Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory

The nonlinear forced vibrations of a microbeam are investigated in this paper, employing the strain gradient elasticity theory. The geometrically nonlinear equation of motion of the microbeam, taking into account the size effect, is obtained employing a variational approach. Specifically, Hamilton's principle is used to derive the nonlinear partial differential equation governing the motion of the system which is then discretized into a set of second-order nonlinear ordinary differential equations (ODEs) by means of the Galerkin technique. A change of variables is then introduced to this set of second-order ODEs, and a new set of ODEs is obtained consisting of first-order nonlinear ordinary differential equations. This new set is solved numerically employing the pseudo-arclength continuation technique which results in the frequency-response curves of the system. The advantage of this method lies in its capability of continuing both stable and unstable solution branches.