Nonlinear vibration of fractional viscoelastic micro-beams

Nonlinear vibration of a fractional viscoelastic micro-beam is investigated in this paper. The Euler–Bernoulli beam theory and the nonlinear von Kármán strain are used to model the beam. The small-scale effects are considered by employing the Modified Couple Stress Theory (MCST). The viscoelastic material of the beam is modeled via the fractional Kelvin–Voigt model. Utilizing the Hamilton's Principle, a partial fractional differential equation is derived as the governing equation of motion. The Finite Difference Method (FDM) and the Galerkin method are used together for solving the partial fractional differential equation. The FDM is utilized to discretize the time domain, and the Galerkin method is employed to discretize the space domain. In this paper, the FDM and the Shooting method are coupled together to find the periodic solution of the fractional micro-beam and draw the corresponding amplitude–frequency curve. The effects of the order of the fractional derivative, viscoelastic model, and the micro-scale are studied numerically here in this study. Numerical simulations suggest that, the effect of the fractional derivative is very strong and must be considered for modeling the viscoelastic behavior; especially when the amplitude is high. Results also show that the effects of the nonlinear viscoelastic part are considerable when the amplitude is high; this may happen when the excitation frequency is near the natural frequency, at which the maximum amplitude occurs.