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The nonlinear dynamic analysis of the fractional viscoelastic nanotube resting on a foundation, with simply-supported boundary conditions, is performed. The existence of a significant internal damping for the structure led to the choice of a fractional Zener model to obtain the governing equation. The solving of this is made with the help of a new variational iteration method, the Laplace transform, Bessel functions theory and binominal series. The effects of the nonlocal parameter, fractional order and viscoelastic foundation on the transverse displacements of the nanostructure are studied. Validation study was performed by comparing the results obtained for the model of the structure with the corresponding ones existing in the literature. The proposed algorithm for solving the integral-differential governing equation is useful in the engineering design of the biological nano-sensors and nanoscale devices resting on a viscoelastic foundation.

Nonlinear vibrations of fractional nonlocal viscoelastic nanotube resting on a Kelvin–Voigt foundation