Internal and combination resonances in variable cross-section arched microbeams for bifurcation-based chaos mass sensing

Microelectromechanical devices are an essential component of recently developed portable sensing and monitoring solutions. Most of these devices incorporate microbeam-like components controlled through electrostatic actuation. The performance of these devices is dependent on a good understanding of their dynamics and the appropriate selection of both the geometry and the excitation forces. In general, constraints associated with the employed microfabrication technology further restrict this selection. This work aims at demonstrating how variations in the cross-section of an arched microbeam affect its dynamic behavior and performance. The study includes the investigation of the possible internal resonances that may appear in these microbeams. A general form cross-section governing equation of the microbeam is derived using Hamilton’s principle and accounting for the geometric and electrostatic nonlinearities. The differential quadrature method is then used to spatially discretize the equation of motion in order to calculate the static response, forced natural frequencies, and associated mode shapes. The Galerkin procedure is employed to discretize the equation of motion for a nonlinear dynamic analysis, by using the calculated mode shapes as test functions. The findings revealed that adequate variations in the thickness of the microbeam along its axis give rise to several potential internal and combination resonances. The frequency response curves of the oscillator around these resonances exhibited a rich dynamics that make of it a high-performance mass sensing device. The introduction of an infinitesimal mass into the microbeam results in a chaotic response attaining amplitudes up to five times larger than its initial state.