Research ArticleNonlinear Vibration Analysis of DamagedMicroplate consideringSize Effect

Jihai Yuan12 Xiangmin Zhang2 and Changping Chen 12

1Department of Civil Engineering Xiamen University Xiamen Fujian 361005 China2School of Civil Engineering and Architecture Xiamen University of Technology Xiamen Fujian 361024 China

Correspondence should be addressed to Changping Chen cpchenhnueducn

Received 18 May 2020 Revised 1 August 2020 Accepted 11 August 2020 Published 26 August 2020

Academic Editor Marco Alfano

Copyright copy 2020 Jihai Yuan et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Since microplates are extensively used in MEMS devices such as microbumps micromirrors and microphones this work aims tostudy nonlinear vibration of an electrically actuatedmicroplate whose four edges are clamped Based on the modified couple stresstheory (MCST) and strain equivalent assumption size effect and damage are taken into consideration in the present model edynamic governing partial differential equations of the microplate system were obtained using Hamiltonrsquos principle and solvedusing the harmonic balance method after they are transformed into ordinary differential equation with regard to time Size effectand damage effect on nonlinear free vibration of the microplate under DC voltage are discussed using frequency-response curveIn the forced vibration analysis the frequency-response curves were also employed for the purpose of highlighting the influence ofdifferent physical parameters such as external excitation damping coefficient material length scale parameter and damagevariable when the system is under AC voltage e results presented in this study may be helpful and useful for the dynamicstability of a electrically actuated microplate system

1 Introduction

Since a microelectromechanical system (MEMS) receiveda considerable amount of attention in recent years nu-merous works related to nonlinear responses and charac-teristics because of intrinsic existence of nonlinearity ofthese microdevices have been carried out in recent yearsUsing the multiple scale method Younis and Nayfeh [1]studied nonlinear vibration of a resonant microbeam underdynamic electrostatic force Based on the same methodAbdel-Rahman and Nayfeh [2] studied response ofa microresonant sensor actuated by superharmonic andsubharmonic electric forces and their results provide ananalytical solution to predict the resonant response Zhangand Meng [3] proposed a simplified model in order to studythe resonant responses and nonlinear dynamics of micro-cantilever under electronic excitation An electromechanicalcoupled nonlinear dynamic response was presented by Xuand Jia [4] who employed the perturbation method todiscuss influence of mechanical and electric parameters on

nonlinear natural frequencies and vibrating amplitudes ofa microbeam Vogl and Nayfeh [5] reported the response ofthe clamped plate with circular shape under primary res-onance excitation using a reduced-order model which isgeneral enough for effectively designing capacitive micro-machined ultrasonic transducers Nayfeh et al [6] developeda novel model for a resonant gas sensor and studied itsnonlinear dynamic characteristics Jia et al [7] conducteda parametric study on forced vibration of microswitches toshow effects of Casimir force residual stress and geo-metrical nonlinearity on the frequency response charac-teristics Kim et al [8] examined resonant behaviors ofa microbeam subjected to axial force and electrostatic forceSaghir and Younis [9] conducted a study on nonlinear vi-bration behavior of rectangular microplate under static anddynamic load ey found an interesting phenomenonwhere the microplate shows a hardening behavior whichswitches to softening as the DC load increases Sheikhlouet al [10] investigated nonlinear resonant behavior of di-aphragm-type micropumps

HindawiShock and VibrationVolume 2020 Article ID 8897987 13 pageshttpsdoiorg10115520208897987

However size-dependent behavior of microstructureshas been experimentally validated [11] With this con-sideration some interesting studies related to vibrationbehavior are presented Using strain gradient theory andshear deformation theory Arefi and Zenkour [12] pre-sented free vibration of a piezoelectric laminated mi-crobeam Based on nonlocal piezoelasticity theory Arefiand Zenkour presented size-dependent bending and vi-bration response of a sandwich piezomagnetic nanobeamwith curvature [13] and without curvature [14] ey alsoinvestigated thermo-electro-magneto-mechanical bend-ing behavior of a Sandwich nanoplate with simply-supported boundary conditions [15] A nonclassicalmicroplate model was proposed by Ke et al [16] whoemployed MCST [17] to study nonlinear free vibration ofannular microplate Using the same theory Ghayesh et al[18] analyzed the nonlinear behavior of a size-dependentresonator under primary and superharmonic excitationsUsing Eringen nonlocal theory Arani and Jafari [19]analyzed nonlinear vibration of sandwich microplatesresting on an elastic matrix With application of differentkinds of materials such as exponentially graded piezo-electric and magnetic Sobhy and Zenkour [20] Arefi andZenkour [21] and Arefi et al [22] investigated size-de-pendent free vibration of sandwich microplate resting ondifferent foundations e nonlinear size-dependentstatic and dynamic behaviors of a MEMS device werepresented by Farokhi and Ghayesh [23] on the basis ofMCST e free vibration characteristics of microplatewith size dependency were displayed by Tahani et al [24]based on the MCSTrecently with consideration of variouseffects such as couple stress components electrostaticattraction and different boundary conditions on bothmode shapes and natural frequencies Unlike the previousresearch Veysi et al [25] presented vibrational behaviorof microdoubly curved shallow shells incorporated withvon-Karman geometric nonlinearity size dependenceand shear deformation by adopting multiple scalemethod

In the meantime during manufacturing and operatingprocess damage that is fatigue damage [26] discrete brittledamage [27] and contact damage [28] may occur due to thevery existence of microcracks or abrasions which may evolveand develop [29] However most of these works ignored thesignificant role of size effect in mechanical behavior ofmicrostructures Actually nonlinear size-dependent be-havior of microstructure considering damage effect is lim-ited in current literatures Hence a better understanding ofdamage and microscale effect on mechanical behavior ofmicrostructure is necessary With this motivation this workaims to study size-dependent nonlinear free vibration andforced vibration behavior for microplates with damageunder electrostatic actuation In this research the damageconstitutive relations are established based on the strainequivalent assumption [30] and MCST e governingdifferential equations were derived via Hamiltonrsquos principleand solved numerically by Galerkinrsquos method and theharmonic balance method (HBM) e influence of various

system parameters on nonlinear vibration of the microplateis studied comprehensively

2 Mathematical Modeling

Consider a movable microplate with mass density ρ overa stationary electrode under electronic force in Figure 1where its width length and thickness are b a and h sep-arately e initial gap and external voltage between theelectrode and the microplate are d and VC accordingly emidsurface of the microplate is regarded as the referenceplane z 0 as shown in this figure

With the introduction of only one extra material pa-rameter material length scale parameter Yang et al [17]proposed MCST which claims the strain energy of anelastomer with volume V is expressed as

U 12

1113946Vσijεij + mijχij1113872 1113873dV (i j x y z) (1)

where stress tensor (σij) couple stress tensor (mij) straintensor (εij) and symmetric curvature tensor (χij) are givenas follows

σij λtr εij1113872 1113873δij + 2μεij (2)

mij 2l2μχij (3)

εij 12

uij + uji1113872 1113873 (4)

χij 12

θij + θji1113872 1113873 (5)

where ui represents displacement vector ere are twoconstitutive relations one is between Cauchyrsquos stress andinfinitesimal engineering strain represented by (2) and theother is between couple stress and symmetric curvaturerepresented (3) e former one is traditional elastic con-stitutive relation with lame constant λ μ e latter one isextra constitutive relation accounting for size dependencyusing material length scale parameter l e gyration vector(θi) is written as

θi 12eijkukj (6)

where eijk is the permutation tensorKirchhoffrsquos displacement components u1 u2 u3 at

a point of plate can be defined as

u1(x y z t) u(x y t) minus zwx

u2(x y z t) v(x y t) minus zwy

u3(x y z t) w(x y t)

(7)

where u v andw donate the displacements of the referenceplane separately For microplate encounters large deflectionthe von Karmanrsquos plate theory is adopted to describe thegeometric nonlinearity of the microplate us the strain-displacement relations can be given as

2 Shock and Vibration

εx ε0x + zκx

εy ε0y + zκy

εxy ε0xy + zκxy

(8)

where stains ε0x ε0y and ε0xy and curvatures κx κy and κxy ata point of the middle surface are expressed as

ε0x

ε0y

ε0xy

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎭

ux +12w

2x

vy +12w

2y

uy + vx + wxwy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

κx

κy

κxy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

minus wxx

minus wyy

minus 2wxy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(9)

Substituting (7) into (6) one has gyration vector

θx(x y) wy

θy(x y) minus wx

θz(x y) 12

vx minus uy1113872 1113873

(10)

Substituting (10) into (2) yields curvature tensor

χxx

χyy

χxy

χxz

χyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

wxy

minus wxy

12

wyy minus wxx1113872 1113873

14

vxx minus uxy1113872 1113873

14

vxy minus uyy1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11)

e stress-strain relations can be given with respect toPoissonrsquos ratio υ and Youngrsquos modulus E as

σx

σy

σxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

Qd

bull

εx

εy

εxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(12)

mxx myy mxy mxz myz1113872 1113873 F χxx χyy χxy χxz χyz1113872 1113873

(13)

where

Qd

Qd11 Q

d12 0

Qd22 0

Qd66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

E

1 minus υ2υE

1 minus υ20

E

1 minus υ20

G

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1 minus D)

F 2l2G(1 minus D)

(14)

where D denotes damage variable varying from 0 to 1 [30] Gis shear modulus

With Cauchyrsquos stress and couple stress expressed abovebending moments Mx My Mxy membrane stress re-sultants Nx Ny Nxy and couple momentsYxx Yyy Yxy Yxz Yyz can be respectively defined as

Mx My Mxy1113960 1113961 1113946h2

minus (h2)z σx σy σxy1113960 1113961dz

Nx Ny Nxy1113960 1113961 1113946h2

minus (h2)

σx σy σxy1113960 1113961dz

Yxx Yyy Yxy Yxz Yyz1113960 1113961 1113946h2

minus (h2)

mxx myy mxy mxz myz1113960 1113961dz

(15)

Substituting (12) and (13) into (15) yields the followingconstitutive equations

Nx

Ny

Nxy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

A11 A12 0

A12 A22 0

0 0 A66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ε0x

ε0y

ε0xy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

Mx

My

Mxy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

D11 D12 0

D12 D22 0

0 0 D66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

κx

κy

κxy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

Yxx Yyy Yxy Yxz Yyz1113960 1113961 G χxx χyy χxy χxz χyz1113960 1113961

(16)

where

o a X

b

y

z

h

d

Movable plate

Stationary electrode

Vc

Figure 1 Electrically actuated microplate

Shock and Vibration 3

Aij hQdij

Dij h3

121113888 1113889Q

dij

Gprime hF

(i j 1 2 6)

(17)

With Hamiltonrsquos principle we have

δ1113946t2

t1

(T minus U + W)dt 0 (18)

Substituting stresses and strains into (1) the virtualstrain energy is obtained as

δU 1113946Vσijδεij1113872 1113873dV + 1113946

Vmijδχij1113872 1113873dV (19)

e virtual kinetic energy is given as

δT 12ρ1113946

Vδ wt( 1113857

2dV (20)

e virtual work done by electric force and dampingforce can be expressed as

δW Bs

Fz minus cwt( 1113857δwdS (21)

where c denotes damping coefficient and Fz represents theelectric force defined as

Fz εvV

2C

2(d minus w)2 (22)

where εv is the dielectric constant of airBy substituting (19)ndash(21) into (18) one has the nonlinear

dynamic equations of the microplate system as

Nxx + Nxyy +12Yxzxy +

12Yyzyy 0

Nyy + Nxyx minus12Yxzxx minus

12Yyzxy 0

minus Nxwx + Nxywy1113872 1113873x

minus Nywy + Nxywx1113872 1113873y

+ minus Mx minus Yxy1113872 1113873xx

+ minus My + Yxy1113872 1113873yy

+ minus 2Mxy + Yxx minus Yyy1113872 1113873xy

+ Fe ρhwtt + cwt

(23)

Boundary conditions are as follows

Nx +12Yxzy 0

or δu 0

Nxy minus12Yxzx minus

12Yyzy 0

or δv 0

12Yxz 0

or δvx 0

Mxx + 2Mxyy + Nxwx + Nxywy + Yxyx + minus Yxx + Yyy1113872 1113873y

0

or δw 0

Mx + Yxy 0

or δwx 0

y 0 b

Ny minus12Yyzx 0

or δv 0

Nxy +12Yyzy +

12Yxzx 0

or δu 0

12Yyz 0

or δuy 0

Myy + Nywy + Nxywx + 2Mxyx minus Yxyy minus Yxx minus Yyy1113872 1113873x

0

or δw 0

My minus Yxy 0

or δwy 0

(24)

e following dimensionless parameters are introduced

4 Shock and Vibration

ξ x

a

η y

b

U u

a

V v

b

W w

h

λ1 h

a

λ2 h

b

Aij Aij

Q11h

l l

h

Dij 12Dij

Q11h3

G1 Gprime

8Ga3

G2 Gprime

8Gb3

G3 Gprime

2Gh3

β G

Q11

Q11 E

1 minus υ2

τ t

T

C ca

2b2

Q11h3T

VC VC

V0

α1 εva

2b2V

20

2d2h4Q11

α2 εva

2b2V

20

d3h3Q11

T ab

h

ρQ11

1113970

ω ΩT

(25)

Shock and Vibration 5

where V0 is the unit voltageWith application of Taylor series expansion into (22) and

ignoring nonlinear terms with respect to w [31] the non-

dimensional governing equations of the microplate arewritten as

A11 Uξξ + λ21WξWξξ1113872 1113873 + A12 Vξη + λ22WηWξη1113872 1113873 + βA66λ22λ21

Uηη + Vξη + λ22WξWηη + λ22WηWξη1113888 1113889

+βλ1

G1 Vξξξη minusλ22λ21

Uξξηη +λ22λ21

Vξηηη minusλ42λ41

Uηηηη1113888 1113889 0

A12 Uξη + λ21WξWξη1113872 1113873 + A22 Vηη + λ22WηWηη1113872 1113873 + βA66 Uξη +λ21λ22

Vξξ + λ21WηWξξ + λ21WξWξη1113888 1113889

minusβλ2

G2λ41λ42

Vξξξξ minusλ21λ22

Uξξξη +λ21λ22

Vξξηη minus Uξηηη1113888 1113889 0

112

D11 + βG31113874 1113875λ21λ22

Wξξξξ +16D12 +

13D66 + 2βG31113874 1113875Wξξηη +

112

D22 + βG31113874 1113875λ22λ21

Wηηηη + Wττ + CWτ

A11 Uξ +12λ21W

2ξ1113874 1113875 + A12 Vη +

12λ22W

2η1113874 11138751113876 1113877

1λ22

Wξξ + A11 Uξξ + λ21WξWξξ1113872 1113873 + A12 Vξη + λ22WηWξη1113872 11138731113960 11139611λ22

Wξ

+ 2βA661λ21

Uη +1λ22

Vξ + WηWξ1113888 1113889Wξη + βA661λ22

Vξξ +1λ21

Uξη + WηWξξ + WξWξη1113888 1113889Wη

+ A12 Uξ +12λ21W

2ξ1113874 1113875 + A22 Vη +

12λ22W

2η1113874 11138751113876 1113877

1λ21

Wηη + A12 Uξη + λ21WξWξη1113872 1113873 + A22 Vηη + λ22WηWηη1113872 11138731113960 11139611λ21

Wη

+ βA661λ21

Uηη +1λ22

Vξη + WξηWη + WξWηη1113888 1113889Wξ + α1 + α2W( 1113857V2C

(26)

Due to the fact that a fully clamped rectangularmicroplate is extensively used in MEMS applications [32]

this kind of boundary condition is also studied in this workand written in dimensionless formulation as follows

A11 Uξ +12W

2ξ1113874 1113875 +

βG3

8h2

abVξξη minus

h2

b2Uξηη1113888 1113889 0 ξ 0 1 V 0 W 0 Vξ 0 Wξ 0

A22 Vη +12λ1W

2η1113874 1113875 minus

βG3

8h2

a2Vξξη minus

h2

abUξηη1113888 1113889 0 η 0 1 U 0 W 0 Uη 0 Wη 0

(27)

3 Solution Methodology

First note that in the boundary conditions in (27) the itemsrelated to size effect are neglected due to the fact that they arenegligible compared with other items [29] With satisfactionof boundary conditions a solution for (26) takes the fol-lowing form

U(ξ η τ) U(τ) cos πξ sin2 πη

V(ξ η τ) V(τ) sin2 πξ cos πη

W(ξ η τ) W(τ) sin2 πξ sin2 πη

(28)

Submitting (28) into (26)fd25 and multiplying them bycos πξ sin2 πη sin2 πξ cos πη and sin2 πξ sin2 πη

6 Shock and Vibration

respectively and integrating them from 0 to 1 with respect toξ and η the nonlinear governing differential equations withrespect to time can be obtained as

minus3π2

16Aminus

11 minus βAminus

66π2λ224λ21

minusGminus

1βλ1

π4λ224λ21

+π4λ42λ41

⎡⎢⎢⎣ ⎤⎥⎥⎦Uminus

+ minus169

Aminus

12 minus169βA

minus

66 +Gminus

1βλ1

64π2

9+16π2λ229λ21

⎡⎢⎢⎣ ⎤⎥⎥⎦Vminus

+π2

6λ21A

minus

11 +2π2

15λ22A

minus

12 minus2π2

15βA

minus

66λ221113890 1113891W

minus 2 0

minus169

Aminus

12 minus169βA

minus

66 +Gminus

2βλ2

16π2λ219λ22

+64π2

9⎡⎢⎢⎣ ⎤⎥⎥⎦U

minus

+ minus3π2

16Aminus

22 minusπ2λ214λ22

βAminus

66 minusGminus

2βλ2

π4λ41λ42

+π4λ214λ22

⎡⎢⎢⎣ ⎤⎥⎥⎦Vminus

+2π2

15λ21A

minus

12 +π2

6λ22A

minus

22 minus2π2

15βA

minus

66λ211113890 1113891W

minus 2 0

3112

Dminus

11 + βGminus

3λ21λ22

+16Dminus

12 +13βD

minus

66 + 2βGminus

3 + 3112

Dminus

22 + βGminus

3λ21λ22

1113890 1113891π4

41113896 1113897W

minus

+14Wminus

ττ +Cminus

4Wminus

τ

π2A

minus

11

3λ22minus4π2βA

minus

66

15λ21+4π2A

minus

12

15λ21Uminus

Wminus

+4π2A

minus

12

15λ22minus4π2βA

minus

66

15λ22+π2A

minus

22

3λ21Vminus

Wminus

minus525π4λ211024λ22

Aminus

11 +25π4

1024Aminus

12

+525π4λ221024λ21

Aminus

22 +50π4

1024βA

minus

66Wminus 3

+14α1 +

916α2W

minus

Vminus 2

C

(29)

For convenience (29) can be expressed as follows

M11Uminus

+ M12Vminus

+ M13Wminus 2

0

M21Uminus

+ M22Vminus

+ M23Wminus 2

0

M31Wminus

+14Wminus

ττ +Cminus

4Wminus

τ M32Uminus

Wminus

+ M33Vminus

Wminus

minus M34Wminus 3

+14α1 +

916α2W

minus

Vminus 2

C

(30)

where

M11 minus3π2

16A11 minus βA66

π2λ224λ21

minusG1βλ1

π4λ224λ21

+π4λ42λ41

1113888 1113889

M12 minus169

A12 minus169βA66 +

G1βλ1

64π2

9+16π2λ229λ21

1113888 1113889

M13 π2

6λ21A11 +

2π2

15λ22A12 minus

2π2

15βA66λ

22

M21 minus169

A12 minus169βA66 +

G2βλ2

16π2λ219λ22

+64π2

91113888 1113889

Shock and Vibration 7

M22 minus3π2

16A22 minus

π2λ214λ22

βA66 minusG2βλ2

π4λ41λ42

+π4λ214λ22

1113888 1113889

M23 2π2

15λ21A12 +

π2

6λ22A22 minus

2π2

15βA66

M31 3112

D11 + βG31113874 1113875λ21λ22

+16D12 +

13D66 + 2βG31113874 1113875 + 3

112

D22 + βG31113874 1113875λ21λ22

1113890 1113891π4

4

M32 π2

A11

3λ22minus4π2βA66

15λ21+4π2A12

15λ21

M33 4π2A12

15λ22minus4π2βA66

15λ22+π2A22

3λ21

M34 minus525π4λ211024λ22

A11 minus25π4

1024A12 minus

525π4λ221024λ21

A22 minus50π4

1024βA66

(31)

After eliminating U(τ) V(τ) the nonlinear differentialequation only related to W(τ) can be obtained

14Wττ +

C

4Wτ + M31W minus φW

314

α1 +916α2W1113874 1113875V

2C

(32)

where

φ M34 +M32M22M13 minus M32M12M23

M12M21 minus M11M22+

M33M11M23 minus M33M13M21

M12M21 minus M11M22 (33)

4 Results and Discussion

In this section the nonlinear size-dependent vibration analysisof the microplate is studied under the influence of size effectand damage effect Free vibration and forced vibration analysisof the microplate under DC voltage and AC voltage are dis-cussed in detail accordinglye frequency-response curves arepresented for various system parameters such as materiallength scale parameter damage variable damping ratio andexternal AC voltage Note that geometric and physical pa-rameters of the microplate are listed in Table 1

41 Free Vibration Analysis under DC Voltage For staticanalysis (32) can be written as

Wττ + C Wτ

4+ M31W minus φW

3

α1 +(916)α2W( 1113857V2d

4

(34)

where VD donates the DC voltage

e harmonic balance method is feasible for findingsolutions of both weakly and strongly nonlinear problems Ageneral solution of (34) can be given as

W(τ) A0 + A1 cos(ωτ + θ) (35)

Submitting (35) into (34) and using harmonic balancemethod (HB) [33] we obtain

ω20A0 minus 4φ A

30 +

32A0A

211113874 1113875 α1V

2d

minus A1ω2

+ ω20A1 minus 4φ 3A

20A1 +

34A311113874 1113875 0

(36)

where A0 is so smaller than A1 that it can be neglected in thefurther step of solution and then the following is obtained

ω20 4M31 minus

916α2V

2d

ω2 ω2

0 minus 3φA21

(37)

8 Shock and Vibration

e variation of amplitude-frequency response of themicroplate with various values of damage variable is pre-sented in Figure 2 where DC voltage Vd and non-dimensional material length scale parameter lh are taken as20 and 02 respectively It should be pointed out that thecurrent value of material length scale parameter is onlya mean to illustrate microscale effect Sophisticated exper-iments need to be conducted to measure material lengthscale parameter as it plays a significant role in microscalestructures As shown in Figure 2 the vibration amplitudebecomes larger with increase of nonlinear frequency whichindicates that the microplate exhibits a hardening-typebehavior It is also observed that the larger the damagevariable is the lower the nonlinear free vibration frequencybecomes Moreover free vibration amplitude of themicroplate midpoint increases as damage variable increasesdue to reduced stiffness of the microplate

Influence of size effect on frequency-response curves isgiven in Figure 3 in which damage variable D and DCvoltage Vd are taken as 02 and 20 separately It is observedthat large material length scale parameter results in smallamplitude and high frequency of the microplate Whenespecially size effect is omitted the classical microplatemodel is recovered As can be observed from Figure 3material length scale parameter increases the differencebetween the results of present microplate model and theclassical microplate model increases indicating that themicroscale effect induce additional rigidity for microstruc-turee extra stiffness contributed by couple stress togetherwith the classical stiffness increased the total stiffness of themicroplate Furthermore our results are consistent with thepredictions obtained by Ansari et al [34] who studied thenonlinear vibrations properties of functionally gradedMindlinrsquos microplates using MCST

42 Forced Vibration Analysis under AC Voltage As fordynamic analysis (32) can be written as14Wττ +

14

C Wτ + M31W minus φW3

14

α1 +916α2W1113874 1113875V

2a cos

2 ωτ

(38)

Similarly the harmonic balance method is employed forsolving (38) and the general solution for this equation can begiven as

W(τ) A1 cos 2ωτ + θ0( 1113857 + A2 sin 2ωτ + θ0( 1113857

A1 cos 2Φ + A2 sin 2Φ(39)

where

cos 2ωτ A3 cos 2Φ + A4 sin 2Φ A3 cos

θ0 A4 sin θ0(40)

Substituting (39) into (38) we obtain

Table 1 Geometric and material properties of the microplate

Property ValueLength a 300 μmWidth b 150 μmickness h 2 μmInitial gap d 2 μmPoissonrsquos ratio υ 006Youngrsquos modulus E 166 times 109 Paβ 02Dielectric constant εv 8854 times 10minus 12 Fm

D = 0D = 02D = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 2 Damage effect on frequency-response curves of themicroplate

lh = 0lh = 02lh = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 3 Size effect on frequency-response curves of themicroplate

Shock and Vibration 9

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A1 + 2ωCA2 minus 3φA1 A

21 + A

221113872 1113873

α12

V2aA3

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A2 minus 2ωCA1 minus 3φA2 A

21 + A

221113872 1113873

α12

V2aA4

(41)

Using A21 + A2

2 A2 and A23 + A2

4 1 the relationshipbetween the amplitude of the microplate midpoint and theexcitation frequency can be achieved as

9φ2A6

minus 6φ 4M31 minus 4ω2minus

932α2V

2a1113874 1113875A

4+ 4M31 minus 4ω2

minus932α2V

2a1113874 1113875

2+ 4ω2

C2

1113890 1113891A2

α214

V4a (42)

e frequency-response curves of the system withdamage variable D nondimensional material length scaleparameter lh and damping coefficient C are respectivelytaken as 02 02 and 02 and are depicted in Figure 4 whichreveals that the system still exhibits a hardening-type be-havior It is observed that the amplitude of microplatemidpoint increases as excitation frequency increases from11 actually from 0 until reaching a limit point (upperjump point) corresponding to the nonlinear resonanceand then the amplitude of microplate midpoint fallssuddenly for another the amplitude of the microplatemidpoint increases as the excitation frequency is decreasedfrom 35 until reaching another limit point (lower jumppoint) and then the amplitude of the microplate midpointrises abruptly e region between the lower jump pointand the upper jump point is regarded as the unstable re-gion is unique behavior is called as jump phenomenonwhich results from the nonlinearity of the dynamic systemAdditionally with the increase of the amplitude of ACvoltage the upper jump point tends to move towards theright side and the amplitude of the microplate midpointincreases In other words the nonlinear response region isenlarged by the greater external load e jump phe-nomenon disappears when the amplitude of AC voltage issmall in this case such as 10 (black line) Without enough

energy input by external excitation the dynamic systemwill remain stable

Figure 5 indicates the effect of damping coefficient on thefrequency-response curve of the microplate In the nu-merical simulation the nondimensional material lengthscale parameter lh damage variableD and amplitude of ACvoltage Va are taken to be 02 02 and 20 accordingly Fromthis figure we can see that by increasing the dampingcoefficient the unstable region decreases and the peak of thecurves goes down It is attributed to the fact that the energyinput by external load is consumed by the dissipation sys-tem e larger the damping coefficient is the more energythe dynamic system expends

