Nonlinear forced vibration of size-dependent functionally graded microbeams with damping effects

The nonlinear dynamics of functionally graded (FG) microbeams have been studied based on the von Karman nonlinear theory, the modified couple stress and Euler-Bernoulli beam theories. The internal damping of materials is taken into account using the Kelvin-Voigt model. The coupled nonlinear mode equations of FG microbeams are obtained using Hamilton's principle and Galerkin's method. The primary resonance and internal resonance are investigated by means of the method of multiple scales and the Runge-Kutta numerical method. The effects of the length scale parameter, volume fraction exponent and internal damping constant on the nonlinear vibration are discussed using the numerical simulation. The numerical results show that periodic, period-n, and chaotic oscillations can be displayed by changing the length scale parameter, volume fraction exponent and internal damping constant. Boundary conditions can also change the nonlinear vibration behavior of FG microbeams. In the present study, the second natural frequency is approximately equal to three times the first one for the FG clamped-clamped microbeam, and a three-to-one internal resonance is detected using the time response curves. The present analysis is validated by comparing the numerical results with existing results and very good agreement is obtained. (C) 2019 Elsevier Inc. All rights reserved.