Nonlinear dynamics of micromechanical resonator arrays for mass sensing

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.

Appendices

Appendix A: Reduced-order model

The non-dimensional equation of motion for beam s is

$$\begin{aligned}&\dfrac{\partial ^4 w_s }{\partial x^4 } + \dfrac{\partial ^2 w_s }{\partial t^2} + c \dfrac{\partial w_s}{\partial t}- \Big [N + \alpha _1 \int _0^1 \! \big ( \dfrac{\partial w}{\partial x} \big )^2 \, \mathrm {d}x \Big ] \dfrac{\partial ^2 w_s }{\partial x^2}\nonumber \\&\quad =\alpha _2 \frac{V_{s,s+1}^2}{\big (1+w_{s+1}-w_{s} \big )^2} - \alpha _2 \frac{V_{s-1,s}^2}{\big ( 1+w_{s}-w_{s-1}\big )^2}. \end{aligned}$$

(28)

By using the Galerkin method and seventh-order Taylor series, Eq. 28 is replaced by a set of equations in the following matrix form:

$$\begin{aligned}&\ddot{{{\varvec{a}}}}^s + {{\varvec{C}}}_0 {\dot{{{\varvec{a}}}}}^s + {{\varvec{K}}}_0 {{\varvec{a}}}^s - \big ( N+ \alpha _1 T_2^s({{\varvec{a}}}^s) \big ) {{\varvec{K}}}_T {{\varvec{a}}}^s \nonumber \\&\quad =\alpha _2 V_{s,s+1}^2 {{\varvec{Q}}}_0 \nonumber \\&\qquad +\,\alpha _2 V_{s,s+1}^2 \bigg [ {{\varvec{Q}}}_1 + {{\varvec{Q}}}_2^s({{\varvec{b}}}^s)+ {{\varvec{Q}}}_3^s({{\varvec{b}}}^s) +{{\varvec{Q}}}_4^s({{\varvec{b}}}^s) + {{\varvec{Q}}}_5^s({{\varvec{b}}}^s) \nonumber \\&\qquad +\,{{\varvec{Q}}}_6^s({{\varvec{b}}}^s) + {{\varvec{Q}}}_7^s({{\varvec{b}}}^s) \bigg ] {{\varvec{b}}}^s - \alpha _2 V_{s-1,s} V_{s-1,s}^2 {{\varvec{P}}}_0 \nonumber \\&\qquad -\, \alpha _2 V_{s-1,s}^2 \bigg [ {{\varvec{P}}}_1+ {{\varvec{P}}}_2^s({{\varvec{b}}}^{s-1}) + {{\varvec{P}}}_3^s({{\varvec{b}}}^{s-1})\nonumber \\&\qquad +\,{{\varvec{P}}}_4^s({{\varvec{b}}}^{s-1}) + {{\varvec{P}}}_5^s({{\varvec{b}}}^{s-1}) + {{\varvec{P}}}_6^s ({{\varvec{b}}}^{s-1})+ {{\varvec{P}}}_7^s({{\varvec{b}}}^{s-1}) \bigg ] {{\varvec{b}}}^{s-1}. \nonumber \\ \end{aligned}$$

(29)