Figure 6 displays damage effect on frequency-responsecurves of the microplate with damping coefficient C non-dimensional material length scale parameter lh and am-plitude of AC voltage Va are set to 02 02 and 20respectively It is observed that the larger the damage var-iable the bigger the peak of the curve and the wider theunstable region and nonlinear response region What ismore unlike the first two cases the unstable region movestowards the left side when the microplate suffers more se-rious damage e system does not possess enough stiffnessto resist external load e reason for this is that the stiffnessof the structure is reduced by damage effect

V_a = 20

V_a = 30

V_a = 10

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 4 e frequency-response curves of the system for different amplitude of the AC voltage VAC

10 Shock and Vibration

Figure 7 highlights the size effect on the frequency-response curves of dynamic system e numerical cal-culations are performed by assumingD 02 C 02 Va 20 It is revealed that the peak ofthe curve goes down slightly and the hardening influencereduces mildly when nondimensional material lengthscale parameter increases In addition the unstable regionmoves to the right side and diminishes as size effect be-comes more obvious Furthermore the distinction be-tween the results predicted by classical theory (lh 0)

and nonclassical theory gets more obvious e reason forthis change was already discussed in Section 41 Besidesthe present result is parallel to that of Ansari et al [35]who investigated the forced vibration of functionallygraded microplate based onMCST It should be noted thathorizontal axis in this work represents nondimensionalexternal frequency while in Ansari et alrsquos work it rep-resents the frequency ratio (the ratio of external frequencyto linear first frequency)

5 Conclusions

e nonlinear size-dependent vibration of a microplate withdamage was explored by employing MCST and the strainequivalent assumption in this research e nonlineargoverning partial differential equations were transformedinto nonlinear ordinary differential equations via Galerkinrsquosscheme and further solved numerically by the harmonicbalance method Numerical results indicate that damageeffect and size dependency both have obvious influences onthe static and dynamic behaviors of the microplate system Itwas concluded that on one hand the hardening-typenonlinear behavior of the microplate system enhances whenit encounters damage on the other hand the system exhibitsa weaker nonlinear behavior greater nondimensional fre-quency and lower amplitude of the microplate midpoint assize effect gets obvious

Data Availability

e Matlab simulation and control program data used tosupport the findings of this study are available from thecorresponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors would like to acknowledge with great gratitudefor the supports of the National Science Foundation ofChina (grant nos 51778551 and 11272270)

References

[1] M I Younis and A H Nayfeh ldquoA study of the nonlinearresponse of a resonant micro-beam to an electric actuationrdquoNonlinear Dynamics vol 31 no 1 pp 91ndash117 2003

C_

= 02

C_

= 04

C_

= 01

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 5e frequency-response curves of the system for differentdamping coefficients C

D = 02

D = 04

D = 0

15 19 23 27 31 3511ω

0

02

04

06

08

A

Figure 6e frequency-response curves of the system for differentdamage variable D

lh = 04

lh = 0

lh = 02

15 19 23 27 31 3511ω

0

01

02

03

04

05

A

Figure 7e frequency-response curves of the system for differentnondimensional material length scale parameters lh

Shock and Vibration 11

[2] E M Abdel-Rahman and A H Nayfeh ldquoSecondary reso-nances of electrically actuated resonant microsensorsrdquoJournal of Micromechanics and Microengineering vol 13no 3 pp 491ndash501 2003

[3] W Zhang and G Meng ldquoNonlinear dynamical system ofmicro-cantilever under combined parametric and forcingexcitations in MEMSrdquo Sensors and Actuators A Physicalvol 119 no 2 pp 291ndash299 2005

[4] L Xu and X Jia ldquoElectromechanical coupled nonlinear dy-namics for microbeamsrdquo Archive of Applied Mechanicsvol 77 no 7 pp 485ndash502 2007

[5] W G Vogl and A H Nayfeh ldquoPrimary resonance excitationof electrically actuated clamped circular platesrdquo NonlinearDynamics vol 47 no 1ndash3 pp 181ndash192 2006

[6] A H Nayfeh H M Ouakad F Choura and E M Abdel-Rahman ldquoNonlinear dynamics of a resonant gas sensorrdquoNonlinear Dynamics vol 59 no 4 pp 607ndash618 2010

[7] X L Jia J Yang S Kitipornchai and C W Lim ldquoResonancefrequency response of geometrically nonlinear micro-switches under electrical actuationrdquo Journal of Sound andVibration vol 331 no 14 pp 3397ndash3411 2012

[8] P Kim S Bae and J Seok ldquoResonant behaviors of a nonlinearcantilever beam with tip mass subject to an axial force andelectrostatic excitationrdquo International Journal of MechanicalSciences vol 64 no 1 pp 232ndash257 2012

[9] S Saghir and M I Younis ldquoAn investigation of the static anddynamic behavior of electrically actuated rectangularmicroplatesrdquo International Journal of Non-linear Mechanicsvol 85 pp 81ndash93 2016

[10] M Sheikhlou R Shabani and G Rezazadeh ldquoNonlinearanalysis of electrostatically actuated diaphragm-type micro-pumpsrdquo Nonlinear Dynamics vol 83 no 1-2 pp 1ndash11 2016

[11] P M Osterberg Electrostatically actuated micromechanicaltest structure for material property measurement PhD dis-sertation Massachusetts Institute of Technology CambridgeMA USA 1995

[12] M Arefi and A M Zenkour ldquoSize dependent electro-elasticanalysis of a sandwich microbeam based on higher ordersinusoidal shear deformation theory and strain gradienttheoryrdquo Journal of Intelligent Material Systems and Structuresvol 29 no 7 pp 27ndash40 2018

[13] M Arefi and A M Zenkour ldquoSize-dependent vibration andbending analyses of the piezomagnetic three-layer nano-beamsrdquo Applied Physics A vol 123 no 3 p 202 2017

[14] M Arefi and A M Zenkour ldquoSize-dependent vibration andelectro-magneto-elastic bending responses of sandwich pie-zomagnetic curved nanobeamsrdquo Steel and Composite Struc-tures vol 29 no 5 pp 579ndash590 2018

[15] M Arefi and A M Zenkour ldquoermo-electro-magneto-mechanical bending behavior of size-dependent sandwichpiezomagnetic nanoplatesrdquo Mechanics Research Communi-cations vol 84 pp 27ndash42 2017

[16] L L Ke J Yang S Kitipornchai M A Bradford andY S Wang ldquoAxisymmetric nonlinear free vibration of size-dependent functionally graded annular microplatesrdquo Com-posites Part B Engineering vol 53 no 7 pp 207ndash217 2013

[17] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[18] M H Ghayesh H Farokhi and M Amabili ldquoNonlinearbehaviour of electrically actuated MEMS resonatorsrdquo

International Journal of Engineering Science vol 71 no 10pp 137ndash155 2013

[19] A G Arani and G S Jafari ldquoNonlinear vibration analysis oflaminated composite Mindlin micronano-plates resting onorthotropic Pasternak medium using DQMrdquo AppliedMathematics and Mechanics vol 36 no 8 pp 1033ndash10442015

[20] M Sobhy and A M Zenkour ldquoA comprehensive study onthe size-dependent hygrothermal analysis of exponentiallygraded microplates on elastic foundationsrdquo Mechanics ofAdvanced Materials and Structures vol 27 no 10pp 816ndash830 2019

[21] M Arefi and A M Zenkour ldquoSize-dependent free vibrationand dynamic analyses of piezo-electro-magnetic sandwichnanoplates resting on viscoelastic foundationrdquo Physica BCondensed Matter vol 521 pp 188ndash197 2017

[22] M Arefi M Kiani and A M Zenkour ldquoSize-dependent freevibration analysis of a three-layered exponentially gradednano-micro-plate with piezomagnetic face sheets resting onPasternakrsquos foundation via MCSTrdquo Journal of SandwichStructures and Materials vol 22 no 1 pp 55ndash86 2020

[23] H Farokhi and M H Ghayesh ldquoSize-dependent behavior ofelectrically actuated microcantilever-based MEMSrdquo In-ternational Journal of Mechanics amp Materials in Designvol 12 no 3 pp 1ndash15 2016

[24] M Tahani A R Askari Y Mohandes and B Hassani ldquoSize-dependent free vibration analysis of electrostatically pre-de-formed rectangular micro-plates based on the modifiedcouple stress theoryfied couple stress theoryrdquo InternationalJournal of Mechanical Sciences vol 94-95 no 22 pp 185ndash1982015

[25] A Veysi R Shabani and G Rezazadeh ldquoNonlinear vibrationsof micro-doubly curved shallow shells based on the modifiedcouple stress theoryfied couple stress theoryrdquo NonlinearDynamics vol 87 no 3 pp 2051ndash2065 2017

[26] B Jalalahmadi F Sadeghi and D Peroulis ldquoA numericalfatigue damage model for life scatter of MEMS devicesrdquoJournal of Microelectromechanical Systems vol 18 no 5pp 1016ndash1031 2009

[27] T S Slack F Sadeghi and D Peroulis ldquoA phenomenologicaldiscrete brittle damage-mechanics model for fatigue of MEMSdevices with application to LIGA Nirdquo Journal of Micro-electromechanical Systems vol 18 no 1 pp 119ndash128 2009

[28] A Basu R P Hennessy G G Adams and N E McGruerldquoHot switching damage mechanisms in MEMS contacts-ev-idence and understandingrdquo Journal of Micromechanics andMicroengineering vol 24 no 10 p 16 2014

[29] C Chen J Yuan and Y Mao ldquoPost-buckling of size-de-pendent micro-plate considering damage effectsrdquo NonlinearDynamics vol 90 no 2 pp 1301ndash1314 2017

[30] J Lemaitre A Course on Damage Mechanics Springer BerlinGermany 1992

[31] C Chen H Hu and L Dai ldquoNonlinear behavior andcharacterization of a piezoelectric laminated microbeamsystemrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1304ndash1315 2013

[32] A R Askari and M Tahani ldquoSize-dependent dynamic pull-inanalysis of geometric non-linear micro-plates based on themodified couple stress theoryrdquo Physica E Low-DimensionalSystems and Nanostructures vol 86 pp 262ndash274 2017

[33] A H Nayfeh and D T Mook Nonlinear OscillationsSpringer International Publishing New York NY USA 1979

12 Shock and Vibration

[34] R Ansari M Faghih Shojaei V Mohammadi R Gholamiand M A Darabi ldquoNonlinear vibrations of functionallygraded Mindlin microplates based on the modified couplestress theoryfied couple stress theoryrdquo Composite Structuresvol 114 no 18 pp 124ndash134 2014

[35] R Ansari R Gholami and A Shahabodini ldquoSize-dependentgeometrically nonlinear forced vibration analysis of func-tionally graded first-order shear deformable microplatesrdquoJournal of Mechanics vol 32 no 5 pp 539ndash554 2016

Shock and Vibration 13

εx ε0x + zκx

εy ε0y + zκy

εxy ε0xy + zκxy

(8)

where stains ε0x ε0y and ε0xy and curvatures κx κy and κxy ata point of the middle surface are expressed as

ε0x

ε0y

ε0xy

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎭

ux +12w

2x

vy +12w

2y

uy + vx + wxwy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

κx

κy

κxy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

minus wxx

minus wyy

minus 2wxy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(9)

Substituting (7) into (6) one has gyration vector

θx(x y) wy

θy(x y) minus wx

θz(x y) 12

vx minus uy1113872 1113873

(10)

Substituting (10) into (2) yields curvature tensor

χxx

χyy

χxy

χxz

χyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

wxy

minus wxy

12

wyy minus wxx1113872 1113873

14

vxx minus uxy1113872 1113873

14

vxy minus uyy1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(11)

e stress-strain relations can be given with respect toPoissonrsquos ratio υ and Youngrsquos modulus E as

σx

σy

σxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

Qd

bull

εx

εy

εxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(12)

mxx myy mxy mxz myz1113872 1113873 F χxx χyy χxy χxz χyz1113872 1113873

(13)

where

Qd

Qd11 Q

d12 0

Qd22 0

Qd66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

E

1 minus υ2υE

1 minus υ20

E

1 minus υ20

G

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1 minus D)

F 2l2G(1 minus D)

(14)

where D denotes damage variable varying from 0 to 1 [30] Gis shear modulus

With Cauchyrsquos stress and couple stress expressed abovebending moments Mx My Mxy membrane stress re-sultants Nx Ny Nxy and couple momentsYxx Yyy Yxy Yxz Yyz can be respectively defined as

Mx My Mxy1113960 1113961 1113946h2

minus (h2)z σx σy σxy1113960 1113961dz

Nx Ny Nxy1113960 1113961 1113946h2

minus (h2)

σx σy σxy1113960 1113961dz

Yxx Yyy Yxy Yxz Yyz1113960 1113961 1113946h2

minus (h2)

mxx myy mxy mxz myz1113960 1113961dz

(15)

Substituting (12) and (13) into (15) yields the followingconstitutive equations

Nx

Ny

Nxy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

A11 A12 0

A12 A22 0

0 0 A66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ε0x

ε0y

ε0xy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

Mx

My

Mxy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

D11 D12 0

D12 D22 0

0 0 D66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

κx

κy

κxy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

Yxx Yyy Yxy Yxz Yyz1113960 1113961 G χxx χyy χxy χxz χyz1113960 1113961

(16)

where

o a X

b

y

z

h

d

Movable plate

Stationary electrode

Vc

Figure 1 Electrically actuated microplate

Shock and Vibration 3

Aij hQdij

Dij h3

121113888 1113889Q

dij

Gprime hF

(i j 1 2 6)

(17)

With Hamiltonrsquos principle we have

δ1113946t2

t1

(T minus U + W)dt 0 (18)

Substituting stresses and strains into (1) the virtualstrain energy is obtained as

δU 1113946Vσijδεij1113872 1113873dV + 1113946

Vmijδχij1113872 1113873dV (19)

e virtual kinetic energy is given as

δT 12ρ1113946

Vδ wt( 1113857

2dV (20)

e virtual work done by electric force and dampingforce can be expressed as

δW Bs

Fz minus cwt( 1113857δwdS (21)

where c denotes damping coefficient and Fz represents theelectric force defined as

Fz εvV

2C

2(d minus w)2 (22)

where εv is the dielectric constant of airBy substituting (19)ndash(21) into (18) one has the nonlinear

dynamic equations of the microplate system as

Nxx + Nxyy +12Yxzxy +

12Yyzyy 0

Nyy + Nxyx minus12Yxzxx minus

12Yyzxy 0

minus Nxwx + Nxywy1113872 1113873x

minus Nywy + Nxywx1113872 1113873y

+ minus Mx minus Yxy1113872 1113873xx

+ minus My + Yxy1113872 1113873yy

+ minus 2Mxy + Yxx minus Yyy1113872 1113873xy

+ Fe ρhwtt + cwt

(23)

Boundary conditions are as follows

Nx +12Yxzy 0

or δu 0

Nxy minus12Yxzx minus

12Yyzy 0

or δv 0

12Yxz 0

or δvx 0

Mxx + 2Mxyy + Nxwx + Nxywy + Yxyx + minus Yxx + Yyy1113872 1113873y

0

or δw 0

Mx + Yxy 0

or δwx 0

y 0 b

Ny minus12Yyzx 0

or δv 0

Nxy +12Yyzy +

12Yxzx 0

or δu 0

12Yyz 0

or δuy 0

Myy + Nywy + Nxywx + 2Mxyx minus Yxyy minus Yxx minus Yyy1113872 1113873x

0

or δw 0

My minus Yxy 0

or δwy 0

(24)

e following dimensionless parameters are introduced

4 Shock and Vibration

ξ x

a

η y

b

U u

a

V v

b

W w

h

λ1 h

a

λ2 h

b

Aij Aij

Q11h

l l

h

Dij 12Dij

Q11h3

G1 Gprime

8Ga3

G2 Gprime

8Gb3

G3 Gprime

2Gh3

β G

Q11

Q11 E

1 minus υ2

τ t

T

C ca

2b2

Q11h3T

VC VC

V0

α1 εva

2b2V

20

2d2h4Q11

α2 εva

2b2V

20

d3h3Q11

T ab

h

ρQ11

1113970

ω ΩT

(25)

Shock and Vibration 5

where V0 is the unit voltageWith application of Taylor series expansion into (22) and

ignoring nonlinear terms with respect to w [31] the non-

dimensional governing equations of the microplate arewritten as

A11 Uξξ + λ21WξWξξ1113872 1113873 + A12 Vξη + λ22WηWξη1113872 1113873 + βA66λ22λ21

Uηη + Vξη + λ22WξWηη + λ22WηWξη1113888 1113889

+βλ1

G1 Vξξξη minusλ22λ21

Uξξηη +λ22λ21

Vξηηη minusλ42λ41

Uηηηη1113888 1113889 0

A12 Uξη + λ21WξWξη1113872 1113873 + A22 Vηη + λ22WηWηη1113872 1113873 + βA66 Uξη +λ21λ22

Vξξ + λ21WηWξξ + λ21WξWξη1113888 1113889

minusβλ2

G2λ41λ42

Vξξξξ minusλ21λ22

Uξξξη +λ21λ22

Vξξηη minus Uξηηη1113888 1113889 0

112

D11 + βG31113874 1113875λ21λ22

Wξξξξ +16D12 +

13D66 + 2βG31113874 1113875Wξξηη +

112

D22 + βG31113874 1113875λ22λ21

Wηηηη + Wττ + CWτ

A11 Uξ +12λ21W

2ξ1113874 1113875 + A12 Vη +

12λ22W

2η1113874 11138751113876 1113877

1λ22

Wξξ + A11 Uξξ + λ21WξWξξ1113872 1113873 + A12 Vξη + λ22WηWξη1113872 11138731113960 11139611λ22

Wξ

+ 2βA661λ21

Uη +1λ22

Vξ + WηWξ1113888 1113889Wξη + βA661λ22

Vξξ +1λ21

Uξη + WηWξξ + WξWξη1113888 1113889Wη

+ A12 Uξ +12λ21W

2ξ1113874 1113875 + A22 Vη +

12λ22W

2η1113874 11138751113876 1113877

1λ21

Wηη + A12 Uξη + λ21WξWξη1113872 1113873 + A22 Vηη + λ22WηWηη1113872 11138731113960 11139611λ21

Wη

+ βA661λ21

Uηη +1λ22

Vξη + WξηWη + WξWηη1113888 1113889Wξ + α1 + α2W( 1113857V2C

(26)

Due to the fact that a fully clamped rectangularmicroplate is extensively used in MEMS applications [32]

this kind of boundary condition is also studied in this workand written in dimensionless formulation as follows

A11 Uξ +12W

2ξ1113874 1113875 +

βG3

8h2

abVξξη minus

h2

b2Uξηη1113888 1113889 0 ξ 0 1 V 0 W 0 Vξ 0 Wξ 0

A22 Vη +12λ1W

2η1113874 1113875 minus

βG3

8h2

a2Vξξη minus

h2

abUξηη1113888 1113889 0 η 0 1 U 0 W 0 Uη 0 Wη 0

(27)

3 Solution Methodology

First note that in the boundary conditions in (27) the itemsrelated to size effect are neglected due to the fact that they arenegligible compared with other items [29] With satisfactionof boundary conditions a solution for (26) takes the fol-lowing form

U(ξ η τ) U(τ) cos πξ sin2 πη

V(ξ η τ) V(τ) sin2 πξ cos πη

W(ξ η τ) W(τ) sin2 πξ sin2 πη

(28)

Submitting (28) into (26)fd25 and multiplying them bycos πξ sin2 πη sin2 πξ cos πη and sin2 πξ sin2 πη

6 Shock and Vibration

respectively and integrating them from 0 to 1 with respect toξ and η the nonlinear governing differential equations withrespect to time can be obtained as

minus3π2

16Aminus

11 minus βAminus

66π2λ224λ21

minusGminus

1βλ1

π4λ224λ21

+π4λ42λ41

⎡⎢⎢⎣ ⎤⎥⎥⎦Uminus

+ minus169

Aminus

12 minus169βA

minus

66 +Gminus

1βλ1

64π2

9+16π2λ229λ21

⎡⎢⎢⎣ ⎤⎥⎥⎦Vminus

+π2

6λ21A

minus

11 +2π2

15λ22A

minus

12 minus2π2

15βA

minus

66λ221113890 1113891W

minus 2 0

minus169

Aminus

12 minus169βA

minus

66 +Gminus

2βλ2

16π2λ219λ22

+64π2

9⎡⎢⎢⎣ ⎤⎥⎥⎦U

minus

+ minus3π2

16Aminus

22 minusπ2λ214λ22

βAminus

66 minusGminus

2βλ2

π4λ41λ42

+π4λ214λ22

⎡⎢⎢⎣ ⎤⎥⎥⎦Vminus

+2π2

15λ21A

minus

12 +π2

6λ22A

minus

22 minus2π2

15βA

minus

66λ211113890 1113891W

minus 2 0

3112

Dminus

11 + βGminus

3λ21λ22

+16Dminus

12 +13βD

minus

66 + 2βGminus

3 + 3112

Dminus

22 + βGminus

3λ21λ22

1113890 1113891π4

41113896 1113897W

minus

+14Wminus

ττ +Cminus

4Wminus

τ

π2A

minus

11

3λ22minus4π2βA

minus

66

15λ21+4π2A

minus

12

15λ21Uminus

Wminus

+4π2A

minus

12

15λ22minus4π2βA

minus

66

15λ22+π2A

minus

22

3λ21Vminus

Wminus

minus525π4λ211024λ22

Aminus

11 +25π4

1024Aminus

12

+525π4λ221024λ21

Aminus

22 +50π4

1024βA

minus

66Wminus 3

+14α1 +

916α2W

minus

Vminus 2

C

(29)

For convenience (29) can be expressed as follows

M11Uminus

+ M12Vminus

+ M13Wminus 2

0

M21Uminus

+ M22Vminus

+ M23Wminus 2

0

M31Wminus

+14Wminus

ττ +Cminus

4Wminus

τ M32Uminus

Wminus

+ M33Vminus

Wminus

minus M34Wminus 3

+14α1 +

916α2W

minus

Vminus 2

C

(30)

where

M11 minus3π2

16A11 minus βA66

π2λ224λ21

minusG1βλ1

π4λ224λ21

+π4λ42λ41

1113888 1113889

M12 minus169

A12 minus169βA66 +

G1βλ1

64π2

9+16π2λ229λ21

1113888 1113889

M13 π2

6λ21A11 +

2π2

15λ22A12 minus

2π2

15βA66λ

22

M21 minus169

A12 minus169βA66 +

G2βλ2

16π2λ219λ22

+64π2

91113888 1113889

Shock and Vibration 7

M22 minus3π2

16A22 minus

π2λ214λ22

βA66 minusG2βλ2

π4λ41λ42

+π4λ214λ22

1113888 1113889

M23 2π2

15λ21A12 +

π2

6λ22A22 minus

2π2

15βA66

M31 3112

D11 + βG31113874 1113875λ21λ22

+16D12 +

13D66 + 2βG31113874 1113875 + 3

112

D22 + βG31113874 1113875λ21λ22

1113890 1113891π4

4

M32 π2

A11

3λ22minus4π2βA66

15λ21+4π2A12

15λ21

M33 4π2A12

15λ22minus4π2βA66

15λ22+π2A22

3λ21

M34 minus525π4λ211024λ22

A11 minus25π4

1024A12 minus

525π4λ221024λ21

A22 minus50π4

1024βA66

(31)

After eliminating U(τ) V(τ) the nonlinear differentialequation only related to W(τ) can be obtained

14Wττ +

C

4Wτ + M31W minus φW

314

α1 +916α2W1113874 1113875V

2C

(32)

where

φ M34 +M32M22M13 minus M32M12M23

M12M21 minus M11M22+

M33M11M23 minus M33M13M21

M12M21 minus M11M22 (33)

4 Results and Discussion

In this section the nonlinear size-dependent vibration analysisof the microplate is studied under the influence of size effectand damage effect Free vibration and forced vibration analysisof the microplate under DC voltage and AC voltage are dis-cussed in detail accordinglye frequency-response curves arepresented for various system parameters such as materiallength scale parameter damage variable damping ratio andexternal AC voltage Note that geometric and physical pa-rameters of the microplate are listed in Table 1

41 Free Vibration Analysis under DC Voltage For staticanalysis (32) can be written as

Wττ + C Wτ

4+ M31W minus φW

3

α1 +(916)α2W( 1113857V2d

4

(34)

where VD donates the DC voltage

e harmonic balance method is feasible for findingsolutions of both weakly and strongly nonlinear problems Ageneral solution of (34) can be given as

W(τ) A0 + A1 cos(ωτ + θ) (35)

Submitting (35) into (34) and using harmonic balancemethod (HB) [33] we obtain

ω20A0 minus 4φ A

30 +

32A0A

211113874 1113875 α1V

2d

minus A1ω2

+ ω20A1 minus 4φ 3A

20A1 +

34A311113874 1113875 0

(36)

where A0 is so smaller than A1 that it can be neglected in thefurther step of solution and then the following is obtained

ω20 4M31 minus

916α2V

2d

ω2 ω2

0 minus 3φA21

(37)

8 Shock and Vibration

e variation of amplitude-frequency response of themicroplate with various values of damage variable is pre-sented in Figure 2 where DC voltage Vd and non-dimensional material length scale parameter lh are taken as20 and 02 respectively It should be pointed out that thecurrent value of material length scale parameter is onlya mean to illustrate microscale effect Sophisticated exper-iments need to be conducted to measure material lengthscale parameter as it plays a significant role in microscalestructures As shown in Figure 2 the vibration amplitudebecomes larger with increase of nonlinear frequency whichindicates that the microplate exhibits a hardening-typebehavior It is also observed that the larger the damagevariable is the lower the nonlinear free vibration frequencybecomes Moreover free vibration amplitude of themicroplate midpoint increases as damage variable increasesdue to reduced stiffness of the microplate

Influence of size effect on frequency-response curves isgiven in Figure 3 in which damage variable D and DCvoltage Vd are taken as 02 and 20 separately It is observedthat large material length scale parameter results in smallamplitude and high frequency of the microplate Whenespecially size effect is omitted the classical microplatemodel is recovered As can be observed from Figure 3material length scale parameter increases the differencebetween the results of present microplate model and theclassical microplate model increases indicating that themicroscale effect induce additional rigidity for microstruc-turee extra stiffness contributed by couple stress togetherwith the classical stiffness increased the total stiffness of themicroplate Furthermore our results are consistent with thepredictions obtained by Ansari et al [34] who studied thenonlinear vibrations properties of functionally gradedMindlinrsquos microplates using MCST

42 Forced Vibration Analysis under AC Voltage As fordynamic analysis (32) can be written as14Wττ +

14

C Wτ + M31W minus φW3

14

α1 +916α2W1113874 1113875V

2a cos

2 ωτ

(38)

Similarly the harmonic balance method is employed forsolving (38) and the general solution for this equation can begiven as

W(τ) A1 cos 2ωτ + θ0( 1113857 + A2 sin 2ωτ + θ0( 1113857

A1 cos 2Φ + A2 sin 2Φ(39)

where

cos 2ωτ A3 cos 2Φ + A4 sin 2Φ A3 cos

θ0 A4 sin θ0(40)

Substituting (39) into (38) we obtain

Table 1 Geometric and material properties of the microplate

Property ValueLength a 300 μmWidth b 150 μmickness h 2 μmInitial gap d 2 μmPoissonrsquos ratio υ 006Youngrsquos modulus E 166 times 109 Paβ 02Dielectric constant εv 8854 times 10minus 12 Fm

