Nonlinear random vibrations of micro-beams with fractional viscoelastic core

Abstract

This work investigates the nonlinear vibration of a fractional viscoelastic micro-beam exposed to random excitation. For the micro-scale effect, the micro-beam model is established based on the Modified Couple Stress Theory (MCST). The fractional nonlinear partial differential equation is obtained using Von-Karman’s nonlinear strains and the Kelvin–Voigt fractional viscoelastic model. The micro-beam is exposed to a random force which its amplitude is considered as the Gaussian white noise. The Finite Difference Method (FDM) and the Galerkin method are applied to discretize the stochastic governing equation of motion. To solve the stochastic partial differential equation, a numerical discretization for the white noise excitation is applied. Moreover, the Statistical Linearization (SL) method and the Spectral Representation (SR) method are also employed and a comparison is done between these methods. The simulations show good agreement between the numerical method and other methods. In addition, it is shown that the proposed method can be used for different boundary conditions of the micro-beam. The impacts of the fractional derivative order, white noise intensity, viscoelastic model, and length-scale parameter on the statistical properties of the micro-beam are shown. The numerical simulations indicate that the high values of fractional-order can significantly affect the variance and the Root Mean Square (RMS) of the transverse amplitude, However, this influence for small values of the fractional-order is negligible; the impact of the fractional-order derivative on the statistical characteristics of the micro-beam is very different from 0<α<0.6, and 0.6<α<1.