The analysis for the dynamic pull-in of a micro-electromechanical system

Abstract

The pull-in behavior is an inherent property of the micro-electromechanical oscillator. The bisection method and an iterative method are introduced to find its pull-in voltage, and the main factors affecting the pull-in voltage are elucidated. The simple and efficient operability is demonstrated through theoretical analysis and result comparison.

Introduction

Nonlinear dynamical systems are usually used to model the behaviors of complex systems such as neuron science, chemical, and mechanical systems. Most nonlinear equations cannot be solved very precisely; therefore, many efforts have been made toward the studying of different numerical analysis methods to calculate approximate solutions that meet certain accuracy requirements. Finding a high-precision solution is an extremely crucial and meaningful research topic. There are many analytical methods for finding approximate solutions and discussing the characteristics of nonlinear systems. Some of these methods are the homotopy analysis method,^1,2 the homotopy perturbation method,^3-7 Newton-Raphson method,^8-11 He’s frequency formula method, ¹² Taylor series method,^13,14 and bisection method. ¹⁵ Riahi and Qattan studied the nonlinear, non-convex optimization problem under system dynamic constraints and applied their analysis to parameter identification of systems governed by general nonlinear differential equations. ¹⁶

Apart from the above methods, the iteration method is intuitive, and it is a simple calculation way to find approximate real roots of nonlinear equations, including transfer issues and vibration problems. It can be considered that the solution is linearly approximated by two hypothetical solutions.^17-19 Because of its simple operation and fast convergence, the iteration method is widely used in numerical simulation, and has great practical implication in numerous real-life challenges in different areas of engineering, such as industrial engineering, civil engineering, electrical engineering, and mechanical engineering.

How to solve a nonlinear equation quickly is very important because a simple calculation is always required in industrial design. In this paper, two methods are introduced to find a high-precision solution of the nonlinear equation describing dynamic pull-in phenomenon. The simple and efficient operability can be seen through theoretical analysis.

Micro-electromechanical system

A micro-electromechanical system (MEMS) is a tiny mechanical module that is driven by electricity, and has multiple uses in many researches on various micro-devices such as detectors, sensors, and capacitors. ²⁰ In these fields, MEMS plays its unique advantages that can make up for or even replace the shortcomings of traditional products. With the development of science and technology, MEMS has been widely used due to its small size, low energy consumption, high integration, and high intelligence. But a serious limitation on the use of these devices lies in the pull-in phenomenon which can cause the failure in the device’s function. ²¹

The pull-in phenomenon analysis of the electrostatic drive device is of great significance to the efficient operation and reliability of the device. The pull-in instability becomes a hot topic in both industry and academy, and many literatures have conducted a lot of analysis on the dynamic pull-in of MEMS models. Tian and her colleagues proposed a fractal MEMS system and found the pull-in instability can be converted to a stable condition.^22,23 He established a variational principle that can be used for both analytical and numerical analyses of the MEMS system. ²⁴

The dynamic differential equation used to describe the movement of the wire as a point mass can be considered as follows

$y^{″}+y+\frac{\theta }{y-1}=0, y^{\prime }\left(0\right)=0, y\left(0\right)=0,\theta >0$

(1)

where y is the dimensionless distance, $\mathrm{y}=\frac{u}{b}$ , and $\theta$ is a voltage-related parameter, $\mathrm{\theta }=\frac{2\times {10}^{-7}{i}_1{i}_{2}h}{\mathrm{k}b}$ , as shown in Figure 1.

Figure 1.

The MEMS system with a current carrying wire.

The system behaves either periodically or unsteadily. When the applied voltage is small, the system is periodic; and when the voltage is larger, the system becomes instable, which is called as the pull-in instability. Figure 2 shows the periodic solutions when $\theta$ takes different values. We see that when $\theta$ does not exceed a critical value, the phase space trajectory closes on itself and the system moves periodically, and when $\theta$ exceeds the critical value, it becomes the pull-in instability. The critical value is $\underset{\_}{\theta }=$ 0.203632188. ²⁵ The threshold value is called as the pull-in voltage. The pull-in behavior is an inherent property of the MEMS oscillator, which occurs when the voltage is larger than its threshold value. It plays an important role in electrostatic drive sensors because of their efficient and reliable operation.

