Nonlinear resonant behaviors of bi-directional functionally graded material microbeams: One-/two-parameter bifurcation analyses

In this paper, the nonlinear resonance behaviors of bi-directional functionally graded (BDFG) microbeams are investigated. The material properties including the material length scale parameter vary along both thickness and axial directions. Employing Hamilton's principle, the differential equations are derived based on vonKarman geometric nonlinearity and Timoshenko beam theory. The modified couple stress theory is adopted to capture the size effects. The continuous system dynamics model is discretized by method of Galerkin scheme along with appropriate eigen functions, resulting in the reduced order model which is a coupled large-dimension system of nonlinear ordinary differential equations. The nonlinear resonance behaviors of two type of BDFG microbeams are explored by performing one- and two-parameter bifurcation analyses. In one-parameter bifurcation analysis, the frequency- and force-response curves are constructed by tracing the period motion of microbeam using the pseudo-arclength continuation technique. Cyclic-fold bifurcation which indicates jump phenomenon is detected in the period motion. The trajectories of cyclic-fold bifurcation points are achieved by implementing the two-parameter bifurcation analysis. The cusp bifurcation of periodic motion implies the occurrence of CF bifurcation. Numerical simulations are performed to examine the influences of the system parameters, e.g. gradient indexes,dimensionless length scale parameter, damping coefficients and aspect ratio on the nonlinear resonance of BDFG microbeams.