Electro-dynamic behavior of an electrically actuated micro-beam: Effects of initial curvature and nonlinear deformation

Computers and Structures 96-97 (2012) 25–33

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Electro-dynamic behavior of an electrically actuated micro-beam: Effects of initial curvature and nonlinear deformation J. Yang a,⇑, Y.J. Hu b, S. Kitipornchai c a

School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, P.O. Box 71, Bundoora, VIC 3083, Australia College of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China c Department of Civil and Architectural Engineering, City University of Hong Kong, Kowloon, Hong Kong b

a r t i c l e

i n f o

Article history: Received 27 June 2011 Accepted 12 January 2012 Available online 7 February 2012 Keywords: Micro-beam Pull-in Electro-dynamic response Initial curvature Nonlinear deformation Arc coordinate system

a b s t r a c t This paper presents a curved beam model for the nonlinear electro-dynamic analysis of micro-beams with initial curvature, taking into account the nonlinear electric force and nonlinear deformation. The governing equations of motion and boundary conditions are derived in an arc coordinate system. Unlike previous studies based on von Karman nonlinearity or considering mid-plane stretching only, the present study does not involve any assumptions on nonlinear deformation. Differential quadrature method and Petzold–Gear BDF method are employed to obtain the nonlinear electro-dynamic behavior of curved micro-beams. The effects of nonlinear deformation, initial gap and initial rise on the nonlinear electrodynamic characteristics are studied. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Electrically actuated microelectromechanical systems (MEMS) are widely used in many engineering applications as microswitches, transistors, accelerometers, pressure sensors, micromirrors, micro-pumps, micro-grippers, and bio-MEMS [1]. MEMS systems are highly nonlinear in nature due to the fact that there exist at least two physical domains, electrical and mechanical, with complicated nonlinear coupling between them, and the geometrical nonlinearity introduced by the large deformation of the system that can take place even under low electrical actuation because of their small sizes. Typical MEMS systems consist of arrays of thin beam-type electrodes in the order of microns where the movable electrode can be modeled as a clamped–clamped or cantilever micro-beam subjected to an applied direct current voltage (DC) or an alternating current (AC) voltage, or a combination of these two. The voltage difference between the electrodes creates an electric force that pulls the movable electrode towards to the fixed electrode. The system is in equilibrium state when the electric force is balanced by the elastic restoring force of the deformed micro-beam but loses equilibrium and becomes unstable with the movable electrode snapping and then adhering to the fixed electrode when the balance cannot be achieved once the applied voltage exceeds a critical value. This phenomenon, known as pull-in instability, has ⇑ Corresponding author. E-mail address: [email protected] (J. Yang). 0045-7949/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2012.01.008

been experimentally observed in previous investigations [2,3]. Accurate prediction of pull-in instability is crucial in MEMS design. In most cases, it is highly desirable to avoid or delay the onset of pull-in while in other cases such as micro-switch applications, in particular, this behavior should be well exploited for better switching performance. So far, extensive analytical and numerical studies on electrostatically actuated micro-beams have been conducted, with or without consideration of the effect of geometrically nonlinear deformation [4–17]. When actuated by a suddenly applied step voltage or a harmonic-like voltage, a micro-beam undergoes dynamic deflection that must be taken into account for more accurate pull-in voltage predictions. Research in this area has received increasing attention due to its practical importance [17–21]. Based on a continuous model, Chao et al. [22] discussed the dynamic pull-in instability of a clamped–clamped micro-beam under a step voltage and found that the dynamic pull-in voltage is around 91–92% of the static value. Chen et al. [23] analyzed the dynamic behavior of microbeams using a hybrid differential transformation/finite difference method. The effect of nonlinear deformation was neglected in these two papers [22,23]. Using the reduced order model, Chaterjee and Pohit [17] investigated the deflection time history and dynamic pull-in behavior of microcantilever beams. Their results showed that large deflection should be considered for microbeams in and around pull-in zone. Xie et al. [24] performed the nonlinear dynamic analysis of a clamped–clamped micro-switch through the use of Galerkin method and the invariant manifold