D = 0D = 02D = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 2 Damage effect on frequency-response curves of themicroplate

lh = 0lh = 02lh = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 3 Size effect on frequency-response curves of themicroplate

Shock and Vibration 9

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A1 + 2ωCA2 minus 3φA1 A

21 + A

221113872 1113873

α12

V2aA3

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A2 minus 2ωCA1 minus 3φA2 A

21 + A

221113872 1113873

α12

V2aA4

(41)

Using A21 + A2

2 A2 and A23 + A2

4 1 the relationshipbetween the amplitude of the microplate midpoint and theexcitation frequency can be achieved as

9φ2A6

minus 6φ 4M31 minus 4ω2minus

932α2V

2a1113874 1113875A

4+ 4M31 minus 4ω2

minus932α2V

2a1113874 1113875

2+ 4ω2

C2

1113890 1113891A2

α214

V4a (42)

e frequency-response curves of the system withdamage variable D nondimensional material length scaleparameter lh and damping coefficient C are respectivelytaken as 02 02 and 02 and are depicted in Figure 4 whichreveals that the system still exhibits a hardening-type be-havior It is observed that the amplitude of microplatemidpoint increases as excitation frequency increases from11 actually from 0 until reaching a limit point (upperjump point) corresponding to the nonlinear resonanceand then the amplitude of microplate midpoint fallssuddenly for another the amplitude of the microplatemidpoint increases as the excitation frequency is decreasedfrom 35 until reaching another limit point (lower jumppoint) and then the amplitude of the microplate midpointrises abruptly e region between the lower jump pointand the upper jump point is regarded as the unstable re-gion is unique behavior is called as jump phenomenonwhich results from the nonlinearity of the dynamic systemAdditionally with the increase of the amplitude of ACvoltage the upper jump point tends to move towards theright side and the amplitude of the microplate midpointincreases In other words the nonlinear response region isenlarged by the greater external load e jump phe-nomenon disappears when the amplitude of AC voltage issmall in this case such as 10 (black line) Without enough

energy input by external excitation the dynamic systemwill remain stable

Figure 5 indicates the effect of damping coefficient on thefrequency-response curve of the microplate In the nu-merical simulation the nondimensional material lengthscale parameter lh damage variableD and amplitude of ACvoltage Va are taken to be 02 02 and 20 accordingly Fromthis figure we can see that by increasing the dampingcoefficient the unstable region decreases and the peak of thecurves goes down It is attributed to the fact that the energyinput by external load is consumed by the dissipation sys-tem e larger the damping coefficient is the more energythe dynamic system expends

Figure 6 displays damage effect on frequency-responsecurves of the microplate with damping coefficient C non-dimensional material length scale parameter lh and am-plitude of AC voltage Va are set to 02 02 and 20respectively It is observed that the larger the damage var-iable the bigger the peak of the curve and the wider theunstable region and nonlinear response region What ismore unlike the first two cases the unstable region movestowards the left side when the microplate suffers more se-rious damage e system does not possess enough stiffnessto resist external load e reason for this is that the stiffnessof the structure is reduced by damage effect

V_a = 20

V_a = 30

V_a = 10

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 4 e frequency-response curves of the system for different amplitude of the AC voltage VAC

10 Shock and Vibration

Figure 7 highlights the size effect on the frequency-response curves of dynamic system e numerical cal-culations are performed by assumingD 02 C 02 Va 20 It is revealed that the peak ofthe curve goes down slightly and the hardening influencereduces mildly when nondimensional material lengthscale parameter increases In addition the unstable regionmoves to the right side and diminishes as size effect be-comes more obvious Furthermore the distinction be-tween the results predicted by classical theory (lh 0)

and nonclassical theory gets more obvious e reason forthis change was already discussed in Section 41 Besidesthe present result is parallel to that of Ansari et al [35]who investigated the forced vibration of functionallygraded microplate based onMCST It should be noted thathorizontal axis in this work represents nondimensionalexternal frequency while in Ansari et alrsquos work it rep-resents the frequency ratio (the ratio of external frequencyto linear first frequency)

5 Conclusions

e nonlinear size-dependent vibration of a microplate withdamage was explored by employing MCST and the strainequivalent assumption in this research e nonlineargoverning partial differential equations were transformedinto nonlinear ordinary differential equations via Galerkinrsquosscheme and further solved numerically by the harmonicbalance method Numerical results indicate that damageeffect and size dependency both have obvious influences onthe static and dynamic behaviors of the microplate system Itwas concluded that on one hand the hardening-typenonlinear behavior of the microplate system enhances whenit encounters damage on the other hand the system exhibitsa weaker nonlinear behavior greater nondimensional fre-quency and lower amplitude of the microplate midpoint assize effect gets obvious

Data Availability

e Matlab simulation and control program data used tosupport the findings of this study are available from thecorresponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors would like to acknowledge with great gratitudefor the supports of the National Science Foundation ofChina (grant nos 51778551 and 11272270)

References

[1] M I Younis and A H Nayfeh ldquoA study of the nonlinearresponse of a resonant micro-beam to an electric actuationrdquoNonlinear Dynamics vol 31 no 1 pp 91ndash117 2003

C_

= 02

C_

= 04

C_

= 01

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 5e frequency-response curves of the system for differentdamping coefficients C

D = 02

D = 04

D = 0

15 19 23 27 31 3511ω

0

02

04

06

08

A

Figure 6e frequency-response curves of the system for differentdamage variable D

lh = 04

lh = 0

lh = 02

15 19 23 27 31 3511ω

0

01

02

03

04

05

A

Figure 7e frequency-response curves of the system for differentnondimensional material length scale parameters lh

Shock and Vibration 11

[2] E M Abdel-Rahman and A H Nayfeh ldquoSecondary reso-nances of electrically actuated resonant microsensorsrdquoJournal of Micromechanics and Microengineering vol 13no 3 pp 491ndash501 2003

[3] W Zhang and G Meng ldquoNonlinear dynamical system ofmicro-cantilever under combined parametric and forcingexcitations in MEMSrdquo Sensors and Actuators A Physicalvol 119 no 2 pp 291ndash299 2005

[4] L Xu and X Jia ldquoElectromechanical coupled nonlinear dy-namics for microbeamsrdquo Archive of Applied Mechanicsvol 77 no 7 pp 485ndash502 2007

[5] W G Vogl and A H Nayfeh ldquoPrimary resonance excitationof electrically actuated clamped circular platesrdquo NonlinearDynamics vol 47 no 1ndash3 pp 181ndash192 2006

[6] A H Nayfeh H M Ouakad F Choura and E M Abdel-Rahman ldquoNonlinear dynamics of a resonant gas sensorrdquoNonlinear Dynamics vol 59 no 4 pp 607ndash618 2010

[7] X L Jia J Yang S Kitipornchai and C W Lim ldquoResonancefrequency response of geometrically nonlinear micro-switches under electrical actuationrdquo Journal of Sound andVibration vol 331 no 14 pp 3397ndash3411 2012

[8] P Kim S Bae and J Seok ldquoResonant behaviors of a nonlinearcantilever beam with tip mass subject to an axial force andelectrostatic excitationrdquo International Journal of MechanicalSciences vol 64 no 1 pp 232ndash257 2012

[9] S Saghir and M I Younis ldquoAn investigation of the static anddynamic behavior of electrically actuated rectangularmicroplatesrdquo International Journal of Non-linear Mechanicsvol 85 pp 81ndash93 2016

[10] M Sheikhlou R Shabani and G Rezazadeh ldquoNonlinearanalysis of electrostatically actuated diaphragm-type micro-pumpsrdquo Nonlinear Dynamics vol 83 no 1-2 pp 1ndash11 2016

[11] P M Osterberg Electrostatically actuated micromechanicaltest structure for material property measurement PhD dis-sertation Massachusetts Institute of Technology CambridgeMA USA 1995

[12] M Arefi and A M Zenkour ldquoSize dependent electro-elasticanalysis of a sandwich microbeam based on higher ordersinusoidal shear deformation theory and strain gradienttheoryrdquo Journal of Intelligent Material Systems and Structuresvol 29 no 7 pp 27ndash40 2018

[13] M Arefi and A M Zenkour ldquoSize-dependent vibration andbending analyses of the piezomagnetic three-layer nano-beamsrdquo Applied Physics A vol 123 no 3 p 202 2017

[14] M Arefi and A M Zenkour ldquoSize-dependent vibration andelectro-magneto-elastic bending responses of sandwich pie-zomagnetic curved nanobeamsrdquo Steel and Composite Struc-tures vol 29 no 5 pp 579ndash590 2018

[15] M Arefi and A M Zenkour ldquoermo-electro-magneto-mechanical bending behavior of size-dependent sandwichpiezomagnetic nanoplatesrdquo Mechanics Research Communi-cations vol 84 pp 27ndash42 2017

[16] L L Ke J Yang S Kitipornchai M A Bradford andY S Wang ldquoAxisymmetric nonlinear free vibration of size-dependent functionally graded annular microplatesrdquo Com-posites Part B Engineering vol 53 no 7 pp 207ndash217 2013

[17] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[18] M H Ghayesh H Farokhi and M Amabili ldquoNonlinearbehaviour of electrically actuated MEMS resonatorsrdquo

International Journal of Engineering Science vol 71 no 10pp 137ndash155 2013

[19] A G Arani and G S Jafari ldquoNonlinear vibration analysis oflaminated composite Mindlin micronano-plates resting onorthotropic Pasternak medium using DQMrdquo AppliedMathematics and Mechanics vol 36 no 8 pp 1033ndash10442015

[20] M Sobhy and A M Zenkour ldquoA comprehensive study onthe size-dependent hygrothermal analysis of exponentiallygraded microplates on elastic foundationsrdquo Mechanics ofAdvanced Materials and Structures vol 27 no 10pp 816ndash830 2019

[21] M Arefi and A M Zenkour ldquoSize-dependent free vibrationand dynamic analyses of piezo-electro-magnetic sandwichnanoplates resting on viscoelastic foundationrdquo Physica BCondensed Matter vol 521 pp 188ndash197 2017

[22] M Arefi M Kiani and A M Zenkour ldquoSize-dependent freevibration analysis of a three-layered exponentially gradednano-micro-plate with piezomagnetic face sheets resting onPasternakrsquos foundation via MCSTrdquo Journal of SandwichStructures and Materials vol 22 no 1 pp 55ndash86 2020

[23] H Farokhi and M H Ghayesh ldquoSize-dependent behavior ofelectrically actuated microcantilever-based MEMSrdquo In-ternational Journal of Mechanics amp Materials in Designvol 12 no 3 pp 1ndash15 2016

[24] M Tahani A R Askari Y Mohandes and B Hassani ldquoSize-dependent free vibration analysis of electrostatically pre-de-formed rectangular micro-plates based on the modifiedcouple stress theoryfied couple stress theoryrdquo InternationalJournal of Mechanical Sciences vol 94-95 no 22 pp 185ndash1982015

[25] A Veysi R Shabani and G Rezazadeh ldquoNonlinear vibrationsof micro-doubly curved shallow shells based on the modifiedcouple stress theoryfied couple stress theoryrdquo NonlinearDynamics vol 87 no 3 pp 2051ndash2065 2017

[26] B Jalalahmadi F Sadeghi and D Peroulis ldquoA numericalfatigue damage model for life scatter of MEMS devicesrdquoJournal of Microelectromechanical Systems vol 18 no 5pp 1016ndash1031 2009

[27] T S Slack F Sadeghi and D Peroulis ldquoA phenomenologicaldiscrete brittle damage-mechanics model for fatigue of MEMSdevices with application to LIGA Nirdquo Journal of Micro-electromechanical Systems vol 18 no 1 pp 119ndash128 2009

[28] A Basu R P Hennessy G G Adams and N E McGruerldquoHot switching damage mechanisms in MEMS contacts-ev-idence and understandingrdquo Journal of Micromechanics andMicroengineering vol 24 no 10 p 16 2014

[29] C Chen J Yuan and Y Mao ldquoPost-buckling of size-de-pendent micro-plate considering damage effectsrdquo NonlinearDynamics vol 90 no 2 pp 1301ndash1314 2017

[30] J Lemaitre A Course on Damage Mechanics Springer BerlinGermany 1992

[31] C Chen H Hu and L Dai ldquoNonlinear behavior andcharacterization of a piezoelectric laminated microbeamsystemrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1304ndash1315 2013

[32] A R Askari and M Tahani ldquoSize-dependent dynamic pull-inanalysis of geometric non-linear micro-plates based on themodified couple stress theoryrdquo Physica E Low-DimensionalSystems and Nanostructures vol 86 pp 262ndash274 2017

[33] A H Nayfeh and D T Mook Nonlinear OscillationsSpringer International Publishing New York NY USA 1979

12 Shock and Vibration

[34] R Ansari M Faghih Shojaei V Mohammadi R Gholamiand M A Darabi ldquoNonlinear vibrations of functionallygraded Mindlin microplates based on the modified couplestress theoryfied couple stress theoryrdquo Composite Structuresvol 114 no 18 pp 124ndash134 2014

[35] R Ansari R Gholami and A Shahabodini ldquoSize-dependentgeometrically nonlinear forced vibration analysis of func-tionally graded first-order shear deformable microplatesrdquoJournal of Mechanics vol 32 no 5 pp 539ndash554 2016

Shock and Vibration 13

Aij hQdij

Dij h3

121113888 1113889Q

dij

Gprime hF

(i j 1 2 6)

(17)

With Hamiltonrsquos principle we have

δ1113946t2

t1

(T minus U + W)dt 0 (18)

Substituting stresses and strains into (1) the virtualstrain energy is obtained as

δU 1113946Vσijδεij1113872 1113873dV + 1113946

Vmijδχij1113872 1113873dV (19)

e virtual kinetic energy is given as

δT 12ρ1113946

Vδ wt( 1113857

2dV (20)

e virtual work done by electric force and dampingforce can be expressed as

δW Bs

Fz minus cwt( 1113857δwdS (21)

where c denotes damping coefficient and Fz represents theelectric force defined as

Fz εvV

2C

2(d minus w)2 (22)

where εv is the dielectric constant of airBy substituting (19)ndash(21) into (18) one has the nonlinear

dynamic equations of the microplate system as

Nxx + Nxyy +12Yxzxy +

12Yyzyy 0

Nyy + Nxyx minus12Yxzxx minus

12Yyzxy 0

minus Nxwx + Nxywy1113872 1113873x

minus Nywy + Nxywx1113872 1113873y

+ minus Mx minus Yxy1113872 1113873xx

+ minus My + Yxy1113872 1113873yy

+ minus 2Mxy + Yxx minus Yyy1113872 1113873xy

+ Fe ρhwtt + cwt

(23)

Boundary conditions are as follows

Nx +12Yxzy 0

or δu 0

Nxy minus12Yxzx minus

12Yyzy 0

or δv 0

12Yxz 0

or δvx 0

Mxx + 2Mxyy + Nxwx + Nxywy + Yxyx + minus Yxx + Yyy1113872 1113873y

0

or δw 0

Mx + Yxy 0

or δwx 0

y 0 b

Ny minus12Yyzx 0

or δv 0

Nxy +12Yyzy +

12Yxzx 0

or δu 0

12Yyz 0

or δuy 0

Myy + Nywy + Nxywx + 2Mxyx minus Yxyy minus Yxx minus Yyy1113872 1113873x

0

or δw 0

My minus Yxy 0

or δwy 0

(24)

e following dimensionless parameters are introduced

4 Shock and Vibration

ξ x

a

η y

b

U u

a

V v

b

W w

h

λ1 h

a

λ2 h

b

Aij Aij

Q11h

l l

h

Dij 12Dij

Q11h3

G1 Gprime

8Ga3

G2 Gprime

8Gb3

G3 Gprime

2Gh3

β G

Q11

Q11 E

1 minus υ2

τ t

T

C ca

2b2

Q11h3T

VC VC

V0

α1 εva

2b2V

20

2d2h4Q11

α2 εva

2b2V

20

d3h3Q11

T ab

h

ρQ11

1113970

ω ΩT

(25)

Shock and Vibration 5

where V0 is the unit voltageWith application of Taylor series expansion into (22) and

ignoring nonlinear terms with respect to w [31] the non-

dimensional governing equations of the microplate arewritten as

A11 Uξξ + λ21WξWξξ1113872 1113873 + A12 Vξη + λ22WηWξη1113872 1113873 + βA66λ22λ21

Uηη + Vξη + λ22WξWηη + λ22WηWξη1113888 1113889

+βλ1

G1 Vξξξη minusλ22λ21

Uξξηη +λ22λ21

Vξηηη minusλ42λ41

Uηηηη1113888 1113889 0

A12 Uξη + λ21WξWξη1113872 1113873 + A22 Vηη + λ22WηWηη1113872 1113873 + βA66 Uξη +λ21λ22

Vξξ + λ21WηWξξ + λ21WξWξη1113888 1113889

minusβλ2

G2λ41λ42

Vξξξξ minusλ21λ22

Uξξξη +λ21λ22

Vξξηη minus Uξηηη1113888 1113889 0

112

D11 + βG31113874 1113875λ21λ22

Wξξξξ +16D12 +

13D66 + 2βG31113874 1113875Wξξηη +

112

D22 + βG31113874 1113875λ22λ21

Wηηηη + Wττ + CWτ

A11 Uξ +12λ21W

2ξ1113874 1113875 + A12 Vη +

12λ22W

2η1113874 11138751113876 1113877

1λ22

Wξξ + A11 Uξξ + λ21WξWξξ1113872 1113873 + A12 Vξη + λ22WηWξη1113872 11138731113960 11139611λ22

Wξ

+ 2βA661λ21

Uη +1λ22

Vξ + WηWξ1113888 1113889Wξη + βA661λ22

Vξξ +1λ21

Uξη + WηWξξ + WξWξη1113888 1113889Wη

+ A12 Uξ +12λ21W

2ξ1113874 1113875 + A22 Vη +

12λ22W

2η1113874 11138751113876 1113877

1λ21

Wηη + A12 Uξη + λ21WξWξη1113872 1113873 + A22 Vηη + λ22WηWηη1113872 11138731113960 11139611λ21

Wη

+ βA661λ21

Uηη +1λ22

Vξη + WξηWη + WξWηη1113888 1113889Wξ + α1 + α2W( 1113857V2C

(26)

Due to the fact that a fully clamped rectangularmicroplate is extensively used in MEMS applications [32]

this kind of boundary condition is also studied in this workand written in dimensionless formulation as follows

A11 Uξ +12W

2ξ1113874 1113875 +

βG3

8h2

abVξξη minus

h2

b2Uξηη1113888 1113889 0 ξ 0 1 V 0 W 0 Vξ 0 Wξ 0

A22 Vη +12λ1W

2η1113874 1113875 minus

βG3

8h2

a2Vξξη minus

h2

abUξηη1113888 1113889 0 η 0 1 U 0 W 0 Uη 0 Wη 0

(27)

3 Solution Methodology

First note that in the boundary conditions in (27) the itemsrelated to size effect are neglected due to the fact that they arenegligible compared with other items [29] With satisfactionof boundary conditions a solution for (26) takes the fol-lowing form

U(ξ η τ) U(τ) cos πξ sin2 πη

V(ξ η τ) V(τ) sin2 πξ cos πη

W(ξ η τ) W(τ) sin2 πξ sin2 πη

(28)

Submitting (28) into (26)fd25 and multiplying them bycos πξ sin2 πη sin2 πξ cos πη and sin2 πξ sin2 πη

6 Shock and Vibration

respectively and integrating them from 0 to 1 with respect toξ and η the nonlinear governing differential equations withrespect to time can be obtained as

minus3π2

16Aminus

11 minus βAminus

66π2λ224λ21

minusGminus

1βλ1

π4λ224λ21

+π4λ42λ41

⎡⎢⎢⎣ ⎤⎥⎥⎦Uminus

+ minus169

Aminus

12 minus169βA

minus

66 +Gminus

1βλ1

64π2

9+16π2λ229λ21

⎡⎢⎢⎣ ⎤⎥⎥⎦Vminus

+π2

6λ21A

minus

11 +2π2

15λ22A

minus

12 minus2π2

15βA

minus

66λ221113890 1113891W

minus 2 0

minus169

Aminus

12 minus169βA

minus

66 +Gminus

2βλ2

16π2λ219λ22

+64π2

9⎡⎢⎢⎣ ⎤⎥⎥⎦U

minus

+ minus3π2

16Aminus

22 minusπ2λ214λ22

βAminus

66 minusGminus

2βλ2

π4λ41λ42

+π4λ214λ22

⎡⎢⎢⎣ ⎤⎥⎥⎦Vminus

+2π2

15λ21A

minus

12 +π2

6λ22A

minus

22 minus2π2

15βA

minus

66λ211113890 1113891W

minus 2 0

3112

Dminus

11 + βGminus

3λ21λ22

+16Dminus

12 +13βD

minus

66 + 2βGminus

3 + 3112

Dminus

22 + βGminus

3λ21λ22

1113890 1113891π4

41113896 1113897W

minus

+14Wminus

ττ +Cminus

4Wminus

τ

π2A

minus

11

3λ22minus4π2βA

minus

66

15λ21+4π2A

minus

12

15λ21Uminus

Wminus

+4π2A

minus

12

15λ22minus4π2βA

minus

66

15λ22+π2A

minus

22

3λ21Vminus

Wminus

minus525π4λ211024λ22

Aminus

11 +25π4

1024Aminus

12

+525π4λ221024λ21

Aminus

22 +50π4

1024βA

minus

66Wminus 3

+14α1 +

916α2W

minus

Vminus 2

C

(29)

For convenience (29) can be expressed as follows

M11Uminus

+ M12Vminus

+ M13Wminus 2

0

M21Uminus

+ M22Vminus

+ M23Wminus 2

0

M31Wminus

+14Wminus

ττ +Cminus

4Wminus

τ M32Uminus

Wminus

+ M33Vminus

Wminus

minus M34Wminus 3

+14α1 +

916α2W

minus

Vminus 2

C

(30)

where

M11 minus3π2

16A11 minus βA66

π2λ224λ21

minusG1βλ1

π4λ224λ21

+π4λ42λ41

1113888 1113889

M12 minus169

A12 minus169βA66 +

G1βλ1

64π2

9+16π2λ229λ21

1113888 1113889

M13 π2

6λ21A11 +

2π2

15λ22A12 minus

2π2

15βA66λ

22

M21 minus169

A12 minus169βA66 +

G2βλ2

16π2λ219λ22

+64π2

91113888 1113889

Shock and Vibration 7

M22 minus3π2

16A22 minus

π2λ214λ22

βA66 minusG2βλ2

π4λ41λ42

+π4λ214λ22

1113888 1113889

M23 2π2

15λ21A12 +

π2

6λ22A22 minus

2π2

15βA66

M31 3112

D11 + βG31113874 1113875λ21λ22

+16D12 +

13D66 + 2βG31113874 1113875 + 3

112

D22 + βG31113874 1113875λ21λ22

1113890 1113891π4

4

M32 π2

A11

3λ22minus4π2βA66

15λ21+4π2A12

15λ21

M33 4π2A12

15λ22minus4π2βA66

15λ22+π2A22

3λ21

M34 minus525π4λ211024λ22

A11 minus25π4

1024A12 minus

525π4λ221024λ21

A22 minus50π4

1024βA66

(31)

After eliminating U(τ) V(τ) the nonlinear differentialequation only related to W(τ) can be obtained

14Wττ +

C

4Wτ + M31W minus φW

314

α1 +916α2W1113874 1113875V

2C

(32)

where

φ M34 +M32M22M13 minus M32M12M23

M12M21 minus M11M22+

M33M11M23 minus M33M13M21

M12M21 minus M11M22 (33)

4 Results and Discussion

In this section the nonlinear size-dependent vibration analysisof the microplate is studied under the influence of size effectand damage effect Free vibration and forced vibration analysisof the microplate under DC voltage and AC voltage are dis-cussed in detail accordinglye frequency-response curves arepresented for various system parameters such as materiallength scale parameter damage variable damping ratio andexternal AC voltage Note that geometric and physical pa-rameters of the microplate are listed in Table 1

41 Free Vibration Analysis under DC Voltage For staticanalysis (32) can be written as

Wττ + C Wτ

4+ M31W minus φW

3

α1 +(916)α2W( 1113857V2d

4

(34)

where VD donates the DC voltage

e harmonic balance method is feasible for findingsolutions of both weakly and strongly nonlinear problems Ageneral solution of (34) can be given as

W(τ) A0 + A1 cos(ωτ + θ) (35)

Submitting (35) into (34) and using harmonic balancemethod (HB) [33] we obtain

ω20A0 minus 4φ A

30 +

32A0A

211113874 1113875 α1V

2d

minus A1ω2

+ ω20A1 minus 4φ 3A

20A1 +

34A311113874 1113875 0

(36)

where A0 is so smaller than A1 that it can be neglected in thefurther step of solution and then the following is obtained

ω20 4M31 minus

916α2V

2d

ω2 ω2

0 minus 3φA21

(37)

8 Shock and Vibration

e variation of amplitude-frequency response of themicroplate with various values of damage variable is pre-sented in Figure 2 where DC voltage Vd and non-dimensional material length scale parameter lh are taken as20 and 02 respectively It should be pointed out that thecurrent value of material length scale parameter is onlya mean to illustrate microscale effect Sophisticated exper-iments need to be conducted to measure material lengthscale parameter as it plays a significant role in microscalestructures As shown in Figure 2 the vibration amplitudebecomes larger with increase of nonlinear frequency whichindicates that the microplate exhibits a hardening-typebehavior It is also observed that the larger the damagevariable is the lower the nonlinear free vibration frequencybecomes Moreover free vibration amplitude of themicroplate midpoint increases as damage variable increasesdue to reduced stiffness of the microplate

Influence of size effect on frequency-response curves isgiven in Figure 3 in which damage variable D and DCvoltage Vd are taken as 02 and 20 separately It is observedthat large material length scale parameter results in smallamplitude and high frequency of the microplate Whenespecially size effect is omitted the classical microplatemodel is recovered As can be observed from Figure 3material length scale parameter increases the differencebetween the results of present microplate model and theclassical microplate model increases indicating that themicroscale effect induce additional rigidity for microstruc-turee extra stiffness contributed by couple stress togetherwith the classical stiffness increased the total stiffness of themicroplate Furthermore our results are consistent with thepredictions obtained by Ansari et al [34] who studied thenonlinear vibrations properties of functionally gradedMindlinrsquos microplates using MCST

42 Forced Vibration Analysis under AC Voltage As fordynamic analysis (32) can be written as14Wττ +

14

C Wτ + M31W minus φW3

14

α1 +916α2W1113874 1113875V

2a cos

2 ωτ

(38)

Similarly the harmonic balance method is employed forsolving (38) and the general solution for this equation can begiven as

W(τ) A1 cos 2ωτ + θ0( 1113857 + A2 sin 2ωτ + θ0( 1113857

A1 cos 2Φ + A2 sin 2Φ(39)

where

cos 2ωτ A3 cos 2Φ + A4 sin 2Φ A3 cos

θ0 A4 sin θ0(40)

Substituting (39) into (38) we obtain

Table 1 Geometric and material properties of the microplate

Property ValueLength a 300 μmWidth b 150 μmickness h 2 μmInitial gap d 2 μmPoissonrsquos ratio υ 006Youngrsquos modulus E 166 times 109 Paβ 02Dielectric constant εv 8854 times 10minus 12 Fm

