Dynamic stability in parametric resonance of axially excited Timoshenko microbeams

The dynamic stability in parametric resonance of a Timoshenko microbeam subject to a time-dependent axial excitation load (comprised of a mean value along with time-dependent variations) is analysed in the subcritical regime. Based on the modified couple stress theory, continuous expressions for the elastic potential and kinetic energies are developed using kinematic and kinetic relations. The continuous model of the system is obtained via use of Hamilton's principal. A model reduction procedure is carried out by applying the Galerkin scheme, in conjunction with an assumed-mode technique, yielding a high-dimensional second-order reduced-order model. A liner analysis is carried out upon the linear part of this model in order to obtain the linear natural frequencies and critical buckling loads. For the system in the subcritical regime, the parametric nonlinear responses are analysed by exciting the system at the principal parametric resonance in the first mode of transverse motion; this analysis is performed via use of a continuation technique, the Floquet theory, and a direct time integration method. Results are shown in the form of parametric frequency-responses, parametric force-responses, time traces, phase-plane diagrams, and fast Fourier transforms. The validity of the numerical simulations is tested via comparing our results, for simpler models for buckling response, with those given in the literature.