Coupled Nonlinear Dynamics of Geometrically Imperfect Shear Deformable Extensible Microbeams

Abstract

This paper aims at analyzing the coupled nonlinear dynamical behavior of geometrically imperfect shear deformable extensible microbeams based on the thirdorder shear deformation and modified couple stress theories. Using Hamilton's principle and taking into account extensibility, the three nonlinear coupled continuous expressions are obtained for an initially slightly curved (i.e., a geometrically imperfect) microbeam, describing the longitudinal, transverse, and rotational motions. A highdimensional Galerkin scheme is employed, together with an assumedmode technique, in order to truncate the continuous system with an infinite number of degrees of freedom into a discretized model with sufficient degrees of freedom. This highdimensional discretized model is solved by means of the pseudoarclength continuation technique for the system at the primary resonance, and also by direct timeintegration to characterize the dynamic response at a fixed forcing amplitude and frequency; stability analysis is conducted via the Floquet theory. Apart from analyzing the nonlinear resonant response, the linear natural frequencies are obtained via an eigenvalue analysis. Results are shown through frequency–response curves, force–response curves, time traces, phaseplane portraits, and fast Fourier transforms (FFTs). The effect of taking into account the lengthscale parameter on the coupled nonlinear dynamic response of the system is also highlighted.