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J. Yang et al. / Computers and Structures 96-97 (2012) 25–33

method. De and Aluru [25] applied full-Lagrangian schemes-based relaxation and Newton method to the nonlinear MEMS dynamic analysis and obtained resonant frequency, frequency response curves and switching speed (pull-in time) for cantilever and clamped–clamped micro-beams. For a micro-resonator under DC and AC excitation, Alsaleem et al. [26] presented numerical and experimental results revealing some nonlinear phenomena such as jumps and hysteresis, dynamic pull-in and escape and primary and secondary resonances. The aforementioned investigations [17–26] are for straight micro-beams only. The electrostatic and electrodynamic behaviors of micro-machined arches, buckled beams and chevron/V-shaped MEMS devices are a topic of active research in recent years due to their unique and attractive features. Zhang et al. [27] analytically studied the snap-through and pull-in instability of an arch-shaped micro-beam under electrostatic loading by using a single mode approximation for displacement function and expanding the electrostatic force into Taylor series up to the cubic order. Das and Batra [28] investigated the symmetry breaking, snap-through and pull-in instability of bi-stable arch-shaped MEMS under static and dynamic electric loads. The Galerkin method and a time integration scheme that adaptively adjusts the time step were used. Their study showed that the instability parameters of an electrically loaded arch with and without the effect of mechanical inertia are quite different. Younis and his co-workers [29,30] reported numerical and experimental results for the nonlinear dynamics of MEMS shallow arches under harmonic electric actuation. Krylov and Nick [31] considered the transient response and dynamic bistability region of an initially curved, shallow, clamped–clamped micro-beam, actuated by electrostatic and inertial forces. It should be pointed out that the above mentioned research works were based on Euler–Bernoulli beam theory and considered moderately large deflection or the mid-plane stretching effect of the microbeam only. Most recently, Hu et al. [32] suggested a curved beam model in an arc coordinate system for the static pull-in instability of micro-beams with an initial curvature which does not involve any assumption on nonlinear deformation and can be used for arbitrary initial curved configuration of the micro-beam. This paper further investigates the geometrically nonlinear electro-dynamic behavior of curved micro-beams with an arbitrary initial configuration using the curved beam model developed in our previous work [32]. The nonlinear governing equations of motion are derived in an arc coordinate system based on the equilibrium of an infinitesimal beam element. These nonlinear partial differential equations are solved by using the differential quadrature method (DQM) in the space domain and the Petzold–Gear BDF method in the time domain. The present analysis is validated through direct comparisons with existing results. The effects of nonlinear deformation, initial gap and initial rise of the curved micro-beam on its deflection time history, dynamic pull-in voltage, collapse mode and nonlinear fundamental frequency are discussed in detail.

Fig. 1. A typical curved MEMS switch.

dinate system is used to account for the effect of large deformation [32–34]. Fig. 2 defines an arc coordinate system used in the present analysis and its relationship with Cartesian coordinates where s 2 [0, l] is the arc coordinate along the beam axis, h0(s) is the angle between the tangential line and the y-axis at an arbitrary point C whose initial displacements along the x- and y-directions are denoted as w0(s) and u0(s), respectively. The domain occupied by the micro-beam at its initial state is C0 : fðx; yÞjx ¼ w0 ðsÞ; y ¼ s þ u0 ðsÞ; 0 6 s 6 lg. Under a time-varying voltage V(t), the induced attractive electric force is both position- and time- dependent and pulls the micro-beam towards the fixed electrode. Consequently, point C(w0, s + u0) on the unformed domain C0 moves to point C0 (w0 + w, s + u0 + u) on the deformed domain C : fðx; yÞjx ¼ w0 þ w; y ¼ s þ u0 þ u; 0 6 s 6 lg in which w = w(s, t), u = u(s, t) and h(s, t) are the additional displacements and angle of point C due to the electric excitation. The geometrical relationship for a curved beam with large deformation is

@u ¼ R1 cos h  cos h0 ; @s

@w ¼ R1 sin h  sin h0 @s

ð1Þ

where (R1  1) = (R1(s)  1) is the axial extension ratio. For an infinitesimal element of the curved micro-beam as shown in Fig. 3, the equations of motion can be obtained as

@ @ @2w ðN sin hÞ  ðQ cos hÞ ¼ m 2 @s @s @t

ð2aÞ

2. Theoretical formulations A typical curved MEMS switch is shown in Fig. 1 where the initially curved upper electrode is modeled as a clamped or cantilever micro-beam with an initial arc length l, width b, thickness h and radius of curvature R, separated by a dielectric space from a fixed flat ground electrode with gap g. The distance between the clamped end and the fixed electrode is denoted by g0 and that between the clamped end and the top of the curved beam at stress free state is the initial rise. For a curved micro-beam undergoing geometrically nonlinear deformation, it is convenient and thus preferred that the arc coor-

Fig. 2. Initial and deformed configurations in rectangular and arc coordinate systems.

