## Suppression of pull-in instability in MEMS using a high-frequency actuation

### Faouzi Lakrad ^{*} , Mohamed Belhaq

### University Hassan II-Casablanca, BP 5366 Maarif, Casablanca, Morocco

### a r t i c l e i n f o

### Article history:

### Received 21 August 2009

### Received in revised form 25 December 2009 Accepted 27 December 2009

### Available online 4 January 2010

### Keywords:

### Pull-in MEMS

### High-frequency voltage

### a b s t r a c t

### We study the effect of a high-frequency AC tension on the pull-in instability induced by a DC tension in a microelectromechanical system. The microstructure is modelled as a sin- gle-degree-of-freedom system. The method of direct partition of motion is used to split the fast and slow dynamics. Analysis of steady-states of the slow dynamic enables us to depict the effect of the AC voltage on the pull-in. The main result of this paper indicates that it is possible to suppress the electrostatically induced pull-in instability for a range of values of the amplitude and the frequency of the high-frequency AC.

### Ó 2010 Elsevier B.V. All rights reserved.

### 1. Introduction

### Electrostatically actuated microelectromechanical systems (MEMS) ﬁnd many applications today in areas such as iner- tial sensors, relays, switches, etc. A serious limitation on the use of these devices lies in the pull-in phenomenon [1–3], which is a structural instability resulting from the interaction between elastic and electrostatic forces. This instability re- sults from the unbalance between the electric actuation and the mechanical restoring force leading the suspended micro- beam to hit the stationary electrode underneath it causing stiction and short circuit problems and hence the failure in the device’s function [4].

### Keeping MEMS devices operating in a stable regime away from the pull-in instability limit has a crucial interest from de- sign and commercialization point of view. Various works [5,6] investigated the static pull-in phenomenon and performed techniques to predict its occurrence by determining the largest DC voltage for which the system operate in a stable behavior.

### The dynamic pull-in was studied under various loads, for instance step voltage [7], AC harmonic voltage [8] and mechanical shock load [9]. It was shown that the dynamic actuation reduces drastically the static pull-in limit. In the framework of con- trolling the pull-in, a control method consisting in adding superharmonics to a reference harmonic excitation was proposed in [10]. In the present work, a high-frequency voltage (HFV) is used to suppress an initially established static pull-in. The proposed method is applied to a simpliﬁed mass-spring-damping system modelling the dynamic of capacitive MEMS. It is worth nothing that the problem of studying the effect of high-frequency excitation on the dynamic of mechanical systems has been examined previously by a number of authors; see for instance [11] and references therein. Recent works focused attention on the effect of high-frequency excitation on natural frequencies [12,13], symmetry breaking [14], on limit cycles [15] and on hysteresis [16,17]. The rest of the paper is organized as follows: In Section 2, after describing the model, an aver- aging technique is performed over the rapid dynamic produced by HFV and an equation governing the slow dynamic of the MEMS device is derived. In Section 3, the effect of a HFV on the occurrence of the pull-in instability is investigated. Section 4 concludes the work.

### 1007-5704/$ - see front matter

Ó### 2010 Elsevier B.V. All rights reserved.

### doi:10.1016/j.cnsns.2009.12.033

*

### Corresponding author.

### E-mail address:

lakrad@hotmail.com### (F. Lakrad).

### Contents lists available at ScienceDirect

## Commun Nonlinear Sci Numer Simulat

### j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c n s n s

### 2. Formulation of the problem and the slow dynamic

### A single-degree-of-freedom model depicted in Fig. 1 is utilized to represent a capacitive MEMS device employing a DC and AC voltages as actuator.

### m € x þ c x _ þ kx ¼ e A

### 2ðd xÞ

^{2}

### V

^{2}

### ð1Þ

### where x is the displacement of the movable mass m, c and k are the damping and stiffness of the system, respectively. The dielectric constant of the gap medium is denoted e , d is the initial capacitor gap width, A is the area of the cross section, and V is the electric load.

