Studying linear and nonlinear vibrations of fractional viscoelastic Timoshenko micro-/nano-beams using the strain gradient theory

In this paper, the vibrational behavior of micro- and nano-scale viscoelastic beams under different types of end conditions in the linear and nonlinear regimes is investigated based on the fractional calculus. To capture the effects of small scale, the modified strain gradient theory is utilized. Also, the beams are modeled based on the Timoshenko beam theory, von Karman nonlinear relations and the fractional Kelvin-Voigt viscoelastic model. Derivation of governing equations is performed using Hamilton's principle. For the linear solution, the generalized differential quadrature and finite difference methods are employed. Moreover, in the nonlinear solution procedure, the Galerkin method is first used to convert the fractional integro-partial differential governing equations into fractional ordinary differential equations which are then arranged in an effective state-space form. The predictor-corrector technique is finally used to solve the set of nonlinear fractional time-dependent equations. Selected numerical results are given on the linear and nonlinear time responses of the fractional viscoelastic small-scale beams to study the effects of fractional-order, viscoelasticity coefficient and length scale parameter.