Dynamic behavior of T-beam resonator with repulsive actuation

Abstract

Electrostatic MEMS transducer driven by repulsive force is an attractive possibility and has advantages of avoiding the pull-in instability, tuning the natural frequency, and achieving high sensitivity by applying high enough voltages. In this work, a “T”-shaped beam, which is formed by attaching a secondary beam perpendicular to a primary cantilever at the tip, is introduced and its nonlinear dynamics is analyzed. A reduced-order model is derived from mode shapes formed from electromechanical coupling effects respectively. Generalized forms of forced Mathieu equation of motion are derived, and then, dynamic behaviors are investigated through the theory of multiple scales. The resonant responses, including both primary and principal parametric resonances, reveal softening behavior originating from quadratic and cubic nonlinearities in the governing equation. The behavior of the T-beam is compared with traditional cantilever structure. The resonance under repulsive force demonstrates that the T-beam has several advantages over a traditional cantilever: Lower natural frequency but higher resonant responses can improve the signal-to-noise ratio; with an attached micropaddle, the T-beam has a larger surface for absorption of targeted analytes for mass sensing. We conclude that an electrostatic MEMS resonator with a “T”-shaped beam is potentially appropriate for the new generations of sensors and actuators.

Appendix

1.1 A. Derivation of voltage-related modes

Using Galerkin’s method to discretize, we approximate the beam deflection as:

$$\begin{aligned} W(x,t){\approx }W_s(x)+\sum _{j=1}^n\phi (x)q(t),\ \ddot{q}(t)=-\omega ^2q(t).\nonumber \\ \end{aligned}$$

(50)

In our case, the mode shape for a T-beam is of the form:

$$\begin{aligned} \phi (x)= & {} c_1\cos ({\beta }x)+c_2\sin ({\beta }x)+c_3\cosh ({\beta }x)\nonumber \\&+c_4\sinh ({\beta }x) \end{aligned}$$

(51)

in which we have $\omega =\beta ^2$.

Plug into boundary conditions and combine them with $W_s(x): @x=0$:

$$\begin{aligned} \phi (x)q(t)= & {} 0 \Rightarrow c_1+c_3=0 \end{aligned}$$

(52)

$$\begin{aligned} \phi '(x)q(t)= & {} 0 \Rightarrow c_2+c_4=0 \end{aligned}$$

(53)

@x=1:

$$\begin{aligned}&\phi ''(x)q(t)+ML_C\phi (x)\ddot{q}(t)+\frac{4}{3}ML^2_C\phi '(x)\ddot{q}(t)\nonumber \\&\quad +V^2\kappa _1\sum _{i=0}^9r_ih^i\left( [W_s(x)+\phi (x)q(t)]^i-[W_s(x)]^i\right) \nonumber \\&\quad +V^2\kappa _2\bigg ([W'_s(x)+\phi '(x)q(t)]\sum _{i=0}^{8}r_{i+1}h^{i+1}[W_s(x)\nonumber \\&\quad +\phi (x)q(t)]^i\nonumber \\&\quad -W'_s(x)\sum _{i=0}^{8}r_{i+1}h^{i+1}[W_s(x)]^i\bigg )=0 \end{aligned}$$

(54)

$$\begin{aligned}&\phi '''(x)q(t)-M\phi (x)\ddot{q}(t)-ML_C\phi '(x)\ddot{q}(t)\nonumber \\&\quad -V^2\kappa _3\sum _{i=0}^9r_ih^i\left( [W_s(x)+\phi (x)q(t)]^i-[W_s(x)]^i\right) \nonumber \\&\quad -V^2\kappa _4\bigg ([W'_s(x)+\phi '(x)q(t)]\sum _{i=0}^{8}r_{i+1}h^{i+1}[W_s(x)\nonumber \\&\quad +\phi (x)q(t)]^i\nonumber \\&\quad -W'_s(x)\sum _{i=0}^{8}r_{i+1}h^{i+1}[W_s(x)]^i\bigg )=0. \end{aligned}$$

