Coupled vibrations of functionally graded Timoshenko microbeams

In this paper, the nonlinear size-dependent forced vibrations of a functionally graded extensible microbeam is investigated based on the modified couple stress and Timoshenko beam theories; the transverse and longitudinal displacements and inertia as well as rotation and rotary inertia are included in the system modelling and simulations. The Mori-Tanaka homogenisation technique is employed to estimate the material properties of the extensible microbeam along the thickness. According to the modified couple stress theory, the elastic potential energy of the Timoshenko microbeam is obtained. The kinetic energy of the system as well as the work due to an external force is also developed. In order to derive the equations of motions, Hamilton's principle is employed retaining the transverse and longitudinal displacements and inertia as well as the rotation and rotary inertia; the Galerkin scheme is used to convert the nonlinear partial differential equations of motions into nonlinear ordinary differential equations. A numerical solution process, based on the pseudo-arclength continuation technique in conjunction with direct time-integration, is used; the nonlinear resonant dynamic responses of the system are obtained and frequency-response and force-response curves are plotted. Moreover, the effect of the length-scale parameter on the nonlinear forced dynamics of the system is analysed by comparing the results obtained using the modified couple stress theory versus the classical continuum theory. Finally, the influence of other system parameters, involving the material gradient index, on the frequency-responses and force-responses is investigated.