Mass sensing based on nonlinear intermodal coupling via 2:1 internal resonance of electrostatically actuated clamped–clamped microbeams

Abstract

Over recent decades, the application of internal resonance-based nonlinear intermodal coupling in electromechanical resonators has shown promise for enhancing functionality and performance. This paper proposes a novel mass sensor by exploiting the frequency shift in the second mode response of an electrostatically actuated clamped–clamped microbeam. The sensor achieves this through intermodal coupling via a 2:1 internal resonance, with external excitation applied to the third mode. Through a systematic investigation using a reduced-order model and finite element analysis, we analyze the influence of adsorbed mass and its position on modal frequencies and establish a feasibility range for the resonance condition. Subsequently, we investigate the influence of adsorbed mass on the dynamics of coupled modes, determining the frequency shift in the second mode response for sensitivity estimation. The influence of critical parameters such as excitation amplitude, damping, frequency gap in the resonance relation, and mass positions on sensitivity is also investigated. Notably, sensitivity exhibits a non-monotonic behavior concerning the non-dimensional parameter ($\alpha _1$) and reaches maximum values at higher $\alpha _1$. The sensor demonstrates a twentyfold increase in sensitivity compared to the fundamental mode. Practical viability is confirmed by evaluating mass sensitivity for experimentally viable beams made of Polysilicon and GaAs. This study presents a comprehensive and systematic methodology for exploring a mass sensor based on modal interaction in electrostatically actuated microbeams.

Appendix A

1.1 A. 1 Coefficient of quadratic terms

$$\begin{aligned} \alpha _{n1}=\Gamma _{22n}, \quad \alpha _{n2}=\Gamma _{23n}+\Gamma _{32n}, \quad \alpha _{n3}=\Gamma _{33n} \end{aligned}$$

where

$$\begin{aligned} \Gamma _{ijn}=\left[ \alpha _1 \int _{0}^{1} w_{\mathrm{{dc}}}^{\prime \prime } \phi _n dx \int _{0}^{1} \phi _i \phi _j dx +\alpha _1 \int _{0}^{1}\phi _i^{\prime \prime } \right. \nonumber \\ \phi _n dx \left. \int _{0}^{1} 2 \phi _j^{\prime } w_{\mathrm{{dc}}}^{\prime } dx +3 \alpha _2 V_{\mathrm{{dc}}}^2 \int _{0}^{1}\frac{\phi _i \phi _j \phi _n}{(1-w_{\mathrm{{dc}}})^4}dx\right] \end{aligned}$$

for $i, j, n=2, 3$.

1.2 A. 2 Coefficient of cubic terms

$$\begin{aligned} \alpha _{n11}=\Gamma _{222n}, \quad \alpha _{n12}=\Gamma _{223n}+\Gamma _{232n}+\Gamma _{322n}, \\ \alpha _{n13}=\Gamma _{233n}+\Gamma _{323n}+\Gamma _{332n}, \quad \alpha _{n14}=\Gamma _{333n}, \end{aligned}$$

where

$$\begin{aligned} \Gamma _{ijkn}=&\left[ \alpha _1 \int _{0}^{1}\phi _i^{\prime \prime }\phi _n dx \int _{0}^{1}\phi _j^{\prime } \phi _k^{\prime }dx \right. \\ {}&\left. + 4 \alpha _2 V_{\mathrm{{dc}}}^2 \int _{0}^{1}\frac{\phi _i \phi _j \phi _k \phi _n}{(1-w_{\mathrm{{dc}}})^5} dx \right] \\ \end{aligned}$$

for i, j, k and n $=$ 2, 3.