D = 0D = 02D = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 2 Damage effect on frequency-response curves of themicroplate

lh = 0lh = 02lh = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 3 Size effect on frequency-response curves of themicroplate

Shock and Vibration 9

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A1 + 2ωCA2 minus 3φA1 A

21 + A

221113872 1113873

α12

V2aA3

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A2 minus 2ωCA1 minus 3φA2 A

21 + A

221113872 1113873

α12

V2aA4

(41)

Using A21 + A2

2 A2 and A23 + A2

4 1 the relationshipbetween the amplitude of the microplate midpoint and theexcitation frequency can be achieved as

9φ2A6

minus 6φ 4M31 minus 4ω2minus

932α2V

2a1113874 1113875A

4+ 4M31 minus 4ω2

minus932α2V

2a1113874 1113875

2+ 4ω2

C2

1113890 1113891A2

α214

V4a (42)

e frequency-response curves of the system withdamage variable D nondimensional material length scaleparameter lh and damping coefficient C are respectivelytaken as 02 02 and 02 and are depicted in Figure 4 whichreveals that the system still exhibits a hardening-type be-havior It is observed that the amplitude of microplatemidpoint increases as excitation frequency increases from11 actually from 0 until reaching a limit point (upperjump point) corresponding to the nonlinear resonanceand then the amplitude of microplate midpoint fallssuddenly for another the amplitude of the microplatemidpoint increases as the excitation frequency is decreasedfrom 35 until reaching another limit point (lower jumppoint) and then the amplitude of the microplate midpointrises abruptly e region between the lower jump pointand the upper jump point is regarded as the unstable re-gion is unique behavior is called as jump phenomenonwhich results from the nonlinearity of the dynamic systemAdditionally with the increase of the amplitude of ACvoltage the upper jump point tends to move towards theright side and the amplitude of the microplate midpointincreases In other words the nonlinear response region isenlarged by the greater external load e jump phe-nomenon disappears when the amplitude of AC voltage issmall in this case such as 10 (black line) Without enough

energy input by external excitation the dynamic systemwill remain stable

Figure 5 indicates the effect of damping coefficient on thefrequency-response curve of the microplate In the nu-merical simulation the nondimensional material lengthscale parameter lh damage variableD and amplitude of ACvoltage Va are taken to be 02 02 and 20 accordingly Fromthis figure we can see that by increasing the dampingcoefficient the unstable region decreases and the peak of thecurves goes down It is attributed to the fact that the energyinput by external load is consumed by the dissipation sys-tem e larger the damping coefficient is the more energythe dynamic system expends

Figure 6 displays damage effect on frequency-responsecurves of the microplate with damping coefficient C non-dimensional material length scale parameter lh and am-plitude of AC voltage Va are set to 02 02 and 20respectively It is observed that the larger the damage var-iable the bigger the peak of the curve and the wider theunstable region and nonlinear response region What ismore unlike the first two cases the unstable region movestowards the left side when the microplate suffers more se-rious damage e system does not possess enough stiffnessto resist external load e reason for this is that the stiffnessof the structure is reduced by damage effect

V_a = 20

V_a = 30

V_a = 10

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 4 e frequency-response curves of the system for different amplitude of the AC voltage VAC

10 Shock and Vibration

Figure 7 highlights the size effect on the frequency-response curves of dynamic system e numerical cal-culations are performed by assumingD 02 C 02 Va 20 It is revealed that the peak ofthe curve goes down slightly and the hardening influencereduces mildly when nondimensional material lengthscale parameter increases In addition the unstable regionmoves to the right side and diminishes as size effect be-comes more obvious Furthermore the distinction be-tween the results predicted by classical theory (lh 0)

and nonclassical theory gets more obvious e reason forthis change was already discussed in Section 41 Besidesthe present result is parallel to that of Ansari et al [35]who investigated the forced vibration of functionallygraded microplate based onMCST It should be noted thathorizontal axis in this work represents nondimensionalexternal frequency while in Ansari et alrsquos work it rep-resents the frequency ratio (the ratio of external frequencyto linear first frequency)

5 Conclusions

e nonlinear size-dependent vibration of a microplate withdamage was explored by employing MCST and the strainequivalent assumption in this research e nonlineargoverning partial differential equations were transformedinto nonlinear ordinary differential equations via Galerkinrsquosscheme and further solved numerically by the harmonicbalance method Numerical results indicate that damageeffect and size dependency both have obvious influences onthe static and dynamic behaviors of the microplate system Itwas concluded that on one hand the hardening-typenonlinear behavior of the microplate system enhances whenit encounters damage on the other hand the system exhibitsa weaker nonlinear behavior greater nondimensional fre-quency and lower amplitude of the microplate midpoint assize effect gets obvious

Data Availability

e Matlab simulation and control program data used tosupport the findings of this study are available from thecorresponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors would like to acknowledge with great gratitudefor the supports of the National Science Foundation ofChina (grant nos 51778551 and 11272270)

References

[1] M I Younis and A H Nayfeh ldquoA study of the nonlinearresponse of a resonant micro-beam to an electric actuationrdquoNonlinear Dynamics vol 31 no 1 pp 91ndash117 2003

C_

= 02

C_

= 04

C_

= 01

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 5e frequency-response curves of the system for differentdamping coefficients C

D = 02

D = 04

D = 0

15 19 23 27 31 3511ω

0

02

04

06

08

A

Figure 6e frequency-response curves of the system for differentdamage variable D

lh = 04

lh = 0

lh = 02

15 19 23 27 31 3511ω

0

01

02

03

04

05

A

Figure 7e frequency-response curves of the system for differentnondimensional material length scale parameters lh

Shock and Vibration 11

[2] E M Abdel-Rahman and A H Nayfeh ldquoSecondary reso-nances of electrically actuated resonant microsensorsrdquoJournal of Micromechanics and Microengineering vol 13no 3 pp 491ndash501 2003

[3] W Zhang and G Meng ldquoNonlinear dynamical system ofmicro-cantilever under combined parametric and forcingexcitations in MEMSrdquo Sensors and Actuators A Physicalvol 119 no 2 pp 291ndash299 2005

[4] L Xu and X Jia ldquoElectromechanical coupled nonlinear dy-namics for microbeamsrdquo Archive of Applied Mechanicsvol 77 no 7 pp 485ndash502 2007

[5] W G Vogl and A H Nayfeh ldquoPrimary resonance excitationof electrically actuated clamped circular platesrdquo NonlinearDynamics vol 47 no 1ndash3 pp 181ndash192 2006

[6] A H Nayfeh H M Ouakad F Choura and E M Abdel-Rahman ldquoNonlinear dynamics of a resonant gas sensorrdquoNonlinear Dynamics vol 59 no 4 pp 607ndash618 2010

[7] X L Jia J Yang S Kitipornchai and C W Lim ldquoResonancefrequency response of geometrically nonlinear micro-switches under electrical actuationrdquo Journal of Sound andVibration vol 331 no 14 pp 3397ndash3411 2012

[8] P Kim S Bae and J Seok ldquoResonant behaviors of a nonlinearcantilever beam with tip mass subject to an axial force andelectrostatic excitationrdquo International Journal of MechanicalSciences vol 64 no 1 pp 232ndash257 2012

[9] S Saghir and M I Younis ldquoAn investigation of the static anddynamic behavior of electrically actuated rectangularmicroplatesrdquo International Journal of Non-linear Mechanicsvol 85 pp 81ndash93 2016

[10] M Sheikhlou R Shabani and G Rezazadeh ldquoNonlinearanalysis of electrostatically actuated diaphragm-type micro-pumpsrdquo Nonlinear Dynamics vol 83 no 1-2 pp 1ndash11 2016

[11] P M Osterberg Electrostatically actuated micromechanicaltest structure for material property measurement PhD dis-sertation Massachusetts Institute of Technology CambridgeMA USA 1995

[12] M Arefi and A M Zenkour ldquoSize dependent electro-elasticanalysis of a sandwich microbeam based on higher ordersinusoidal shear deformation theory and strain gradienttheoryrdquo Journal of Intelligent Material Systems and Structuresvol 29 no 7 pp 27ndash40 2018

[13] M Arefi and A M Zenkour ldquoSize-dependent vibration andbending analyses of the piezomagnetic three-layer nano-beamsrdquo Applied Physics A vol 123 no 3 p 202 2017

[14] M Arefi and A M Zenkour ldquoSize-dependent vibration andelectro-magneto-elastic bending responses of sandwich pie-zomagnetic curved nanobeamsrdquo Steel and Composite Struc-tures vol 29 no 5 pp 579ndash590 2018

[15] M Arefi and A M Zenkour ldquoermo-electro-magneto-mechanical bending behavior of size-dependent sandwichpiezomagnetic nanoplatesrdquo Mechanics Research Communi-cations vol 84 pp 27ndash42 2017

[16] L L Ke J Yang S Kitipornchai M A Bradford andY S Wang ldquoAxisymmetric nonlinear free vibration of size-dependent functionally graded annular microplatesrdquo Com-posites Part B Engineering vol 53 no 7 pp 207ndash217 2013

[17] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[18] M H Ghayesh H Farokhi and M Amabili ldquoNonlinearbehaviour of electrically actuated MEMS resonatorsrdquo

International Journal of Engineering Science vol 71 no 10pp 137ndash155 2013

[19] A G Arani and G S Jafari ldquoNonlinear vibration analysis oflaminated composite Mindlin micronano-plates resting onorthotropic Pasternak medium using DQMrdquo AppliedMathematics and Mechanics vol 36 no 8 pp 1033ndash10442015

[20] M Sobhy and A M Zenkour ldquoA comprehensive study onthe size-dependent hygrothermal analysis of exponentiallygraded microplates on elastic foundationsrdquo Mechanics ofAdvanced Materials and Structures vol 27 no 10pp 816ndash830 2019

[21] M Arefi and A M Zenkour ldquoSize-dependent free vibrationand dynamic analyses of piezo-electro-magnetic sandwichnanoplates resting on viscoelastic foundationrdquo Physica BCondensed Matter vol 521 pp 188ndash197 2017

[22] M Arefi M Kiani and A M Zenkour ldquoSize-dependent freevibration analysis of a three-layered exponentially gradednano-micro-plate with piezomagnetic face sheets resting onPasternakrsquos foundation via MCSTrdquo Journal of SandwichStructures and Materials vol 22 no 1 pp 55ndash86 2020

[23] H Farokhi and M H Ghayesh ldquoSize-dependent behavior ofelectrically actuated microcantilever-based MEMSrdquo In-ternational Journal of Mechanics amp Materials in Designvol 12 no 3 pp 1ndash15 2016

[24] M Tahani A R Askari Y Mohandes and B Hassani ldquoSize-dependent free vibration analysis of electrostatically pre-de-formed rectangular micro-plates based on the modifiedcouple stress theoryfied couple stress theoryrdquo InternationalJournal of Mechanical Sciences vol 94-95 no 22 pp 185ndash1982015

[25] A Veysi R Shabani and G Rezazadeh ldquoNonlinear vibrationsof micro-doubly curved shallow shells based on the modifiedcouple stress theoryfied couple stress theoryrdquo NonlinearDynamics vol 87 no 3 pp 2051ndash2065 2017

[26] B Jalalahmadi F Sadeghi and D Peroulis ldquoA numericalfatigue damage model for life scatter of MEMS devicesrdquoJournal of Microelectromechanical Systems vol 18 no 5pp 1016ndash1031 2009

[27] T S Slack F Sadeghi and D Peroulis ldquoA phenomenologicaldiscrete brittle damage-mechanics model for fatigue of MEMSdevices with application to LIGA Nirdquo Journal of Micro-electromechanical Systems vol 18 no 1 pp 119ndash128 2009

[28] A Basu R P Hennessy G G Adams and N E McGruerldquoHot switching damage mechanisms in MEMS contacts-ev-idence and understandingrdquo Journal of Micromechanics andMicroengineering vol 24 no 10 p 16 2014

[29] C Chen J Yuan and Y Mao ldquoPost-buckling of size-de-pendent micro-plate considering damage effectsrdquo NonlinearDynamics vol 90 no 2 pp 1301ndash1314 2017

[30] J Lemaitre A Course on Damage Mechanics Springer BerlinGermany 1992

[31] C Chen H Hu and L Dai ldquoNonlinear behavior andcharacterization of a piezoelectric laminated microbeamsystemrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1304ndash1315 2013

[32] A R Askari and M Tahani ldquoSize-dependent dynamic pull-inanalysis of geometric non-linear micro-plates based on themodified couple stress theoryrdquo Physica E Low-DimensionalSystems and Nanostructures vol 86 pp 262ndash274 2017

[33] A H Nayfeh and D T Mook Nonlinear OscillationsSpringer International Publishing New York NY USA 1979

12 Shock and Vibration

[34] R Ansari M Faghih Shojaei V Mohammadi R Gholamiand M A Darabi ldquoNonlinear vibrations of functionallygraded Mindlin microplates based on the modified couplestress theoryfied couple stress theoryrdquo Composite Structuresvol 114 no 18 pp 124ndash134 2014

[35] R Ansari R Gholami and A Shahabodini ldquoSize-dependentgeometrically nonlinear forced vibration analysis of func-tionally graded first-order shear deformable microplatesrdquoJournal of Mechanics vol 32 no 5 pp 539ndash554 2016

Shock and Vibration 13

ξ x

a

η y

b

U u

a

V v

b

W w

h

λ1 h

a

λ2 h

b

Aij Aij

Q11h

l l

h

Dij 12Dij

Q11h3

G1 Gprime

8Ga3

G2 Gprime

8Gb3

G3 Gprime

2Gh3

β G

Q11

Q11 E

1 minus υ2

τ t

T

C ca

2b2

Q11h3T

VC VC

V0

α1 εva

2b2V

20

2d2h4Q11

α2 εva

2b2V

20

d3h3Q11

T ab

h

ρQ11

1113970

ω ΩT

(25)

Shock and Vibration 5

where V0 is the unit voltageWith application of Taylor series expansion into (22) and

ignoring nonlinear terms with respect to w [31] the non-

dimensional governing equations of the microplate arewritten as

A11 Uξξ + λ21WξWξξ1113872 1113873 + A12 Vξη + λ22WηWξη1113872 1113873 + βA66λ22λ21

Uηη + Vξη + λ22WξWηη + λ22WηWξη1113888 1113889

+βλ1

G1 Vξξξη minusλ22λ21

Uξξηη +λ22λ21

Vξηηη minusλ42λ41

Uηηηη1113888 1113889 0

A12 Uξη + λ21WξWξη1113872 1113873 + A22 Vηη + λ22WηWηη1113872 1113873 + βA66 Uξη +λ21λ22

Vξξ + λ21WηWξξ + λ21WξWξη1113888 1113889

minusβλ2

G2λ41λ42

Vξξξξ minusλ21λ22

Uξξξη +λ21λ22

Vξξηη minus Uξηηη1113888 1113889 0

112

D11 + βG31113874 1113875λ21λ22

Wξξξξ +16D12 +

13D66 + 2βG31113874 1113875Wξξηη +

112

D22 + βG31113874 1113875λ22λ21

Wηηηη + Wττ + CWτ

A11 Uξ +12λ21W

2ξ1113874 1113875 + A12 Vη +

12λ22W

2η1113874 11138751113876 1113877

1λ22

Wξξ + A11 Uξξ + λ21WξWξξ1113872 1113873 + A12 Vξη + λ22WηWξη1113872 11138731113960 11139611λ22

Wξ

+ 2βA661λ21

Uη +1λ22

Vξ + WηWξ1113888 1113889Wξη + βA661λ22

Vξξ +1λ21

Uξη + WηWξξ + WξWξη1113888 1113889Wη

+ A12 Uξ +12λ21W

2ξ1113874 1113875 + A22 Vη +

12λ22W

2η1113874 11138751113876 1113877

1λ21

Wηη + A12 Uξη + λ21WξWξη1113872 1113873 + A22 Vηη + λ22WηWηη1113872 11138731113960 11139611λ21

Wη

+ βA661λ21

Uηη +1λ22

Vξη + WξηWη + WξWηη1113888 1113889Wξ + α1 + α2W( 1113857V2C

(26)

Due to the fact that a fully clamped rectangularmicroplate is extensively used in MEMS applications [32]

this kind of boundary condition is also studied in this workand written in dimensionless formulation as follows

A11 Uξ +12W

2ξ1113874 1113875 +

βG3

8h2

abVξξη minus

h2

b2Uξηη1113888 1113889 0 ξ 0 1 V 0 W 0 Vξ 0 Wξ 0

A22 Vη +12λ1W

2η1113874 1113875 minus

βG3

8h2

a2Vξξη minus

h2

abUξηη1113888 1113889 0 η 0 1 U 0 W 0 Uη 0 Wη 0

(27)

3 Solution Methodology

First note that in the boundary conditions in (27) the itemsrelated to size effect are neglected due to the fact that they arenegligible compared with other items [29] With satisfactionof boundary conditions a solution for (26) takes the fol-lowing form

U(ξ η τ) U(τ) cos πξ sin2 πη

V(ξ η τ) V(τ) sin2 πξ cos πη

W(ξ η τ) W(τ) sin2 πξ sin2 πη

(28)

Submitting (28) into (26)fd25 and multiplying them bycos πξ sin2 πη sin2 πξ cos πη and sin2 πξ sin2 πη

6 Shock and Vibration

respectively and integrating them from 0 to 1 with respect toξ and η the nonlinear governing differential equations withrespect to time can be obtained as

minus3π2

16Aminus

11 minus βAminus

66π2λ224λ21

minusGminus

1βλ1

π4λ224λ21

+π4λ42λ41

⎡⎢⎢⎣ ⎤⎥⎥⎦Uminus

+ minus169

Aminus

12 minus169βA

minus

66 +Gminus

1βλ1

64π2

9+16π2λ229λ21

⎡⎢⎢⎣ ⎤⎥⎥⎦Vminus

+π2

6λ21A

minus

11 +2π2

15λ22A

minus

12 minus2π2

15βA

minus

66λ221113890 1113891W

minus 2 0

minus169

Aminus

12 minus169βA

minus

66 +Gminus

2βλ2

16π2λ219λ22

+64π2

9⎡⎢⎢⎣ ⎤⎥⎥⎦U

minus

+ minus3π2

16Aminus

22 minusπ2λ214λ22

βAminus

66 minusGminus

2βλ2

π4λ41λ42

+π4λ214λ22

⎡⎢⎢⎣ ⎤⎥⎥⎦Vminus

+2π2

15λ21A

minus

12 +π2

6λ22A

minus

22 minus2π2

15βA

minus

66λ211113890 1113891W

minus 2 0

3112

Dminus

11 + βGminus

3λ21λ22

+16Dminus

12 +13βD

minus

66 + 2βGminus

3 + 3112

Dminus

22 + βGminus

3λ21λ22

1113890 1113891π4

41113896 1113897W

minus

+14Wminus

ττ +Cminus

4Wminus

τ

π2A

minus

11

3λ22minus4π2βA

minus

66

15λ21+4π2A

minus

12

15λ21Uminus

Wminus

+4π2A

minus

12

15λ22minus4π2βA

minus

66

15λ22+π2A

minus

22

3λ21Vminus

Wminus

minus525π4λ211024λ22

Aminus

11 +25π4

1024Aminus

12

+525π4λ221024λ21

Aminus

22 +50π4

1024βA

minus

66Wminus 3

+14α1 +

916α2W

minus

Vminus 2

C

(29)

For convenience (29) can be expressed as follows

M11Uminus

+ M12Vminus

+ M13Wminus 2

0

M21Uminus

+ M22Vminus

+ M23Wminus 2

0

M31Wminus

+14Wminus

ττ +Cminus

4Wminus

τ M32Uminus

Wminus

+ M33Vminus

Wminus

minus M34Wminus 3

+14α1 +

916α2W

minus

Vminus 2

C

(30)

where

M11 minus3π2

16A11 minus βA66

π2λ224λ21

minusG1βλ1

π4λ224λ21

+π4λ42λ41

1113888 1113889

M12 minus169

A12 minus169βA66 +

G1βλ1

64π2

9+16π2λ229λ21

1113888 1113889

M13 π2

6λ21A11 +

2π2

15λ22A12 minus

2π2

15βA66λ

22

M21 minus169

A12 minus169βA66 +

G2βλ2

16π2λ219λ22

+64π2

91113888 1113889

Shock and Vibration 7

M22 minus3π2

16A22 minus

π2λ214λ22

βA66 minusG2βλ2

π4λ41λ42

+π4λ214λ22

1113888 1113889

M23 2π2

15λ21A12 +

π2

6λ22A22 minus

2π2

15βA66

M31 3112

D11 + βG31113874 1113875λ21λ22

+16D12 +

13D66 + 2βG31113874 1113875 + 3

112

D22 + βG31113874 1113875λ21λ22

1113890 1113891π4

4

M32 π2

A11

3λ22minus4π2βA66

15λ21+4π2A12

15λ21

M33 4π2A12

15λ22minus4π2βA66

15λ22+π2A22

3λ21

M34 minus525π4λ211024λ22

A11 minus25π4

1024A12 minus

525π4λ221024λ21

A22 minus50π4

1024βA66

(31)

After eliminating U(τ) V(τ) the nonlinear differentialequation only related to W(τ) can be obtained

14Wττ +

C

4Wτ + M31W minus φW

314

α1 +916α2W1113874 1113875V

2C

(32)

where

φ M34 +M32M22M13 minus M32M12M23

M12M21 minus M11M22+

M33M11M23 minus M33M13M21

M12M21 minus M11M22 (33)

4 Results and Discussion

In this section the nonlinear size-dependent vibration analysisof the microplate is studied under the influence of size effectand damage effect Free vibration and forced vibration analysisof the microplate under DC voltage and AC voltage are dis-cussed in detail accordinglye frequency-response curves arepresented for various system parameters such as materiallength scale parameter damage variable damping ratio andexternal AC voltage Note that geometric and physical pa-rameters of the microplate are listed in Table 1

41 Free Vibration Analysis under DC Voltage For staticanalysis (32) can be written as

Wττ + C Wτ

4+ M31W minus φW

3

α1 +(916)α2W( 1113857V2d

4

(34)

where VD donates the DC voltage

e harmonic balance method is feasible for findingsolutions of both weakly and strongly nonlinear problems Ageneral solution of (34) can be given as

W(τ) A0 + A1 cos(ωτ + θ) (35)

Submitting (35) into (34) and using harmonic balancemethod (HB) [33] we obtain

ω20A0 minus 4φ A

30 +

32A0A

211113874 1113875 α1V

2d

minus A1ω2

+ ω20A1 minus 4φ 3A

20A1 +

34A311113874 1113875 0

(36)

where A0 is so smaller than A1 that it can be neglected in thefurther step of solution and then the following is obtained

ω20 4M31 minus

916α2V

2d

ω2 ω2

0 minus 3φA21

(37)

8 Shock and Vibration

e variation of amplitude-frequency response of themicroplate with various values of damage variable is pre-sented in Figure 2 where DC voltage Vd and non-dimensional material length scale parameter lh are taken as20 and 02 respectively It should be pointed out that thecurrent value of material length scale parameter is onlya mean to illustrate microscale effect Sophisticated exper-iments need to be conducted to measure material lengthscale parameter as it plays a significant role in microscalestructures As shown in Figure 2 the vibration amplitudebecomes larger with increase of nonlinear frequency whichindicates that the microplate exhibits a hardening-typebehavior It is also observed that the larger the damagevariable is the lower the nonlinear free vibration frequencybecomes Moreover free vibration amplitude of themicroplate midpoint increases as damage variable increasesdue to reduced stiffness of the microplate

Influence of size effect on frequency-response curves isgiven in Figure 3 in which damage variable D and DCvoltage Vd are taken as 02 and 20 separately It is observedthat large material length scale parameter results in smallamplitude and high frequency of the microplate Whenespecially size effect is omitted the classical microplatemodel is recovered As can be observed from Figure 3material length scale parameter increases the differencebetween the results of present microplate model and theclassical microplate model increases indicating that themicroscale effect induce additional rigidity for microstruc-turee extra stiffness contributed by couple stress togetherwith the classical stiffness increased the total stiffness of themicroplate Furthermore our results are consistent with thepredictions obtained by Ansari et al [34] who studied thenonlinear vibrations properties of functionally gradedMindlinrsquos microplates using MCST

42 Forced Vibration Analysis under AC Voltage As fordynamic analysis (32) can be written as14Wττ +

14

C Wτ + M31W minus φW3

14

α1 +916α2W1113874 1113875V

2a cos

2 ωτ

(38)

Similarly the harmonic balance method is employed forsolving (38) and the general solution for this equation can begiven as

W(τ) A1 cos 2ωτ + θ0( 1113857 + A2 sin 2ωτ + θ0( 1113857

A1 cos 2Φ + A2 sin 2Φ(39)

where

cos 2ωτ A3 cos 2Φ + A4 sin 2Φ A3 cos

θ0 A4 sin θ0(40)

Substituting (39) into (38) we obtain

Table 1 Geometric and material properties of the microplate

Property ValueLength a 300 μmWidth b 150 μmickness h 2 μmInitial gap d 2 μmPoissonrsquos ratio υ 006Youngrsquos modulus E 166 times 109 Paβ 02Dielectric constant εv 8854 times 10minus 12 Fm

D = 0D = 02D = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 2 Damage effect on frequency-response curves of themicroplate

lh = 0lh = 02lh = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 3 Size effect on frequency-response curves of themicroplate

Shock and Vibration 9

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A1 + 2ωCA2 minus 3φA1 A

21 + A

221113872 1113873

α12

V2aA3

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A2 minus 2ωCA1 minus 3φA2 A

21 + A

221113872 1113873

α12

V2aA4

(41)

Using A21 + A2

2 A2 and A23 + A2

4 1 the relationshipbetween the amplitude of the microplate midpoint and theexcitation frequency can be achieved as

9φ2A6

minus 6φ 4M31 minus 4ω2minus

932α2V

2a1113874 1113875A

4+ 4M31 minus 4ω2

minus932α2V

2a1113874 1113875

2+ 4ω2

C2

1113890 1113891A2

α214

V4a (42)

e frequency-response curves of the system withdamage variable D nondimensional material length scaleparameter lh and damping coefficient C are respectivelytaken as 02 02 and 02 and are depicted in Figure 4 whichreveals that the system still exhibits a hardening-type be-havior It is observed that the amplitude of microplatemidpoint increases as excitation frequency increases from11 actually from 0 until reaching a limit point (upperjump point) corresponding to the nonlinear resonanceand then the amplitude of microplate midpoint fallssuddenly for another the amplitude of the microplatemidpoint increases as the excitation frequency is decreasedfrom 35 until reaching another limit point (lower jumppoint) and then the amplitude of the microplate midpointrises abruptly e region between the lower jump pointand the upper jump point is regarded as the unstable re-gion is unique behavior is called as jump phenomenonwhich results from the nonlinearity of the dynamic systemAdditionally with the increase of the amplitude of ACvoltage the upper jump point tends to move towards theright side and the amplitude of the microplate midpointincreases In other words the nonlinear response region isenlarged by the greater external load e jump phe-nomenon disappears when the amplitude of AC voltage issmall in this case such as 10 (black line) Without enough

energy input by external excitation the dynamic systemwill remain stable

Figure 5 indicates the effect of damping coefficient on thefrequency-response curve of the microplate In the nu-merical simulation the nondimensional material lengthscale parameter lh damage variableD and amplitude of ACvoltage Va are taken to be 02 02 and 20 accordingly Fromthis figure we can see that by increasing the dampingcoefficient the unstable region decreases and the peak of thecurves goes down It is attributed to the fact that the energyinput by external load is consumed by the dissipation sys-tem e larger the damping coefficient is the more energythe dynamic system expends

