Nonlinear free and forced vibrations of fractional modeled viscoelastic FGM micro-beam

Abstract

In this paper, nonlinear vibrations of fractional viscoelastic functionally graded micro-beam are studied based on the fractional calculus. The Modified Couple Stress Theory (MCST) is utilized for considering the effect of the micro-scale. The micro-beam is modeled based on the Euler-Bernoulli theory and the Von Karman’s nonlinear strain relations. The viscoelastic part of the micro-beam is considered using the fractional Kelvin-Voigt viscoelastic model. Functionally graded properties vary along the thickness based on a power-law function. The small-scale effects of the micro-beam are modeled using the MCST. Hamiltons principle is used to derive the governing equation of motion. For the solution, the Finite Difference Method (FDM) and the Finite Element Method (FEM) are employed. The FDM is used for discretizing the time domain, and the FEM is used for discretizing the space domain. Effects of the fractional-order, microstructure parameters, and Functionally Graded Material (FGM) properties on the time response of the viscoelastic micro-beam are analyzed numerically. Numerical results show that the fractional-order modeling of the viscoelastic micro-beam can change the behavior of the structure, and increasing the fractional-order, can increase the damping of the system. Numerical simulations also suggest that the effect of the fractional derivative order on the responses of free and forced vibrations is different because there is an important correlation between the fractional-order derivative and the excitation frequency. The results of this paper can be used for modeling the damping of the viscoelastic structures.