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J. Yang et al. / Computers and Structures 96-97 (2012) 25–33

CF :

uð0Þ ¼ 0;

NðlÞ ¼ 0; CC :

wð0Þ ¼ 0;

Q ðlÞ ¼ 0; uð0Þ ¼ 0;

uðlÞ ¼ 0;

hð0Þ ¼ h0 ð0Þ; ð6Þ

dh ðlÞ ¼ 0 ds wð0Þ ¼ 0;

wðlÞ ¼ 0;

hð0Þ ¼ h0 ð0Þ;

ð7Þ

hðlÞ ¼ h0 ðlÞ

Introducing the following dimensionless quantities

ðS; U; W; U 0 ; W 0 ; G; BÞ ¼ ðk1 ; k2 ; k3 Þ ¼

ðs; u; w; u0 ; w0 ; g; bÞ ; l

ðNl; Ql; MÞl ; EI sffiffiffiffiffiffiffiffi

dh0  EI ;t ¼ t ; 4 dS ml   a GþU ; Fe ¼ 1 þ 0:65 B ðG þ UÞ2

v0 ¼

a¼

e0 bV 2 l 2EI

;

g¼

Al I

2

ð8Þ

Eq. (2) can be re-written in dimensionless form as

Fig. 3. An infinitesimal beam element in the arc coordinate system.

2

@ @ @2W ðk1 sin hÞ  ðk2 cos hÞ ¼ 2 @S @S @t

ð9aÞ

@ @ @2U ðk1 cos hÞ þ ðk2 sin hÞ þ F e ¼ 2 @S @S @t

ð9bÞ

@k3  k2 ¼ 0 @S

ð9cÞ

@ @ @ u ðN cos hÞ þ ðQ sin hÞ þ F e ¼ m 2 @s @s @t

ð2bÞ

Similarly, Eqs. (1) and (4b) become

@M Q ¼0 @s

ð2cÞ

@U ¼ R1 cos h  cos h0 @S

ð10aÞ

@W ¼ R1 sin h  sin h0 @S

ð10bÞ

@h  k3  v0 ¼ 0 @S

ð10cÞ

where N, M, Q are the axial force, bending moment and shear force, m is the mass density of the beam per unit length and Fe is the electric force per unit length [35]

Fe ¼

e0 bV 2  2ðg þ uÞ

2

g þ u 1 þ 0:65 b

ð3Þ

From Eq. (4a), R1 can be expressed in terms of k1 as

in which e0 = 8.85  1012 C2N1m2 is the permittivity of vacuum and the second term represents the fringing field effect. The constitutive law for linear elastic materials is

N ¼ EAðR1  1Þ   @h @h0  M ¼ EI @s @s

R1 ¼ 1 þ k1 =g

Substituting Eq. (11) into Eqs. (10a) and (10b) yields the following equations

k1 cos h ¼

  @U þ cos h0  cos h g @S

ð12aÞ

k1 sin h ¼

  @W þ sin h0  sin h g @S

ð12bÞ

ð4aÞ ð4bÞ

Here, A, EA and EI are the cross-sectional area, extensional rigidity and flexural rigidity of the micro-beam, respectively. The effective modulus E is the Young’s modulus for a narrow beam (b < 5h) and E/(1  m2) for a wide beam (b P 5h), m is Poisson’s ratio. The initial angle h0 defines the original configuration (w0, u0) of the micro-beam and satisfies

du0 ¼ cos h0 ðsÞ  cos hb ds

ð5aÞ

dw0 ¼ sin h0 ðsÞ  sin hb ds

ð5bÞ

where hb = 0 for a vertical micro-beam or hb ¼ p2 for the horizontal micro-beam considered in this paper. The micro-beam is assumed to be clamped at both ends (C–C) or clamped on the left end and free on the right end (C–F). The associated boundary conditions require

ð11Þ

Combining Eqs. (9c) and (10c) gives

k2 ¼

@2h @S2



@ v0 @S

ð13Þ

Eliminating R1 from Eqs. (10a) and (10b), one can also obtain

  @W @U þ sin h0 ¼ þ cos h0 tan h @S @S

ð14Þ

Substituting Eqs. (12) and (13) into Eqs. (9a) and (9b) leads to the following equations

  @ @W @ þ sin h0  sin h  g @S @S @S

! ! @ v0 @2W cos h ¼ 2  2 @S @t @S

@2h

ð15aÞ

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J. Yang et al. / Computers and Structures 96-97 (2012) 25–33

  @ @U @ þ cos h0  cos h þ g @S @S @S

! @ v0 @2U sin h þ  F ¼ e @S @t2 @S2

@2h

!