The components of the matrices are given by

$$\begin{aligned}&{{{\varvec{C}}}_0}_{ij} = c_i \delta _{ij}, \quad {{{\varvec{K}}}_0}_{ij}= \lambda _i^4 \delta _{ij}, \quad \\&T_2^s({{\varvec{a}}}^s)= \sum _{k=1}^{N_m} \sum _{l=1}^{N_m} \left( \int _0^1 \, \phi _k' \phi _l' \! \mathrm {d}x\right) {{\varvec{a}}}_k^s {{\varvec{a}}}_l^s,\\&{{\varvec{K}}}_{Tij} = \int _0^1 \, \phi _j'' \phi _i \! \mathrm {d}x , \quad {{\varvec{Q}}}_{0i}^s = \int _0^1 \, \phi _i \! \mathrm {d}x ,\\&{{\varvec{Q}}}_{1ij}^s= -2\int _0^1 \, \phi _i \phi _j \! \mathrm {d}x , \quad \\&{{\varvec{Q}}}_{2ij}^s= 3\sum _{k=1}^{N_m} \left( \int _0^1 \, \phi _i \phi _j \phi _k \! \mathrm {d}x \right) {{\varvec{b}}}_k^{s},\\&{{\varvec{Q}}}_{3ij}^s=-4\sum _{k=1}^{N_m} \sum _{l=1}^{N_m} \left( \int _0^1 \, \phi _i \phi _j \phi _k \phi _l \! \mathrm {d}x \right) {{\varvec{b}}}_k^{s}{{\varvec{b}}}_l^s, \\&{{\varvec{Q}}}_{4ij}^s=5\sum _{k=1}^{N_m} \sum _{l=1}^{N_m}\sum _{m=1}^{N_m} \left( \int _0^1 \, \phi _i \phi _j \phi _k \phi _l \phi _m\! \mathrm {d}x \right) {{\varvec{b}}}_k^{s}{{\varvec{b}}}_l^s {{\varvec{b}}}_m^s, \\&{{\varvec{Q}}}_{5ij}^s=-6\sum _{k=1}^{N_m} \sum _{l=1}^{N_m}\sum _{m=1}^{N_m}\sum _{n=1}^{N_m} \\&\qquad \qquad \times \,\left( \int _0^1 \, \phi _i \phi _j \phi _k \phi _l \phi _m \phi _n \! \mathrm {d}x \right) {{\varvec{b}}}_k^{s}{{\varvec{b}}}_l^s {{\varvec{b}}}_m^s {{\varvec{b}}}_n^s, \\&{{\varvec{Q}}}_{6ij}^s=7\sum _{k=1}^{N_m} \sum _{l=1}^{N_m}\sum _{m=1}^{N_m}\sum _{n=1}^{N_m}\sum _{o=1}^{N_m} \left( \int _0^1 \, \phi _i \phi _j \phi _k \phi _l \phi _m \phi _n \phi _o \! \mathrm {d}x \right) \\&\qquad \qquad \times \,{{\varvec{b}}}_k^{s}{{\varvec{b}}}_l^s {{\varvec{b}}}_m^s {{\varvec{b}}}_n^s {{\varvec{b}}}_o^s, \\&{{\varvec{Q}}}_{7ij}^s=-8\sum _{k=1}^{N_m} \sum _{l=1}^{N_m}\sum _{m=1}^{N_m}\sum _{n=1}^{N_m}\sum _{o=1}^{N_m} \sum _{p=1}^{N_m}\\&\qquad \qquad \times \,\left( \int _0^1 \, \phi _i \phi _j \phi _k \phi _l \phi _m \phi _n \phi _o \phi _p \! \mathrm {d}x \right) {{\varvec{b}}}_k^{s}{{\varvec{b}}}_l^s {{\varvec{b}}}_m^s {{\varvec{b}}}_n^s {{\varvec{b}}}_o^s{{\varvec{b}}}_p^s, \\&{{\varvec{P}}}_{0i} = \int _0^1 \, \phi _i \! \mathrm {d}x , \, \,{{\varvec{P}}}_{1ij}= -2\int _0^1 \, \phi _i \phi _j \! \mathrm {d}x ,\, \\&{{\varvec{P}}}_{2ij}^s= 3\sum _{k=1}^{N_m} \left( \int _0^1 \, \phi _i \phi _j \phi _k \! \mathrm {d}x \right) {{\varvec{b}}}_k^{s-1}, \\&{{\varvec{P}}}_{3ij}^s=-4\sum _{k=1}^{N_m} \sum _{l=1}^{N_m} \left( \int _0^1 \, \phi _i \phi _j \phi _k \phi _l \! \mathrm {d}x \right) {{\varvec{b}}}_k^{s-1}{{\varvec{b}}}_l^{s-1},\\&{{\varvec{P}}}_{4ij}^s=5\sum _{k=1}^{N_m} \sum _{l=1}^{N_m}\sum _{m=1}^{N_m} \left( \int _0^1 \, \phi _i \phi _j \phi _k \phi _l \phi _m\! \mathrm {d}x \right) \\&\qquad \qquad \times \,{{\varvec{b}}}_k^{s-1}{{\varvec{b}}}_l^{s-1} {{\varvec{b}}}_m^{s-1}, \\&{{\varvec{P}}}_{5ij}^s=-6\sum _{k=1}^{N_m} \sum _{l=1}^{N_m}\sum _{m=1}^{N_m}\sum _{n=1}^{N_m} \left( \int _0^1 \, \phi _i \phi _j \phi _k \phi _l \phi _m \phi _n \! \mathrm {d}x \right) \\&\qquad \qquad \times \,{{\varvec{b}}}_k^{s-1}{{\varvec{b}}}_l^{s-1} {{\varvec{b}}}_m^{s-1} {{\varvec{b}}}_n^{s-1},\\&{{\varvec{P}}}_{6ij}^s=7\sum _{k=1}^{N_m} \sum _{l=1}^{N_m}\sum _{m=1}^{N_m}\sum _{n=1}^{N_m}\sum _{o=1}^{N_m} \left( \int _0^1 \, \phi _i \phi _j \phi _k \phi _l \phi _m \phi _n \phi _o \! \mathrm {d}x \right) \\&\qquad \qquad \times \,{{\varvec{b}}}_k^{s-1}{{\varvec{b}}}_l^{s-1} {{\varvec{b}}}_m^{s-1} {{\varvec{b}}}_n^{s-1} {{\varvec{b}}}_o^{s-1}, \\&{{\varvec{P}}}_{7ij}^s=-8\sum _{k=1}^{N_m} \sum _{l=1}^{N_m}\sum _{m=1}^{N_m}\sum _{n=1}^{N_m}\sum _{o=1}^{N_m} \sum _{p=1}^{N_m}\\&\qquad \qquad \times \,\left( \int _0^1 \, \phi _i \phi _j \phi _k \phi _l \phi _m \phi _n \phi _o \phi _p \! \mathrm {d}x \right) \\&\qquad \qquad \times \,{{\varvec{b}}}_k^{s-1}{{\varvec{b}}}_l^{s-1} {{\varvec{b}}}_m^{s-1} {{\varvec{b}}}_n^{s-1} {{\varvec{b}}}_o^{s-1}{{\varvec{b}}}_p^{s-1}, \\&\qquad \qquad \text {with } i,j=1,\ldots ,N_m. \end{aligned}$$

