Investigating internal resonances and 3:1 modal interaction in an electrostatically actuated clamped-hinged microbeam

In this paper, we investigated the realization of all possible internal resonance conditions between the first three modes of an electrostatically actuated, straight, clamped-hinged microbeam. We first obtained the static displacement and the first three natural frequencies around the deflected shape of the beam by using Galerkin based reduced order model and the finite element method. We then found all six possible commensurable relations between these frequencies as a function of two non-dimensional parameters (\(\alpha _1\) and \(\alpha _2 V_{\mathrm{dc}}^2\)) that depend on beam dimensions, material properties, and external forcing. Furthermore, we also examined the governing equations to check the feasibility of dynamic modal interaction. We found that although dynamical modal coupling was theoretically possible for all resonance conditions upon external excitation, the 3:1 internal resonance between the first two modes seemed to be the most experimentally feasible. Hence, we carried out a detailed forced vibrations analysis corresponding to this condition by solving the governing nonlinear equations using numerical time integration and the method of multiple scales. The system exhibited dynamically rich internal resonance behavior that can be controlled with the two non-dimensional parameters (\(\alpha _1\) and \(\alpha _2 V_{\mathrm{dc}}^2\)) and external damping. Specifically, we observed saddle node bifurcations, jump conditions, and internal loops in the amplitude response curves as a function of excitation frequency and amplitude. Overall, our work provides a systematic methodology and simple rules for in-depth exploration of internal resonance in microbeams. The findings of this paper could also assist in the development of sensors based on internal resonance.