Critical speed and free vibration analysis of spinning 3D single-walled carbon nanotubes resting on elastic foundations

In this article, the influences of critical speed on the free vibration behavior of spinning 3D single-walled carbon nanotubes (SWCNT) are investigated using modified couple stress theory (MCST). Moreover, the surrounding elastic medium of SWCNT has been considered as a model of Winkler, characterized by the spring. Taking into consideration the first-order shear deformation theory (FSDT), the rotating SWCNT is modeled and its equations of motion are derived using the Hamilton principle. The formulations include Coriolis, centrifugal and initial hoop tension effects due to rotation of the SWCNT. The accuracy of the presented model is validated by some cases in the literature. The novelty of this study is considering the effects of rotation and MCST, in addition to considering the various boundary conditions of SWCNT. The generalized differential quadrature method (GDQM) is used to discretize the model and to approximate the equation of motion. Then investigation has been made on critical speed and natural frequency of the rotating SWCNT due to the influence of initial hoop tension, material length scale parameter, constant of spring, frequency mode number, angular velocity, length-to-radius ratio, radius-to-thickness ratio and boundary conditions.