Appendix B: Averaging method for a two-beam array

The Galerkin method with the fundamental mode is used as follows

$$\begin{aligned} w_1(x,t)= & {} \phi _1(x) a_{11}(t) \nonumber \\ w_2(x,t)= & {} \phi _1(x) a_{21}(t), \end{aligned}$$

(30)

and the first-order Taylor series for the electrostatic forces is as follows

$$\begin{aligned} \dfrac{1}{(1+w_{s+1}-w_s)^2}= 1-2(w_{s+1}-w_s), \nonumber \\ \dfrac{1}{(1+w_s-w_{s-1})^2}= 1-2(w_s-w_{s-1}). \end{aligned}$$

(31)

Eq. (15) becomes

$$\begin{aligned}&\ddot{ a}_{11} {+} c {\dot{a}}_{11} {+} \omega _1 a_{11} {+} \beta _{11} a_{11}^3 {+} \big (\beta _{12} \cos \varOmega t {+} \beta _{13} \cos ^2\varOmega t \big ) a_{11} \nonumber \nonumber \\&\quad +\,\big ( \delta _{11} +\delta _{12}\cos \varOmega t + \delta _{13} \cos ^2 \varOmega t \big ) a_{21} \nonumber \\&\quad +\, \big (\gamma _{11} + \gamma _{12} \cos \varOmega t + \gamma _{13} \cos ^2\varOmega t \big ) =0 \end{aligned}$$

(32)

$$\begin{aligned}&\ddot{a}_{21} {+} c{\dot{a}}_{21} {+} \omega _2 a_{21} {+} \beta _{21} a_{21}^3 {+}\,\big (\beta _{22} \cos \varOmega t {+} \beta _{23} \cos ^2\varOmega t \big ) a_{21} \nonumber \\&\quad +\,\big ( \delta _{21} +\delta _{22}\cos \varOmega t + \delta _{23} \cos ^2 \varOmega t \big ) a_{11} \nonumber \\&\quad +\,\big (\gamma _{21} + \gamma _{22} \cos \varOmega t + \gamma _{23} \cos ^2\varOmega t \big ) =0 \end{aligned}$$

(33)

where

$$\begin{aligned} \beta _{s1}= & {} 151.35 \alpha _1; \, \\ \beta _{s2}= & {} -4\alpha _2\big (V_{\mathrm{dc}_{s,s+1}}V_{\mathrm{ac}_{s,s+1}} + V_{\mathrm{dc}_{s-1,s}}V_{\mathrm{ac}_{s-1,s}} \big ); \\ \beta _{s3}= & {} -2\alpha _2 \big (V_{\mathrm{ac}_{s-1,s}}^2 {+} V_{\mathrm{ac}_{s,s+1}}^2 \big );\quad \delta _{11}{=}\delta _{21} {=}2\alpha _2 V_{\mathrm{dc}_{12}}^2; \\ \delta _{12}= & {} \delta _{22}=4\alpha _2 V_{\mathrm{dc}_{12}}V_{\mathrm{ac}_{12}}; \\ \delta _{13}= & {} \delta _{23} = 2\alpha _2V_{\mathrm{ac}_{12}}^2; \,\\ \gamma _{s1}= & {} 0.83 \alpha _2 \big ( V_{\mathrm{dc}_{s,s+1}}^2-V_{\mathrm{dc}_{s-1,s}}^2 \big ); \\ \gamma _{s2}= & {} 1.66 \alpha _2 \big ( V_{\mathrm{dc}_{s,s+1}}V_{\mathrm{ac}_{s,s+1}} - V_{\mathrm{dc}_{s-1,s}}V_{\mathrm{ac}_{s-1,s}} \big ); \\ \gamma _{s3}= & {} 0.83 \alpha _2 \big (V_{\mathrm{ac}_{s,s+1}}^2 - V_{\mathrm{ac}_{s-1,s}}^2 \big ) \\ \end{aligned}$$

and $s=1,2$.

Let the relation between $\varOmega$ and $\omega _s$ be

$$\begin{aligned} \varOmega =\omega _s+\varepsilon \sigma _{s}, \end{aligned}$$

(34)

where $\omega _s$ is determined by

$$\begin{aligned} \omega _s= \dfrac{\lambda _s^4- 2\alpha _2 \big ( V_{\mathrm{dc}_{s,s+12}}^2 + V_{\mathrm{dc}_{s-1,s}}^2 \big )}{1+ \delta _{s_0}(s) m \phi _1(x_0)^2}. \end{aligned}$$

(35)

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.

Table 4 Asymmetric actuation voltages of the three-beam array

Fig. 19

Design # 4. Responses of the three-beam array without added mass. a Beam #1; b Beam #2; c Beam #3