On the nonlinear resonant dynamics of Timoshenko microbeams: effects of axial load and geometric imperfection

The nonlinear size-dependent dynamics of a geometrically imperfect Timoshenko microbeam subject to an axial load is investigated numerically based on the modified couple stress theory. Increasing the compressive axial load leads to an increase in the deflection (bending) amplitude of the microbeam. A distributed harmonic excitation load in the transverse direction is then applied to the microbeam and the nonlinear dynamics over the new deflection is analyzed. More specifically, the kinetic energy of the system, the size-dependent strain potential energy, and the works due to the external excitations and damping are constructed as functions of the displacement field. These expressions are then inserted in Hamilton's principle, a variational energy technique, in order to obtain the nonlinear partial differential equations of motion. The Galerkin scheme is then applied to the resultant equations, yielding a set of second-order nonlinear ordinary differential equations. These equations are solved numerically via a direct time integration method as well as a continuation technique. Results are plotted in the form of deflection-axial load diagrams, frequency-responses, force-responses, time traces, phase-plane portraits, and fast Fourier transforms. The importance of taking into account small-size effects, through use of the modified couple stress theory, is highlighted.