On the response of MEMS resonators under generic electrostatic loadings: experiments and applications

Abstract

We present an investigation of the dynamic behavior of an electrostatically actuated clamped–clamped microbeam, under the simultaneous excitation of primary and subharmonic resonance. The simultaneous excitation of primary and subharmonic resonances of similar strength is experimentally investigated by using different combinations of AC and DC voltages. It is observed that the response of the resonator is governed by a mixed effect of both excitations. Subharmonic-dominated response shows sharp amplitude transitions and smaller monostable regime, while primary-dominated response shows gradual amplitude transition and larger monostable regime. Two potential applications are experimentally demonstrated. The first is a resonator-based MEMS AND logic gate based on AC only subharmonic excitation. The second is a charge sensor based on the transition from subharmonic-dominated response to primary-dominated response, which is potentially capable of detecting a small amount of electric charges.

Appendix

$$\begin{aligned}&\alpha _q =-3\beta V_{\mathrm{eff}} \left( {\mathop \int \limits _0^1 \frac{\phi _j ^{3}}{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \nonumber \\&\quad -\,\alpha _1 \left( {\mathop \int \limits _0^1 \phi _j w_s ^{\prime \prime }\left( {\mathop \int \limits _0^1 \left( {\phi _j ^{\prime }} \right) ^{2}\hbox {d}x} \right) \hbox {d}x} \right) \end{aligned}$$

(A.1)

$$\begin{aligned}&\quad -\,2\alpha _1 \left( {\mathop \int \limits _0^1 \phi _j \phi _j ^{\prime \prime }\left( {\mathop \int \limits _0^1 \phi _j ^{\prime }w_s ^{\prime }\hbox {d}x} \right) \hbox {d}x} \right) \end{aligned}$$

(A.2)

$$\begin{aligned}&\alpha _c =-4\beta V_{\mathrm{eff}} \left( {\mathop \int \limits _0^1 \frac{\phi _j ^{4}}{\left( {1-w_s } \right) ^{5}}\hbox {d}x} \right) \nonumber \\&\quad -\,\alpha _1 \left( {\mathop \int \limits _0^1 \phi _j \phi _j ^{\prime \prime }\left( {\mathop \int \limits _0^1 \left( {\phi _j ^{\prime }} \right) ^{2}\hbox {d}x} \right) \hbox {d}x} \right) \end{aligned}$$

(A.3)

$$\begin{aligned}&F_p =2\beta V_\mathrm{AC} V_\mathrm{DC} \left( {\mathop \int \limits _0^1 \frac{\phi _j }{\left( {1-w_s } \right) ^{2}}\hbox {d}x} \right) ;\nonumber \\&\quad F_s =\beta \frac{V_\mathrm{AC}^2 }{2}\left( {\mathop \int \limits _0^1 \frac{\phi _j }{\left( {1-w_s } \right) ^{2}}\hbox {d}x} \right) \end{aligned}$$

(A.4)

$$\begin{aligned}&F_{\mathrm{ppar}1} =4\beta V_\mathrm{AC} V_\mathrm{DC} \left( {\mathop \int \limits _0^1 \frac{\phi _j ^{2}}{\left( {1-w_s } \right) ^{3}}\hbox {d}x} \right) ;\nonumber \\&\quad F_{\mathrm{spar}1} =\beta V_\mathrm{AC}^2 \left( {\mathop \int \limits _0^1 \frac{\phi _j ^{2}}{\left( {1-w_s } \right) ^{3}}\hbox {d}x} \right) \end{aligned}$$

(A.5)

$$\begin{aligned}&F_{\mathrm{ppar}2} =6\beta V_\mathrm{AC} V_\mathrm{DC} \left( {\mathop \int \limits _0^1 \frac{\phi _j ^{3}}{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) ;\nonumber \\&\quad F_{\mathrm{spar}2} =\beta \frac{3 V_\mathrm{AC}^2 }{2}\left( {\mathop \int \limits _0^1 \frac{\phi _j ^{3}}{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$

(A.6)

$$\begin{aligned}&F_{\mathrm{ppar}3} =8\beta V_\mathrm{AC} V_\mathrm{DC} \left( {\mathop \int \limits _0^1 \frac{\phi _j ^{4}}{\left( {1-w_s } \right) ^{5}}\hbox {d}x} \right) ;\nonumber \\&\quad F_{\mathrm{spar}3} =2\beta V_\mathrm{AC}^2 \left( {\mathop \int \limits _0^1 \frac{\phi _j ^{4}}{\left( {1-w_s } \right) ^{5}}\hbox {d}x} \right) \end{aligned}$$

(A.7)