Theoretical and experimental investigations of the crossover phenomenon in micromachined arch resonator: part I—linear problem

Abstract

We investigate, experimentally and theoretically, the linear mode coupling between the first symmetric and antisymmetric modes of an electrothermally tuned and electrostatically actuated micromachined arch resonator. The arch is excited using an antisymmetric partial electrode to activate both modes of vibrations. Theoretically, we explore the static and dynamic behavior using the Galerkin method. When tuning the electrothermal voltage, the first symmetric frequency increases while the first antisymmetric frequency decreases until they cross. The results show linear coupling and hybridization of both modes near crossing only in the presence of the perturbation from the electrostatic force using the partial electrode. We show the linear merging of both modes at crossing. Also, the eigenfrequency variation around crossing shows a ratio of 2:1 between the second symmetric mode and the first symmetric/antisymmetric modes, which can lead to simultaneous 1:1 and 2:1 internal resonances.

The pull-in instability and eigenfrequency variations of a graphene resonator under electrostatic loading

“…Once W o (j) is solved from equation ( 21) and substituted into equation (22), the eigenfrequencies are obtained. In equation (21), the electrostatic force keeps the nonlinear form of a 4 =(1 À W o ) 2 in order to achieve a higher accuracy in the equilibrium computation.…”

“…In equation (21), the electrostatic force keeps the nonlinear form of a 4 =(1 À W o ) 2 in order to achieve a higher accuracy in the equilibrium computation. While, the electrostatic force has to be linearized in equation (22) to obtain the eigenfrequency and this linearization process is given in equation (15). The accuracy of equation (15) depends on the expansion number M of the Taylor series.…”

“…The weak mode coupling in essence reduces the modal interactions, which lessens the system nonlinearities and makes the above two nonlinearities the dominant ones. It should be kept in mind that the mode coupling of a structure under an electrostatic loading can be significantly enhanced by its initial deflection [22,23].…”

“…Here, the continuum model of a circular membrane is applied to the graphene-based resonator under a large built-in stress. Since a suspended structure and an actuating electrode form a parallel capacitor, the electrostatic force provides a convenient, efficient and fast-response actuation mechanism in many NEMS devices [20][21][22][23]. The electrostatic force is physically nonlinear as it is inversely proportional to the gap distance between the structure and electrode.…”

Nonlinear size-dependent modeling and dynamics of nanocrystalline arc resonators

“…Hence, different internal resonances were deeply studied numerically and experimentally as tuning the ratio between different modes (Hajjaj et al 2018a(Hajjaj et al , 2019aAlfosail et al 2019). The multiple scales techniques were numerically and widely used to characterize the different instability regions during the nonlinear interaction (Alfosail et al 2019;Hajjaj et al 2019b). Internal resonance was used recently in various potential applications, like sensing (Hacker and Gottlieb 2012;Zhang et al 2018a;Xia et al 2020), synchronization (Pu et al 2018), and communication (Antonio et al 2012;Hajjaj et al 2018b).…”