### The tension is given by the following AC and DC voltages:

### V ¼ V

0### þ U cosð X

^{}

### tÞ ð2Þ

### where V

0### is a DC voltage, whereas U and X

^{}

### are the amplitude and the frequency of the AC actuation, respectively.

### The square of this tension is given by V

^{2}

### ¼ V

^{2}

_{0}

### þ U

^{2}

### 2

### !

### þ 2V

0### U cosð X

^{}

### tÞ þ U

^{2}

### 2 cosð2 X

^{}

### tÞ ð3Þ

### Thus, the AC tension contributes positively in the electrostatic force acting on the movable electrode. As a consequence, the static pull-in threshold is affected. In order to preclude this effect, a new DC tension V

1### is added and it is chosen to vanish this undesirable contribution. Thus the electric tension can be written as follows:

### V ¼ V

0### þ V

1### þ U cosð X

^{}

### tÞ ð4Þ

### with V

1### ¼ V

0### 0:5 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4V

^{2}

_{0}

### 2U

^{2}

### q

### . It is worth noting that V

1### exists only for U 6 ﬃﬃﬃ p 2

### V

0### . Consequently, the square of the tension is given by

### V

^{2}

### ¼ V

^{2}

_{0}

### U

^{2}

_{1}

### cosð X

^{}

### tÞ þ U

^{2}

_{2}

### cosð2 X

^{}

### tÞ ð5Þ

### where U

^{2}

_{1}

### ¼ U

### ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4V

^{2}

_{0}

### 2U

^{2}

### q

### and U

^{2}

_{2}

### ¼ U

^{2}

### =2. We set X ¼

^{x}

_{d}

### ; s ¼ x t; x ¼

### ﬃﬃﬃ

k m### q

### ; n ¼

_{2m}

^{c}

_{x}

### and X ¼

^{X}

_{x}

^{}

### . Thus the nondimensional equation of motion is given by

### X

^{00}

### þ 2nX

^{0}

### þ X ¼ a

### ð1 XÞ

^{2}

### þ b

### ð1 XÞ

^{2}

### cosð X s Þ þ c

### ð1 XÞ

^{2}

### cosð2 X s Þ ð6Þ

### The primes denote the derivatives with respect to the normalized time s , and

### a ¼ e A

### 2m x

^{2}

### d

^{3}

### V

^{2}

_{0}

### ð7Þ

### b ¼ e A

### 2m x

^{2}

### d

^{3}

### U

^{2}

_{1}

### ð8Þ

### c ¼ e A

### 2m x

^{2}

### d

^{3}

### U

^{2}

_{2}

### ð9Þ

### These coefﬁcients are related by the following expression:

### b

^{2}

### ¼ 8 c ð a c Þ ð10Þ

### When ﬁxing two parameters of the electric actuation the third one is determined. Fig. 2 shows b versus c for various values of

### a . The allowed parameters in the plane ðb; c Þ constitute symmetric ellipsis, with respect to b, that are increasing when the electrostatic effect a increases. The upper bound value of this latter is given by the pull-in instability threshold.

** Stationary electrode**

**d** **x**

** V **

** m **

** c ** **k**

Fig. 1.

### A single-degree-of-freedom model of a MEMS device.

### Indeed, even if the DC voltage V

1### is not introduced, Eqs. (6) and (10) are still valid. The difference is in the deﬁnition of the parameters a ; b and c which are given by

### a ¼ e A

### 2m x

^{2}

### d

^{3}

### V

^{2}

_{0}

### þ U

^{2}

### 2

### !

### ð11Þ

### b ¼ e A

### m x

^{2}

### d

^{3}

### V

0### U ð12Þ

### c ¼ e A

### 4m x

^{2}

### d

^{3}

### U

^{2}

### ð13Þ

### Eq. (6) contains a slow dynamic which describes the main motions at time-scale of natural vibrations of the microstructure and a fast dynamic at time scale of the high-frequency voltage. To obtain the main equation governing the slow dynamic of the device, we implement the method of direct partition of motion [11] by introducing two different time-scales: a fast time T

0### ¼ g

^{1=2}

### s and a slow time T

1### ¼ s , and we split up the displacement of the mass Xð s Þ into a slow part ZðT

1### Þ and a fast part /ðT

0### ; T

1### Þ as follows:

### Xð s Þ ¼ ZðT

1### Þ þ /ðT

0### ; T

1### Þ ZðT

1### Þ þ g

^{2}

### /ðT ^{~}

0### ; T

1### Þ ð14Þ Here the positive parameter g is introduced to measure the smallness of other parameters ð0 < g 1Þ. As a consequence the high frequency is expressed as X ¼ g

