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Nonlinear coupled vibration of electrostatically actuated clamped–clamped microbeams under higher-order modes excitation

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Abstract

Nonlinear modal interactions have recently become the focus of intense research in micro-resonators for their use to improve oscillator performance and probe the frontiers of fundamental physics. Understanding and controlling nonlinear coupling between vibrational modes is critical for the development of advanced micromechanical devices. This article aims to theoretically investigate the influence of antisymmetry mode on nonlinear dynamic characteristics of electrically actuated microbeam via considering nonlinear modal interactions. Under higher-order modes excitation, two nonlinear coupled flexural modes to describe microbeam-based resonators are obtained by using Hamilton’s principle and Galerkin method. Then, the Method of Multiple Scales is applied to determine the response and stability of the system for small amplitude vibration. Through Hopf bifurcation analysis, the bifurcation sets for antisymmetry mode vibration are theoretically derived, and the mechanism of energy transfer between antisymmetry mode and symmetry mode is detailed studied. The pseudo-trajectory processing method is introduced to investigate the influence of external drive on amplitude and bifurcation behavior. Results show that nonlinear modal interactions can transit vibration energy from one mode to nearby mode. In what follows, an effective way is proposed to suppress midpoint displacement of the microbeam and to reduce the possibility of large deflection. The quantitative relationship between vibrational modes is also obtained. The displacement of one mode can be predicted by detecting another mode, which shows great potential of developing parameter design in MEMS. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

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Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant Nos. 11372210, 11772218 and 11702192) and Tianjin Research Program of Application Foundation and Advanced Technology (16JCQNJC04700).

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Correspondence to Qichang Zhang.

Appendix

Appendix

$$\begin{aligned} a_{2r}= & {} -\left[ \alpha _1 \int _0^1 {{\phi }_2^{\prime \prime } \phi _2 \hbox {d}x} \int _0^1 {2{\phi }_3^{\prime } {w}_\mathrm{dc}^{\prime } \hbox {d}x} \right. \nonumber \\&\left. +\,6\alpha _2 V_\mathrm{dc}^2 \int _0^1 {\frac{\phi _3 \phi _2^2 \hbox {d}x}{(1-w_\mathrm{dc})^{4}}} \right] \end{aligned}$$
(32)
$$\begin{aligned} a_{2s}= & {} -\left[ \alpha _1 \int _0^1 {{\phi }_2^{\prime 2} \hbox {d}x} \int _0^1 {{\phi }_2^{\prime \prime } \phi _2 \hbox {d}x}\right. \nonumber \\&\left. +\,4\alpha _2 V_\mathrm{dc}^2 \int _0^1 {\frac{\phi _2^4 \hbox {d}x}{(1-w_\mathrm{dc})^{5}}} \right] \end{aligned}$$
(33)
$$\begin{aligned} a_{2t}= & {} -\left[ \alpha _1 \int _0^1 {{\phi }_3^{\prime 2} \hbox {d}x} \int _0^1 {{\phi }_2^{\prime \prime } \phi _2 \hbox {d}x}\right. \nonumber \\&\left. +\,12\alpha _2 V_\mathrm{dc}^2 \int _0^1 {\frac{\phi _2^2 \phi _3^2 \hbox {d}x}{(1-w_\mathrm{dc})^{5}}} \right] \end{aligned}$$
(34)
$$\begin{aligned} a_{3r}= & {} -\left[ \alpha _1 \int _0^1 {{w}_\mathrm{dc}^{\prime \prime } \phi _3 \hbox {d}x} \int _0^1 {{\phi }_2^{\prime } {\phi }_2^{\prime } \hbox {d}x}\right. \nonumber \\&\left. +\,3\alpha _2 V_\mathrm{dc}^2 \int _0^1 {\frac{\phi _2 \phi _2 \phi _3 \hbox {d}x}{(1-w_\mathrm{dc})^{4}}} \right] \end{aligned}$$
(35)
$$\begin{aligned} a_{3s}= & {} -\left[ \alpha _1 \int _0^1 {{w}_\mathrm{dc}^{\prime \prime } \phi _3 \hbox {d}x} \int _0^1 {{\phi }_3^{\prime } {\phi }_3^{\prime } \hbox {d}x} \right. \nonumber \\&+\,\alpha _1 \int _0^1 {{\phi }_3^{\prime \prime } \phi _3 \hbox {d}x} \int _0^1 {2{\phi }_3^{\prime } {w}_\mathrm{dc}^{\prime } \hbox {d}x}\nonumber \\&\left. +\,3\alpha _2 V_\mathrm{dc}^2 \int _0^1 {\frac{\phi _3 \phi _3 \phi _3 \hbox {d}x}{(1-w_\mathrm{dc})^{4}}} \right] \end{aligned}$$
(36)
$$\begin{aligned} a_{3t}= & {} -\left[ \alpha _1 \int _0^1 {{\phi }_3^{\prime } {\phi }_3^{\prime } \hbox {d}x} \int _0^1 {{\phi }_3^{\prime \prime } \phi _3 \hbox {d}x} \right. \nonumber \\&\left. +\,4\alpha _2 V_\mathrm{dc}^2 \int _0^1 {\frac{\phi _3 \phi _3 \phi _3 \phi _3 \hbox {d}x}{(1-w_\mathrm{dc})^{5}}} \right] \end{aligned}$$
(37)
$$\begin{aligned} a_{3p}= & {} -\left[ \alpha _1 \int _0^1 {{\phi }_2^{\prime } {\phi }_2^{\prime } \hbox {d}x} \int _0^1 {{\phi }_3^{\prime \prime } \phi _3 \hbox {d}x} \right. \nonumber \\&\left. +\,12\alpha _2 V_\mathrm{dc}^2 \int _0^1 {\frac{\phi _2 \phi _2 \phi _3 \phi _3 \hbox {d}x}{(1-w_\mathrm{dc})^{5}}} \right] \end{aligned}$$
(38)

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Li, L., Zhang, Q., Wang, W. et al. Nonlinear coupled vibration of electrostatically actuated clamped–clamped microbeams under higher-order modes excitation. Nonlinear Dyn 90, 1593–1606 (2017). https://doi.org/10.1007/s11071-017-3751-3

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