Dynamics of functionally graded micro-cantilevers

The nonlinear size-dependent dynamics of a functionally graded micro-cantilever is investigated when subject to a base excitation resulting in large-amplitude oscillations. A geometric nonlinearities due to large changes in the curvature is taken into account. Employing the Mori-Tanaka homogenisation technique (for the material properties), the modified couple stress theory (MCST) is used to formulate the potential and kinetic energies of the system in terms of the transverse and axial motions. A dynamic energy balance is performed between the energy terms, yielding the continuous models for the axial and transverse displacements. The inextensibility condition results in the size-dependent model of the functionally graded micro-cantilever involving inertial and stiffness nonlinear terms. The resultant model is discretised based on a weighted-residual technique yielding a high-dimensional truncated model (required for accurate simulations). A parameter-continuation scheme together with a time integration method is introduced to the truncated model so as to determine the resonances with stable and unstable solution branches with special consideration to the effect of different system parameters, such as material gradient index and the length-scale effect on the nonlinear dynamics of the functionally graded micro-cantilever.