Forced Vibration of Electrically Actuated FGM Micro-Switches

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ProcediaEngi Engineering 14(201 (2011) 280–287 Procedia ineering 00 1) 000–000

Proceedia Engineeering www w.elsevier.com/loocate/procedia

The Twellfth East Assia-Pacific Conference C on Structurral Engineerring and Coonstruction

Forcced Vibrration off Electri cally Acctuated FGM F M MicroSwittches X. L. JIA A1a, J. YAN NG2, S. KIT TIPORNCH HAI1 and C.W C LIM1 1 2

Depaartment of Buildin ng and Constructiion, City Universsity of Hong Kong g, China

School of Aeerospace, Mechannical and Manufa acturing Engineerring, RMIT Univeersity, PO Box 71 1, Bundoora, VIC C 3083 Australia

Abstract

This paper presents an analytical stu udy on the fforced vibration of micro--switches undder combined d d axial residuaal stress. The micro-switch considered inn this study iss electrostaticc, intermolecular forces and wo material ph hases. The nonnlinear partiall made of nonn-homogeneouus functionallly graded matterial with tw differential equation whiich describes the forced vvibration of th he micro-beam m is derived based on thee and are solved d framework oof von Karmaan-type geomeetric nonlineaarity and Euler-Bernoulli beeam theory, an using the m method of averraging. The modulations m off the amplitud de of clamped-clamped miccro-switch aree obtained. T The present reesults are vallidated througgh direct com mparisons witth published experimentall ment has been achieved. A parametric stu udy is conduccted to show the effects off results. Excellent agreem mposition andd AC harmon nic force on tthe frequency y response ch haracteristics oof the micro-material com switch. © 2011 Published by Elseevier Ltd. Forced vibratiion, Frequency y response, M Micro-beam, Fu unctionally grraded materiall. Keywords: F

1.

Introdu uction

Nonlineaar dynamic behavior b has been experim mentally obseerved in man ny micro- annd nano-scalee mechanical devices (Turrner et al. 1998; Craigheead 2000), which w necessitates the neeeds of a fulll g understandinng of this beehavior in engineering dessign and real applications.. Tilmans andd Legtenberg studied the ddynamic probblem of a reson nator using m modified Ritz method m and th heir theoreticaal results weree compared w with experim mental results (Tilmans annd Legtenberrg 1994). Yo ounis and Naayfeh (2003)) employed m multiple scale method to an nalyze the ressponse of the resonant miccro-beam undder an electricc actuation. A mathematicaal model wass proposed annd used to ex xamine the micro-beam dyynamics using g

a

Corresponding author & Presenter: xiaoljia@stu udent.cityu.edu.hhk

1877–7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.07.034

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2

DQM-FDM M method by Najar et al. (2006). To simultaneoussly meet all material, ecoonomical and d performancee requirementts for an MEM MS structural layer, Witvro ouw and Mehtta (2005) propposed a novell non-homogeenous functionally graded layer polycryystalline-SiGee (poly-SiGe) layer in MEM MS. Materiall properties off the two-phasse MEMS were assumed too vary continu uously in the th hickness direcction (Ke et all. 2009). The size-dependeent static and free vibratioon behavior of o ultra-thin films f made off functionally y he basis of a ggeneralized reffine theory (Lü C F et al. 20009). graded mateerials was inveestigated on th The objecctive of this paper p is to preesent a forced vibration anaalysis for non--homogeneouss functionally y graded mateerial (FGM) micro-switche m es under comb mbined intermo olecular Casm mir and electrrostatic forcess within the frramework of von v Karman-ttype geometri c nonlinearity y and Euler- Bernoulli B beam m theory, with h an emphasiss on the effeccts of materiaal compositionn and AC harrmonic force on the nonlinnear dynamicc characteristiics of the micrro-switches. The T method off averaging is used to obtain n the frequenccy-response. 2.

Theoreetical formulaation

2.1. Governing equation Shown inn Figure 1 is the t structure of o a typical miicro-switch wh here the key components c innclude a fixed d electrode m modeled as a ground g plane and a movabble electrode modeled as a micro-beam m of length L,, width b, andd thickness h, separated by y a dielectric spacer with an a initial gap g0. The axiall force due to o residual straain from fabrication process is denotedd by Na and iss positive forr a tensile forrce. Upon thee application of an appliedd voltage V wh hich consists of a DC com mponent V0 and d a small timee-varying AC C he micro-beam m deflects to owards the fix xed electrodee and vibratess component ΔV (t ) = VACC cos(Ωt ) , th under the acction of distribbuted electrosttatic force Fe and Casmir fo orce Fc. A rectangular coord rdinate system m is used wherre the origin iss located at th he left end of tthe movable electrode and the t deflection is denoted by y w.

