A mathematical control for the pseudo-pull-in stability arising in a micro-electromechanical system

Abstract

A micro-electromechanical system’s reliability depends strongly upon its periodic motion; however, the pull-in instability might arise when the applied voltage is high enough. No criterion is available to judge the reliability condition in a simple but effective way. This paper tackles this challenge mathematically. The sufficient conditions for the periodic motion and the pull-in instability of a micro-electromechanical system with a current-carrying conductor are established, respectively. There is an uncertain area, where the system behaves firstly periodic motion, and after a certain number of cycles, the pull-in instability occurs. This new phenomenon is called as a pseudo-pull-in stability (pseudo-periodic motion); its main factors affecting the pseudo-periodic motion are elucidated.

Introduction

The last two decades have witnessed a skyrocketing interest in the micro/nanoelectromechanical system, sparked by its wide advanced applications in nanotechnology, micro-devices, micro-sensors, micro-actuators, and micro-power generators,^1-6 and it has been caught a renewed attention to the pull-in stability from both industry and academia because the pull-in instability is an unwanted phenomenon,^7-11 which causes unreliability. Much effort was made to avoid the pull-in instability. Tian et al. revealed that the pull-in instability can be eliminated in a fractal space;^12,13 K.L. Wang ¹⁴ and C.H. He ⁵ established a variational principle for the fractal MEMS system; Zhang et al. outlined the main factors affecting the pull-in instability. ¹⁵ Additionally, equilibrium analysis ¹⁶ and nanofluid dynamics ¹⁷ were also involved for the MEMS research. Due to the development of nanotechnology, especially the emergence of graphene and nanotube, nanoelectromechanical systems have the systems ever smaller with more special applications.^18-22 Now, the NEMS systems consist of single- and multilayer graphene sheets, and molecular scale electromechanical systems are the thinnest resonators, and their dynamical properties become ever more complex.

The electromechanical system operates periodically when the voltage is smaller than a threshold value, beyond which the pull-in instability occurs. ²³ The pull-in voltage has led to skyrocketing interest in analytical methods,^24-29 numerical methods, ³⁰ and experimental methods, ³¹ and a fast insight into the system’s operating properties is needed for practical applications.

This paper considers a MEMS system as illustrated in Figure 1, where the current-carrying conductor produces a force, which can be calculated as ²³

$f=\frac{{\mu }_0{i}_1{i}_{2}l}{2\pi r}=\frac{{\mu }_0{i}_1{i}_{2}l}{2\pi \left(b-U\right)}$

(1)

where ${\mu }_0$ is the magnetic coefficient, $i_1$ and $i_2$ are the electronic currents in the adjacent wires, respectively, $l$ is the wire’s length, $r$ is the distance between the two wires, and the distances $b$ and $U$ are shown in Figure 1.

Figure 1.

MEMS with a current-carrying wire.

The spring system produces a nonlinear restoring force

$F=kU+k_1{U}^3$

(2)

where $k$ is the spring coefficient and $k_1$ is the coefficient for the nonlinear term.

According to Newton’s second law, the system’s governing equation can be written as

$m\frac{d^{2}U}{d{T}^2}+kU+k_1{U}^3-\frac{{\mu }_0{i}_1{i}_{2}l}{2\pi \left(b-U\right)}=0$

(3)

where $m$ is the mass of rod attached with the spring system.

Equation (3) can be simplified as

$\frac{d^{2}u}{d{t}^2}+u+\epsilon u^3=\frac{K}{1-u}$

(4)

with the following initial conditions

$u\left(0\right)=0,\frac{du}{dt}\left(0\right)=0$

(5)

where $u=U/b$ and $t=T{\omega }_0$ , ${\omega }_0^2=k/m$ , $\epsilon =k_1{b}^2/\left(m{\omega }_0^2\right)$ , and $K={\mu }_0{i}_1{i}_{2}l/\left(2\pi k{b}^2\right)$ .

Equation (4) is a nonlinear oscillator with a singular nonlinearity and it is similar to the well-known Helmholtz–Duffing oscillator;^32,33 it becomes Duffing oscillator^32,33 when $K=0$ , and it has periodic solution for a small $K$ . When $K$ increases to a threshold value, the pull-in instability ²³ occurs. The threshold value is relative to the pull-in voltage. When $\epsilon =0$ , the threshold value is $K*=0.203632199$ ; ²³ the system has periodic motion when $K_1<K*$ ; and it becomes pull-in instability when $K_2>K*$ . However, the threshold value is very difficult to be found, and in practical applications, there is an uncertain area $K_2<K*<K_1$ , beyond which the system operates either periodically or unstably. This uncertain area plays an important role in the theoretical analysis and practical applications.

