Nonlinear free vibration of damped and undamped bi-directional functionally graded beams using a cubic-quintic nonlinear model

摘要

In this paper, a novel higher order cubic-quintic nonlinear model is proposed for the nonlinear free vibration of damped and undamped bi-directional functionally graded (2D FG) beams. To the best of the researchers' knowledge, no study has focused on damping characteristics of 2D FG beams. Thus, the present paper extends the previous studies in this field. It is assumed that the material properties of the beam change in the axial and lateral directions simultaneously according to exponential and power-law functions, respectively. A new neutral surface is defined to remove the stretching and bending couplings effect. The variational iteration method (VIM) and the Hamiltonian approach (HA) are applied to obtain closed-form analytical solutions for the nonlinear vibration of damped and undamped 2D FG beams, respectively. Based on the results, an increase in the material grading indices can similarly decrease the damping coefficient for an underdamped 2D FG beam, and more time is required for dying out the oscillations. Further, compared to the homogeneous beams, a significant difference is observed between the cubic and cubic-quintic models indicating the importance of applying higher order nonlinear models for the nonlinear analysis of 2D FG beams.