Nonlinear analysis and characteristics of electrically-coupled microbeams under mechanical shock

Abstract

Several studies have shown that MEMS devices deploying electrically-actuated vibrating beams, such as resonant sensors and RF filters may fail to operate when undergoing mechanical shocks due to the pull-in instability. To this end, we investigate the possibility to overcome or exploit this issue by considering different microsystem designs based on the application of interest. This objective is carried out through developing a nonlinear reduced-order model to simulate the dynamic response of single and dual microbeams under varying electric actuation and shock loads. The actuation of the single-beam system is made via a fixed electrode (uncoupled actuation) while the dual-beam system, composed of two movable microbeams, is actuated by applying a voltage among them (coupled actuation). We use the Galerkin method to discretize the governing equations in space and the Runge–Kutta method to integrate the resulting nonlinear ordinary differential equations. We first perform the static analysis to determine the pull-in voltage. We formulate the coupled eigenvalue problem to compute the natural frequencies of the microsystems under investigation for different applied DC voltages. Then, we introduce the AC excitation and generate the frequency-response curves. Finally, we analyze the impact of the mechanical shock (represented by an impact pulse acceleration) on the microsystems’ dynamic behavior. The present results are in good agreement with those obtained from previously-published theoretical and experimental studies. We observe a significant reduction in the static pull-in voltage and switching time when considering the dual-beam system in comparison with the single-beam case. The frequency-response curves show expanded dynamic pull-in bandwidth when operating the dual-beam system near the primary resonance. We notice that the dual-beam systems are more robust in terms of resistance to mechanical shock. This shows the suitability of such design for the operation and reliability of MEMS devices in harsh environments characterized by high mechanical shock levels. On the other hand, single-beam systems seem to be more attractive for use as microswitches which are intended to trigger a signal once receiving a mechanical shock or abrupt change in acceleration to activate safety functionalities, such as airbag systems.

Appendix

The schematic representation of the microsystem under investigation is shown in Fig. 1. The parameters $l_i$, $b_i$, $h_i$, $I_i$, and $\rho$ denote the length, width, thickness, the second area moment of the cross section, and the mass density of the microbeam. The subscript i refers to microbeam i. E is the Young’s modulus. The microbeams are electrically-actuated and subjected to mechanical shock. The displacements of the microbeams are represented by $w_i$. We express the kinetic energy of the dual beam microsystem as

$$\begin{aligned} T=\frac{1}{2}\int _{0}^{l_1} \rho b_1 h_1 \dot{w}_1^2 dx_1 + \frac{1}{2}\int _{0}^{l_2} \rho b_2 h_2 \dot{w}_2^2 dx_2. \end{aligned}$$

(32)

The potential energy is given by

$$\begin{aligned} V=\int _{0}^{l_1} E I_1 (w_1^{''})^2 dx_1 + \int _{0}^{l_2} E I_2 (w_2^{''})^2 dx_2. \end{aligned}$$

(33)

Here, the prime and the dot denote the spatial and time derivatives, respectively. The variation of the works done by the electric excitation, the mechanical shock (represented by an impact acceleration pulse of a half-sine waveform), and the linear damping are expressed as

$$\begin{aligned} \delta W_{F_e}&= \int _{0}^{l_1} \frac{\epsilon b_1 (V_{DC}+v_{AC}(t))^2}{2(d-w_1+w_2)^2}\delta w_1 dx_1-\int _{0}^{l_2} U(x_2-\delta )\frac{\epsilon b_2 (V_{DC}+v_{AC}(t))^2}{2(d-w_1+w_2)^2}\delta w_2 dx_2 \\ \delta W_{F_{sh}}&= \int _{0}^{l_1}\rho b_1 h_1 a_0 {\mathrm {g}}(t) \delta w_1 dx_1 + \int _{0}^{l_2}\rho b_2 h_2 a_0 {\mathrm {g}}(t) \delta w_2 dx_2 \\ \delta W_D&= -\,c_1 \int _{0}^{l_1} \dot{w_1} \delta w_1 dx_1 -c_2 \int _{0}^{l_2} \dot{w_2} \delta w_2 dx_2. \end{aligned}$$

(34)

Substituting Eqs. (32)–(34) into the generalized Hamilton’s principle given by

$$\begin{aligned} \int _{t_1}^{t_2} (\delta T - \delta V) dt + \int _{t_1}^{t_2} (\delta W_{F_e}+\delta W_{F_{sh}}+\delta W_D) dt = 0. \end{aligned}$$

(35)

leads to the nonlinear coupled equations of motion of the dual beam microsystem given by Eqs. (3) and (4).