Pseudo-pull-in stability (pseudo-periodic motion)

Periodic criterion

The period or the frequency of a nonlinear oscillator can be easily obtained for various non-secular cases, and the singular term in equation (4) makes it extremely difficult when an analytical method, for example, the variational iteration method ³⁴ or the homotopy perturbation method,^35-40 is used.

The periodic criterion of equation (4) can be approximately obtained according a general sinusoidal vibration of a linear spring, that is, the direction of acceleration is opposite to that of displacement during all periodic motion. We consider a general differential equation

$\frac{d^{2}u}{d{t}^2}+h\left(u\right)=\frac{d^{2}u}{d{t}^2}+\left[\frac{h\left(u\right)}{u}\right]u=0$

(6)

For the MEMS system, $h\left(u\right)$ can be expressed as

$h\left(u\right)=u+\epsilon u^3-\frac{K}{1-u}$

(7)

In equation (6), $d^{2}u/d{t}^2$ is the acceleration and $u$ is the displacement, so periodic criterion is

$\frac{h\left(u\right)}{u}>0\hspace{0.17em}\mathrm{f}\mathrm{o}\mathrm{r}\hspace{0.17em}\mathrm{a}\mathrm{l}\mathrm{l}\hspace{0.17em}t>0$

(8)

For Duffing equation, $h\left(u\right)=u+\epsilon u^3$ , we have $h\left(u\right)/u=1+\epsilon u^2>0$ , so it has a periodic solution for $\epsilon >0$ .

Pull-in instability criterion

Equation (8) is the sufficient condition for periodic solution of equation (6). Similarly, we have the following non-periodic criterion, that is

$\frac{h\left(u\right)}{u}<0\hspace{0.17em}\mathrm{f}\mathrm{o}\mathrm{r}\hspace{0.17em}\mathrm{a}\mathrm{l}\mathrm{l}\hspace{0.17em}t>0$

(9)

For our MEMS system, the non-periodic criterion is the pull-in instability criterion. So, we have the following sufficient condition for pull-in instability

$1+\epsilon u^2-\frac{K}{u\left(1-u\right)}<0$

(10)

If equation (10) is satisfied for all $t>0$ , the system becomes instable. In order to find the pull-in instable area, we integrate equation (10) from 0 to 1 by adding a weighting function of $u^3$ , and this results in

$\begin{array}{l}{\int }_0^1{u}^3\left(1+\epsilon u^2-\frac{K}{u\left(1-u\right)}\right)du={\int }_0^1\left(u^3+\epsilon u^5-\frac{K{u}^2}{\left(1-u\right)}\right)du\\ =\frac{1}{4}+\frac{1}{6}\epsilon -\frac{3}{2}K<0\end{array}$

(11)

The pull-in stability criterion is

$K>K_1=\frac{1}{6}+\frac{1}{9}\epsilon$

(12)

Equation (12) is the sufficient condition for pull-in instability. For example, when $\epsilon =1$ , we have $K_1=1/6+\epsilon /9=0.2777$ , and if we choose $K=0.35>K_1$ , we can predict a pull-in instability phenomenon as shown in Figure 2(d).

Figure 2.

From pull-in stability to pseudo-pull-in stability and finally to pull-in instability: (a) phase diagram; (b) periodic motion; (c) pseudo-periodic motion; and (d) pull-in instability.

The frequency formulation

He’s frequency formulation ⁴¹ has been considered as the simplest approach to nonlinear oscillators. Here, He–Liu’s modification ⁴² is used

${\omega }^2=\frac{{\int }_0^{A}w\left(u\right)h\left(u\right)du}{{\int }_0^{A}w\left(u\right)udu}$

(13)

where $w\left(u\right)$ is a weighting function. When $w\left(u\right)=1$ , it turns out to be He’s frequency formulation. ⁴¹ There are many other frequency formulations in literature,^43-47 but He–Liu’s formulation ⁴² given in equation (13) is much simple and relatively accurate. Generally, h(u)/u should be larger than zero. If h(u) is an odd function, it is easy to prove h(u)/u>0. However, when h(u) is not an odd function, equation (13) is still approximately valid when h(u)/u>0, but the even term will greatly affect the frequency property like that for the Toda oscillator. ⁴⁸ In equation (4), the singular term can be expressed as $1/\left(1-u\right)=1+u+u^2+u^3+\cdots$ , both odd and even terms are involved, and the later will lead to a non-periodic solution which will be discussed in the next section. El-Dib suggested some frequency formulations for nonlinear non-conservative oscillators.^49,50