Figure 1: A beaam model for a micro-switch. m

In this ppaper, a micrro-beam with h a functionaally graded poly-SiGe p lay yer structure proposed by y (Witvrouw aand Mehta 20005) is consid dered where ggermanium (G Ge) and silicon n (Si) are useed as materiall phase 1 andd material phase 2, respectiv vely. The voluume fraction of o germanium m V1 and that of silicon V2 are related bby (Reddy 20000)

2z + h ⎞ V1 + V2 = 1, V2 ( z ) = ⎛⎜ ⎟ ⎝ 2h ⎠

n

(1))

where superrscript n is a power law index i that chharacterizes th he volume fraaction profile through the thickness. T The mass densiity is assumed d to change aloong the thickn ness

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2z + h ⎞ ρ ( z) = ( ρ 2 − ρ1 ) ⎛⎜ ⎟ + ρ1 ⎝ 2h ⎠

(2)

K is predicted by (Mori and Tanaka 1973)

The effective bulk modulus

K − K1 V2 = K 2 − K1 1 + (1 − V2 )( K 2 − K1 ) /( K1 + 3 4 μ1 ) where the bulk modulus

Ki =

(3)

Ki and shear modulus μi can be expressed as follows

Ei Ei , μi = , i = 1, 2 3(1 − 2ν i ) 2(1 +ν i )

(4)

Table 1: Material properties of germanium (Ge) and silicon (Si)

Material

Ei (GPa)

Ge Si

132 173

νi

ρi ( Kg i m −3 ) 3

0.26 0.26

5.33×10 2.33×103

μi (GPa)

K i (GPa)

52.38 68.65

91.67 120.14

The material properties of both germanium and silicon (Witvrouw and Mehta 2005) are listed in Table 1. The effective Young’s modulus E can be deduced from Equations (1)- (4)

⎡ ⎤ V2 ( z )( K 2 − K1 ) 3(1 − 2ν i ) ⎢ + K1 ⎥ E( z) = ⎣1 + (1 − V2 ( z ))( K 2 − K1 ) /( K1 + 3 4 μ1 ) ⎦

(5)

The electrostatic force and Casimir force per unit length can be written as

= Fe

ε 0bV 2

2( g 0 − w) 2

+

0.65ε 0V 2 π 2 hcb , Fc = 2( g 0 − w) 240( g 0 − w) 4

(6)

where ε 0 = 8.854 ×10 C N m is the permittivity of vacuum, = h 1.055 × 10 Js is Planck’s 8 −1 constant divided by 2π and c = 3 ×10 ms is the speed of light. Based on the principle of virtual work, the non-dimensional dynamic equation of an FGM micro-beam accounting for a viscous damping cd per unit length can be expressed as (Jia et al. submitted) -12

2

-1

−2

−34

2 ∂ 2 w ∂ 2 w 1 ⎡ ⎛ ∂w ⎞ ∂2w ⎤ + 4 − N a 2 − 2 ∫ ⎢ζ 1 ⎜ − ζ ⎥dx 2 ⎟ ∂t ∂x ∂x ∂x 0 ⎣⎢ ⎝ ∂x ⎠ ∂x 2 ⎦⎥

ρl L4 ∂ 2 w cd L4 ∂w ∂ 4 w k

∂t 2

+

k

B [V0 + VAC cos(Ωt ) ] fB [V0 + VAC cos(Ωt ) ] Rc = + + 2 (1 − w) 1− w (1 − w) 4 2

2

(7)

in which h 2 h − 2

ρl = ∫ ρ bdz,

ζ 2 = k2 g0 k

h 2 h − 2

k1 = ∫

ˆ , Ebdz

h 2 h − 2

k2 = ∫

ˆ Ebzdz ,

h 2 h − 2

ˆ 2 dz , k3 = ∫ Ebz

ζ 1 = k1 g02 2k , (8)

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(

)

= Eˆ E / 1 −ν for a wide where the effective modulus Eˆ = E for a narrow beam ( b < 5h ) and beam ( b ≥ 5h ). The associated boundary conditions of the micro-beam are w = 0, ∂w ∂x = 0 ( x = 0,1) . And the following dimensionless quantities in Equation (7) are w=

2

g w x ε bL4 k2 π 2 hcbL4 N a L2 , x = , k= k3 − 2 , B = 0 3 , f = 0.65 0 , Rc = , N = a g0 L b k1 2 g0 k 240 g 05 k k