Figure 6 displays damage effect on frequency-responsecurves of the microplate with damping coefficient C non-dimensional material length scale parameter lh and am-plitude of AC voltage Va are set to 02 02 and 20respectively It is observed that the larger the damage var-iable the bigger the peak of the curve and the wider theunstable region and nonlinear response region What ismore unlike the first two cases the unstable region movestowards the left side when the microplate suffers more se-rious damage e system does not possess enough stiffnessto resist external load e reason for this is that the stiffnessof the structure is reduced by damage effect

V_a = 20

V_a = 30

V_a = 10

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 4 e frequency-response curves of the system for different amplitude of the AC voltage VAC

10 Shock and Vibration

Figure 7 highlights the size effect on the frequency-response curves of dynamic system e numerical cal-culations are performed by assumingD 02 C 02 Va 20 It is revealed that the peak ofthe curve goes down slightly and the hardening influencereduces mildly when nondimensional material lengthscale parameter increases In addition the unstable regionmoves to the right side and diminishes as size effect be-comes more obvious Furthermore the distinction be-tween the results predicted by classical theory (lh 0)

and nonclassical theory gets more obvious e reason forthis change was already discussed in Section 41 Besidesthe present result is parallel to that of Ansari et al [35]who investigated the forced vibration of functionallygraded microplate based onMCST It should be noted thathorizontal axis in this work represents nondimensionalexternal frequency while in Ansari et alrsquos work it rep-resents the frequency ratio (the ratio of external frequencyto linear first frequency)

5 Conclusions

e nonlinear size-dependent vibration of a microplate withdamage was explored by employing MCST and the strainequivalent assumption in this research e nonlineargoverning partial differential equations were transformedinto nonlinear ordinary differential equations via Galerkinrsquosscheme and further solved numerically by the harmonicbalance method Numerical results indicate that damageeffect and size dependency both have obvious influences onthe static and dynamic behaviors of the microplate system Itwas concluded that on one hand the hardening-typenonlinear behavior of the microplate system enhances whenit encounters damage on the other hand the system exhibitsa weaker nonlinear behavior greater nondimensional fre-quency and lower amplitude of the microplate midpoint assize effect gets obvious

Data Availability

e Matlab simulation and control program data used tosupport the findings of this study are available from thecorresponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors would like to acknowledge with great gratitudefor the supports of the National Science Foundation ofChina (grant nos 51778551 and 11272270)

References

[1] M I Younis and A H Nayfeh ldquoA study of the nonlinearresponse of a resonant micro-beam to an electric actuationrdquoNonlinear Dynamics vol 31 no 1 pp 91ndash117 2003

C_

= 02

C_

= 04

C_

= 01

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 5e frequency-response curves of the system for differentdamping coefficients C

D = 02

D = 04

D = 0

15 19 23 27 31 3511ω

0

02

04

06

08

A

Figure 6e frequency-response curves of the system for differentdamage variable D

lh = 04

lh = 0

lh = 02

15 19 23 27 31 3511ω

0

01

02

03

04

05

A

Figure 7e frequency-response curves of the system for differentnondimensional material length scale parameters lh

Shock and Vibration 11

[2] E M Abdel-Rahman and A H Nayfeh ldquoSecondary reso-nances of electrically actuated resonant microsensorsrdquoJournal of Micromechanics and Microengineering vol 13no 3 pp 491ndash501 2003

[3] W Zhang and G Meng ldquoNonlinear dynamical system ofmicro-cantilever under combined parametric and forcingexcitations in MEMSrdquo Sensors and Actuators A Physicalvol 119 no 2 pp 291ndash299 2005

[4] L Xu and X Jia ldquoElectromechanical coupled nonlinear dy-namics for microbeamsrdquo Archive of Applied Mechanicsvol 77 no 7 pp 485ndash502 2007

[5] W G Vogl and A H Nayfeh ldquoPrimary resonance excitationof electrically actuated clamped circular platesrdquo NonlinearDynamics vol 47 no 1ndash3 pp 181ndash192 2006

[6] A H Nayfeh H M Ouakad F Choura and E M Abdel-Rahman ldquoNonlinear dynamics of a resonant gas sensorrdquoNonlinear Dynamics vol 59 no 4 pp 607ndash618 2010

[7] X L Jia J Yang S Kitipornchai and C W Lim ldquoResonancefrequency response of geometrically nonlinear micro-switches under electrical actuationrdquo Journal of Sound andVibration vol 331 no 14 pp 3397ndash3411 2012

[8] P Kim S Bae and J Seok ldquoResonant behaviors of a nonlinearcantilever beam with tip mass subject to an axial force andelectrostatic excitationrdquo International Journal of MechanicalSciences vol 64 no 1 pp 232ndash257 2012

[9] S Saghir and M I Younis ldquoAn investigation of the static anddynamic behavior of electrically actuated rectangularmicroplatesrdquo International Journal of Non-linear Mechanicsvol 85 pp 81ndash93 2016

[10] M Sheikhlou R Shabani and G Rezazadeh ldquoNonlinearanalysis of electrostatically actuated diaphragm-type micro-pumpsrdquo Nonlinear Dynamics vol 83 no 1-2 pp 1ndash11 2016

[11] P M Osterberg Electrostatically actuated micromechanicaltest structure for material property measurement PhD dis-sertation Massachusetts Institute of Technology CambridgeMA USA 1995

[12] M Arefi and A M Zenkour ldquoSize dependent electro-elasticanalysis of a sandwich microbeam based on higher ordersinusoidal shear deformation theory and strain gradienttheoryrdquo Journal of Intelligent Material Systems and Structuresvol 29 no 7 pp 27ndash40 2018

[13] M Arefi and A M Zenkour ldquoSize-dependent vibration andbending analyses of the piezomagnetic three-layer nano-beamsrdquo Applied Physics A vol 123 no 3 p 202 2017

[14] M Arefi and A M Zenkour ldquoSize-dependent vibration andelectro-magneto-elastic bending responses of sandwich pie-zomagnetic curved nanobeamsrdquo Steel and Composite Struc-tures vol 29 no 5 pp 579ndash590 2018

[15] M Arefi and A M Zenkour ldquoermo-electro-magneto-mechanical bending behavior of size-dependent sandwichpiezomagnetic nanoplatesrdquo Mechanics Research Communi-cations vol 84 pp 27ndash42 2017

[16] L L Ke J Yang S Kitipornchai M A Bradford andY S Wang ldquoAxisymmetric nonlinear free vibration of size-dependent functionally graded annular microplatesrdquo Com-posites Part B Engineering vol 53 no 7 pp 207ndash217 2013

[17] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[18] M H Ghayesh H Farokhi and M Amabili ldquoNonlinearbehaviour of electrically actuated MEMS resonatorsrdquo

International Journal of Engineering Science vol 71 no 10pp 137ndash155 2013

[19] A G Arani and G S Jafari ldquoNonlinear vibration analysis oflaminated composite Mindlin micronano-plates resting onorthotropic Pasternak medium using DQMrdquo AppliedMathematics and Mechanics vol 36 no 8 pp 1033ndash10442015

[20] M Sobhy and A M Zenkour ldquoA comprehensive study onthe size-dependent hygrothermal analysis of exponentiallygraded microplates on elastic foundationsrdquo Mechanics ofAdvanced Materials and Structures vol 27 no 10pp 816ndash830 2019

[21] M Arefi and A M Zenkour ldquoSize-dependent free vibrationand dynamic analyses of piezo-electro-magnetic sandwichnanoplates resting on viscoelastic foundationrdquo Physica BCondensed Matter vol 521 pp 188ndash197 2017

[22] M Arefi M Kiani and A M Zenkour ldquoSize-dependent freevibration analysis of a three-layered exponentially gradednano-micro-plate with piezomagnetic face sheets resting onPasternakrsquos foundation via MCSTrdquo Journal of SandwichStructures and Materials vol 22 no 1 pp 55ndash86 2020

[23] H Farokhi and M H Ghayesh ldquoSize-dependent behavior ofelectrically actuated microcantilever-based MEMSrdquo In-ternational Journal of Mechanics amp Materials in Designvol 12 no 3 pp 1ndash15 2016

[24] M Tahani A R Askari Y Mohandes and B Hassani ldquoSize-dependent free vibration analysis of electrostatically pre-de-formed rectangular micro-plates based on the modifiedcouple stress theoryfied couple stress theoryrdquo InternationalJournal of Mechanical Sciences vol 94-95 no 22 pp 185ndash1982015

[25] A Veysi R Shabani and G Rezazadeh ldquoNonlinear vibrationsof micro-doubly curved shallow shells based on the modifiedcouple stress theoryfied couple stress theoryrdquo NonlinearDynamics vol 87 no 3 pp 2051ndash2065 2017

[26] B Jalalahmadi F Sadeghi and D Peroulis ldquoA numericalfatigue damage model for life scatter of MEMS devicesrdquoJournal of Microelectromechanical Systems vol 18 no 5pp 1016ndash1031 2009

[27] T S Slack F Sadeghi and D Peroulis ldquoA phenomenologicaldiscrete brittle damage-mechanics model for fatigue of MEMSdevices with application to LIGA Nirdquo Journal of Micro-electromechanical Systems vol 18 no 1 pp 119ndash128 2009

[28] A Basu R P Hennessy G G Adams and N E McGruerldquoHot switching damage mechanisms in MEMS contacts-ev-idence and understandingrdquo Journal of Micromechanics andMicroengineering vol 24 no 10 p 16 2014

[29] C Chen J Yuan and Y Mao ldquoPost-buckling of size-de-pendent micro-plate considering damage effectsrdquo NonlinearDynamics vol 90 no 2 pp 1301ndash1314 2017

[30] J Lemaitre A Course on Damage Mechanics Springer BerlinGermany 1992

[31] C Chen H Hu and L Dai ldquoNonlinear behavior andcharacterization of a piezoelectric laminated microbeamsystemrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1304ndash1315 2013

[32] A R Askari and M Tahani ldquoSize-dependent dynamic pull-inanalysis of geometric non-linear micro-plates based on themodified couple stress theoryrdquo Physica E Low-DimensionalSystems and Nanostructures vol 86 pp 262ndash274 2017

[33] A H Nayfeh and D T Mook Nonlinear OscillationsSpringer International Publishing New York NY USA 1979

12 Shock and Vibration

[34] R Ansari M Faghih Shojaei V Mohammadi R Gholamiand M A Darabi ldquoNonlinear vibrations of functionallygraded Mindlin microplates based on the modified couplestress theoryfied couple stress theoryrdquo Composite Structuresvol 114 no 18 pp 124ndash134 2014

[35] R Ansari R Gholami and A Shahabodini ldquoSize-dependentgeometrically nonlinear forced vibration analysis of func-tionally graded first-order shear deformable microplatesrdquoJournal of Mechanics vol 32 no 5 pp 539ndash554 2016

Shock and Vibration 13

where V0 is the unit voltageWith application of Taylor series expansion into (22) and

ignoring nonlinear terms with respect to w [31] the non-

dimensional governing equations of the microplate arewritten as

A11 Uξξ + λ21WξWξξ1113872 1113873 + A12 Vξη + λ22WηWξη1113872 1113873 + βA66λ22λ21

Uηη + Vξη + λ22WξWηη + λ22WηWξη1113888 1113889

+βλ1

G1 Vξξξη minusλ22λ21

Uξξηη +λ22λ21

Vξηηη minusλ42λ41

Uηηηη1113888 1113889 0

A12 Uξη + λ21WξWξη1113872 1113873 + A22 Vηη + λ22WηWηη1113872 1113873 + βA66 Uξη +λ21λ22

Vξξ + λ21WηWξξ + λ21WξWξη1113888 1113889

minusβλ2

G2λ41λ42

Vξξξξ minusλ21λ22

Uξξξη +λ21λ22

Vξξηη minus Uξηηη1113888 1113889 0

112

D11 + βG31113874 1113875λ21λ22

Wξξξξ +16D12 +

13D66 + 2βG31113874 1113875Wξξηη +

112

D22 + βG31113874 1113875λ22λ21

Wηηηη + Wττ + CWτ

A11 Uξ +12λ21W

2ξ1113874 1113875 + A12 Vη +

12λ22W

2η1113874 11138751113876 1113877

1λ22

Wξξ + A11 Uξξ + λ21WξWξξ1113872 1113873 + A12 Vξη + λ22WηWξη1113872 11138731113960 11139611λ22

Wξ

+ 2βA661λ21

Uη +1λ22

Vξ + WηWξ1113888 1113889Wξη + βA661λ22

Vξξ +1λ21

Uξη + WηWξξ + WξWξη1113888 1113889Wη

+ A12 Uξ +12λ21W

2ξ1113874 1113875 + A22 Vη +

12λ22W

2η1113874 11138751113876 1113877

1λ21

Wηη + A12 Uξη + λ21WξWξη1113872 1113873 + A22 Vηη + λ22WηWηη1113872 11138731113960 11139611λ21

Wη

+ βA661λ21

Uηη +1λ22

Vξη + WξηWη + WξWηη1113888 1113889Wξ + α1 + α2W( 1113857V2C

(26)

Due to the fact that a fully clamped rectangularmicroplate is extensively used in MEMS applications [32]

this kind of boundary condition is also studied in this workand written in dimensionless formulation as follows

A11 Uξ +12W

2ξ1113874 1113875 +

βG3

8h2

abVξξη minus

h2

b2Uξηη1113888 1113889 0 ξ 0 1 V 0 W 0 Vξ 0 Wξ 0

A22 Vη +12λ1W

2η1113874 1113875 minus

βG3

8h2

a2Vξξη minus

h2

abUξηη1113888 1113889 0 η 0 1 U 0 W 0 Uη 0 Wη 0

(27)

3 Solution Methodology

First note that in the boundary conditions in (27) the itemsrelated to size effect are neglected due to the fact that they arenegligible compared with other items [29] With satisfactionof boundary conditions a solution for (26) takes the fol-lowing form

U(ξ η τ) U(τ) cos πξ sin2 πη

V(ξ η τ) V(τ) sin2 πξ cos πη

W(ξ η τ) W(τ) sin2 πξ sin2 πη

(28)

Submitting (28) into (26)fd25 and multiplying them bycos πξ sin2 πη sin2 πξ cos πη and sin2 πξ sin2 πη

6 Shock and Vibration

respectively and integrating them from 0 to 1 with respect toξ and η the nonlinear governing differential equations withrespect to time can be obtained as

minus3π2

16Aminus

11 minus βAminus

66π2λ224λ21

minusGminus

1βλ1

π4λ224λ21

+π4λ42λ41

⎡⎢⎢⎣ ⎤⎥⎥⎦Uminus

+ minus169

Aminus

12 minus169βA

minus

66 +Gminus

1βλ1

64π2

9+16π2λ229λ21

⎡⎢⎢⎣ ⎤⎥⎥⎦Vminus

+π2

6λ21A

minus

11 +2π2

15λ22A

minus

12 minus2π2

15βA

minus

66λ221113890 1113891W

minus 2 0

minus169

Aminus

12 minus169βA

minus

66 +Gminus

2βλ2

16π2λ219λ22

+64π2

9⎡⎢⎢⎣ ⎤⎥⎥⎦U

minus

+ minus3π2

16Aminus

22 minusπ2λ214λ22

βAminus

66 minusGminus

2βλ2

π4λ41λ42

+π4λ214λ22

⎡⎢⎢⎣ ⎤⎥⎥⎦Vminus

+2π2

15λ21A

minus

12 +π2

6λ22A

minus

22 minus2π2

15βA

minus

66λ211113890 1113891W

minus 2 0

3112

Dminus

11 + βGminus

3λ21λ22

+16Dminus

12 +13βD

minus

66 + 2βGminus

3 + 3112

Dminus

22 + βGminus

3λ21λ22

1113890 1113891π4

41113896 1113897W

minus

+14Wminus

ττ +Cminus

4Wminus

τ

π2A

minus

11

3λ22minus4π2βA

minus

66

15λ21+4π2A

minus

12

15λ21Uminus

Wminus

+4π2A

minus

12

15λ22minus4π2βA

minus

66

15λ22+π2A

minus

22

3λ21Vminus

Wminus

minus525π4λ211024λ22

Aminus

11 +25π4

1024Aminus

12

+525π4λ221024λ21

Aminus

22 +50π4

1024βA

minus

66Wminus 3

+14α1 +

916α2W

minus

Vminus 2

C

(29)

For convenience (29) can be expressed as follows

M11Uminus

+ M12Vminus

+ M13Wminus 2

0

M21Uminus

+ M22Vminus

+ M23Wminus 2

0

M31Wminus

+14Wminus

ττ +Cminus

4Wminus

τ M32Uminus

Wminus

+ M33Vminus

Wminus

minus M34Wminus 3

+14α1 +

916α2W

minus

Vminus 2

C

(30)

where

M11 minus3π2

16A11 minus βA66

π2λ224λ21

minusG1βλ1

π4λ224λ21

+π4λ42λ41

1113888 1113889

M12 minus169

A12 minus169βA66 +

G1βλ1

64π2

9+16π2λ229λ21

1113888 1113889

M13 π2

6λ21A11 +

2π2

15λ22A12 minus

2π2

15βA66λ

22

M21 minus169

A12 minus169βA66 +

G2βλ2

16π2λ219λ22

+64π2

91113888 1113889

Shock and Vibration 7

M22 minus3π2

16A22 minus

π2λ214λ22

βA66 minusG2βλ2

π4λ41λ42

+π4λ214λ22

1113888 1113889

M23 2π2

15λ21A12 +

π2

6λ22A22 minus

2π2

15βA66

M31 3112

D11 + βG31113874 1113875λ21λ22

+16D12 +

13D66 + 2βG31113874 1113875 + 3

112

D22 + βG31113874 1113875λ21λ22

1113890 1113891π4

4

M32 π2

A11

3λ22minus4π2βA66

15λ21+4π2A12

15λ21

M33 4π2A12

15λ22minus4π2βA66

15λ22+π2A22

3λ21

M34 minus525π4λ211024λ22

A11 minus25π4

1024A12 minus

525π4λ221024λ21

A22 minus50π4

1024βA66

(31)

After eliminating U(τ) V(τ) the nonlinear differentialequation only related to W(τ) can be obtained

14Wττ +

C

4Wτ + M31W minus φW

314

α1 +916α2W1113874 1113875V

2C

(32)

where

φ M34 +M32M22M13 minus M32M12M23

M12M21 minus M11M22+

M33M11M23 minus M33M13M21

M12M21 minus M11M22 (33)

4 Results and Discussion

In this section the nonlinear size-dependent vibration analysisof the microplate is studied under the influence of size effectand damage effect Free vibration and forced vibration analysisof the microplate under DC voltage and AC voltage are dis-cussed in detail accordinglye frequency-response curves arepresented for various system parameters such as materiallength scale parameter damage variable damping ratio andexternal AC voltage Note that geometric and physical pa-rameters of the microplate are listed in Table 1

41 Free Vibration Analysis under DC Voltage For staticanalysis (32) can be written as

Wττ + C Wτ

4+ M31W minus φW

3

α1 +(916)α2W( 1113857V2d

4

(34)

where VD donates the DC voltage

e harmonic balance method is feasible for findingsolutions of both weakly and strongly nonlinear problems Ageneral solution of (34) can be given as

W(τ) A0 + A1 cos(ωτ + θ) (35)

Submitting (35) into (34) and using harmonic balancemethod (HB) [33] we obtain

ω20A0 minus 4φ A

30 +

32A0A

211113874 1113875 α1V

2d

minus A1ω2

+ ω20A1 minus 4φ 3A

20A1 +

34A311113874 1113875 0

(36)

where A0 is so smaller than A1 that it can be neglected in thefurther step of solution and then the following is obtained

ω20 4M31 minus

916α2V

2d

ω2 ω2

0 minus 3φA21

(37)

8 Shock and Vibration

e variation of amplitude-frequency response of themicroplate with various values of damage variable is pre-sented in Figure 2 where DC voltage Vd and non-dimensional material length scale parameter lh are taken as20 and 02 respectively It should be pointed out that thecurrent value of material length scale parameter is onlya mean to illustrate microscale effect Sophisticated exper-iments need to be conducted to measure material lengthscale parameter as it plays a significant role in microscalestructures As shown in Figure 2 the vibration amplitudebecomes larger with increase of nonlinear frequency whichindicates that the microplate exhibits a hardening-typebehavior It is also observed that the larger the damagevariable is the lower the nonlinear free vibration frequencybecomes Moreover free vibration amplitude of themicroplate midpoint increases as damage variable increasesdue to reduced stiffness of the microplate

Influence of size effect on frequency-response curves isgiven in Figure 3 in which damage variable D and DCvoltage Vd are taken as 02 and 20 separately It is observedthat large material length scale parameter results in smallamplitude and high frequency of the microplate Whenespecially size effect is omitted the classical microplatemodel is recovered As can be observed from Figure 3material length scale parameter increases the differencebetween the results of present microplate model and theclassical microplate model increases indicating that themicroscale effect induce additional rigidity for microstruc-turee extra stiffness contributed by couple stress togetherwith the classical stiffness increased the total stiffness of themicroplate Furthermore our results are consistent with thepredictions obtained by Ansari et al [34] who studied thenonlinear vibrations properties of functionally gradedMindlinrsquos microplates using MCST

42 Forced Vibration Analysis under AC Voltage As fordynamic analysis (32) can be written as14Wττ +

14

C Wτ + M31W minus φW3

14

α1 +916α2W1113874 1113875V

2a cos

2 ωτ

(38)

Similarly the harmonic balance method is employed forsolving (38) and the general solution for this equation can begiven as

W(τ) A1 cos 2ωτ + θ0( 1113857 + A2 sin 2ωτ + θ0( 1113857

A1 cos 2Φ + A2 sin 2Φ(39)

where

cos 2ωτ A3 cos 2Φ + A4 sin 2Φ A3 cos

θ0 A4 sin θ0(40)

Substituting (39) into (38) we obtain

Table 1 Geometric and material properties of the microplate

Property ValueLength a 300 μmWidth b 150 μmickness h 2 μmInitial gap d 2 μmPoissonrsquos ratio υ 006Youngrsquos modulus E 166 times 109 Paβ 02Dielectric constant εv 8854 times 10minus 12 Fm

D = 0D = 02D = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 2 Damage effect on frequency-response curves of themicroplate

lh = 0lh = 02lh = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 3 Size effect on frequency-response curves of themicroplate

Shock and Vibration 9

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A1 + 2ωCA2 minus 3φA1 A

21 + A

221113872 1113873

α12

V2aA3

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A2 minus 2ωCA1 minus 3φA2 A

21 + A

221113872 1113873

α12

V2aA4

(41)

Using A21 + A2

2 A2 and A23 + A2

4 1 the relationshipbetween the amplitude of the microplate midpoint and theexcitation frequency can be achieved as

9φ2A6

minus 6φ 4M31 minus 4ω2minus

932α2V

2a1113874 1113875A

4+ 4M31 minus 4ω2

minus932α2V

2a1113874 1113875

2+ 4ω2

C2

1113890 1113891A2

α214

V4a (42)

e frequency-response curves of the system withdamage variable D nondimensional material length scaleparameter lh and damping coefficient C are respectivelytaken as 02 02 and 02 and are depicted in Figure 4 whichreveals that the system still exhibits a hardening-type be-havior It is observed that the amplitude of microplatemidpoint increases as excitation frequency increases from11 actually from 0 until reaching a limit point (upperjump point) corresponding to the nonlinear resonanceand then the amplitude of microplate midpoint fallssuddenly for another the amplitude of the microplatemidpoint increases as the excitation frequency is decreasedfrom 35 until reaching another limit point (lower jumppoint) and then the amplitude of the microplate midpointrises abruptly e region between the lower jump pointand the upper jump point is regarded as the unstable re-gion is unique behavior is called as jump phenomenonwhich results from the nonlinearity of the dynamic systemAdditionally with the increase of the amplitude of ACvoltage the upper jump point tends to move towards theright side and the amplitude of the microplate midpointincreases In other words the nonlinear response region isenlarged by the greater external load e jump phe-nomenon disappears when the amplitude of AC voltage issmall in this case such as 10 (black line) Without enough

energy input by external excitation the dynamic systemwill remain stable

Figure 5 indicates the effect of damping coefficient on thefrequency-response curve of the microplate In the nu-merical simulation the nondimensional material lengthscale parameter lh damage variableD and amplitude of ACvoltage Va are taken to be 02 02 and 20 accordingly Fromthis figure we can see that by increasing the dampingcoefficient the unstable region decreases and the peak of thecurves goes down It is attributed to the fact that the energyinput by external load is consumed by the dissipation sys-tem e larger the damping coefficient is the more energythe dynamic system expends

Figure 6 displays damage effect on frequency-responsecurves of the microplate with damping coefficient C non-dimensional material length scale parameter lh and am-plitude of AC voltage Va are set to 02 02 and 20respectively It is observed that the larger the damage var-iable the bigger the peak of the curve and the wider theunstable region and nonlinear response region What ismore unlike the first two cases the unstable region movestowards the left side when the microplate suffers more se-rious damage e system does not possess enough stiffnessto resist external load e reason for this is that the stiffnessof the structure is reduced by damage effect

V_a = 20

V_a = 30

V_a = 10

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 4 e frequency-response curves of the system for different amplitude of the AC voltage VAC

10 Shock and Vibration

Figure 7 highlights the size effect on the frequency-response curves of dynamic system e numerical cal-culations are performed by assumingD 02 C 02 Va 20 It is revealed that the peak ofthe curve goes down slightly and the hardening influencereduces mildly when nondimensional material lengthscale parameter increases In addition the unstable regionmoves to the right side and diminishes as size effect be-comes more obvious Furthermore the distinction be-tween the results predicted by classical theory (lh 0)

and nonclassical theory gets more obvious e reason forthis change was already discussed in Section 41 Besidesthe present result is parallel to that of Ansari et al [35]who investigated the forced vibration of functionallygraded microplate based onMCST It should be noted thathorizontal axis in this work represents nondimensionalexternal frequency while in Ansari et alrsquos work it rep-resents the frequency ratio (the ratio of external frequencyto linear first frequency)

5 Conclusions

e nonlinear size-dependent vibration of a microplate withdamage was explored by employing MCST and the strainequivalent assumption in this research e nonlineargoverning partial differential equations were transformedinto nonlinear ordinary differential equations via Galerkinrsquosscheme and further solved numerically by the harmonicbalance method Numerical results indicate that damageeffect and size dependency both have obvious influences onthe static and dynamic behaviors of the microplate system Itwas concluded that on one hand the hardening-typenonlinear behavior of the microplate system enhances whenit encounters damage on the other hand the system exhibitsa weaker nonlinear behavior greater nondimensional fre-quency and lower amplitude of the microplate midpoint assize effect gets obvious

Data Availability

e Matlab simulation and control program data used tosupport the findings of this study are available from thecorresponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors would like to acknowledge with great gratitudefor the supports of the National Science Foundation ofChina (grant nos 51778551 and 11272270)