ð15bÞ Eqs. (14) and (15) constitute the dimensionless nonlinear governing equations of motion from which the linear governing equations can be easily recovered if the nonlinear terms in Eq. (15) are neglected. It is also worth noting that by setting h0 = 0, Eqs. (14) and (15) reduce to the nonlinear governing equations for straight micro-beams without initial curvature.The arbitrary initial curvature is defined by h0 which satisfies

dU 0 ¼ cos h0 ðSÞ  cos hb ds

ð16aÞ

ð1Þ

Aik W k þ sin h0i ¼

k¼1

g

n X



ð16bÞ

Wð0Þ ¼ 0;

hð0Þ ¼ h0 ð0Þ;

@hð1Þ ¼ 0; Fð1Þ ¼ k1 ð1Þ sin hð1Þ  k2 ð1Þ cos hð1Þ ¼ 0; @S Pð1Þ ¼ k1 ð1Þ cos hð1Þ þ k2 ð1Þ sin hð1Þ ¼ 0

g

n X

ð1Þ Aik

n X

þ

! ð1Þ

Aik U k þ cos h0i tan hi

ð22aÞ

! ð1Þ Akj W j

þ sin h0k  sin hk

j¼1 n X

n X

ð1Þ

Aik

ð2Þ

Akj hj 

j¼1

ð1Þ Aik

n X

n X

! ð1Þ

Akj

j¼1

n X

!

@2W

v0j cos hk ¼ 2 i @t

ð22bÞ

! ð1Þ Akj U j

þ cos h0k  cos hk

n X

n X

j¼1

k¼1

ð17Þ

n X k¼1

k¼1

The boundary conditions (6) and (7) can be expressed in dimensionless form as

Uð0Þ ¼ 0;

n X

k¼1

dW 0 ¼ sin h0 ðSÞ  sin hb ds

CF :

where {Um, Wm, hm} = {U(Sm, t), W(Sm, t), h(Sm, t)}, n is the total number of nodes distributed along the S-axis, lm(S) are the Lagrange ðkÞ interpolation polynomials, and Aim are the weighting coefficients dependent on the coordinates of the discrete points only and can be calculated through recursive formula [36]. Applying the above DQM approximation to Eqs. (14) and (15), a set of ordinary differential equations can be obtained

ð1Þ Aik

ð2Þ Akj hj



j¼1

k¼1

! ð1Þ Akj

j¼1

!

@2U

v0j sin hk þ F ei ¼ 2 i @t

ð22cÞ

in which, i = 1, 2, . . . , n, and

CC :

Uð0Þ ¼ 0;

Uð1Þ ¼ 0;

Wð0Þ ¼ 0;

Wð1Þ ¼ 0;

hð0Þ ¼ h0 ð0Þ;

hð1Þ ¼ h0 ð1Þ

ð18Þ

where F and P are the horizontal and vertical force components. For a micro-beam free at the right end (S = 1), F(1) = 0 and P(1) = 0. The following initial conditions are considered in the present analysis

UðS; 0Þ ¼ WðS; 0Þ ¼

@UðS; 0Þ @WðS; 0Þ ¼ ¼0 @t @t

ð19Þ

It should be noted that unlike other models within the framework of von Kármán nonlinearity where the equations of motion are established in the pre-deformed configuration hence are suitable for moderately large deformation only, the equations of motion in the present model are derived based on the deformed configuration without involving any assumption and can be used for any geometrically nonlinear problems. Also, due to the use of the arc coordinate system, the present nonlinear model is capable of describing the initial curved configuration of the micro-beam much more easily and conveniently. 3. Solution method

F ei ¼



a 2

ðG þ U i Þ

1 þ 0:65

G þ Ui B

 ð23Þ

The boundary conditions (17) and (18) can be approximated as follows

C  F : U 1 ¼ 0; W 1 ¼ 0; h1 ¼ h01 ; n X ð1Þ Ank hk ¼ 0; k1n sin hn  k2n cos hn ¼ 0; k1n cos hn  k2n sin hn ¼ 0 CC :

U 1 ¼ 0;

U n ¼ 0;

W n ¼ 0;

W 1 ¼ 0;

h1 ¼ h01 ;

ð25Þ

hn ¼ h0n

where h01 ¼ h0 ð0Þ; h0n ¼ h0 ð1Þ; k1n ¼ k1 ð1Þ; k2n ¼ k2 ð1Þ. Eqs. (22) and (24), (25) form a set of nonlinear differential equations in time from which the nonlinear dynamic behavior of curved micro-beams with given initial curvature can be numerically obtained. Combining Eqs. (22) and (24), (25) and putting them into a matrix form yields the following first order ordinary differential equation system

_ ¼0 Dð/; /Þ Analytical solutions of nonlinear governing Eqs. (14) and (15) under the associate boundary conditions in Eq. (17), (18) is very difficult if not impossible. The differential quadrature method (DQM) [32–34,36–38] is employed to numerically solve this problem. The fundamental idea behind DQM is to approximate an unknown function and its derivative at any discrete point as the linear weighted sums of its values at all of the discrete points chosen in the solution domain. According to DQM rules, the functions U, W, h and their kth derivatives with respect to S are approximated by

fU; W; hg ¼

n X

lm ðSÞfU m ; W m ; hm g

ð20Þ

m¼1

@

k

@Sk

fU; W; hgjS¼Si ¼

n X m¼1

ðkÞ

Aim fU m ; W m ; hm gði ¼ 1; 2;    ; nÞ

ð21Þ

ð24Þ

k¼1

ð26Þ

where unknown vector / is composed of all unknown variables at the discrete points, i.e.,

