Dynamic stability of single-walled carbon nanotube embedded in a viscoelastic medium under the influence of the axially harmonic load

The nonlinear model of a single-walled carbon nanotube (SWCNT) modeled as a nanobeam embedded in a Kelvin-Voigt viscoelastic medium is developed by using the nonlocal continuum theory. It is assumed that the nanobeam vibrates under the influence of the longitudinal magnetic field and time-varying axial load. Based on the nonlocal Euler-Bernoulli beam theory, Maxwell's equations and von Karman nonlinear strain-displacements relation, we obtain the nonlinear partial differential equations of transversal motion of the embedded nanobeam with different boundary conditions. The relationship between nonlinear amplitude and frequency of variable axial load in the presence of the longitudinal magnetic field is derived by using the perturbation method of multiple scales. An approximate analytical solution for nonlinear frequency and instability regions for the linear case of vibration is also considered in this paper. In order to analyze nonlinear dynamical stability regions of SWCNT, the incremental harmonic balance (IHB) method is introduced for obtaining iterative relationship of frequency and amplitude of time-varying axial load. It is showed that the nonlocal parameter, magnetic field effects and stiffness coefficient of the viscoelastic medium have significant effects on vibration and stability behavior of nanobeam and therefore receive substantial attention. In addition, from the presented numerical results one can see the influence of the small scale, magnetic field and foundation coefficients on the frequency-response curve, nonlinear frequency and instability regions for the linear and nonlinear cases.