2.2. Galerkin procedure In order to transform Equation (7) into a finite-degree-of-freedom system with ordinary differential equations in time by Galerkin procedure, the deflection is expressed by m

w ( x ,τ ) = ∑ yi (τ )φi ( x )

(10)

i =1

where yi (τ ) is the kth generalized coordinate and φi ( x ) is the 1 kth linear undamped mode shape of the straight micro-beam which satisfies orthogonality condition φiφ j dx = δ ij where δ = 0 if i ≠ j and δ = 1 if i = j . Multiplying Equation (7) by φ j ( x ) [1 − w( x )]0 4 to include the complete contribution of the nonlinear force and integrating from x = 0 to 1, a set of ordinary differential equations in time can be obtained. Assuming that the first mode is dominant while the effect of other modes is insignificant and can be neglected, one has

∫

1

τ = t T , T = ( ρl L4 k μ1 ) 2 , ξ = cd T 2ρl

(11)

Finally, Equation (7) can be expressed as

y + y + ( −4 χ 3 yy + 6 χ 4 y 2 y − 4 χ 5 y 3 y + χ 6 y 4 y ) + 2ξ ( y − 4 χ 3 yy + 6 χ 4 y 2 y − 4 χ 5 y 3 y + χ 6 y 4 y )

+ ( ε 2 y 2 + ε 3 y 3 + ε 4 y 4 + ε 5 y 5 + ε 6 y 6 + ε 7 y 7 ) =F0 + (η0 + η1 y + η2 y 2 + η3 y 3 ) cos(Ωt ) 2

2

(12)

2

where the term involving VAC is dropped since normally VAC  V0 for resonance of micro-beam. The normalized fundamental frequency ω1 = 1 , and the expression of all parameters in Equation (12) are 1

1

1

0

0

1

1

1

χ1 = ∫ (φ1′) dx, χ2 = ∫ φ1′′dx, χ3 = ∫ φ13dx, χ4 = ∫ φ14 dx, χ5 = ∫ φ15 dx, χ6 = ∫ φ16 dx, 2

0

1

1

0

0

0

1

0

1

0

1

χ7 = ∫ φ1( 4)φ1dx, χ8 = ∫ φ1( 4)φ12 dx, χ9 = ∫ φ1( 4)φ13dx, χ10 = ∫ φ1( 4)φ14 dx, χ11 = ∫ φ1( 4)φ15dx, 1

χ12 = ∫ φ1′φ′ 1dx, 0 1

χ17 = ∫ φ1dx

0

1

χ13 = ∫ φ1′φ′ 12 dx, 0

0

1

χ14 = ∫ φ ′′φ13dx, 0

0

1

χ15 = ∫ φ1′φ′ 14 dx, 0

1

χ16 = ∫ φ ′′φ15dx, 0

0

(13)

μ1 = χ7 − Na χ12 + (2 + 3 f ) BV02 , μ2 =−4 χ8 + 4 N a χ13 + ζ 2 χ 2 χ12 − (1 + 3 f ) χ3 BV02 , μ3 = 6 χ9 − 6 N a χ14 − ζ 1 χ1χ12 − 4ζ 2 χ 2 χ13 + f χ 4 BV02 , μ4 = 4 χ10 + 4 N a χ15 + 4ζ 1χ1χ13 + 6ζ 2 χ 2 χ14 , μ5 = χ11 − N a χ16 − 6ζ 1 χ1χ14 − 4ζ 2 χ 2 χ15 , μ6 = 4ζ 1χ1χ15 + ζ 2 χ2 χ16 , μ7 = −ζ 1χ1χ16

(14)

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F0 = Rc χ17 μ1 , η= 2 (1 + f ) χ17 BV0VAC μ1 , η1 = −2 ( 2 + 3 f ) BV0VAC μ1 , 0

η= 2 (1 + 3 f ) χ3 BV0VAC μ1 , η3 = −2χ 4 fBV0VAC μ1 2 = ε k μ= 1, 2,...7 ) k μ1 ( k

(15) (9)

2.3. Average method The method of averaging is employed in this study to analyze the dynamic equation (Nayfeh 1993). It is noted that near-resonant behavior is the principal operation regime of the proposed system, hence a detuning parameter σ1 is introduced as 2 Ω= ω12 (1 + εσ 1 )

(17)

y + Ω2 y = − ( −4χ3 yy + 6χ4 y2 y − 4χ5 y3 y + χ6 y4 y ) − 2ξ ( y − 4χ3 yy + 6χ4 y2 y − 4χ5 y3 y + χ6 y4 y ) +εσ1ω12 y − ( ε2 y2 + ε3 y3 + ε4 y4 + ε5 y5 + ε6 y6 + ε7 y7 ) + F0 + (η0 +η1 y +η2 y2 +η3 y3 ) cos(Ωτ ) (18) To facilitate the perturbation method of averaging, the following constrained coordinate transformation is introduced

y =− AΩ2 cosψ − A Ω sinψ + AΩθ cosψ y = A cosψ , y =− AΩ sinψ , 

(19)

in which ψ =Ωτ + θ . Substituting Equation (19) for y , y and y in Equation (18), and applying the method of averaging yields the frequency-response equation