^{1=2}

### . The fast motion and its derivatives are assumed to be 2 p -periodic functions of the fast time T

0### with zero mean value with respect to it. Thus, hXð s Þi ¼ ZðT

1### Þ where h:i ¼

_{2}

^{1}

_{p}

### R

2p0

### ð:ÞdT

0### deﬁnes the fast time-aver- aging operator. Introducing D

^{n}

_{m}

### ¼

_{@T}

^{@}

^{n}n

m

### , yields d

### d s ^{¼} g

^{1=2}

### D

0### þ D

1### ð15Þ

### d

^{2}

### d s

^{2}

^{¼} g

^{1}

### D

^{2}0

### þ g

^{1=2}

### 2D

0### D

1### þ D

^{2}1

### ð16Þ

### Substituting Eqs. (15) and (16) into (6) yields to the following equation:

### g ðD

^{2}

_{0}

### /Þ þ ~ 2 g

^{3=2}

### ðD

0### D

1### /Þ þ ~ g

^{2}

### ðD

^{2}

_{1}

### /Þ þ ðD ~

^{2}

_{1}

### ZÞ þ 2n½ g

^{3=2}

### ðD

0### /Þ þ ~ g

^{2}

### ðD

1### /Þ þ ðD ~

1### ZÞ þ Z þ g

^{2}

### / ^{~}

### ¼ a

### ð1 Z g

^{2}

### /Þ ^{~}

^{2}

### þ b

### ð1 Z g

^{2}

### /Þ ^{~}

^{2}

### cosðT

0### Þ þ c

### ð1 Z g

^{2}

### /Þ ^{~}

^{2}

### cosð2T

0### Þ ð17Þ

### In what follows we set b ¼ g ^{~} b; c ¼ g ^{~} c and n ¼ g ^{~} n. The parameters with tildes are of order Oð1Þ. The dominant terms depend- ing on T

0### up to the order Oð g Þ in Eq. (17) are

### ðD

^{2}

_{0}

### /Þ ¼ ~

### b ~ cosðT

0### Þ

### ð1 ZÞ

^{2}

### þ ~ c cosð2T

0### Þ

### ð1 ZÞ

^{2}

### ð18Þ

### Thus up to this leading order, the fast motion is given by

### 0 0.05 0.1 0.15 0.2 0.25 0.3

### −0.5

### −0.4

### −0.3

### −0.2

### −0.1 0 0.1 0.2 0.3 0.4 0.5

### γ

### β

### α=0.3

### α=0.2 α=0.149

Fig. 2.b

### versus c for various values of a .

### /ðT ~

0### ; T

1### Þ ¼ ~ b cosðT

0### Þ

### ð1 ZÞ

^{2}

### ~ c cosð2T

0### Þ

### 4ð1 ZÞ

^{2}

### þ Oð g Þ ð19Þ

### Now averaging Eq. (17) over a period of the fast time scale T

0### leads to the following equation:

### ðD

^{2}

_{1}

### ZÞ þ 2nðD

1### ZÞ þ Z ¼ a

### ð1 Z g

^{2}

### /Þ ^{~}

^{2}

### * +

### þ b

### ð1 Z g

^{2}

### /Þ ^{~}

^{2}

### cosðT

0### Þ

### * +

### þ c

### ð1 Z g

^{2}

### /Þ ^{~}

^{2}

### cosð2T

0### Þ

### * +

### ð20Þ

### where up to order Oð g

^{3}

### Þ

### a

### ð1 Z g

^{2}

### /Þ ^{~}

^{2}

### * +

### ¼ a

### ð1 ZÞ

^{2}

### ð21Þ

### b

### ð1 Z g

^{2}

### /Þ ^{~}

^{2}

### cosðT

0### Þ

### * +

### ¼ b ~

^{2}

### ð1 ZÞ

^{5}

### ð22Þ

### c

### ð1 Z g

^{2}

### /Þ ^{~}

^{2}

### cosð2T

0### Þ

### * +

### ¼ ~ c

^{2}

### 4ð1 ZÞ

^{5}

### ð23Þ

### Consequently, the equation of the slow dynamics obtained from Eq. (20) can be written as ðD

^{2}

_{1}

### ZÞ þ 2nðD

1### ZÞ þ Z ¼ a

### ð1 ZÞ

^{2}

### 4b

^{2}

### þ c

^{2}

### 4 X

^{2}

### ð1 ZÞ

^{5}

### ð24Þ

### The last term of the right hand side of Eq. (24) is the effect of the HFV on the slow motion of the mass. It is worth noting that the zeros of the slow dynamic (24) are the averaged positions around which the periodic solutions of Eq. (6) oscillate. To obtain these ﬁxed points the following sixth order algebraic equation should be solved:

