Free vibration analysis of fractional viscoelastic Euler-Bernoulli nano-beam

Fractional calculus is a branch of mathematics which in recent decades has been of great interest to scientists in various disciplines, including engineering. One of the applications of this branch in engineering, is in modeling the viscoelastic materials using fractional differentiation. In this article, by inserting fractional calculus as a viscoelastic material compatibility equations in nonlocal beam theory, a viscoelastic Euler-Bernoulli nano-beam with different boundary conditions at two ends, has been modeled. Material properties of a carbon nanotube is considered and two methods, pure numerical and numerical-analytical have been used for solving obtained equations in time domain. Main method is completely numerical and operator matrices used in it to discrete equations in time and spatial domain. Second method is introduced for validation of pervious method’s answers. In this method equation of system reduced to an ordinary differential equation using Galerkin and obtained equation solved using a numerical direct integrator method. Finally, in a case study, the effects of fractional order, viscoelasticity coefficient and nanlocal theory coefficient on the time response of the viscoelastic Euler-Bernoulli nano-beam with different boundary conditions have been studied.