(55)

Simplify the above to be:

@x=1:

$$\begin{aligned}&\phi ''(x)q(t)-\omega ^2ML_C\phi (x){q}(t)-\frac{4}{3}\omega ^2ML^2_C\phi '(x){q}(t)\nonumber \\&\qquad +V^2\kappa _1\sum _{i=0}^9r_ih^i\left( [W_s(x)+\phi (x)q(t)]^i-[W_s(x)]^i\right) \nonumber \\&\qquad +V^2\kappa _2W'_s(x)\sum _{i=0}^{8}r_{i+1}h^{i+1}\times \end{aligned}$$

(56)

$$\begin{aligned}&\bigg ([W_s(x)+\phi (x)q(t)]^i-[W_s(x)]^i\bigg )\nonumber \\&\qquad +V^2\kappa _2\phi '(x)q(t)\sum _{i=0}^{8}r_{i+1}h^{i+1}\times \nonumber \\&\bigg [W_s(x)+\phi (x)q(t)\bigg ]^i\end{aligned}$$

(57)

$$\begin{aligned}&\quad =0 \end{aligned}$$

(58)

$$\begin{aligned}&\phi '''(x)q(t)+\omega ^2M\phi (x){q}(t)+\omega ^2ML_C\phi '(x){q}(t)\nonumber \\&\qquad -V^2\kappa _3\sum _{i=0}^9r_ih^i\left( [W_s(x)+\phi (x)q(t)]^i-[W_s(x)]^i\right) \nonumber \\&\qquad -V^2\kappa _4W'_s(x)\sum _{i=0}^{8}r_{i+1}h^{i+1} \times \end{aligned}$$

(59)

$$\begin{aligned}&\bigg ([W_s(x)+\phi (x)q(t)]^i-[W_s(x)]^i\bigg )\nonumber \\&\qquad -V^2\kappa _4\phi '(x)q(t)\sum _{i=0}^{8}r_{i+1}h^{i+1}\nonumber \\&\qquad \times \bigg [W_s(x)+\phi (x)q(t)\bigg ]^i \end{aligned}$$

(60)

$$\begin{aligned}&\quad =0. \end{aligned}$$

(61)

Drop nonlinear terms of $\phi (x)q(t)$ and q(t) is canceled:

@x=1:

$$\begin{aligned}&\phi ''(x)-\omega ^2ML_C\phi (x)-\frac{4}{3}\omega ^2ML^2_C\phi '(x)\nonumber \\&\quad +V^2\kappa _1\phi (x)\sum _{i=1}^9r_ih^i\genfrac(){0.0pt}1{i}{1}[W_s(x)]^{i-1}\nonumber \\&\quad +V^2\kappa _2W'_s(x)\phi (x)\sum _{i=1}^{8}r_{i+1}h^{i+1}\genfrac(){0.0pt}1{i}{1}[W_s(x)]^{i-1}\nonumber \\&\quad +V^2\kappa _2\phi '(x)r_{1}h=0 \end{aligned}$$

(62)

$$\begin{aligned}&\phi '''(x)+\omega ^2M\phi (x)+\omega ^2ML_C\phi '(x)\nonumber \\&\quad -V^2\kappa _3\phi (x)\sum _{i=1}^9r_ih^i\genfrac(){0.0pt}1{i}{1}[W_s(x)]^{i-1}\nonumber \\&\quad -V^2\kappa _4W'_s(x)\phi (x)\sum _{i=1}^{8}r_{i+1}h^{i+1}\genfrac(){0.0pt}1{i}{1}[W_s(x)]^{i-1}\nonumber \\&\quad -V^2\kappa _4\phi '(x)r_{1}h=0. \end{aligned}$$

(63)

Table 7 Voltage-related mode shapes

The algebraic equations above can be solved, and therefore, voltage-related natural frequencies are obtained as given in Table 7. They are quite close to each other, most probably from dropping too many nonlinear terms, and both static and dynamic results based on them hardly have any difference.

