Analytical solution for nonlinear forced vibrations of functionally graded micro resonators

Abstract

The objective of this study is to investigate the size-dependent nonlinear forced vibrations of homogeneous and functionally graded micro resonators with axially immovable ends. According to the power law distribution, for the functionally graded micro resonators, the material properties are changed in the thickness direction. Considering the damping and mid-plane stretching effects, the governing nonlinear partial differential equations of homogeneous and functionally graded micro resonators are achieved based on the nonlocal strain gradient theory. The nonlinear ordinary differential equations, including nonlinear external forces such as electrostatic actuation and Casimir force, are derived for homogeneous and functionally graded micro resonators by employing Galerkin approach. The static DC and harmonic AC voltages are combined to apply as the electrostatic actuator voltage. The frequency responses of the homogeneous and functionally graded micro resonators are studied by applying the multiple scales method. Finally, the effects of damping coefficient, amplitude of excitation, and nonlocal strain gradient parameters are investigated for the frequency responses of the homogeneous and functionally graded micro resonators.