_ W _ T; / ¼ ½U; W; h; U; U ¼ ½U 1 ; . . . ; U n ;

_ W; _ h; € W € T _ U; /_ ¼ ½U;

W ¼ ½W 1 ; . . . ; W n ;

  @U 1 @U n ; . . . ; ; U_ ¼ @t @t " # 2 2 € ¼ @ U1 ; . . . ; @ Un ; U @t 2 @t 2

h ¼ ½h1 ; . . . ; hn ;

  _ ¼ @W 1 ; . . . ; @W n ; W @t @t " # 2 2 € ¼ @ W1 ; . . . ; @ Wn : W @t2 @t2

ð27Þ

The Petzold–Gear BDF method [39] is used to solve Eq. (26). As a special case, the nonlinear electrostatic equation can be obtained by neglecting the inertia terms on the right-hand-side of Eq. (22) and the Newton–Raphson method can be used to solve this equation to find the static pull-in voltage and pull-in deflection.

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J. Yang et al. / Computers and Structures 96-97 (2012) 25–33

2

4.1. Validation study

0

Present Ref. [25]

-2 -4 -6 -8 -10 -12 -14

0

3

6

9 12 Time (µ m)

15

18

Fig. 4. Tip deflection time history of a cantilever micro-beam under a step voltage of 0.5 V.

0 Mid-span deflection (nm)

To illustrate the convergence of the combined DQM and Petzold–Gear BDF method, Table 1 gives the tip deflections at t = 1, 3, 5, 7 ls with two different time steps (Dt = 1  103 ls, 0.5  103 ls) and varying total number of discrete points (n = 7, 11, 17, 21). As can be seen, the result is relatively more affected by the time step than the number of discrete points. The present method exhibits excellent convergence and the maximum relative error between the results with n = 7, Dt = 1  103 ls and those with n = 21, Dt = 0.5  103 ls is less than 8%0. In what follows, n = 21 and Dt = 0.5  103 ls are used. The present analysis is validated through a direct comparison with the nonlinear dynamic responses given by De and Aluru [25] using full-Lagrangian schemes. A clamped micro-beam and a cantilever micro-beam are considered. Both are straight beams having the same dimensions and material properties: l = 80 lm, h = 0.5 lm, b = 10 lm, E = 169 GPa, m = 2231 kg/m3, m = 0.3. The initial gap between the beam and the ground electrode is g = g0 = 0.7 lm. Fig. 4 shows the tip deflection time history of the cantilever micro-beam under a suddenly applied step voltage of 0.5 V. The dynamic deflection at the center of the clamped micro-beam under a step voltage of 2 V is plotted in Fig. 5. The solid lines and dots represent the present results and those by De and Aluru [25]. Good agreement is observed in both examples although our peak values are slightly lower than theirs. Table 2 lists the pull-in voltages of curved cantilever microbeams (R = 40000 lm, l = 100–500 lm, E = 153 GPa, b = 40 lm, h = 2.1 lm, g0 = 2.4 lm, e0 = 8.85 pF/m, m = 2231 kg/m3). A comparison between our results, the measured data reported by Gupta [5], and the analytical solutions by Wei et al. [7] and Hu et al. [8] shows that the present pull-in voltages are very close to the data obtained from the measurement, especially at small beam length (l 6 150 lm). Fig. 6 further displays the time histories of the tip deflection of the cantilever micro-beam under different step voltages in the vicinity of the state where the beam starts to collapse then adheres to the ground electrode (i.e. the dynamic pull-in state). When the system approaches pull-in instability, its effective stiffness is close to zero or, in other words, on the verge of losing elastic restoring capability. Around the critical pull-in point, a tiny voltage increase can cause very large displacement. It is found that dynamic pull-in instability occurs at 2.22 V. Chaterjee and Pohit [17] and De and Aluru [25] discussed the dynamic pull-in of this cantilever microbeam before and reported its pull-in voltage to be 2.25 and 2.12 V, respectively. In other words, our result is 4.5% higher than De and Aluru’s prediction using full-Lagrangian schemes and 1.35% lower than Chaterjee and Pohit’s solution using reduced order model. The dynamic mid-span deflection curves of the clamped micro-beam are given in Fig. 7 from which it is seen that pull-in

Tip deflection (nm)

4. Numerical results

Present Ref. [25]

-1 -2 -3 -4 -5

0

1

2

Time ( µ s)

3

Fig. 5. Mid-span deflection time history of a clamped micro-beam under a step voltage of 2 V.

