Characterisation of the internal resonances of a clamped-clamped beam MEMS resonator

Abstract

Nonlinear coupling between modes through internal resonance has many applications in MEMS resonators. The primary objective of this work is to systematically study all possible internal resonance conditions between the first three modes and the feasibility of mode interaction in an electrostatically excited, straight, clamped-clamped beam. The static displacement and the first three natural frequencies of the beam are obtained for an applied DC voltage by using Galerkin based reduced order model and finite element method. All six possible commensurable relations between the first three frequencies of the beam are obtained by sweeping the non-dimensional parameters ($$\alpha _1$$ and $$\alpha _2 V_\mathrm{dc}^2$$) which depend on beam dimensions, material properties and external forcing. It is also demonstrated by a further examination of the dynamical equations that only one resonance condition is capable of exhibiting modal coupling when externally excited. A detailed analysis is carried out for this feasible resonance condition by altering $$\alpha _1$$ and $$\alpha _2 V_\mathrm{dc}^2$$ and solving the relevant nonlinear coupled equations by using both numerical time integration and the method of multiple scales. From this study, we observe that the lower mode is automatically excited after driving amplitude for the higher mode reaches a critical value—a sign of mode interaction initiation. We also find that amplitude of the higher mode is saturated after interaction with the lower mode for a combination of $$\alpha _1$$ and $$\alpha _2 V_\mathrm{dc}^2$$. Moreover, we see that the frequency bandwidth of modal interaction increases with excitation amplitude for all combinations of $$\alpha _1$$ and $$\alpha _2 V_\mathrm{dc}^2$$. Finally, the role of external damping on the amplitude-frequency response curve of both modes during the mode interaction is also investigated.