Vibration analysis of viscoelastic carbon nanotubes

The vibration behaviour of a simply supported carbon nanotube is studied using the non-local viscoelasticity theory, including the thermal and foundation effects. The complex frequency in closed-form expression is obtained in terms of the damping ratio. The frequency is associated with the non-local parameter, damping coefficient and thermal and foundation effects. According to the analysis, the natural frequency significantly decreases by increasing the damping coefficient, especially at a small non-local parameter. The critical damping coefficient decreases when the thermal effect is considered in the analysis, especially at a higher non-local parameter. However, the situation is reversed when the foundation effect is considered. In addition, by increasing the non-local parameter, the mode shape increases. However, the mode shape apparently decreases by increasing the damping coefficient.