1.2 B. Derivation of Mathieu’s equation [Eq. (23)] based on one-mode approximation

One mechanical mode approximation is applied after checking the voltage-related modes. From static deflection $q_s$, we have:

$$\begin{aligned} \begin{aligned}&\int _{0}^{1}\phi _{1}(x)\phi ''''_{1}(x)dx{q}_{s}\\&\qquad +{\alpha }({V}_{DC}^2)\sum _{i=0}^{9}p_{i}{h}^{i}\int _{0}^{1}(\phi _{1}(x))^{i+1}(q_s)^i{dx}\\&\quad =K_{11}{q}_{s}+{\alpha }({V}_{DC}^2)\sum _{i=0}^{9}p_{i}{h}^{i}\int _{0}^{1}(\phi _{1}(x))^{i+1}(q_s)^i{dx}\\&\quad =0. \end{aligned} \end{aligned}$$

(64)

Plug $V^2={V}_{DC}^2+2V_{DC}V_{AC}cos{\omega {t}}$ and $q_{1}(t)=q_{total}(t)=q_s+q(t)$ into Eq. (21):

$$\begin{aligned} \begin{aligned}&\int _{0}^{1}(\phi _{1}(x))^{2}dx\ddot{q}(t)+c\int _{0}^{1}(\phi _{1}(x))^{2}dx{\dot{q}}(t)\\&\qquad +\int _{0}^{1}\phi _{1}(x)\phi ''''_{1}(x)dx(q_s+q(t))\\&\qquad +{\alpha }({V}_{DC}^2)\sum _{i=0}^{9}p_{i}{h}^{i}\int _{0}^{1}(\phi _{1}(x))^{i+1}(q_s+q(t))^i{dx}\\&\qquad +{\alpha }(2V_{DC}V_{AC}cos{\omega {t}})\\&\qquad \times \sum _{i=0}^{9}p_{i}{h}^{i}\int _{0}^{1}(\phi _{1}(x))^{i+1}(q_s+q(t))^i{dx}\\&\quad =M_{11}\ddot{q}(t)+2\zeta {\omega _0}M_{11}{\dot{q}}(t)+K_{11}(q_s+q(t))\\&\qquad +{\alpha }({V}_{DC}^2)\times \sum _{i=0}^{9}p_{i}{h}^{i}\int _{0}^{1}(\phi _{1}(x))^{i+1}{dx}(q_s+q(t))^i\\&\qquad +{\alpha }(2V_{DC}V_{AC}cos{\omega {t}})\\&\qquad \times \sum _{i=0}^{9}p_{i}{h}^{i}\int _{0}^{1}(\phi _{1}(x))^{i+1}{dx}(q_s+q(t))^i=0. \end{aligned} \end{aligned}$$

(65)

Subtract Eq. (64) from (65) and do some manipulating with respect to $\tau =\omega _0{t}$:

$$\begin{aligned} \begin{aligned}&{\omega _0}^2\ddot{q}(\tau )+2\zeta {\omega _0}\times {\omega _0}{\dot{q}}(\tau )+\frac{K_{11}}{M_{11}}(q(\tau ))\\&\quad +\frac{\alpha }{M_{11}}({V}_{DC}^2+2V_{DC}V_{AC}cos{\frac{\omega }{\omega _0}{\tau }})\sum _{i=0}^{9}p_{i}{h}^{i}\\&\quad \times \int _{0}^{1}(\phi _{1}(x))^{i+1}(q_s+q(\tau ))^i{dx}\\&\quad -\frac{\alpha }{M_{11}}({V}_{DC}^2)\sum _{i=0}^{9}p_{i}{h}^{i}\int _{0}^{1}(\phi _{1}(x))^{i+1}(q_s)^i{dx}=0. \end{aligned} \end{aligned}$$

(66)

Finally, rearranging based on order of $(q(t))^k (k=0,1,2,...)$ in the expansion of $(q_s+q(t))^i$ and manipulating the equation with respect to $q(\tau )=q(\omega _0{t})$, we will get Eq. (23).