Parametric instability of microbeams in supercritical regime
The supercritical parametric instability of a microbeam subject to a time-dependent axial load is examined. The axial load is comprised of a constant mean value along with harmonic fluctuations. The mean value is increased from zero and is set to a value in the supercritical regime; the nonlinear parametric instability over the buckled state, due to the axial load variations, is examined. From the modelling perspective, based on the modified couple stress theory, the potential energy of the system is obtained in terms of the system parameters and displacement field. Moreover, the kinetic energy is formulated as a function of system parameters and the displacement field. The continuous model is developed by means of Hamilton's principle and then truncated into a reduced-order model via a weighted-residual technique. Three different numerical techniques, i.e. the pseudo-arclength continuation method, a direct time-integration scheme, and an eigenvalue extraction, are employed to solve the high-dimensional reduced-order model. A stability analysis is also conducted via the Floquet theory. The nonlinear size-dependent parametric response of the system over the buckled configuration is presented in the form of frequency-response diagrams, force-response curves, time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.
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