencies, especially for higher modes of vibration and short nanotubes. The frequency is also compared with the previous study based on Euler beam model. In addition, if the ratio of length to diameter increased to 60, the influence of the shear deformation and rotary inertia on the mode shape and the resonant frequencies can be neglected. However, the influence is very obvious when the ratio decreased to 20. As the flow velocity of the fluid increases in the vicinity of 2π, the SWCNT reveals the divergence instability. It regains stability when the flow velocity reaches about 9. As the velocity increases further, the SWCNT undergoes a coupled-mode flutter and results in a larger amplitude.