Size-dependent parametric dynamics of imperfect microbeams

The nonlinear parametric dynamics of a geometrically imperfect microbeam subject to a time-dependent axial load is investigated in this paper. Based on the Euler–Bernoulli beam theory and the modified couple stress theory, continuous models for kinetic and potential energies are developed and balanced via use of Hamilton's principle. A model reduction procedure is carried out by applying the Galerkin scheme coupled with an assumed-mode technique, yielding a high-dimensional second-order reduced-order model. A linear analysis is performed upon the linear part of the reduced-order model in order to obtain the linear size-dependent natural frequencies. A nonlinear analysis is performed on the reduced-order model using the pseudo-arclength continuation method and a direct time-integration technique, yielding generalised coordinates, and hence the system parametric response. It is shown that, the steady-state frequency-response curves possess a trivial solution, both stable and unstable, throughout the solution space, separated by period-doubling bifurcation points, from which non-trivial solution branches bifurcate.