Abstract
This paper investigates the mass sensing capability of an array of a few identical electrostatically actuated microbeams, as a first step toward the implementation of arrays of thousands of such resonant sensors. A reduced-order model is considered, and Taylor series are used to simplify the nonlinear electrostatic force. Then, the harmonic balance method associated with the asymptotic numerical method, as well as time integration or averaging methods, is applied to this model, and its results are compared. In this paper, two- and three-beam arrays are studied. The predicted responses exhibit complex branches of solutions with additional loops due to the influence of adjacent beams. Moreover, depending on the applied voltages, the solutions with and without added mass exhibit large differences in amplitude which can be used for detection. For symmetric configurations, the symmetry breaking induced by an added mass is exploited to improve mass sensing.
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The authors are indebted to the Institute Carnot Ingénierie@Lyon for its support and funding of the NEMROD project.
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Appendices
Appendix A: Reduced-order model
The non-dimensional equation of motion for beam s is
By using the Galerkin method and seventh-order Taylor series, Eq. 28 is replaced by a set of equations in the following matrix form:
The components of the matrices are given by
Appendix B: Averaging method for a two-beam array
The Galerkin method with the fundamental mode is used as follows
and the first-order Taylor series for the electrostatic forces is as follows
Eq. (15) becomes
where
and \(s=1,2\).
Let the relation between \(\varOmega \) and \(\omega _s\) be
where \(\omega _s\) is determined by
Appendix C: Three-beam array with asymmetric voltages
The voltages used for the asymmetric three-beam array are given in Table 4, and the responses are plotted in Fig. 19.
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Baguet, S., Nguyen, VN., Grenat, C. et al. Nonlinear dynamics of micromechanical resonator arrays for mass sensing. Nonlinear Dyn 95, 1203–1220 (2019). https://doi.org/10.1007/s11071-018-4624-0
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DOI: https://doi.org/10.1007/s11071-018-4624-0