Mechanical behavior of a rectangular capacitive micro-plate subjected to an electrostatic load

Abstract

This study aims at understanding the mechanical behavior and static response of an electrically actuated micro-plate, considering the effects of different boundary conditions. The equations of motion of rectangular micro-plate-based micro-electro-mechanical systems are derived in terms of partial differential equations, exploiting the classical plate theory and von Kármán geometric nonlinearity. Two different boundary conditions are considered, i.e., simply supported and clamped with different in-plane conditions. The Galerkin procedure is employed to obtain a second-order nonlinear ordinary differential equation in time with quadratic, cubic, quartic, and higher nonlinear terms. The attention is mainly focused on the method of elimination of singularity in the electrostatic force. Therefore, two methods are implemented to treat singularity. By using the method of multiple time scales, the transient behavior of the system is obtained. Moreover, a discussion is made on how different design parameters affect the static response of micro-plates. In order to validate the obtained results, a numerical method by using Matlab^®/Simulink^® is employed.

Appendix A

Constants of Eq. (27) for simply supported boundary conditions with zero tangential force and zero normal displacement are as follow

$$ \begin{aligned} B_{1} &= - (2880\beta^{4} \lambda V(t)^{2}\\ &\quad -\, 1440\pi^{4} - 1440\pi^{4} \beta^{4} - 2880\pi^{4} \beta^{2} )/1440\beta^{4} \hfill \\ B_{2} &= - 30720\lambda V(t)^{2} /1440\pi^{2} \hfill \\ B_{3} &= - \frac{{3240\beta^{4} \lambda V(t)^{2} + 1080\pi^{4} \eta^{2} \upsilon^{2} - 3240\pi^{4} \eta^{2} \beta^{4} - 4320\pi^{4} \eta^{2} \upsilon \beta^{2} + 1080\pi^{4} \eta^{2} \upsilon^{2} \beta^{4} - 3240\pi^{4} \eta^{2} }}{{1440\beta^{4} }} \hfill \\ B_{4} &= - 32768\lambda V(t)^{2} /1440\pi^{2} \hfill \\ B_{5} &= - 3375\lambda V(t)^{2} /1440 \hfill \\ B_{6} &= - 23040\lambda V(t)^{2} /5760\pi^{2} \hfill \\ \end{aligned} $$

Constants of Eq. (28) for simply supported boundary conditions with zero tangential force and zero normal displacement are as follows

$$ \begin{aligned} A_{1} &= - 0.1.44101238 \hfill \\ A_{2} & = 0.5625 \hfill \\ A_{3} &= \frac{0.00070361933}{{\beta^{4} }}(2.76880088 \cdot 10^{5} \beta^{2}\\ &\quad +\, 1.384400440 \cdot 10^{5} \beta^{4} + 1.38440044 \cdot 10^{5} ) \hfill \\ A_{4} &= \frac{0.00070361933}{{\beta^{4} }}( - 3.9898763711 \cdot 10^{5} \beta^{2} \\ &\quad -\, 1.9949381851 \cdot 10^{5} \beta^{4} - 1.9949381851 \cdot 10^{5} ) \hfill \\ A_{5} &= \frac{0.00070361933}{{\beta^{4} }}(1.5574504951 \cdot 10^{5} \beta^{2}\\ &\quad +\, 7.7872524474 \cdot 10^{5} \beta^{4} + 4.153201319 \cdot 10^{5} \beta^{2} \eta^{2} \upsilon \hfill \\ &\quad + 3.11490099 \cdot 10^{5} \eta^{2} + 7.787252474 \cdot 10^{5}\\ &\quad -\, 1.0383003301 \cdot 10^{5} \beta^{4} \eta^{2} \upsilon^{2} - 1.0383003301 \cdot 10^{5} \eta^{2} \upsilon^{2} \hfill \\ & \quad + 3.11490099 \cdot 10^{5} \beta^{4} \eta^{2} ) \hfill \\ A_{6} &= \frac{0.00070361933}{{\beta^{4} }}( - 5984814556 \cdot 10^{5} \beta^{2} \eta^{2} \upsilon\\ &\quad -\, 4.787851645 \cdot 10^{5} \eta^{2} + 1.795444367 \cdot 10^{5} \beta^{4} \eta^{2} \upsilon^{2} \hfill \\ &\quad + 1.795444367 \cdot 10^{5} \eta^{2} \upsilon^{2} - 4.787851645 \cdot 10^{5} \beta^{4} \eta^{2} ) \hfill \\ A_{7} &= \frac{0.00070361933}{{\beta^{4} }}(2.336175742 \cdot 10^{5} \beta^{2} \eta^{2} \upsilon\\ &\quad +\, 1.946813118 \cdot 10^{5} \eta^{2} - 7.787252474 \cdot 10^{5} \beta^{4} \eta^{2} \upsilon^{2} \hfill \\ &\quad - 7.787252474 \cdot 10^{5} \eta^{2} \upsilon^{2} + 1.946813118 \cdot 10^{5} \beta^{4} \eta^{2} ) \hfill \\ A_{8} &= - 1.621138938\lambda \hfill \\ \end{aligned} $$