Vibration Analysis of Piezoelectric Carbon Nanotube Considering Surface Effects, Located in the Magnetic Field and Resting on Nonlinear Viscoelastic Foundation

In this study, the free and forced vibrations of piezoelectric carbon nanotubes with the effect of surface effects placed in a magnetic field situated on a viscoelastic foundation with nonlinear damping and stiffness elements under the influence of external harmonic force are investigated. The nonlocal theory is used to illustrate the effects of the nanoscale in the theoretical model and the equations of motion of the system are extracted using the dynamic equilibrium conditions of the element. To reduce the order of the obtained dynamical equations, Galerkin method is used. Considering the boundary conditions of the problem, which are both simple support (SS) or clamped (CC), the nonlinear time differential equations of the system and its coefficients are obtained. After that, using the multiple time scales method, an analytical closed-form solution aimed at amplitude-frequency response curves for forced vibration of a nonlinear system is extracted. Moreover, the effect of CNT surface effect, voltage applied to the system, and nonlinear viscoelastic foundation stiffness on the results of frequencies and dynamic response curves will be discovered. It can be resulted that the highest frequency is related to the form that considers all surface effects and the lowest frequency is related to the form that does not consider any of the surface effects. Also, existence of the non-local parameter reduces the maximum range of fluctuations. Finally, the obtained results are validated with the expected ones