Nonlinear size-dependent dynamics of an imperfect shear deformable microplate

The nonlinear size-dependent dynamics of a geometrically imperfect shear deformable microplate is investigated based on the modified couple stress theory. An initial imperfection in the out-of-plane direction is taken into account. The third-order shear deformation theory is employed to model the system by retaining in-plane displacements and inertia. The governing equations of motion of the system are obtained employing an energy method through use of the Lagrange equations, which upon employing an assumed-mode technique, yields a set of second-order nonlinear ordinary differential equations with coupled terms. A high-dimensional discretised system is considered and made double-dimensional via a change of variables in order to obtain a set of first-order nonlinear ordinary differential equations. The resulting equations are solved using a direct time-integration technique, resulting in time-dependent generalised coordinates for the in-plane and out-of-plane displacements and two rotations. From these generalised coordinates, phase-plane portraits and fast Fourier transforms (FFTs) are also obtained. Moreover, the frequency–response and force–response curves of the system are obtained using a continuation technique; stability analysis is conducted via the Floquet theory. The effect of the initial imperfection as well as the length-scale parameter on the system response is also examined.