takes place at 16.4 V. This value is only 3.8% bigger than the result given by De and Aluru [25] which is 15.7 V. Further validation is conducted in Table 3 which shows the dynamic pull-in voltage for a bell-shaped silicon arch micro-beam clamped at both ends with base length l0 = 1 mm, width b = 30 lm, thickness h = 2.4 lm, initial gap g0 = 10.1 lm, arch rise ˆ = 3 lm, m = 2231 kg/m3, e0 = 8.85 pF/m. The elastic constants H are the same as those used in Ref. [40]. The radius of curvature 2

^

2

Hþ0:5hÞ of this micro-beam is deduced as R ¼ ð0:5l0 Þ 2ð . The finite ele^ Hþh

ment result by Das and Batra [19], the analytical solution based on the reduced-order model (ROM) and the experimental data given

Table 1 Dynamic tip deflections of a cantilever micro-beam. Time (ls)

Dt (ls) 1 3 5 7

n=7

n = 11

n = 17

n = 21

1e3

0.5e3

1e3

0.5e3

1e3

0.5e3

1e3

0.5e3

1.43535 9.58419 11.5661 4.18036

1.43535 9.58419 11.5661 4.18036

1.43315 9.56285 11.5766 4.21382

1.42984 9.56285 11.5766 4.21382

1.43368 9.56271 11.5776 4.21398

1.43035 9.56271 11.5776 4.21398

1.43370 9.56268 11.5759 4.21408

1.43037 9.56268 11.5776 4.21408

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J. Yang et al. / Computers and Structures 96-97 (2012) 25–33

Table 2 Pull-in voltages (V) of curved cantilever micro-beams: Comparisons with existing numerical results and experimentally measured data. Wei et al. [7]

Hu et al. [8]

Gupta [5]

Present

100 125 150 175 200 225 250 300 400 500

81.09 66.72 48.18 36.74 29.48 24.30 20.79 16.12 11.69 9.690

75.51 49.67 35.65 27.25 21.76 18.01 15.30 11.82 8.360 6.770

72.07 48.60 35.82 27.89 22.55 18.79 15.95 12.61 9.100 7.270

74.4 49.0 35.2 26.9 21.4 17.8 15.1 11.6 8.3 6.7

Mid-span deflection (nm)

Length (lm)

0

-200

-400 step voltage : 16.3 V : 16.4 V : 16.5 V : 16.7 V

-600

-800

0.0

0.5

1.0 1.5 Time (µ m)

0

-200 -300

Table 3 Dynamic pull-in voltage of a clamped–clamped bell-shaped micro-beam.

-400 step voltage : 2.21 V : 2.22 V : 2.23 V : 2.24 V

-500 -600 -700 -800

2.5

Fig. 7. Mid-span deflection time history of a clamped micro-beam in the vicinity of pull-in state.

Experiment [40]

ROM solution [40]

FEM [19]

Present

106.0

111.0

92.0

102.1

2.5

0

5

10 Time (µ s)

15

1

20

Fig. 6. Tip deflection time history of a cantilever micro-beam in the vicinity of pullin state.

by Krylov et al. [40] are also provided. As can be seen, our result is in excellent agreement with the experimental result. 4.2. Electro-dynamic characteristics of curved beam

Mid-span deflection (µ m)

Tip deflection (nm)

-100

2.0

2.0

2

1.5

3

4

1.0 5 0.5 6

This section gives nonlinear electro-dynamic results for curved micro-beams. Unless otherwise stated, the parameters used are: h = 14.4 lm, b = 50 lm, g0 = 2 lm, R = 200 lm, l = 2000 lm, E = 169 GPa, m = 2231 kg/m3, m = 0.3. It is assumed that the mi

cro-beam, with an initial curvature defined by h0 ðsÞ ¼ p2  2Rl  Rs , is subjected to a suddenly applied step voltage. As the curvature 1/R considered in the following examples is small, the initial rise (the distance between the mid-span arc height and the clamped ends) is low compared with the arc span. Hence, the arc span can be approximately taken as the arc length l. Figs. 8 and 9 show, respectively, the mid-span deflection time histories of clamped curved micro-beams with g0 = 0 and 1 lm under a suddenly applied step voltage of different magnitudes. The mid-span deflection increases significantly as the step voltage increases. Before the voltage reaches a critical value, the micro-beam undergoes periodic motion but a further voltage increase beyond this point, even very small, will make the micro-beam abruptly collapse onto the ground electrode. This phenomenon is known as dynamic pull-in and the critical voltage value is called dynamic pull-in voltage which is 17.8 V for the micro-beam with g0 = 0 lm and is 35.5 V for the micro-beam with g0 = 1 lm in this example. This is due to the fact that a bigger g0 corresponds to a larger initial gap which in turn results in a much smaller electric force at the same applied voltage, indicating that a much higher voltage is required for the micro-beam with a bigger g0 to pull in.