References

[1] M I Younis and A H Nayfeh ldquoA study of the nonlinearresponse of a resonant micro-beam to an electric actuationrdquoNonlinear Dynamics vol 31 no 1 pp 91ndash117 2003

C_

= 02

C_

= 04

C_

= 01

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 5e frequency-response curves of the system for differentdamping coefficients C

D = 02

D = 04

D = 0

15 19 23 27 31 3511ω

0

02

04

06

08

A

Figure 6e frequency-response curves of the system for differentdamage variable D

lh = 04

lh = 0

lh = 02

15 19 23 27 31 3511ω

0

01

02

03

04

05

A

Figure 7e frequency-response curves of the system for differentnondimensional material length scale parameters lh

Shock and Vibration 11

[2] E M Abdel-Rahman and A H Nayfeh ldquoSecondary reso-nances of electrically actuated resonant microsensorsrdquoJournal of Micromechanics and Microengineering vol 13no 3 pp 491ndash501 2003

[3] W Zhang and G Meng ldquoNonlinear dynamical system ofmicro-cantilever under combined parametric and forcingexcitations in MEMSrdquo Sensors and Actuators A Physicalvol 119 no 2 pp 291ndash299 2005

[4] L Xu and X Jia ldquoElectromechanical coupled nonlinear dy-namics for microbeamsrdquo Archive of Applied Mechanicsvol 77 no 7 pp 485ndash502 2007

[5] W G Vogl and A H Nayfeh ldquoPrimary resonance excitationof electrically actuated clamped circular platesrdquo NonlinearDynamics vol 47 no 1ndash3 pp 181ndash192 2006

[6] A H Nayfeh H M Ouakad F Choura and E M Abdel-Rahman ldquoNonlinear dynamics of a resonant gas sensorrdquoNonlinear Dynamics vol 59 no 4 pp 607ndash618 2010

[7] X L Jia J Yang S Kitipornchai and C W Lim ldquoResonancefrequency response of geometrically nonlinear micro-switches under electrical actuationrdquo Journal of Sound andVibration vol 331 no 14 pp 3397ndash3411 2012

[8] P Kim S Bae and J Seok ldquoResonant behaviors of a nonlinearcantilever beam with tip mass subject to an axial force andelectrostatic excitationrdquo International Journal of MechanicalSciences vol 64 no 1 pp 232ndash257 2012

[9] S Saghir and M I Younis ldquoAn investigation of the static anddynamic behavior of electrically actuated rectangularmicroplatesrdquo International Journal of Non-linear Mechanicsvol 85 pp 81ndash93 2016

[10] M Sheikhlou R Shabani and G Rezazadeh ldquoNonlinearanalysis of electrostatically actuated diaphragm-type micro-pumpsrdquo Nonlinear Dynamics vol 83 no 1-2 pp 1ndash11 2016

[11] P M Osterberg Electrostatically actuated micromechanicaltest structure for material property measurement PhD dis-sertation Massachusetts Institute of Technology CambridgeMA USA 1995

[12] M Arefi and A M Zenkour ldquoSize dependent electro-elasticanalysis of a sandwich microbeam based on higher ordersinusoidal shear deformation theory and strain gradienttheoryrdquo Journal of Intelligent Material Systems and Structuresvol 29 no 7 pp 27ndash40 2018

[13] M Arefi and A M Zenkour ldquoSize-dependent vibration andbending analyses of the piezomagnetic three-layer nano-beamsrdquo Applied Physics A vol 123 no 3 p 202 2017

[14] M Arefi and A M Zenkour ldquoSize-dependent vibration andelectro-magneto-elastic bending responses of sandwich pie-zomagnetic curved nanobeamsrdquo Steel and Composite Struc-tures vol 29 no 5 pp 579ndash590 2018

[15] M Arefi and A M Zenkour ldquoermo-electro-magneto-mechanical bending behavior of size-dependent sandwichpiezomagnetic nanoplatesrdquo Mechanics Research Communi-cations vol 84 pp 27ndash42 2017

[16] L L Ke J Yang S Kitipornchai M A Bradford andY S Wang ldquoAxisymmetric nonlinear free vibration of size-dependent functionally graded annular microplatesrdquo Com-posites Part B Engineering vol 53 no 7 pp 207ndash217 2013

[17] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[18] M H Ghayesh H Farokhi and M Amabili ldquoNonlinearbehaviour of electrically actuated MEMS resonatorsrdquo

International Journal of Engineering Science vol 71 no 10pp 137ndash155 2013

[19] A G Arani and G S Jafari ldquoNonlinear vibration analysis oflaminated composite Mindlin micronano-plates resting onorthotropic Pasternak medium using DQMrdquo AppliedMathematics and Mechanics vol 36 no 8 pp 1033ndash10442015

[20] M Sobhy and A M Zenkour ldquoA comprehensive study onthe size-dependent hygrothermal analysis of exponentiallygraded microplates on elastic foundationsrdquo Mechanics ofAdvanced Materials and Structures vol 27 no 10pp 816ndash830 2019

[21] M Arefi and A M Zenkour ldquoSize-dependent free vibrationand dynamic analyses of piezo-electro-magnetic sandwichnanoplates resting on viscoelastic foundationrdquo Physica BCondensed Matter vol 521 pp 188ndash197 2017

[22] M Arefi M Kiani and A M Zenkour ldquoSize-dependent freevibration analysis of a three-layered exponentially gradednano-micro-plate with piezomagnetic face sheets resting onPasternakrsquos foundation via MCSTrdquo Journal of SandwichStructures and Materials vol 22 no 1 pp 55ndash86 2020

[23] H Farokhi and M H Ghayesh ldquoSize-dependent behavior ofelectrically actuated microcantilever-based MEMSrdquo In-ternational Journal of Mechanics amp Materials in Designvol 12 no 3 pp 1ndash15 2016

[24] M Tahani A R Askari Y Mohandes and B Hassani ldquoSize-dependent free vibration analysis of electrostatically pre-de-formed rectangular micro-plates based on the modifiedcouple stress theoryfied couple stress theoryrdquo InternationalJournal of Mechanical Sciences vol 94-95 no 22 pp 185ndash1982015

[25] A Veysi R Shabani and G Rezazadeh ldquoNonlinear vibrationsof micro-doubly curved shallow shells based on the modifiedcouple stress theoryfied couple stress theoryrdquo NonlinearDynamics vol 87 no 3 pp 2051ndash2065 2017

[26] B Jalalahmadi F Sadeghi and D Peroulis ldquoA numericalfatigue damage model for life scatter of MEMS devicesrdquoJournal of Microelectromechanical Systems vol 18 no 5pp 1016ndash1031 2009

[27] T S Slack F Sadeghi and D Peroulis ldquoA phenomenologicaldiscrete brittle damage-mechanics model for fatigue of MEMSdevices with application to LIGA Nirdquo Journal of Micro-electromechanical Systems vol 18 no 1 pp 119ndash128 2009

[28] A Basu R P Hennessy G G Adams and N E McGruerldquoHot switching damage mechanisms in MEMS contacts-ev-idence and understandingrdquo Journal of Micromechanics andMicroengineering vol 24 no 10 p 16 2014

[29] C Chen J Yuan and Y Mao ldquoPost-buckling of size-de-pendent micro-plate considering damage effectsrdquo NonlinearDynamics vol 90 no 2 pp 1301ndash1314 2017

[30] J Lemaitre A Course on Damage Mechanics Springer BerlinGermany 1992

[31] C Chen H Hu and L Dai ldquoNonlinear behavior andcharacterization of a piezoelectric laminated microbeamsystemrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1304ndash1315 2013

[32] A R Askari and M Tahani ldquoSize-dependent dynamic pull-inanalysis of geometric non-linear micro-plates based on themodified couple stress theoryrdquo Physica E Low-DimensionalSystems and Nanostructures vol 86 pp 262ndash274 2017

[33] A H Nayfeh and D T Mook Nonlinear OscillationsSpringer International Publishing New York NY USA 1979

12 Shock and Vibration

[34] R Ansari M Faghih Shojaei V Mohammadi R Gholamiand M A Darabi ldquoNonlinear vibrations of functionallygraded Mindlin microplates based on the modified couplestress theoryfied couple stress theoryrdquo Composite Structuresvol 114 no 18 pp 124ndash134 2014

[35] R Ansari R Gholami and A Shahabodini ldquoSize-dependentgeometrically nonlinear forced vibration analysis of func-tionally graded first-order shear deformable microplatesrdquoJournal of Mechanics vol 32 no 5 pp 539ndash554 2016

Shock and Vibration 13

respectively and integrating them from 0 to 1 with respect toξ and η the nonlinear governing differential equations withrespect to time can be obtained as

minus3π2

16Aminus

11 minus βAminus

66π2λ224λ21

minusGminus

1βλ1

π4λ224λ21

+π4λ42λ41

⎡⎢⎢⎣ ⎤⎥⎥⎦Uminus

+ minus169

Aminus

12 minus169βA

minus

66 +Gminus

1βλ1

64π2

9+16π2λ229λ21

⎡⎢⎢⎣ ⎤⎥⎥⎦Vminus

+π2

6λ21A

minus

11 +2π2

15λ22A

minus

12 minus2π2

15βA

minus

66λ221113890 1113891W

minus 2 0

minus169

Aminus

12 minus169βA

minus

66 +Gminus

2βλ2

16π2λ219λ22

+64π2

9⎡⎢⎢⎣ ⎤⎥⎥⎦U

minus

+ minus3π2

16Aminus

22 minusπ2λ214λ22

βAminus

66 minusGminus

2βλ2

π4λ41λ42

+π4λ214λ22

⎡⎢⎢⎣ ⎤⎥⎥⎦Vminus

+2π2

15λ21A

minus

12 +π2

6λ22A

minus

22 minus2π2

15βA

minus

66λ211113890 1113891W

minus 2 0

3112

Dminus

11 + βGminus

3λ21λ22

+16Dminus

12 +13βD

minus

66 + 2βGminus

3 + 3112

Dminus

22 + βGminus

3λ21λ22

1113890 1113891π4

41113896 1113897W

minus

+14Wminus

ττ +Cminus

4Wminus

τ

π2A

minus

11

3λ22minus4π2βA

minus

66

15λ21+4π2A

minus

12

15λ21Uminus

Wminus

+4π2A

minus

12

15λ22minus4π2βA

minus

66

15λ22+π2A

minus

22

3λ21Vminus

Wminus

minus525π4λ211024λ22

Aminus

11 +25π4

1024Aminus

12

+525π4λ221024λ21

Aminus

22 +50π4

1024βA

minus

66Wminus 3

+14α1 +

916α2W

minus

Vminus 2

C

(29)

For convenience (29) can be expressed as follows

M11Uminus

+ M12Vminus

+ M13Wminus 2

0

M21Uminus

+ M22Vminus

+ M23Wminus 2

0

M31Wminus

+14Wminus

ττ +Cminus

4Wminus

τ M32Uminus

Wminus

+ M33Vminus

Wminus

minus M34Wminus 3

+14α1 +

916α2W

minus

Vminus 2

C

(30)

where

M11 minus3π2

16A11 minus βA66

π2λ224λ21

minusG1βλ1

π4λ224λ21

+π4λ42λ41

1113888 1113889

M12 minus169

A12 minus169βA66 +

G1βλ1

64π2

9+16π2λ229λ21

1113888 1113889

M13 π2

6λ21A11 +

2π2

15λ22A12 minus

2π2

15βA66λ

22

M21 minus169

A12 minus169βA66 +

G2βλ2

16π2λ219λ22

+64π2

91113888 1113889

Shock and Vibration 7

M22 minus3π2

16A22 minus

π2λ214λ22

βA66 minusG2βλ2

π4λ41λ42

+π4λ214λ22

1113888 1113889

M23 2π2

15λ21A12 +

π2

6λ22A22 minus

2π2

15βA66

M31 3112

D11 + βG31113874 1113875λ21λ22

+16D12 +

13D66 + 2βG31113874 1113875 + 3

112

D22 + βG31113874 1113875λ21λ22

1113890 1113891π4

4

M32 π2

A11

3λ22minus4π2βA66

15λ21+4π2A12

15λ21

M33 4π2A12

15λ22minus4π2βA66

15λ22+π2A22

3λ21

M34 minus525π4λ211024λ22

A11 minus25π4

1024A12 minus

525π4λ221024λ21

A22 minus50π4

1024βA66

(31)

After eliminating U(τ) V(τ) the nonlinear differentialequation only related to W(τ) can be obtained

14Wττ +

C

4Wτ + M31W minus φW

314

α1 +916α2W1113874 1113875V

2C

(32)

where

φ M34 +M32M22M13 minus M32M12M23

M12M21 minus M11M22+

M33M11M23 minus M33M13M21

M12M21 minus M11M22 (33)

4 Results and Discussion

In this section the nonlinear size-dependent vibration analysisof the microplate is studied under the influence of size effectand damage effect Free vibration and forced vibration analysisof the microplate under DC voltage and AC voltage are dis-cussed in detail accordinglye frequency-response curves arepresented for various system parameters such as materiallength scale parameter damage variable damping ratio andexternal AC voltage Note that geometric and physical pa-rameters of the microplate are listed in Table 1

41 Free Vibration Analysis under DC Voltage For staticanalysis (32) can be written as

Wττ + C Wτ

4+ M31W minus φW

3

α1 +(916)α2W( 1113857V2d

4

(34)

where VD donates the DC voltage

e harmonic balance method is feasible for findingsolutions of both weakly and strongly nonlinear problems Ageneral solution of (34) can be given as

W(τ) A0 + A1 cos(ωτ + θ) (35)

Submitting (35) into (34) and using harmonic balancemethod (HB) [33] we obtain

ω20A0 minus 4φ A

30 +

32A0A

211113874 1113875 α1V

2d

minus A1ω2

+ ω20A1 minus 4φ 3A

20A1 +

34A311113874 1113875 0

(36)

where A0 is so smaller than A1 that it can be neglected in thefurther step of solution and then the following is obtained

ω20 4M31 minus

916α2V

2d

ω2 ω2

0 minus 3φA21

(37)

8 Shock and Vibration

e variation of amplitude-frequency response of themicroplate with various values of damage variable is pre-sented in Figure 2 where DC voltage Vd and non-dimensional material length scale parameter lh are taken as20 and 02 respectively It should be pointed out that thecurrent value of material length scale parameter is onlya mean to illustrate microscale effect Sophisticated exper-iments need to be conducted to measure material lengthscale parameter as it plays a significant role in microscalestructures As shown in Figure 2 the vibration amplitudebecomes larger with increase of nonlinear frequency whichindicates that the microplate exhibits a hardening-typebehavior It is also observed that the larger the damagevariable is the lower the nonlinear free vibration frequencybecomes Moreover free vibration amplitude of themicroplate midpoint increases as damage variable increasesdue to reduced stiffness of the microplate

Influence of size effect on frequency-response curves isgiven in Figure 3 in which damage variable D and DCvoltage Vd are taken as 02 and 20 separately It is observedthat large material length scale parameter results in smallamplitude and high frequency of the microplate Whenespecially size effect is omitted the classical microplatemodel is recovered As can be observed from Figure 3material length scale parameter increases the differencebetween the results of present microplate model and theclassical microplate model increases indicating that themicroscale effect induce additional rigidity for microstruc-turee extra stiffness contributed by couple stress togetherwith the classical stiffness increased the total stiffness of themicroplate Furthermore our results are consistent with thepredictions obtained by Ansari et al [34] who studied thenonlinear vibrations properties of functionally gradedMindlinrsquos microplates using MCST

42 Forced Vibration Analysis under AC Voltage As fordynamic analysis (32) can be written as14Wττ +

14

C Wτ + M31W minus φW3

14

α1 +916α2W1113874 1113875V

2a cos

2 ωτ

(38)

Similarly the harmonic balance method is employed forsolving (38) and the general solution for this equation can begiven as

W(τ) A1 cos 2ωτ + θ0( 1113857 + A2 sin 2ωτ + θ0( 1113857

A1 cos 2Φ + A2 sin 2Φ(39)

where

cos 2ωτ A3 cos 2Φ + A4 sin 2Φ A3 cos

θ0 A4 sin θ0(40)

Substituting (39) into (38) we obtain

Table 1 Geometric and material properties of the microplate

Property ValueLength a 300 μmWidth b 150 μmickness h 2 μmInitial gap d 2 μmPoissonrsquos ratio υ 006Youngrsquos modulus E 166 times 109 Paβ 02Dielectric constant εv 8854 times 10minus 12 Fm

D = 0D = 02D = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 2 Damage effect on frequency-response curves of themicroplate

lh = 0lh = 02lh = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 3 Size effect on frequency-response curves of themicroplate

Shock and Vibration 9

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A1 + 2ωCA2 minus 3φA1 A

21 + A

221113872 1113873

α12

V2aA3

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A2 minus 2ωCA1 minus 3φA2 A

21 + A

221113872 1113873

α12

V2aA4

(41)

Using A21 + A2

2 A2 and A23 + A2

4 1 the relationshipbetween the amplitude of the microplate midpoint and theexcitation frequency can be achieved as

9φ2A6

minus 6φ 4M31 minus 4ω2minus

932α2V

2a1113874 1113875A

4+ 4M31 minus 4ω2

minus932α2V

2a1113874 1113875

2+ 4ω2

C2

1113890 1113891A2

α214

V4a (42)

e frequency-response curves of the system withdamage variable D nondimensional material length scaleparameter lh and damping coefficient C are respectivelytaken as 02 02 and 02 and are depicted in Figure 4 whichreveals that the system still exhibits a hardening-type be-havior It is observed that the amplitude of microplatemidpoint increases as excitation frequency increases from11 actually from 0 until reaching a limit point (upperjump point) corresponding to the nonlinear resonanceand then the amplitude of microplate midpoint fallssuddenly for another the amplitude of the microplatemidpoint increases as the excitation frequency is decreasedfrom 35 until reaching another limit point (lower jumppoint) and then the amplitude of the microplate midpointrises abruptly e region between the lower jump pointand the upper jump point is regarded as the unstable re-gion is unique behavior is called as jump phenomenonwhich results from the nonlinearity of the dynamic systemAdditionally with the increase of the amplitude of ACvoltage the upper jump point tends to move towards theright side and the amplitude of the microplate midpointincreases In other words the nonlinear response region isenlarged by the greater external load e jump phe-nomenon disappears when the amplitude of AC voltage issmall in this case such as 10 (black line) Without enough

energy input by external excitation the dynamic systemwill remain stable

Figure 5 indicates the effect of damping coefficient on thefrequency-response curve of the microplate In the nu-merical simulation the nondimensional material lengthscale parameter lh damage variableD and amplitude of ACvoltage Va are taken to be 02 02 and 20 accordingly Fromthis figure we can see that by increasing the dampingcoefficient the unstable region decreases and the peak of thecurves goes down It is attributed to the fact that the energyinput by external load is consumed by the dissipation sys-tem e larger the damping coefficient is the more energythe dynamic system expends

Figure 6 displays damage effect on frequency-responsecurves of the microplate with damping coefficient C non-dimensional material length scale parameter lh and am-plitude of AC voltage Va are set to 02 02 and 20respectively It is observed that the larger the damage var-iable the bigger the peak of the curve and the wider theunstable region and nonlinear response region What ismore unlike the first two cases the unstable region movestowards the left side when the microplate suffers more se-rious damage e system does not possess enough stiffnessto resist external load e reason for this is that the stiffnessof the structure is reduced by damage effect

V_a = 20

V_a = 30

V_a = 10

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 4 e frequency-response curves of the system for different amplitude of the AC voltage VAC

10 Shock and Vibration

Figure 7 highlights the size effect on the frequency-response curves of dynamic system e numerical cal-culations are performed by assumingD 02 C 02 Va 20 It is revealed that the peak ofthe curve goes down slightly and the hardening influencereduces mildly when nondimensional material lengthscale parameter increases In addition the unstable regionmoves to the right side and diminishes as size effect be-comes more obvious Furthermore the distinction be-tween the results predicted by classical theory (lh 0)

and nonclassical theory gets more obvious e reason forthis change was already discussed in Section 41 Besidesthe present result is parallel to that of Ansari et al [35]who investigated the forced vibration of functionallygraded microplate based onMCST It should be noted thathorizontal axis in this work represents nondimensionalexternal frequency while in Ansari et alrsquos work it rep-resents the frequency ratio (the ratio of external frequencyto linear first frequency)

5 Conclusions

e nonlinear size-dependent vibration of a microplate withdamage was explored by employing MCST and the strainequivalent assumption in this research e nonlineargoverning partial differential equations were transformedinto nonlinear ordinary differential equations via Galerkinrsquosscheme and further solved numerically by the harmonicbalance method Numerical results indicate that damageeffect and size dependency both have obvious influences onthe static and dynamic behaviors of the microplate system Itwas concluded that on one hand the hardening-typenonlinear behavior of the microplate system enhances whenit encounters damage on the other hand the system exhibitsa weaker nonlinear behavior greater nondimensional fre-quency and lower amplitude of the microplate midpoint assize effect gets obvious

Data Availability

e Matlab simulation and control program data used tosupport the findings of this study are available from thecorresponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors would like to acknowledge with great gratitudefor the supports of the National Science Foundation ofChina (grant nos 51778551 and 11272270)

References

[1] M I Younis and A H Nayfeh ldquoA study of the nonlinearresponse of a resonant micro-beam to an electric actuationrdquoNonlinear Dynamics vol 31 no 1 pp 91ndash117 2003

C_

= 02

C_

= 04

C_

= 01

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 5e frequency-response curves of the system for differentdamping coefficients C

D = 02

D = 04

D = 0

15 19 23 27 31 3511ω

0

02

04

06

08

A

Figure 6e frequency-response curves of the system for differentdamage variable D

lh = 04

lh = 0

lh = 02

15 19 23 27 31 3511ω

0

01

02

03

04

05

A

Figure 7e frequency-response curves of the system for differentnondimensional material length scale parameters lh

Shock and Vibration 11

[2] E M Abdel-Rahman and A H Nayfeh ldquoSecondary reso-nances of electrically actuated resonant microsensorsrdquoJournal of Micromechanics and Microengineering vol 13no 3 pp 491ndash501 2003

[3] W Zhang and G Meng ldquoNonlinear dynamical system ofmicro-cantilever under combined parametric and forcingexcitations in MEMSrdquo Sensors and Actuators A Physicalvol 119 no 2 pp 291ndash299 2005

[4] L Xu and X Jia ldquoElectromechanical coupled nonlinear dy-namics for microbeamsrdquo Archive of Applied Mechanicsvol 77 no 7 pp 485ndash502 2007

[5] W G Vogl and A H Nayfeh ldquoPrimary resonance excitationof electrically actuated clamped circular platesrdquo NonlinearDynamics vol 47 no 1ndash3 pp 181ndash192 2006

[6] A H Nayfeh H M Ouakad F Choura and E M Abdel-Rahman ldquoNonlinear dynamics of a resonant gas sensorrdquoNonlinear Dynamics vol 59 no 4 pp 607ndash618 2010

[7] X L Jia J Yang S Kitipornchai and C W Lim ldquoResonancefrequency response of geometrically nonlinear micro-switches under electrical actuationrdquo Journal of Sound andVibration vol 331 no 14 pp 3397ndash3411 2012

[8] P Kim S Bae and J Seok ldquoResonant behaviors of a nonlinearcantilever beam with tip mass subject to an axial force andelectrostatic excitationrdquo International Journal of MechanicalSciences vol 64 no 1 pp 232ndash257 2012

[9] S Saghir and M I Younis ldquoAn investigation of the static anddynamic behavior of electrically actuated rectangularmicroplatesrdquo International Journal of Non-linear Mechanicsvol 85 pp 81ndash93 2016

[10] M Sheikhlou R Shabani and G Rezazadeh ldquoNonlinearanalysis of electrostatically actuated diaphragm-type micro-pumpsrdquo Nonlinear Dynamics vol 83 no 1-2 pp 1ndash11 2016

[11] P M Osterberg Electrostatically actuated micromechanicaltest structure for material property measurement PhD dis-sertation Massachusetts Institute of Technology CambridgeMA USA 1995

[12] M Arefi and A M Zenkour ldquoSize dependent electro-elasticanalysis of a sandwich microbeam based on higher ordersinusoidal shear deformation theory and strain gradienttheoryrdquo Journal of Intelligent Material Systems and Structuresvol 29 no 7 pp 27ndash40 2018

[13] M Arefi and A M Zenkour ldquoSize-dependent vibration andbending analyses of the piezomagnetic three-layer nano-beamsrdquo Applied Physics A vol 123 no 3 p 202 2017

[14] M Arefi and A M Zenkour ldquoSize-dependent vibration andelectro-magneto-elastic bending responses of sandwich pie-zomagnetic curved nanobeamsrdquo Steel and Composite Struc-tures vol 29 no 5 pp 579ndash590 2018

[15] M Arefi and A M Zenkour ldquoermo-electro-magneto-mechanical bending behavior of size-dependent sandwichpiezomagnetic nanoplatesrdquo Mechanics Research Communi-cations vol 84 pp 27ndash42 2017

[16] L L Ke J Yang S Kitipornchai M A Bradford andY S Wang ldquoAxisymmetric nonlinear free vibration of size-dependent functionally graded annular microplatesrdquo Com-posites Part B Engineering vol 53 no 7 pp 207ndash217 2013

[17] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[18] M H Ghayesh H Farokhi and M Amabili ldquoNonlinearbehaviour of electrically actuated MEMS resonatorsrdquo

International Journal of Engineering Science vol 71 no 10pp 137ndash155 2013

[19] A G Arani and G S Jafari ldquoNonlinear vibration analysis oflaminated composite Mindlin micronano-plates resting onorthotropic Pasternak medium using DQMrdquo AppliedMathematics and Mechanics vol 36 no 8 pp 1033ndash10442015

[20] M Sobhy and A M Zenkour ldquoA comprehensive study onthe size-dependent hygrothermal analysis of exponentiallygraded microplates on elastic foundationsrdquo Mechanics ofAdvanced Materials and Structures vol 27 no 10pp 816ndash830 2019

[21] M Arefi and A M Zenkour ldquoSize-dependent free vibrationand dynamic analyses of piezo-electro-magnetic sandwichnanoplates resting on viscoelastic foundationrdquo Physica BCondensed Matter vol 521 pp 188ndash197 2017

[22] M Arefi M Kiani and A M Zenkour ldquoSize-dependent freevibration analysis of a three-layered exponentially gradednano-micro-plate with piezomagnetic face sheets resting onPasternakrsquos foundation via MCSTrdquo Journal of SandwichStructures and Materials vol 22 no 1 pp 55ndash86 2020

[23] H Farokhi and M H Ghayesh ldquoSize-dependent behavior ofelectrically actuated microcantilever-based MEMSrdquo In-ternational Journal of Mechanics amp Materials in Designvol 12 no 3 pp 1ndash15 2016

[24] M Tahani A R Askari Y Mohandes and B Hassani ldquoSize-dependent free vibration analysis of electrostatically pre-de-formed rectangular micro-plates based on the modifiedcouple stress theoryfied couple stress theoryrdquo InternationalJournal of Mechanical Sciences vol 94-95 no 22 pp 185ndash1982015

[25] A Veysi R Shabani and G Rezazadeh ldquoNonlinear vibrationsof micro-doubly curved shallow shells based on the modifiedcouple stress theoryfied couple stress theoryrdquo NonlinearDynamics vol 87 no 3 pp 2051ndash2065 2017