${\left(\frac{1+\sqrt{1-4\theta }}{2}\right)}^2+2\theta \ln \left|1-\frac{1+\sqrt{1-4\theta }}{2}\right|=0$

(2)

where $\underset{\_}{\theta }$ is a positive root of equation (2).

Figure 2.

Phase trajectories for different values of $\theta$ . The transcendental equation describing the pull-in phenomenon is.

We need to solve this nonlinear equation to discuss the effect of the MEMS oscillator parameter on the pull-in voltage. For this purpose, we use two methods, one is the bisection method and the other is iterative method.

An approximate solution

The bisection method can be used to find the approximate root of an algebraic equation. First, we use it to solve equation (2).

Let

$g\left(\theta \right)={\left(\frac{1+\sqrt{1-4\theta }}{2}\right)}^2+2\theta \ln \left|1-\frac{1+\sqrt{1-4\theta }}{2}\right|$

(3)

The bisection method begins with two guesses ${\theta }_0$ and ${\theta }_1$ , which satisfied $g\left({\theta }_0\right)g\left({\theta }_1\right)>0$ . The approximate solution is $\underset{\_}{\theta }=\frac{{\theta }_0+{\theta }_1}{2}$ . Using the function image as shown in Figure 3, we choose the interval $\left[{\theta }_0, {\theta }_1\right]$ of the approximate solution and compute $\underset{\_}{\theta }$ . Repeat the calculation until a high-precision approximate solution is obtained. For example, we set ${\theta }_0=0.18$ and ${\theta }_1=0.22$ , the results are $g\left({\theta }_0\right)=0.0639$ , $g\left({\theta }_1\right)=-0.0389$ , and $\underset{\_}{\theta }=0.2$ . The process is shown in Table 1. When going to the fifth step, we have $\underset{\_}{\theta }=$ 0.20375 with a relative error of 0.06%. The bisection method is simple to calculate, but it strongly depends upon its initial guess. A suitable choice of initial guesses converges quickly, while a bad choice requires a long calculation. So, we try to use another method.

Figure 3.

The graph of equation (3).

Table 1.

The bisection method process.

Step	${\theta }_0$	${\theta }_1$	$\underset{\_}{\theta }=\frac{{\theta }_0+{\theta }_1}{2}$	$g\left(\underset{\_}{\theta }\right)$
1	0.18	0.22	0.2	0.009234485
2	0.2	0.22	0.21	−0.015668578
3	0.2	0.21	0.205	−0.003421842
4	0.2	0.205	0.2025	0.002855480
5	0.2025	0.205	0.20375	−0.000295930

Then we use the iterative method to solve equation (2), let

$\frac{1+\sqrt{1-4\theta }}{2}={\cos }^{2}x$

(4)

The nonlinear equation (2) can be rewritten as

${\cos }^{4}x+{\sin }^{2}2x\ln \left|\sin x\right|=0$

(5)

Let

$f\left(x\right)={\cos }^{4}x+{\sin }^{2}2x\ln \left|\sin x\right|$

(6)

We choose an initial guess $x_0=0.5$ , and compute $x_{k+1} \left(k=\mathrm{0,1,2}\cdots \right)$ , where $x_{k+1}=x_k-\frac{f\left(x_k\right)}{f^{\prime }\left(x_k\right)}$ . When going to the second step, there is an error of 0.036% from Table 2. It is worth noting that a high-precision result is obtained after second iteration and the method can be used to find an approximate solution that meeting the engineering requirements for accuracy in industrial designing.

Table 2.

The iterative method process.

Step	$x_k$	${\theta }_k$	Relative error (%)
0	0.5	0.177018355	13.069561
1	$0.554289$	0.200285550	1.643472
2	$0.562596$	0.203559045	0.035919
3	$0.562781$	0.203630973	0.000597

Conclusion

In engineering and scientific research, a simple method with relatively high-precision is most popular. In this paper, the basic idea is to quickly find a high-accuracy solution of the nonlinear equation. The bisection method is introduced and compared with the iterative method. For the transcendental equation describing pull-in phenomenon, the variable $\theta$ is replaced by x to obtain an equation of the trigonometric function, which is then solved by the iterative method. The simple and efficient operability of the method can be seen through theoretical analysis.