### Zð1 ZÞ

^{5}

### ¼ a ð1 ZÞ

^{3}

### 4b

^{2}

### þ c

^{2}

### 4 X

^{2}

^{ð25Þ}

### For b ¼ 0 and c ¼ 0 the standard case of the static tension is obtained i.e., Zð1 ZÞ

^{2}

### ¼ a . The static pull-in in this case is given by a

p### ¼

_{27}

^{4}

### which corresponds to the steady state displacement Z

p### ¼

^{1}

_{3}

### .

### Up to the order Oð g

^{3}

### Þ, the solution of Eq. (6) is given by Xð s Þ ¼ Z b cosð X s Þ

### X

^{2}

### ð1 ZÞ

^{2}

### c cosð2 X s Þ

### 4 X

^{2}

### ð1 ZÞ

^{2}

### þ Oð g

^{3}

### Þ ð26Þ

### 3. Pull-in suppression

### In this section, we analyze the effect of HFV on the suppression of the pull-in instability using the slow dynamic Eq. (24) and numerical simulations of Eq. (6) using Runge–Kutta method . We begin by analyzing the response of the system to a DC voltage in the absence of AC voltage i.e., b ¼ 0. The damping coefﬁcient is taken n ¼ 0:01 in all computations. In Fig. 3, the

### 0 100 200 300 400 500 600 700 800

### 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

### τ

### X

Fig. 3.

### Time history of Eq.

(6)### for a

¼### 0:1, and

b¼### 0; c

¼### 0.

### normalized displacement to the gap Xð s Þ is shown for a ¼ 0:1. We note that the steady-state amplitude is near X ¼ 0:133. In Fig. 4, the time history of Eq. (6) is shown for a ¼ 0:15 which corresponds to a post pull-in threshold value. This instability is characterized by the slope of the displacement approaching inﬁnity.

### The effect of the HFV on the pull-in instability is studied in the case where the mass is in the post pull-in state caused by an electrostatic force a ¼ 0:15 (see Fig. 4). The shift of the averaged solution from the zero solution are computed using the following relation:

### shift ¼ maxðXð s ÞÞ þ minðXð s ÞÞ

### 2 ð27Þ

### Fig. 5 shows, for X ¼ 7, the stationary solutions Z of the slow dynamic (25) and the shift of the averaged solutions of Eq. (6) versus the amplitude c of the HFV. It can be seen that the HFV creates a stable position in the safe range i.e., X < 1. This ﬁgure shows that it is possible to suppress the pull-in for c 2 ½0:0269; 0:1279.

### In Fig. 6, the quasi-static solutions given by Eq. (25) (lines) and the averaged positions of Eq. (6) (circles) are shown versus X for c ¼ 0:1. For X < 9, there is one stable solution in the range Z < 1=3 which disappears by a saddle-node bifurcation. In Fig. 7 is shown the good agreement between the numerical solution of Eq. (6) and the analytical solution (26) validating the used approach.

### Fig. 8 shows in the plane of the frequency and amplitude of the HFV ð X ; c Þ the zone where the electrostatically induced pull-in instability can be suppressed (the grey zone). The grey zone is becoming larger, in terms of the amplitude of the HFV

### 0.02 0.04 0 .06 0.08 0.1 0.12 0.14

### 0.28 0.3 0.32 0.34 0.36 0.38 0.4

### γ

### Z

Fig. 5.

### The averaged position of vibration versus c for X

¼### 7 and a

¼### 0:15

:ðÞ### numerical integration of Eq.

(6). Zeros of Eq.(25): (–) stable and (- - ) unstable.### 0 1 2 3 4 5

### 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

### τ

### X

Fig. 4.

### Time history of Eq.

(6)### for a

¼### 0:15, and

b¼### 0; c

¼### 0.

### 3 4 5 6 7 8 9 0

### 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

### Ω

### Z

Fig. 6.

### The averaged position of vibration versus X for c

¼### 0:1 and a

¼### 0:15. Numerical integration of Eq.

(6): () numerical integration of Eq.(6). Zeros of Eq.(25): (–) stable and (- - ) unstable.

### 2040 2042 2044 2046 2048 2050

### 0.306 0.308 0.31 0.312 0.314 0.316 0.318

### τ

### X

Fig. 7.

### Time history of Eq.

(6)### for a

¼### 0:149; X

¼### 10 and c

¼### 0:0624: () Numerical integration of Eq.

(6)### and lines analytical solution

(26).### 4 5 6 7 8 9 10

### 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

### Ω

### γ

Fig. 8.

### Suppression zone of the pull-in instability in the plane

ð### X

;### c

Þ### for a

¼### 0:15: lines using Eq.

(25)### and () using Eq.

(6).### c , for decreasing X . Note that our results are valid only for X ¼ Oð g

^{1=2}