[26] B Jalalahmadi F Sadeghi and D Peroulis ldquoA numericalfatigue damage model for life scatter of MEMS devicesrdquoJournal of Microelectromechanical Systems vol 18 no 5pp 1016ndash1031 2009

[27] T S Slack F Sadeghi and D Peroulis ldquoA phenomenologicaldiscrete brittle damage-mechanics model for fatigue of MEMSdevices with application to LIGA Nirdquo Journal of Micro-electromechanical Systems vol 18 no 1 pp 119ndash128 2009

[28] A Basu R P Hennessy G G Adams and N E McGruerldquoHot switching damage mechanisms in MEMS contacts-ev-idence and understandingrdquo Journal of Micromechanics andMicroengineering vol 24 no 10 p 16 2014

[29] C Chen J Yuan and Y Mao ldquoPost-buckling of size-de-pendent micro-plate considering damage effectsrdquo NonlinearDynamics vol 90 no 2 pp 1301ndash1314 2017

[30] J Lemaitre A Course on Damage Mechanics Springer BerlinGermany 1992

[31] C Chen H Hu and L Dai ldquoNonlinear behavior andcharacterization of a piezoelectric laminated microbeamsystemrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1304ndash1315 2013

[32] A R Askari and M Tahani ldquoSize-dependent dynamic pull-inanalysis of geometric non-linear micro-plates based on themodified couple stress theoryrdquo Physica E Low-DimensionalSystems and Nanostructures vol 86 pp 262ndash274 2017

[33] A H Nayfeh and D T Mook Nonlinear OscillationsSpringer International Publishing New York NY USA 1979

12 Shock and Vibration

[34] R Ansari M Faghih Shojaei V Mohammadi R Gholamiand M A Darabi ldquoNonlinear vibrations of functionallygraded Mindlin microplates based on the modified couplestress theoryfied couple stress theoryrdquo Composite Structuresvol 114 no 18 pp 124ndash134 2014

[35] R Ansari R Gholami and A Shahabodini ldquoSize-dependentgeometrically nonlinear forced vibration analysis of func-tionally graded first-order shear deformable microplatesrdquoJournal of Mechanics vol 32 no 5 pp 539ndash554 2016

Shock and Vibration 13

M22 minus3π2

16A22 minus

π2λ214λ22

βA66 minusG2βλ2

π4λ41λ42

+π4λ214λ22

1113888 1113889

M23 2π2

15λ21A12 +

π2

6λ22A22 minus

2π2

15βA66

M31 3112

D11 + βG31113874 1113875λ21λ22

+16D12 +

13D66 + 2βG31113874 1113875 + 3

112

D22 + βG31113874 1113875λ21λ22

1113890 1113891π4

4

M32 π2

A11

3λ22minus4π2βA66

15λ21+4π2A12

15λ21

M33 4π2A12

15λ22minus4π2βA66

15λ22+π2A22

3λ21

M34 minus525π4λ211024λ22

A11 minus25π4

1024A12 minus

525π4λ221024λ21

A22 minus50π4

1024βA66

(31)

After eliminating U(τ) V(τ) the nonlinear differentialequation only related to W(τ) can be obtained

14Wττ +

C

4Wτ + M31W minus φW

314

α1 +916α2W1113874 1113875V

2C

(32)

where

φ M34 +M32M22M13 minus M32M12M23

M12M21 minus M11M22+

M33M11M23 minus M33M13M21

M12M21 minus M11M22 (33)

4 Results and Discussion

In this section the nonlinear size-dependent vibration analysisof the microplate is studied under the influence of size effectand damage effect Free vibration and forced vibration analysisof the microplate under DC voltage and AC voltage are dis-cussed in detail accordinglye frequency-response curves arepresented for various system parameters such as materiallength scale parameter damage variable damping ratio andexternal AC voltage Note that geometric and physical pa-rameters of the microplate are listed in Table 1

41 Free Vibration Analysis under DC Voltage For staticanalysis (32) can be written as

Wττ + C Wτ

4+ M31W minus φW

3

α1 +(916)α2W( 1113857V2d

4

(34)

where VD donates the DC voltage

e harmonic balance method is feasible for findingsolutions of both weakly and strongly nonlinear problems Ageneral solution of (34) can be given as

W(τ) A0 + A1 cos(ωτ + θ) (35)

Submitting (35) into (34) and using harmonic balancemethod (HB) [33] we obtain

ω20A0 minus 4φ A

30 +

32A0A

211113874 1113875 α1V

2d

minus A1ω2

+ ω20A1 minus 4φ 3A

20A1 +

34A311113874 1113875 0

(36)

where A0 is so smaller than A1 that it can be neglected in thefurther step of solution and then the following is obtained

ω20 4M31 minus

916α2V

2d

ω2 ω2

0 minus 3φA21

(37)

8 Shock and Vibration

e variation of amplitude-frequency response of themicroplate with various values of damage variable is pre-sented in Figure 2 where DC voltage Vd and non-dimensional material length scale parameter lh are taken as20 and 02 respectively It should be pointed out that thecurrent value of material length scale parameter is onlya mean to illustrate microscale effect Sophisticated exper-iments need to be conducted to measure material lengthscale parameter as it plays a significant role in microscalestructures As shown in Figure 2 the vibration amplitudebecomes larger with increase of nonlinear frequency whichindicates that the microplate exhibits a hardening-typebehavior It is also observed that the larger the damagevariable is the lower the nonlinear free vibration frequencybecomes Moreover free vibration amplitude of themicroplate midpoint increases as damage variable increasesdue to reduced stiffness of the microplate

Influence of size effect on frequency-response curves isgiven in Figure 3 in which damage variable D and DCvoltage Vd are taken as 02 and 20 separately It is observedthat large material length scale parameter results in smallamplitude and high frequency of the microplate Whenespecially size effect is omitted the classical microplatemodel is recovered As can be observed from Figure 3material length scale parameter increases the differencebetween the results of present microplate model and theclassical microplate model increases indicating that themicroscale effect induce additional rigidity for microstruc-turee extra stiffness contributed by couple stress togetherwith the classical stiffness increased the total stiffness of themicroplate Furthermore our results are consistent with thepredictions obtained by Ansari et al [34] who studied thenonlinear vibrations properties of functionally gradedMindlinrsquos microplates using MCST

42 Forced Vibration Analysis under AC Voltage As fordynamic analysis (32) can be written as14Wττ +

14

C Wτ + M31W minus φW3

14

α1 +916α2W1113874 1113875V

2a cos

2 ωτ

(38)

Similarly the harmonic balance method is employed forsolving (38) and the general solution for this equation can begiven as

W(τ) A1 cos 2ωτ + θ0( 1113857 + A2 sin 2ωτ + θ0( 1113857

A1 cos 2Φ + A2 sin 2Φ(39)

where

cos 2ωτ A3 cos 2Φ + A4 sin 2Φ A3 cos

θ0 A4 sin θ0(40)

Substituting (39) into (38) we obtain

Table 1 Geometric and material properties of the microplate

Property ValueLength a 300 μmWidth b 150 μmickness h 2 μmInitial gap d 2 μmPoissonrsquos ratio υ 006Youngrsquos modulus E 166 times 109 Paβ 02Dielectric constant εv 8854 times 10minus 12 Fm

D = 0D = 02D = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 2 Damage effect on frequency-response curves of themicroplate

lh = 0lh = 02lh = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 3 Size effect on frequency-response curves of themicroplate

Shock and Vibration 9

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A1 + 2ωCA2 minus 3φA1 A

21 + A

221113872 1113873

α12

V2aA3

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A2 minus 2ωCA1 minus 3φA2 A

21 + A

221113872 1113873

α12

V2aA4

(41)

Using A21 + A2

2 A2 and A23 + A2

4 1 the relationshipbetween the amplitude of the microplate midpoint and theexcitation frequency can be achieved as

9φ2A6

minus 6φ 4M31 minus 4ω2minus

932α2V

2a1113874 1113875A

4+ 4M31 minus 4ω2

minus932α2V

2a1113874 1113875

2+ 4ω2

C2

1113890 1113891A2

α214

V4a (42)

e frequency-response curves of the system withdamage variable D nondimensional material length scaleparameter lh and damping coefficient C are respectivelytaken as 02 02 and 02 and are depicted in Figure 4 whichreveals that the system still exhibits a hardening-type be-havior It is observed that the amplitude of microplatemidpoint increases as excitation frequency increases from11 actually from 0 until reaching a limit point (upperjump point) corresponding to the nonlinear resonanceand then the amplitude of microplate midpoint fallssuddenly for another the amplitude of the microplatemidpoint increases as the excitation frequency is decreasedfrom 35 until reaching another limit point (lower jumppoint) and then the amplitude of the microplate midpointrises abruptly e region between the lower jump pointand the upper jump point is regarded as the unstable re-gion is unique behavior is called as jump phenomenonwhich results from the nonlinearity of the dynamic systemAdditionally with the increase of the amplitude of ACvoltage the upper jump point tends to move towards theright side and the amplitude of the microplate midpointincreases In other words the nonlinear response region isenlarged by the greater external load e jump phe-nomenon disappears when the amplitude of AC voltage issmall in this case such as 10 (black line) Without enough

energy input by external excitation the dynamic systemwill remain stable

Figure 5 indicates the effect of damping coefficient on thefrequency-response curve of the microplate In the nu-merical simulation the nondimensional material lengthscale parameter lh damage variableD and amplitude of ACvoltage Va are taken to be 02 02 and 20 accordingly Fromthis figure we can see that by increasing the dampingcoefficient the unstable region decreases and the peak of thecurves goes down It is attributed to the fact that the energyinput by external load is consumed by the dissipation sys-tem e larger the damping coefficient is the more energythe dynamic system expends

Figure 6 displays damage effect on frequency-responsecurves of the microplate with damping coefficient C non-dimensional material length scale parameter lh and am-plitude of AC voltage Va are set to 02 02 and 20respectively It is observed that the larger the damage var-iable the bigger the peak of the curve and the wider theunstable region and nonlinear response region What ismore unlike the first two cases the unstable region movestowards the left side when the microplate suffers more se-rious damage e system does not possess enough stiffnessto resist external load e reason for this is that the stiffnessof the structure is reduced by damage effect

V_a = 20

V_a = 30

V_a = 10

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 4 e frequency-response curves of the system for different amplitude of the AC voltage VAC

10 Shock and Vibration

Figure 7 highlights the size effect on the frequency-response curves of dynamic system e numerical cal-culations are performed by assumingD 02 C 02 Va 20 It is revealed that the peak ofthe curve goes down slightly and the hardening influencereduces mildly when nondimensional material lengthscale parameter increases In addition the unstable regionmoves to the right side and diminishes as size effect be-comes more obvious Furthermore the distinction be-tween the results predicted by classical theory (lh 0)

and nonclassical theory gets more obvious e reason forthis change was already discussed in Section 41 Besidesthe present result is parallel to that of Ansari et al [35]who investigated the forced vibration of functionallygraded microplate based onMCST It should be noted thathorizontal axis in this work represents nondimensionalexternal frequency while in Ansari et alrsquos work it rep-resents the frequency ratio (the ratio of external frequencyto linear first frequency)

5 Conclusions

e nonlinear size-dependent vibration of a microplate withdamage was explored by employing MCST and the strainequivalent assumption in this research e nonlineargoverning partial differential equations were transformedinto nonlinear ordinary differential equations via Galerkinrsquosscheme and further solved numerically by the harmonicbalance method Numerical results indicate that damageeffect and size dependency both have obvious influences onthe static and dynamic behaviors of the microplate system Itwas concluded that on one hand the hardening-typenonlinear behavior of the microplate system enhances whenit encounters damage on the other hand the system exhibitsa weaker nonlinear behavior greater nondimensional fre-quency and lower amplitude of the microplate midpoint assize effect gets obvious

Data Availability

e Matlab simulation and control program data used tosupport the findings of this study are available from thecorresponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors would like to acknowledge with great gratitudefor the supports of the National Science Foundation ofChina (grant nos 51778551 and 11272270)

References

[1] M I Younis and A H Nayfeh ldquoA study of the nonlinearresponse of a resonant micro-beam to an electric actuationrdquoNonlinear Dynamics vol 31 no 1 pp 91ndash117 2003

C_

= 02

C_

= 04

C_

= 01

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 5e frequency-response curves of the system for differentdamping coefficients C

D = 02

D = 04

D = 0

15 19 23 27 31 3511ω

0

02

04

06

08

A

Figure 6e frequency-response curves of the system for differentdamage variable D

lh = 04

lh = 0

lh = 02

15 19 23 27 31 3511ω

0

01

02

03

04

05

A

Figure 7e frequency-response curves of the system for differentnondimensional material length scale parameters lh

Shock and Vibration 11

[2] E M Abdel-Rahman and A H Nayfeh ldquoSecondary reso-nances of electrically actuated resonant microsensorsrdquoJournal of Micromechanics and Microengineering vol 13no 3 pp 491ndash501 2003

[3] W Zhang and G Meng ldquoNonlinear dynamical system ofmicro-cantilever under combined parametric and forcingexcitations in MEMSrdquo Sensors and Actuators A Physicalvol 119 no 2 pp 291ndash299 2005

[4] L Xu and X Jia ldquoElectromechanical coupled nonlinear dy-namics for microbeamsrdquo Archive of Applied Mechanicsvol 77 no 7 pp 485ndash502 2007

[5] W G Vogl and A H Nayfeh ldquoPrimary resonance excitationof electrically actuated clamped circular platesrdquo NonlinearDynamics vol 47 no 1ndash3 pp 181ndash192 2006

[6] A H Nayfeh H M Ouakad F Choura and E M Abdel-Rahman ldquoNonlinear dynamics of a resonant gas sensorrdquoNonlinear Dynamics vol 59 no 4 pp 607ndash618 2010

[7] X L Jia J Yang S Kitipornchai and C W Lim ldquoResonancefrequency response of geometrically nonlinear micro-switches under electrical actuationrdquo Journal of Sound andVibration vol 331 no 14 pp 3397ndash3411 2012

[8] P Kim S Bae and J Seok ldquoResonant behaviors of a nonlinearcantilever beam with tip mass subject to an axial force andelectrostatic excitationrdquo International Journal of MechanicalSciences vol 64 no 1 pp 232ndash257 2012

[9] S Saghir and M I Younis ldquoAn investigation of the static anddynamic behavior of electrically actuated rectangularmicroplatesrdquo International Journal of Non-linear Mechanicsvol 85 pp 81ndash93 2016

[10] M Sheikhlou R Shabani and G Rezazadeh ldquoNonlinearanalysis of electrostatically actuated diaphragm-type micro-pumpsrdquo Nonlinear Dynamics vol 83 no 1-2 pp 1ndash11 2016

[11] P M Osterberg Electrostatically actuated micromechanicaltest structure for material property measurement PhD dis-sertation Massachusetts Institute of Technology CambridgeMA USA 1995

[12] M Arefi and A M Zenkour ldquoSize dependent electro-elasticanalysis of a sandwich microbeam based on higher ordersinusoidal shear deformation theory and strain gradienttheoryrdquo Journal of Intelligent Material Systems and Structuresvol 29 no 7 pp 27ndash40 2018

[13] M Arefi and A M Zenkour ldquoSize-dependent vibration andbending analyses of the piezomagnetic three-layer nano-beamsrdquo Applied Physics A vol 123 no 3 p 202 2017

[14] M Arefi and A M Zenkour ldquoSize-dependent vibration andelectro-magneto-elastic bending responses of sandwich pie-zomagnetic curved nanobeamsrdquo Steel and Composite Struc-tures vol 29 no 5 pp 579ndash590 2018

[15] M Arefi and A M Zenkour ldquoermo-electro-magneto-mechanical bending behavior of size-dependent sandwichpiezomagnetic nanoplatesrdquo Mechanics Research Communi-cations vol 84 pp 27ndash42 2017

[16] L L Ke J Yang S Kitipornchai M A Bradford andY S Wang ldquoAxisymmetric nonlinear free vibration of size-dependent functionally graded annular microplatesrdquo Com-posites Part B Engineering vol 53 no 7 pp 207ndash217 2013

[17] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[18] M H Ghayesh H Farokhi and M Amabili ldquoNonlinearbehaviour of electrically actuated MEMS resonatorsrdquo

International Journal of Engineering Science vol 71 no 10pp 137ndash155 2013

[19] A G Arani and G S Jafari ldquoNonlinear vibration analysis oflaminated composite Mindlin micronano-plates resting onorthotropic Pasternak medium using DQMrdquo AppliedMathematics and Mechanics vol 36 no 8 pp 1033ndash10442015

[20] M Sobhy and A M Zenkour ldquoA comprehensive study onthe size-dependent hygrothermal analysis of exponentiallygraded microplates on elastic foundationsrdquo Mechanics ofAdvanced Materials and Structures vol 27 no 10pp 816ndash830 2019

[21] M Arefi and A M Zenkour ldquoSize-dependent free vibrationand dynamic analyses of piezo-electro-magnetic sandwichnanoplates resting on viscoelastic foundationrdquo Physica BCondensed Matter vol 521 pp 188ndash197 2017

[22] M Arefi M Kiani and A M Zenkour ldquoSize-dependent freevibration analysis of a three-layered exponentially gradednano-micro-plate with piezomagnetic face sheets resting onPasternakrsquos foundation via MCSTrdquo Journal of SandwichStructures and Materials vol 22 no 1 pp 55ndash86 2020

[23] H Farokhi and M H Ghayesh ldquoSize-dependent behavior ofelectrically actuated microcantilever-based MEMSrdquo In-ternational Journal of Mechanics amp Materials in Designvol 12 no 3 pp 1ndash15 2016

[24] M Tahani A R Askari Y Mohandes and B Hassani ldquoSize-dependent free vibration analysis of electrostatically pre-de-formed rectangular micro-plates based on the modifiedcouple stress theoryfied couple stress theoryrdquo InternationalJournal of Mechanical Sciences vol 94-95 no 22 pp 185ndash1982015

[25] A Veysi R Shabani and G Rezazadeh ldquoNonlinear vibrationsof micro-doubly curved shallow shells based on the modifiedcouple stress theoryfied couple stress theoryrdquo NonlinearDynamics vol 87 no 3 pp 2051ndash2065 2017

[26] B Jalalahmadi F Sadeghi and D Peroulis ldquoA numericalfatigue damage model for life scatter of MEMS devicesrdquoJournal of Microelectromechanical Systems vol 18 no 5pp 1016ndash1031 2009

[27] T S Slack F Sadeghi and D Peroulis ldquoA phenomenologicaldiscrete brittle damage-mechanics model for fatigue of MEMSdevices with application to LIGA Nirdquo Journal of Micro-electromechanical Systems vol 18 no 1 pp 119ndash128 2009

[28] A Basu R P Hennessy G G Adams and N E McGruerldquoHot switching damage mechanisms in MEMS contacts-ev-idence and understandingrdquo Journal of Micromechanics andMicroengineering vol 24 no 10 p 16 2014

[29] C Chen J Yuan and Y Mao ldquoPost-buckling of size-de-pendent micro-plate considering damage effectsrdquo NonlinearDynamics vol 90 no 2 pp 1301ndash1314 2017

[30] J Lemaitre A Course on Damage Mechanics Springer BerlinGermany 1992

[31] C Chen H Hu and L Dai ldquoNonlinear behavior andcharacterization of a piezoelectric laminated microbeamsystemrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1304ndash1315 2013

[32] A R Askari and M Tahani ldquoSize-dependent dynamic pull-inanalysis of geometric non-linear micro-plates based on themodified couple stress theoryrdquo Physica E Low-DimensionalSystems and Nanostructures vol 86 pp 262ndash274 2017

[33] A H Nayfeh and D T Mook Nonlinear OscillationsSpringer International Publishing New York NY USA 1979

12 Shock and Vibration

[34] R Ansari M Faghih Shojaei V Mohammadi R Gholamiand M A Darabi ldquoNonlinear vibrations of functionallygraded Mindlin microplates based on the modified couplestress theoryfied couple stress theoryrdquo Composite Structuresvol 114 no 18 pp 124ndash134 2014

[35] R Ansari R Gholami and A Shahabodini ldquoSize-dependentgeometrically nonlinear forced vibration analysis of func-tionally graded first-order shear deformable microplatesrdquoJournal of Mechanics vol 32 no 5 pp 539ndash554 2016

Shock and Vibration 13

e variation of amplitude-frequency response of themicroplate with various values of damage variable is pre-sented in Figure 2 where DC voltage Vd and non-dimensional material length scale parameter lh are taken as20 and 02 respectively It should be pointed out that thecurrent value of material length scale parameter is onlya mean to illustrate microscale effect Sophisticated exper-iments need to be conducted to measure material lengthscale parameter as it plays a significant role in microscalestructures As shown in Figure 2 the vibration amplitudebecomes larger with increase of nonlinear frequency whichindicates that the microplate exhibits a hardening-typebehavior It is also observed that the larger the damagevariable is the lower the nonlinear free vibration frequencybecomes Moreover free vibration amplitude of themicroplate midpoint increases as damage variable increasesdue to reduced stiffness of the microplate

Influence of size effect on frequency-response curves isgiven in Figure 3 in which damage variable D and DCvoltage Vd are taken as 02 and 20 separately It is observedthat large material length scale parameter results in smallamplitude and high frequency of the microplate Whenespecially size effect is omitted the classical microplatemodel is recovered As can be observed from Figure 3material length scale parameter increases the differencebetween the results of present microplate model and theclassical microplate model increases indicating that themicroscale effect induce additional rigidity for microstruc-turee extra stiffness contributed by couple stress togetherwith the classical stiffness increased the total stiffness of themicroplate Furthermore our results are consistent with thepredictions obtained by Ansari et al [34] who studied thenonlinear vibrations properties of functionally gradedMindlinrsquos microplates using MCST

42 Forced Vibration Analysis under AC Voltage As fordynamic analysis (32) can be written as14Wττ +

14

C Wτ + M31W minus φW3

14

α1 +916α2W1113874 1113875V

2a cos

2 ωτ

(38)

Similarly the harmonic balance method is employed forsolving (38) and the general solution for this equation can begiven as

W(τ) A1 cos 2ωτ + θ0( 1113857 + A2 sin 2ωτ + θ0( 1113857

A1 cos 2Φ + A2 sin 2Φ(39)

where

cos 2ωτ A3 cos 2Φ + A4 sin 2Φ A3 cos

θ0 A4 sin θ0(40)

Substituting (39) into (38) we obtain

Table 1 Geometric and material properties of the microplate

Property ValueLength a 300 μmWidth b 150 μmickness h 2 μmInitial gap d 2 μmPoissonrsquos ratio υ 006Youngrsquos modulus E 166 times 109 Paβ 02Dielectric constant εv 8854 times 10minus 12 Fm

D = 0D = 02D = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 2 Damage effect on frequency-response curves of themicroplate

lh = 0lh = 02lh = 04

4 45 5 55 6 65 735ω

0

02

04

06

08

1

A1

Figure 3 Size effect on frequency-response curves of themicroplate

Shock and Vibration 9

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A1 + 2ωCA2 minus 3φA1 A

21 + A

221113872 1113873

α12

V2aA3

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A2 minus 2ωCA1 minus 3φA2 A

21 + A

221113872 1113873

α12

V2aA4

(41)

Using A21 + A2

2 A2 and A23 + A2

4 1 the relationshipbetween the amplitude of the microplate midpoint and theexcitation frequency can be achieved as

9φ2A6

minus 6φ 4M31 minus 4ω2minus

932α2V

2a1113874 1113875A

4+ 4M31 minus 4ω2

minus932α2V

2a1113874 1113875

2+ 4ω2

C2

1113890 1113891A2

α214

V4a (42)

e frequency-response curves of the system withdamage variable D nondimensional material length scaleparameter lh and damping coefficient C are respectivelytaken as 02 02 and 02 and are depicted in Figure 4 whichreveals that the system still exhibits a hardening-type be-havior It is observed that the amplitude of microplatemidpoint increases as excitation frequency increases from11 actually from 0 until reaching a limit point (upperjump point) corresponding to the nonlinear resonanceand then the amplitude of microplate midpoint fallssuddenly for another the amplitude of the microplatemidpoint increases as the excitation frequency is decreasedfrom 35 until reaching another limit point (lower jumppoint) and then the amplitude of the microplate midpointrises abruptly e region between the lower jump pointand the upper jump point is regarded as the unstable re-gion is unique behavior is called as jump phenomenonwhich results from the nonlinearity of the dynamic systemAdditionally with the increase of the amplitude of ACvoltage the upper jump point tends to move towards theright side and the amplitude of the microplate midpointincreases In other words the nonlinear response region isenlarged by the greater external load e jump phe-nomenon disappears when the amplitude of AC voltage issmall in this case such as 10 (black line) Without enough

energy input by external excitation the dynamic systemwill remain stable

Figure 5 indicates the effect of damping coefficient on thefrequency-response curve of the microplate In the nu-merical simulation the nondimensional material lengthscale parameter lh damage variableD and amplitude of ACvoltage Va are taken to be 02 02 and 20 accordingly Fromthis figure we can see that by increasing the dampingcoefficient the unstable region decreases and the peak of thecurves goes down It is attributed to the fact that the energyinput by external load is consumed by the dissipation sys-tem e larger the damping coefficient is the more energythe dynamic system expends

Figure 6 displays damage effect on frequency-responsecurves of the microplate with damping coefficient C non-dimensional material length scale parameter lh and am-plitude of AC voltage Va are set to 02 02 and 20respectively It is observed that the larger the damage var-iable the bigger the peak of the curve and the wider theunstable region and nonlinear response region What ismore unlike the first two cases the unstable region movestowards the left side when the microplate suffers more se-rious damage e system does not possess enough stiffnessto resist external load e reason for this is that the stiffnessof the structure is reduced by damage effect

V_a = 20

V_a = 30

V_a = 10

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 4 e frequency-response curves of the system for different amplitude of the AC voltage VAC

10 Shock and Vibration

Figure 7 highlights the size effect on the frequency-response curves of dynamic system e numerical cal-culations are performed by assumingD 02 C 02 Va 20 It is revealed that the peak ofthe curve goes down slightly and the hardening influencereduces mildly when nondimensional material lengthscale parameter increases In addition the unstable regionmoves to the right side and diminishes as size effect be-comes more obvious Furthermore the distinction be-tween the results predicted by classical theory (lh 0)

and nonclassical theory gets more obvious e reason forthis change was already discussed in Section 41 Besidesthe present result is parallel to that of Ansari et al [35]who investigated the forced vibration of functionallygraded microplate based onMCST It should be noted thathorizontal axis in this work represents nondimensionalexternal frequency while in Ansari et alrsquos work it rep-resents the frequency ratio (the ratio of external frequencyto linear first frequency)

5 Conclusions

e nonlinear size-dependent vibration of a microplate withdamage was explored by employing MCST and the strainequivalent assumption in this research e nonlineargoverning partial differential equations were transformedinto nonlinear ordinary differential equations via Galerkinrsquosscheme and further solved numerically by the harmonicbalance method Numerical results indicate that damageeffect and size dependency both have obvious influences onthe static and dynamic behaviors of the microplate system Itwas concluded that on one hand the hardening-typenonlinear behavior of the microplate system enhances whenit encounters damage on the other hand the system exhibitsa weaker nonlinear behavior greater nondimensional fre-quency and lower amplitude of the microplate midpoint assize effect gets obvious