0.0 0

10

20

30 40 Time (µ s)

step voltage 1: 10.0 V 2: 15.0 V 3: 17.5 V 4: 17.7 V 5: 17.8 V 6: 17.9 V 50

60

Fig. 8. Mid-span deflection time history of a clamped curved micro-beam (l = 2000 lm, R = 200 lm) under a suddenly applied step voltage: g0 = 0 lm.

The progressive dynamic pull-in process is clearly demonstrated in Fig. 10 where the deformed configurations at different times are given for the curved micro-beam with g0 = 0 lm excited by the dynamic pull-in voltage. The thick horizontal line at y = 0 represents the ground electrode. The micro-beam gradually deforms towards and finally touches down on the ground electrode about 41.04 ls after the application of the pull-in voltage. After that, the micro-beam keeps the touch-down configuration even when the voltage is further increased. Similar result has been observed for the case with g0 = 1 lm which is omitted for brevity. Table 4 compares the static and dynamic pull-in voltages of clamped curved micro-beams of different total arc lengths (l = 2000, 6000, 8000 lm) with or without the effect of nonlinear deformation. The initial rise corresponding to these arc lengths is 2.5, 22.5 and 40 lm, respectively. A longer micro-beam has a bigger initial rise hence its pull-in voltage is also higher. The dynamic pull-in voltage is less than the static pull-in voltage. The

31

J. Yang et al. / Computers and Structures 96-97 (2012) 25–33

1

2.5 2.0

3.0

1.5

2

2.5 2.0

3

1.5

4

1.0

5

0.5 0.0

step voltage 1: 10.0 V 2: 30.0 V 3: 33.3 V 4: 33.4 V 5: 33.5 V 6: 33.7 V

6 0

10

20

30 Time (µ s)

40

50

Initial configuration Linear analysis Nonlinear analysis

1.0

y-axis (µm)

Mid-span deflection (µ m)

3.5

0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5

60

Ground Electrode 0

500

1000 x-axis (µm)

1500

2000

(a)

Fig. 9. Mid-span deflection time history of a clamped curved micro-beam (l = 2000 lm, R = 200 lm) under a suddenly applied voltage: g0 = 1 lm.

2.5

0 µs

2.0

10 µ s

30 y-axis (µm)

y-axis (µm)

40

1.5

Initial configuration Linear analysis Nonlinear analysis

10

20 µs 1.0

20

35 µ s

0

40 µ s

0.5

Ground Electrode -10

41.04 µs 0.0

0

2000

Ground Electrode

0

500

1000 x-axis (µm)

1500

4000 x-axis (µm)

percentage difference between these two pull-in voltages, which increases as the beam length increases, is around 7–10% for all beams when nonlinear deformation is not included and for beams with l 6 6000 lm when nonlinear deformation is considered in the analysis. This is consistent with the results reported by Chao et al. [22] for DC driven micro-beams. The percentage difference for long micro-beam (l = 8000 lm) is considerably bigger and reaches as high as 32.3%. It can also be seen that the effect of nonlinear deformation on the pull-in voltage is insignificant and negligible for the short micro-beam (l = 2000 lm) but becomes much more pronounced for long micro-beams. Geometrically linear analysis under-estimates pull-in voltage although it gives almost the same result as the geometrically nonlinear analysis for the short microbeam (l = 2000 lm). For the long micro-beam (l = 8000 lm),

Fig. 11. Pull-in configuration of a clamped curved micro-beam (R = 200 lm, g0 = 2 lm) under step pull-in voltage: (a) l = 2000 lm; (b) l = 8000 lm.

however, the linear static and dynamic pull-in voltages are far below and only 43.1% and 52.6% of the results estimated by geometrically nonlinear analysis. These results indicate that nonlinear deformation cannot be ignored and must be taken into account in the analysis, especially for long curved micro-beams. Fig. 11 further displays the initial configuration and deformed configuration at pull-in state for the shortest and longest clamped curved micro-beams (R = 200 lm, g0 = 2 lm, l = 2000, 8000 lm). As can be seen, the collapse modes predicted by both linear and nonlinear analyses are very similar for l = 2000 lm but are quite different for l = 8000 lm. Under the dynamic pull-in voltage, the center of the shortest micro-beam deflects downwards and collapses onto the ground electrode. According to the linear analysis, this collapse mode happens to the longest micro-beam as well but the geometrically nonlinear analysis gives a different scenario where the beam

Table 4 Static and dynamic pull-in voltages with and without nonlinear deformation effect.