⎛ Ω2 ( 8A +12A3χ4 + A5 χ6 ) ⎞ ⎛ 48ε A3 + 40ε A5 + 35ε A7 − 64 A(Ω2 − ω2 ) − 288A3Ω2 χ − 40 A5Ω2 χ ⎞2 5 7 1 4 6 ⎜ ⎟ +⎜ 3 1 (20) ⎟ = 2 2 ⎜ ⎟ 60 η 48 A η + 2 Q 4 η A η + ( ) 0 2 ⎝ ⎠ 0 2 ⎝ ⎠ 2

where the parameter 3.

ξ is related to the quality factor Q by ξ = Ω 2Q .

Result and discussion

3.1. Comparison and Verfication The geometric and material parameters of the homogeneous poly-silicon micro-beams considered in this section are given in Table 2. The condition for resonance is dA dΩ = 0 in Equation (20), where the magnitude of response is at its peak Ar and we define the normalized nonlinear resonance frequency as Ω r ω1 . In Figure 2, results for Ω r ω1 obtained from present analysis are compared with the existing theoretical and experimental ones reported by Tilmans and Legtenberg (1994). The D.C. polarization voltage is V0 = 2V for the first three points. The other points were measured under V0 = 1V . The quality factors Q = 592 were extracted using an equation derived from their analysis, whose theoretical results are not quantitatively in good agreement with the experiments. The discrepancy is due to the variations in the sealing pressure of micro-beam encapsulation which can lead to a wrong measurement of Q ( Tilmans and Legtenberg 1994). Here, the quality factor is determined to be Q = 269 by matching the Ωr ω1 value obtained from Equation (20) to the experimental value. Using this value, our results are in good agreement with experimental data.

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Table 2: Geometric and material parameters of the poly-silicon micro-beam

L (μm) 210

b (μm) 100

g0 (μm) 1.18

h (μm) 1.5

E (GPa) 151

 

Ω r / ω1

1.015

Experiment (Tilmans and Legtenberg 1994) Theory (Tilmans and Legtenberg 1994) Present

1.010

1.005

1.000 0.0

0.1

0.2

0.3

0.4 0.5 VAC(V)

0.6

0.7

0.8

Figure 2: Comparison of the normalized resonance frequency vs. the A.C. drive voltage.

The frequency response for FGM micro-beams with different volume fraction index is shown in Figure 3. The jump phenomenon occurs and it is due to the hardening effect from nonlinear deformation. It is found that decreasing volume fraction index n will enhance the hardening effect of the nonlinearity, decrease the resonance amplitude Ar defined as the peak point of the frequency-response curve, and it will slightly decrease the normalized nonlinear resonance frequency. Figure 4 shows the frequency-response curves at different AC voltage VAC when n=5. The jump phenomenon occurs when VAC is sufficiently high. Increasing the amplitude of the AC voltage from 0.01V to 0.2V moves the left and right parts of the frequency-response curve further away from each other, and pushes both the resonant amplitude Ar and normalized nonlinear resonance frequency to higher values.

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3.2. FGM Micro-beams 0.10

 

0.08

V 0 =23V

V A C =0.01V

A

0.06 0.04

n=0 n=1 n=5 n=20

0.02 0.00 0.998

1.000

1.002

1.004

Ω / ω1 Figure 3: Frequency-response curves for various volume fraction index n.

0.12 0.10

V0=1V

n=0

VAC=0.2

A

0.08 VAC=0.1

0.06 0.04

VAC=0.05

0.02 0.00 0.98

VAC=0.01 0.99

1.00

1.01

1.02

Ω / ω1 Figure 4: Frequency-response curves for various AC voltages

4.

VAC .

Conclusions

Forced vibration of micro-switches under combined electrostatic, intermolecular forces and axial residual stress is investigated using Galerkin procedure and the method of averaging. An analytical study which accounts for both force nonlinearity and geometric nonlinearity has been conducted to examine the effects of material composition and AC harmonic force on the frequency response characteristics. It is found that (1) the stiffness of micro-beam will increase as the volume fraction n decreases and the AC voltage increases; and (2) a higher volume fraction n and AC voltage can result in higher resonance amplitude and normalized nonlinear resonance frequency.

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