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
Ó2010 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
2V
2ð1Þ
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
0þ U cosð X
tÞ ð2Þ
where V
0is 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
2¼ V
20þ U
22
!
þ 2V
0U cosð X
tÞ þ U
22 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
1is added and it is chosen to vanish this undesirable contribution. Thus the electric tension can be written as follows:
V ¼ V
0þ V
1þ U cosð X
tÞ ð4Þ
with V
1¼ V
00:5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4V
202U
2q
. It is worth noting that V
1exists only for U 6 ffiffiffi p 2
V
0. Consequently, the square of the tension is given by
V
2¼ V
20U
21cosð X
tÞ þ U
22cosð2 X
tÞ ð5Þ
where U
21¼ U
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4V
202U
2q
and U
22¼ U
2=2. We set X ¼
xd; s ¼ x t; x ¼
ffiffiffi
k mq
; n ¼
2mcxand X ¼
Xx. Thus the nondimensional equation of motion is given by
X
00þ 2nX
0þ X ¼ a
ð1 XÞ
2þ b
ð1 XÞ
2cosð X s Þ þ c
ð1 XÞ
2cosð2 X s Þ ð6Þ
The primes denote the derivatives with respect to the normalized time s , and
a ¼ e A
2m x
2d
3V
20ð7Þ
b ¼ e A
2m x
2d
3U
21ð8Þ
c ¼ e A
2m x
2d
3U
22ð9Þ
These coefficients are related by the following expression:
b
2¼ 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
1is 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
2d
3V
20þ U
22
!
ð11Þ
b ¼ e A
m x
2d
3V
0U ð12Þ
c ¼ e A
4m x
2d
3U
2ð13Þ
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
0¼ g
1=2s and a slow time T
1¼ s , and we split up the displacement of the mass Xð s Þ into a slow part ZðT
1Þ and a fast part /ðT
0; T
1Þ as follows:
Xð s Þ ¼ ZðT
1Þ þ /ðT
0; T
1Þ ZðT
1Þ þ g
2/ðT ~
0; T
1Þ 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
0with zero mean value with respect to it. Thus, hXð s Þi ¼ ZðT
1Þ where h:i ¼
21pR
2p0
ð:ÞdT
0defines the fast time-aver- aging operator. Introducing D
nm¼
@T@nnm
, yields d
d s ¼ g
1=2D
0þ D
1ð15Þ
d
2d s
2¼ g
1D
20þ g
1=22D
0D
1þ D
21ð16Þ
Substituting Eqs. (15) and (16) into (6) yields to the following equation:
g ðD
20/Þ þ ~ 2 g
3=2ðD
0D
1/Þ þ ~ g
2ðD
21/Þ þ ðD ~
21ZÞ þ 2n½ g
3=2ðD
0/Þ þ ~ g
2ðD
1/Þ þ ðD ~
1ZÞ þ Z þ g
2/ ~
¼ a
ð1 Z g
2/Þ ~
2þ b
ð1 Z g
2/Þ ~
2cosðT
0Þ þ c
ð1 Z g
2/Þ ~
2cosð2T
0Þ 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
0up to the order Oð g Þ in Eq. (17) are
ðD
20/Þ ¼ ~
b ~ cosðT
0Þ
ð1 ZÞ
2þ ~ c cosð2T
0Þ
ð1 ZÞ
2ð18Þ
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
1Þ ¼ ~ b cosðT
0Þ
ð1 ZÞ
2~ c cosð2T
0Þ
4ð1 ZÞ
2þ Oð g Þ ð19Þ
Now averaging Eq. (17) over a period of the fast time scale T
0leads to the following equation:
ðD
21ZÞ þ 2nðD
1ZÞ þ Z ¼ a
ð1 Z g
2/Þ ~
2* +
þ b
ð1 Z g
2/Þ ~
2cosðT
0Þ
* +
þ c
ð1 Z g
2/Þ ~
2cosð2T
0Þ
* +
ð20Þ
where up to order Oð g
3Þ
a
ð1 Z g
2/Þ ~
2* +
¼ a
ð1 ZÞ
2ð21Þ
b
ð1 Z g
2/Þ ~
2cosðT
0Þ
* +
¼ b ~
2ð1 ZÞ
5ð22Þ
c
ð1 Z g
2/Þ ~
2cosð2T
0Þ
* +
¼ ~ c
24ð1 ZÞ
5ð23Þ
Consequently, the equation of the slow dynamics obtained from Eq. (20) can be written as ðD
21ZÞ þ 2nðD
1ZÞ þ Z ¼ a
ð1 ZÞ
24b
2þ c
24 X
2ð1 ZÞ
5ð24Þ
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
5¼ a ð1 ZÞ
34b
2þ c
24 X
2ð25Þ
For b ¼ 0 and c ¼ 0 the standard case of the static tension is obtained i.e., Zð1 ZÞ
2¼ a . The static pull-in in this case is given by a
p¼
274which corresponds to the steady state displacement Z
p¼
13.
Up to the order Oð g
3Þ, the solution of Eq. (6) is given by Xð s Þ ¼ Z b cosð X s Þ
X
2ð1 ZÞ
2c cosð2 X s Þ
4 X
2ð1 ZÞ
2þ Oð g
3Þ 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
¼0:1, and
b¼0; c
¼0.
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
¼7 and a
¼0:15
:ðÞnumerical 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
¼0:15, and
b¼0; c
¼0.
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
¼0:1 and a
¼0:15. 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
¼0:149; X
¼10 and c
¼0:0624: () 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.