2000 6000 8000

8000

(b)

2000

Fig. 10. Progressive deformed configurations of a clamped curved micro-beam (l = 2000 lm, R = 200 lm, g0 = 0 lm) under step pull-in voltage.

Beam length l (lm)

6000

Static pull-in voltage (V)

Dynamic pull-in voltage (V)

Linear

Nonlinear

Linear

Nonlinear

55.1 65.3 77.3

55.2 96.2 179.3

50.3 60.1 71.3

50.4 87.1 135.5

32

J. Yang et al. / Computers and Structures 96-97 (2012) 25–33

90

step voltage 1: 10.0 V 2: 15.0 V 3: 17.7 V 4: 17.8 V 5: 17.9 V

60 Mid-span velocity

30 0 -30

3

2

1

-60 -90

4

5

-120 -150 -180 -2500

-2000

-1500 -1000 -500 Mid-span deflection (nm)

0

(a) 100

Mid-span velocity

50

0

step voltage 1: 10.0 V 2: 30.0 V 3: 33.3 V 4: 33.5 V 5: 33.7 V

-50

3

5. Conclusions

2

Geometrically nonlinear electro-dynamic analysis of curved micro-beams has been conducted based on the curved beam model and nonlinear governing equations derived in an arc coordinates. The differential quadrature method is employed to convert the partial differential equations into a set of ordinary differential equations in time. The Petzold–Gear BDF method is then used to solve these equations to obtain the nonlinear dynamic characteristics of the electrically actuated curved micro-beams. Numerical results show that

1

4 5

-100 -150

-3000

-2500

-2000 -1500 -1000 -500 Mid-span deflection (nm)

0

(b) Fig. 12. Phase portraits of a clamped curved micro-beam (l = 2000 lm; R = 200 lm) under various suddenly applied step voltage: (a) g0 = 0 lm; (b) g0 = 1 lm.

60 Fundamental frequency (KHz)

Fig. 12 gives the phase portraits of a clamped curved microbeam (l = 2000 lm, R = 200 lm) with g0 = 0, 1 lm and subjected to a suddenly applied step voltage of different magnitudes. Before pull-in, the micro-beam is in stable and periodic motion hence its phase portrait is seen to be convergent, periodic and symmetric. Once pull-in occurs, the phase portrait becomes unstable and divergent. This allows the dynamic pull-in voltage to be quantitatively determined from the phase portrait which is the voltage making the phase portrait start to diverge. Appling the Fast Fourier Transform (FFT) technique to the time history of the mid-span deflection of a curved micro-beam, its nonlinear fundamental frequency under a given step voltage can be obtained. Fig. 13 presents the nonlinear fundamental frequencies of clamped curved micro-beams (R = 200 lm, g0 = 0, 1 lm, l = 1500, 2000 lm) under various suddenly applied step voltages. The fundamental frequency decreases slightly at relatively small voltages but experiences a big and abrupt drop as the voltage is close to the dynamic pull-in voltage. Given the same voltage, a shorter micro-beam with a bigger initial gap parameter g0 has higher nonlinear fundamental frequency than a longer micro-beam with a smaller initial gap parameter g0.

50

1: g0 = 0 µ m

(1) Geometrically nonlinear deformation becomes more important as the beam length increases and must be taken into consideration for long curved micro-beams. Linear analysis may significantly under-predict the dynamic pull-in voltage and give incorrect collapse mode. (2) The dynamic pull-in voltage is slightly lower than the static pull-in voltage for short micro-beam but is considerably lower for long micro-beams. (3) A curved micro-beam with a bigger initial gap parameter g0 has a larger dynamic pull-in voltage and higher nonlinear fundamental frequency. (4) The dynamic pull-in voltage increases but nonlinear fundamental frequency decreases as the beam length increases.

2: g0 = 1 µ m 40

1b

a: l = 2000 µ m b: l = 1500 µ m

Acknowledgement

2b 30

20

1a 0

5

10

15 20 25 Step voltage (V)

2a 30

35

Fig. 13. Nonlinear fundamental frequencies of clamped curved micro-beams (R = 200 lm) under various suddenly applied step voltages.

starts to quickly moves down to the ground electrode near one quarter and three-quarters of the beam length from the left end.

The work described in this paper was supported by a research grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 113809]. The corresponding author (Dr. J. Yang) is grateful for the financial support of the Research Leave Award from RMIT university. The co-author (Dr. Y.J. Hu) would like to acknowledge the support from the National Science Foundation for Distinguished Young Scholars of China (Project No. 11002084) and the Innovation Program of Shanghai Municipal Education Commission (Project No. 12YZ092). References [1] Senturia S. Mircrosystem design. Kluwe, MA: Norwell; 2001. [2] Taylor GI. The coalescence of closely spaced drops when they are at different electric potentials. Proc Royal Soc A 1968;306:423–34.

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