Equation (13) is valid for an approximate solution in the form $u=A\cos \left(\omega \mathrm{t}+\sigma \right)$ , while initial conditions given in equation (5) admit an approximate solution in the form $u=A{\sin }^2\left(\omega t+\sigma \right)=0.5\mathrm{A}\left(1-\cos \left(\mathrm{\Omega }t+2\sigma \right)\right)$ , where $\mathrm{\Omega }=2\omega$ . So for nonlinear vibration systems with the zero conditions, equation (13) has to be modified as

$\mathrm{\Omega }=2\omega$

(14)

For our MEMS system, we choose the weighting function as $w\left(u\right)=u\left(1-u\right)$ and $A=1$ , so we have

${\omega }^2=\frac{{\int }_0^{1}u\left(1-u\right)\left(u+\epsilon u^3-\frac{K}{1-u}\right)du}{{\int }_0^1\left(1-u\right)u^{2}du}=1+\frac{2}{5}\epsilon -6K$

(15)

For the periodic solution, it requires that

$1+\frac{2}{5}\epsilon -6K>0$

(16)

therefore, we have the following sufficient condition for periodic solution for the MEMS system

$K<K_2=\frac{1}{6}+\frac{1}{15}\epsilon$

(17)

For example, when $\epsilon =1$ , $K_2=1/6+\epsilon /15=0.2333$ , and if we choose $K=0.2<K_2$ , we can predict a periodic motion as shown in Figure 2(b). When $\epsilon =1$ and $K=0.2$ , the period can be calculated as

$T=\frac{2\pi }{\mathrm{\Omega }}=\frac{2\pi }{2\omega }=\frac{\pi }{\sqrt{1+\frac{2}{5}\epsilon -6K}}=7.0248$

(18)

while the exact period is 6.84744, the relative error 2.59% is acceptable for practical applications.

The pseudo-pull-in stability (pseudo-periodic motion)

The even terms involved inexplicitly in equation (4) will lead to a non-periodic solution, and instability arises. ⁵¹ Nonorthogonality analysis ⁵² is widely used for the coherence resonance and stochastic bifurcation behaviors of an oscillator. ⁵³

When $K\in \left[K_2,K_1\right]$ , there is an uncertain domain; where the system might be in a periodic motion or in a pull-in instability, there might not be a distinctly separating area to distinguish the periodic motion from the pull-in instability, and a pseudo-pull-in stability might occur. This is a novel phenomenon that the MEMS system operates periodically, but after some amount of periodic motion, it tends to the pull-in instability.

We choose $K=0.262452\in \left[K_2,K_1\right]$ , and the pseudo-periodic motion occurs at the initial time, but after two and half cycles, the system becomes the pull-in instability (see Figure 2(c)). If we choose $\epsilon =100$ and $K=7.299689$ , the pseudo-periodic motion becomes pull-in instability after 29 and half cycles (see Figure 3).

Figure 3.

Pull-in instability after a long pseudo-periodic motion.

Discussion

When $K<K_2$ , the spring force has overwhelming superiority, and a periodic solution is predicted; when $K>K_1$ , the electromagnetic force has overwhelming superiority, and the pull-in instability happens; however, when $K\in \left[K_2,K_1\right]$ , the nonlinear term $K/\left(1-u\right)$ might consume some amount of energy in each cycle, and when the consumed energy reaches a threshold, it cannot operate periodically anymore, and pull-in instability occurs. When $\epsilon$ is large, the spring potential is high, and it requires a long time to consume its energy as seen for the case $\epsilon =100$ (see Figure 3).

This paper finds the missing piece of the puzzle when it comes to the pull-in voltage; it clearly elucidates the pseudo-pull-in stability, which was not found in previous literature. The concept is going to change everything you ever thought was real about the traditional MEMS system. Now is the time to tackle the urgent issue on the energy crisis; like other energy harvesting technologies,^54,55 MEMS systems^56,57 can tackle this challenge effectively.

Conclusion

This paper has opened the path for a new way to study pseudo-pull-in stability, and there leaves much space to further improve the accuracy of the pseudo-pull-in stability, and the homotopy perturbation method or the variational iteration method might provide an unprecedented effective tool for accurate analysis of the periodic properties.

To be concluded, this article reveals, for the first time ever, the pseudo-pull-in stability, and it oscillates for some amount of periodic motion, but it will become pull-in instable finally. This phenomenon is important for practical applications; as the pull-in instability is not wanted during its operation, the pseudo-pull-in stability would delude the engineers that the system would operate in a normal working condition. This paper adopts simple mathematics concepts to deal with an unsolved problem in MEMS systems with great success. The essence of mathematics is to make the complicated pull-in instability much simpler,if an even higher accuracy is needed to predict the pull-in instability, the homotopy perturbation method^58,59 has to be used.