Analytical method of predicating the instabilities of a micro arch-shaped beam under electrostatic loading |
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Authors: | Yin Zhang Yisong Wang Zhihong Li |
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Affiliation: | (1) State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Science, Beijing, 100190, China;(2) National Key Laboratory of Nano/Micro Fabrication Technology, Institute of Microelectronics, Peking University, Beijing, 100871, China; |
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Abstract: | An arch-shaped beam with different configurations under electrostatic loading experiences either the direct pull-in instability
or the snap-through first and then the pull-in instability. When the pull-in instability occurs, the system collides with
the electrode and adheres to it, which usually causes the system failure. When the snap-through instability occurs, the system
experiences a discontinuous displacement to flip over without colliding with the electrode. The snap-through instability is
an ideal actuation mechanism because of the following reasons: (1) after snap-through the system regains the stability and
capability of withstanding further loading; (2) the system flips back when the loading is reduced, i.e. the system can be
used repetitively; and (3) when approaching snap-through instability the system effective stiffness reduces toward zero, which
leads to a fast flipping-over response. To differentiate these two types of instability responses for an arch-shaped beam
is vital for the actuator design. For an arch-shaped beam under electrostatic loading, the nonlinear terms of the mid-plane
stretching and the electrostatic loading make the analytical solution extremely difficult if not impossible and the related
numerical solution is rather complex. Using the one mode expansion approximation and the truncation of the higher-order terms
of the Taylor series, we present an analytical solution here. However, the one mode approximation and the truncation error
of the Taylor series can cause serious error in the solution. Therefore, an error-compensating mechanism is also proposed.
The analytical results are compared with both the experimental data and the numerical multi-mode analysis. The analytical
method presented here offers a simple yet efficient solution approach by retaining good accuracy to analyze the instability
of an arch-shaped beam under electrostatic loading. |
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