Data Availability

e Matlab simulation and control program data used tosupport the findings of this study are available from thecorresponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors would like to acknowledge with great gratitudefor the supports of the National Science Foundation ofChina (grant nos 51778551 and 11272270)

References

[1] M I Younis and A H Nayfeh ldquoA study of the nonlinearresponse of a resonant micro-beam to an electric actuationrdquoNonlinear Dynamics vol 31 no 1 pp 91ndash117 2003

C_

= 02

C_

= 04

C_

= 01

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 5e frequency-response curves of the system for differentdamping coefficients C

D = 02

D = 04

D = 0

15 19 23 27 31 3511ω

0

02

04

06

08

A

Figure 6e frequency-response curves of the system for differentdamage variable D

lh = 04

lh = 0

lh = 02

15 19 23 27 31 3511ω

0

01

02

03

04

05

A

Figure 7e frequency-response curves of the system for differentnondimensional material length scale parameters lh

Shock and Vibration 11

[2] E M Abdel-Rahman and A H Nayfeh ldquoSecondary reso-nances of electrically actuated resonant microsensorsrdquoJournal of Micromechanics and Microengineering vol 13no 3 pp 491ndash501 2003

[3] W Zhang and G Meng ldquoNonlinear dynamical system ofmicro-cantilever under combined parametric and forcingexcitations in MEMSrdquo Sensors and Actuators A Physicalvol 119 no 2 pp 291ndash299 2005

[4] L Xu and X Jia ldquoElectromechanical coupled nonlinear dy-namics for microbeamsrdquo Archive of Applied Mechanicsvol 77 no 7 pp 485ndash502 2007

[5] W G Vogl and A H Nayfeh ldquoPrimary resonance excitationof electrically actuated clamped circular platesrdquo NonlinearDynamics vol 47 no 1ndash3 pp 181ndash192 2006

[6] A H Nayfeh H M Ouakad F Choura and E M Abdel-Rahman ldquoNonlinear dynamics of a resonant gas sensorrdquoNonlinear Dynamics vol 59 no 4 pp 607ndash618 2010

[7] X L Jia J Yang S Kitipornchai and C W Lim ldquoResonancefrequency response of geometrically nonlinear micro-switches under electrical actuationrdquo Journal of Sound andVibration vol 331 no 14 pp 3397ndash3411 2012

[8] P Kim S Bae and J Seok ldquoResonant behaviors of a nonlinearcantilever beam with tip mass subject to an axial force andelectrostatic excitationrdquo International Journal of MechanicalSciences vol 64 no 1 pp 232ndash257 2012

[9] S Saghir and M I Younis ldquoAn investigation of the static anddynamic behavior of electrically actuated rectangularmicroplatesrdquo International Journal of Non-linear Mechanicsvol 85 pp 81ndash93 2016

[10] M Sheikhlou R Shabani and G Rezazadeh ldquoNonlinearanalysis of electrostatically actuated diaphragm-type micro-pumpsrdquo Nonlinear Dynamics vol 83 no 1-2 pp 1ndash11 2016

[11] P M Osterberg Electrostatically actuated micromechanicaltest structure for material property measurement PhD dis-sertation Massachusetts Institute of Technology CambridgeMA USA 1995

[12] M Arefi and A M Zenkour ldquoSize dependent electro-elasticanalysis of a sandwich microbeam based on higher ordersinusoidal shear deformation theory and strain gradienttheoryrdquo Journal of Intelligent Material Systems and Structuresvol 29 no 7 pp 27ndash40 2018

[13] M Arefi and A M Zenkour ldquoSize-dependent vibration andbending analyses of the piezomagnetic three-layer nano-beamsrdquo Applied Physics A vol 123 no 3 p 202 2017

[14] M Arefi and A M Zenkour ldquoSize-dependent vibration andelectro-magneto-elastic bending responses of sandwich pie-zomagnetic curved nanobeamsrdquo Steel and Composite Struc-tures vol 29 no 5 pp 579ndash590 2018

[15] M Arefi and A M Zenkour ldquoermo-electro-magneto-mechanical bending behavior of size-dependent sandwichpiezomagnetic nanoplatesrdquo Mechanics Research Communi-cations vol 84 pp 27ndash42 2017

[16] L L Ke J Yang S Kitipornchai M A Bradford andY S Wang ldquoAxisymmetric nonlinear free vibration of size-dependent functionally graded annular microplatesrdquo Com-posites Part B Engineering vol 53 no 7 pp 207ndash217 2013

[17] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[18] M H Ghayesh H Farokhi and M Amabili ldquoNonlinearbehaviour of electrically actuated MEMS resonatorsrdquo

International Journal of Engineering Science vol 71 no 10pp 137ndash155 2013

[19] A G Arani and G S Jafari ldquoNonlinear vibration analysis oflaminated composite Mindlin micronano-plates resting onorthotropic Pasternak medium using DQMrdquo AppliedMathematics and Mechanics vol 36 no 8 pp 1033ndash10442015

[20] M Sobhy and A M Zenkour ldquoA comprehensive study onthe size-dependent hygrothermal analysis of exponentiallygraded microplates on elastic foundationsrdquo Mechanics ofAdvanced Materials and Structures vol 27 no 10pp 816ndash830 2019

[21] M Arefi and A M Zenkour ldquoSize-dependent free vibrationand dynamic analyses of piezo-electro-magnetic sandwichnanoplates resting on viscoelastic foundationrdquo Physica BCondensed Matter vol 521 pp 188ndash197 2017

[22] M Arefi M Kiani and A M Zenkour ldquoSize-dependent freevibration analysis of a three-layered exponentially gradednano-micro-plate with piezomagnetic face sheets resting onPasternakrsquos foundation via MCSTrdquo Journal of SandwichStructures and Materials vol 22 no 1 pp 55ndash86 2020

[23] H Farokhi and M H Ghayesh ldquoSize-dependent behavior ofelectrically actuated microcantilever-based MEMSrdquo In-ternational Journal of Mechanics amp Materials in Designvol 12 no 3 pp 1ndash15 2016

[24] M Tahani A R Askari Y Mohandes and B Hassani ldquoSize-dependent free vibration analysis of electrostatically pre-de-formed rectangular micro-plates based on the modifiedcouple stress theoryfied couple stress theoryrdquo InternationalJournal of Mechanical Sciences vol 94-95 no 22 pp 185ndash1982015

[25] A Veysi R Shabani and G Rezazadeh ldquoNonlinear vibrationsof micro-doubly curved shallow shells based on the modifiedcouple stress theoryfied couple stress theoryrdquo NonlinearDynamics vol 87 no 3 pp 2051ndash2065 2017

[26] B Jalalahmadi F Sadeghi and D Peroulis ldquoA numericalfatigue damage model for life scatter of MEMS devicesrdquoJournal of Microelectromechanical Systems vol 18 no 5pp 1016ndash1031 2009

[27] T S Slack F Sadeghi and D Peroulis ldquoA phenomenologicaldiscrete brittle damage-mechanics model for fatigue of MEMSdevices with application to LIGA Nirdquo Journal of Micro-electromechanical Systems vol 18 no 1 pp 119ndash128 2009

[28] A Basu R P Hennessy G G Adams and N E McGruerldquoHot switching damage mechanisms in MEMS contacts-ev-idence and understandingrdquo Journal of Micromechanics andMicroengineering vol 24 no 10 p 16 2014

[29] C Chen J Yuan and Y Mao ldquoPost-buckling of size-de-pendent micro-plate considering damage effectsrdquo NonlinearDynamics vol 90 no 2 pp 1301ndash1314 2017

[30] J Lemaitre A Course on Damage Mechanics Springer BerlinGermany 1992

[31] C Chen H Hu and L Dai ldquoNonlinear behavior andcharacterization of a piezoelectric laminated microbeamsystemrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1304ndash1315 2013

[32] A R Askari and M Tahani ldquoSize-dependent dynamic pull-inanalysis of geometric non-linear micro-plates based on themodified couple stress theoryrdquo Physica E Low-DimensionalSystems and Nanostructures vol 86 pp 262ndash274 2017

[33] A H Nayfeh and D T Mook Nonlinear OscillationsSpringer International Publishing New York NY USA 1979

12 Shock and Vibration

[34] R Ansari M Faghih Shojaei V Mohammadi R Gholamiand M A Darabi ldquoNonlinear vibrations of functionallygraded Mindlin microplates based on the modified couplestress theoryfied couple stress theoryrdquo Composite Structuresvol 114 no 18 pp 124ndash134 2014

[35] R Ansari R Gholami and A Shahabodini ldquoSize-dependentgeometrically nonlinear forced vibration analysis of func-tionally graded first-order shear deformable microplatesrdquoJournal of Mechanics vol 32 no 5 pp 539ndash554 2016

Shock and Vibration 13

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A1 + 2ωCA2 minus 3φA1 A

21 + A

221113872 1113873

α12

V2aA3

4M31 minus 4ω2minus

932α2V

2A1113874 1113875A2 minus 2ωCA1 minus 3φA2 A

21 + A

221113872 1113873

α12

V2aA4

(41)

Using A21 + A2

2 A2 and A23 + A2

4 1 the relationshipbetween the amplitude of the microplate midpoint and theexcitation frequency can be achieved as

9φ2A6

minus 6φ 4M31 minus 4ω2minus

932α2V

2a1113874 1113875A

4+ 4M31 minus 4ω2

minus932α2V

2a1113874 1113875

2+ 4ω2

C2

1113890 1113891A2

α214

V4a (42)

e frequency-response curves of the system withdamage variable D nondimensional material length scaleparameter lh and damping coefficient C are respectivelytaken as 02 02 and 02 and are depicted in Figure 4 whichreveals that the system still exhibits a hardening-type be-havior It is observed that the amplitude of microplatemidpoint increases as excitation frequency increases from11 actually from 0 until reaching a limit point (upperjump point) corresponding to the nonlinear resonanceand then the amplitude of microplate midpoint fallssuddenly for another the amplitude of the microplatemidpoint increases as the excitation frequency is decreasedfrom 35 until reaching another limit point (lower jumppoint) and then the amplitude of the microplate midpointrises abruptly e region between the lower jump pointand the upper jump point is regarded as the unstable re-gion is unique behavior is called as jump phenomenonwhich results from the nonlinearity of the dynamic systemAdditionally with the increase of the amplitude of ACvoltage the upper jump point tends to move towards theright side and the amplitude of the microplate midpointincreases In other words the nonlinear response region isenlarged by the greater external load e jump phe-nomenon disappears when the amplitude of AC voltage issmall in this case such as 10 (black line) Without enough

energy input by external excitation the dynamic systemwill remain stable

Figure 5 indicates the effect of damping coefficient on thefrequency-response curve of the microplate In the nu-merical simulation the nondimensional material lengthscale parameter lh damage variableD and amplitude of ACvoltage Va are taken to be 02 02 and 20 accordingly Fromthis figure we can see that by increasing the dampingcoefficient the unstable region decreases and the peak of thecurves goes down It is attributed to the fact that the energyinput by external load is consumed by the dissipation sys-tem e larger the damping coefficient is the more energythe dynamic system expends

Figure 6 displays damage effect on frequency-responsecurves of the microplate with damping coefficient C non-dimensional material length scale parameter lh and am-plitude of AC voltage Va are set to 02 02 and 20respectively It is observed that the larger the damage var-iable the bigger the peak of the curve and the wider theunstable region and nonlinear response region What ismore unlike the first two cases the unstable region movestowards the left side when the microplate suffers more se-rious damage e system does not possess enough stiffnessto resist external load e reason for this is that the stiffnessof the structure is reduced by damage effect

V_a = 20

V_a = 30

V_a = 10

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 4 e frequency-response curves of the system for different amplitude of the AC voltage VAC

10 Shock and Vibration

Figure 7 highlights the size effect on the frequency-response curves of dynamic system e numerical cal-culations are performed by assumingD 02 C 02 Va 20 It is revealed that the peak ofthe curve goes down slightly and the hardening influencereduces mildly when nondimensional material lengthscale parameter increases In addition the unstable regionmoves to the right side and diminishes as size effect be-comes more obvious Furthermore the distinction be-tween the results predicted by classical theory (lh 0)

and nonclassical theory gets more obvious e reason forthis change was already discussed in Section 41 Besidesthe present result is parallel to that of Ansari et al [35]who investigated the forced vibration of functionallygraded microplate based onMCST It should be noted thathorizontal axis in this work represents nondimensionalexternal frequency while in Ansari et alrsquos work it rep-resents the frequency ratio (the ratio of external frequencyto linear first frequency)

5 Conclusions

e nonlinear size-dependent vibration of a microplate withdamage was explored by employing MCST and the strainequivalent assumption in this research e nonlineargoverning partial differential equations were transformedinto nonlinear ordinary differential equations via Galerkinrsquosscheme and further solved numerically by the harmonicbalance method Numerical results indicate that damageeffect and size dependency both have obvious influences onthe static and dynamic behaviors of the microplate system Itwas concluded that on one hand the hardening-typenonlinear behavior of the microplate system enhances whenit encounters damage on the other hand the system exhibitsa weaker nonlinear behavior greater nondimensional fre-quency and lower amplitude of the microplate midpoint assize effect gets obvious

Data Availability

e Matlab simulation and control program data used tosupport the findings of this study are available from thecorresponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors would like to acknowledge with great gratitudefor the supports of the National Science Foundation ofChina (grant nos 51778551 and 11272270)

References

[1] M I Younis and A H Nayfeh ldquoA study of the nonlinearresponse of a resonant micro-beam to an electric actuationrdquoNonlinear Dynamics vol 31 no 1 pp 91ndash117 2003

C_

= 02

C_

= 04

C_

= 01

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 5e frequency-response curves of the system for differentdamping coefficients C

D = 02

D = 04

D = 0

15 19 23 27 31 3511ω

0

02

04

06

08

A

Figure 6e frequency-response curves of the system for differentdamage variable D

lh = 04

lh = 0

lh = 02

15 19 23 27 31 3511ω

0

01

02

03

04

05

A

Figure 7e frequency-response curves of the system for differentnondimensional material length scale parameters lh

Shock and Vibration 11

[2] E M Abdel-Rahman and A H Nayfeh ldquoSecondary reso-nances of electrically actuated resonant microsensorsrdquoJournal of Micromechanics and Microengineering vol 13no 3 pp 491ndash501 2003

[3] W Zhang and G Meng ldquoNonlinear dynamical system ofmicro-cantilever under combined parametric and forcingexcitations in MEMSrdquo Sensors and Actuators A Physicalvol 119 no 2 pp 291ndash299 2005

[4] L Xu and X Jia ldquoElectromechanical coupled nonlinear dy-namics for microbeamsrdquo Archive of Applied Mechanicsvol 77 no 7 pp 485ndash502 2007

[5] W G Vogl and A H Nayfeh ldquoPrimary resonance excitationof electrically actuated clamped circular platesrdquo NonlinearDynamics vol 47 no 1ndash3 pp 181ndash192 2006

[6] A H Nayfeh H M Ouakad F Choura and E M Abdel-Rahman ldquoNonlinear dynamics of a resonant gas sensorrdquoNonlinear Dynamics vol 59 no 4 pp 607ndash618 2010

[7] X L Jia J Yang S Kitipornchai and C W Lim ldquoResonancefrequency response of geometrically nonlinear micro-switches under electrical actuationrdquo Journal of Sound andVibration vol 331 no 14 pp 3397ndash3411 2012

[8] P Kim S Bae and J Seok ldquoResonant behaviors of a nonlinearcantilever beam with tip mass subject to an axial force andelectrostatic excitationrdquo International Journal of MechanicalSciences vol 64 no 1 pp 232ndash257 2012

[9] S Saghir and M I Younis ldquoAn investigation of the static anddynamic behavior of electrically actuated rectangularmicroplatesrdquo International Journal of Non-linear Mechanicsvol 85 pp 81ndash93 2016

[10] M Sheikhlou R Shabani and G Rezazadeh ldquoNonlinearanalysis of electrostatically actuated diaphragm-type micro-pumpsrdquo Nonlinear Dynamics vol 83 no 1-2 pp 1ndash11 2016

[11] P M Osterberg Electrostatically actuated micromechanicaltest structure for material property measurement PhD dis-sertation Massachusetts Institute of Technology CambridgeMA USA 1995

[12] M Arefi and A M Zenkour ldquoSize dependent electro-elasticanalysis of a sandwich microbeam based on higher ordersinusoidal shear deformation theory and strain gradienttheoryrdquo Journal of Intelligent Material Systems and Structuresvol 29 no 7 pp 27ndash40 2018

[13] M Arefi and A M Zenkour ldquoSize-dependent vibration andbending analyses of the piezomagnetic three-layer nano-beamsrdquo Applied Physics A vol 123 no 3 p 202 2017

[14] M Arefi and A M Zenkour ldquoSize-dependent vibration andelectro-magneto-elastic bending responses of sandwich pie-zomagnetic curved nanobeamsrdquo Steel and Composite Struc-tures vol 29 no 5 pp 579ndash590 2018

[15] M Arefi and A M Zenkour ldquoermo-electro-magneto-mechanical bending behavior of size-dependent sandwichpiezomagnetic nanoplatesrdquo Mechanics Research Communi-cations vol 84 pp 27ndash42 2017

[16] L L Ke J Yang S Kitipornchai M A Bradford andY S Wang ldquoAxisymmetric nonlinear free vibration of size-dependent functionally graded annular microplatesrdquo Com-posites Part B Engineering vol 53 no 7 pp 207ndash217 2013

[17] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[18] M H Ghayesh H Farokhi and M Amabili ldquoNonlinearbehaviour of electrically actuated MEMS resonatorsrdquo

International Journal of Engineering Science vol 71 no 10pp 137ndash155 2013

[19] A G Arani and G S Jafari ldquoNonlinear vibration analysis oflaminated composite Mindlin micronano-plates resting onorthotropic Pasternak medium using DQMrdquo AppliedMathematics and Mechanics vol 36 no 8 pp 1033ndash10442015

[20] M Sobhy and A M Zenkour ldquoA comprehensive study onthe size-dependent hygrothermal analysis of exponentiallygraded microplates on elastic foundationsrdquo Mechanics ofAdvanced Materials and Structures vol 27 no 10pp 816ndash830 2019

[21] M Arefi and A M Zenkour ldquoSize-dependent free vibrationand dynamic analyses of piezo-electro-magnetic sandwichnanoplates resting on viscoelastic foundationrdquo Physica BCondensed Matter vol 521 pp 188ndash197 2017

[22] M Arefi M Kiani and A M Zenkour ldquoSize-dependent freevibration analysis of a three-layered exponentially gradednano-micro-plate with piezomagnetic face sheets resting onPasternakrsquos foundation via MCSTrdquo Journal of SandwichStructures and Materials vol 22 no 1 pp 55ndash86 2020

[23] H Farokhi and M H Ghayesh ldquoSize-dependent behavior ofelectrically actuated microcantilever-based MEMSrdquo In-ternational Journal of Mechanics amp Materials in Designvol 12 no 3 pp 1ndash15 2016

[24] M Tahani A R Askari Y Mohandes and B Hassani ldquoSize-dependent free vibration analysis of electrostatically pre-de-formed rectangular micro-plates based on the modifiedcouple stress theoryfied couple stress theoryrdquo InternationalJournal of Mechanical Sciences vol 94-95 no 22 pp 185ndash1982015

[25] A Veysi R Shabani and G Rezazadeh ldquoNonlinear vibrationsof micro-doubly curved shallow shells based on the modifiedcouple stress theoryfied couple stress theoryrdquo NonlinearDynamics vol 87 no 3 pp 2051ndash2065 2017

[26] B Jalalahmadi F Sadeghi and D Peroulis ldquoA numericalfatigue damage model for life scatter of MEMS devicesrdquoJournal of Microelectromechanical Systems vol 18 no 5pp 1016ndash1031 2009

[27] T S Slack F Sadeghi and D Peroulis ldquoA phenomenologicaldiscrete brittle damage-mechanics model for fatigue of MEMSdevices with application to LIGA Nirdquo Journal of Micro-electromechanical Systems vol 18 no 1 pp 119ndash128 2009

[28] A Basu R P Hennessy G G Adams and N E McGruerldquoHot switching damage mechanisms in MEMS contacts-ev-idence and understandingrdquo Journal of Micromechanics andMicroengineering vol 24 no 10 p 16 2014

[29] C Chen J Yuan and Y Mao ldquoPost-buckling of size-de-pendent micro-plate considering damage effectsrdquo NonlinearDynamics vol 90 no 2 pp 1301ndash1314 2017

[30] J Lemaitre A Course on Damage Mechanics Springer BerlinGermany 1992

[31] C Chen H Hu and L Dai ldquoNonlinear behavior andcharacterization of a piezoelectric laminated microbeamsystemrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1304ndash1315 2013

[32] A R Askari and M Tahani ldquoSize-dependent dynamic pull-inanalysis of geometric non-linear micro-plates based on themodified couple stress theoryrdquo Physica E Low-DimensionalSystems and Nanostructures vol 86 pp 262ndash274 2017

[33] A H Nayfeh and D T Mook Nonlinear OscillationsSpringer International Publishing New York NY USA 1979

12 Shock and Vibration

[34] R Ansari M Faghih Shojaei V Mohammadi R Gholamiand M A Darabi ldquoNonlinear vibrations of functionallygraded Mindlin microplates based on the modified couplestress theoryfied couple stress theoryrdquo Composite Structuresvol 114 no 18 pp 124ndash134 2014

[35] R Ansari R Gholami and A Shahabodini ldquoSize-dependentgeometrically nonlinear forced vibration analysis of func-tionally graded first-order shear deformable microplatesrdquoJournal of Mechanics vol 32 no 5 pp 539ndash554 2016

Shock and Vibration 13

Figure 7 highlights the size effect on the frequency-response curves of dynamic system e numerical cal-culations are performed by assumingD 02 C 02 Va 20 It is revealed that the peak ofthe curve goes down slightly and the hardening influencereduces mildly when nondimensional material lengthscale parameter increases In addition the unstable regionmoves to the right side and diminishes as size effect be-comes more obvious Furthermore the distinction be-tween the results predicted by classical theory (lh 0)

and nonclassical theory gets more obvious e reason forthis change was already discussed in Section 41 Besidesthe present result is parallel to that of Ansari et al [35]who investigated the forced vibration of functionallygraded microplate based onMCST It should be noted thathorizontal axis in this work represents nondimensionalexternal frequency while in Ansari et alrsquos work it rep-resents the frequency ratio (the ratio of external frequencyto linear first frequency)

5 Conclusions

e nonlinear size-dependent vibration of a microplate withdamage was explored by employing MCST and the strainequivalent assumption in this research e nonlineargoverning partial differential equations were transformedinto nonlinear ordinary differential equations via Galerkinrsquosscheme and further solved numerically by the harmonicbalance method Numerical results indicate that damageeffect and size dependency both have obvious influences onthe static and dynamic behaviors of the microplate system Itwas concluded that on one hand the hardening-typenonlinear behavior of the microplate system enhances whenit encounters damage on the other hand the system exhibitsa weaker nonlinear behavior greater nondimensional fre-quency and lower amplitude of the microplate midpoint assize effect gets obvious

Data Availability

e Matlab simulation and control program data used tosupport the findings of this study are available from thecorresponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors would like to acknowledge with great gratitudefor the supports of the National Science Foundation ofChina (grant nos 51778551 and 11272270)

References

[1] M I Younis and A H Nayfeh ldquoA study of the nonlinearresponse of a resonant micro-beam to an electric actuationrdquoNonlinear Dynamics vol 31 no 1 pp 91ndash117 2003

C_

= 02

C_

= 04

C_

= 01

15 19 23 27 31 3511ω

0

02

04

06

08

1

A

Figure 5e frequency-response curves of the system for differentdamping coefficients C

D = 02

D = 04

D = 0

15 19 23 27 31 3511ω

0

02

04

06

08

A

Figure 6e frequency-response curves of the system for differentdamage variable D

lh = 04

lh = 0

lh = 02

15 19 23 27 31 3511ω

0

01

02

03

04

05

A

Figure 7e frequency-response curves of the system for differentnondimensional material length scale parameters lh

Shock and Vibration 11

[2] E M Abdel-Rahman and A H Nayfeh ldquoSecondary reso-nances of electrically actuated resonant microsensorsrdquoJournal of Micromechanics and Microengineering vol 13no 3 pp 491ndash501 2003

[3] W Zhang and G Meng ldquoNonlinear dynamical system ofmicro-cantilever under combined parametric and forcingexcitations in MEMSrdquo Sensors and Actuators A Physicalvol 119 no 2 pp 291ndash299 2005

[4] L Xu and X Jia ldquoElectromechanical coupled nonlinear dy-namics for microbeamsrdquo Archive of Applied Mechanicsvol 77 no 7 pp 485ndash502 2007

[5] W G Vogl and A H Nayfeh ldquoPrimary resonance excitationof electrically actuated clamped circular platesrdquo NonlinearDynamics vol 47 no 1ndash3 pp 181ndash192 2006

[6] A H Nayfeh H M Ouakad F Choura and E M Abdel-Rahman ldquoNonlinear dynamics of a resonant gas sensorrdquoNonlinear Dynamics vol 59 no 4 pp 607ndash618 2010

[7] X L Jia J Yang S Kitipornchai and C W Lim ldquoResonancefrequency response of geometrically nonlinear micro-switches under electrical actuationrdquo Journal of Sound andVibration vol 331 no 14 pp 3397ndash3411 2012

[8] P Kim S Bae and J Seok ldquoResonant behaviors of a nonlinearcantilever beam with tip mass subject to an axial force andelectrostatic excitationrdquo International Journal of MechanicalSciences vol 64 no 1 pp 232ndash257 2012

[9] S Saghir and M I Younis ldquoAn investigation of the static anddynamic behavior of electrically actuated rectangularmicroplatesrdquo International Journal of Non-linear Mechanicsvol 85 pp 81ndash93 2016

[10] M Sheikhlou R Shabani and G Rezazadeh ldquoNonlinearanalysis of electrostatically actuated diaphragm-type micro-pumpsrdquo Nonlinear Dynamics vol 83 no 1-2 pp 1ndash11 2016

[11] P M Osterberg Electrostatically actuated micromechanicaltest structure for material property measurement PhD dis-sertation Massachusetts Institute of Technology CambridgeMA USA 1995

[12] M Arefi and A M Zenkour ldquoSize dependent electro-elasticanalysis of a sandwich microbeam based on higher ordersinusoidal shear deformation theory and strain gradienttheoryrdquo Journal of Intelligent Material Systems and Structuresvol 29 no 7 pp 27ndash40 2018

[13] M Arefi and A M Zenkour ldquoSize-dependent vibration andbending analyses of the piezomagnetic three-layer nano-beamsrdquo Applied Physics A vol 123 no 3 p 202 2017

[14] M Arefi and A M Zenkour ldquoSize-dependent vibration andelectro-magneto-elastic bending responses of sandwich pie-zomagnetic curved nanobeamsrdquo Steel and Composite Struc-tures vol 29 no 5 pp 579ndash590 2018

[15] M Arefi and A M Zenkour ldquoermo-electro-magneto-mechanical bending behavior of size-dependent sandwichpiezomagnetic nanoplatesrdquo Mechanics Research Communi-cations vol 84 pp 27ndash42 2017

[16] L L Ke J Yang S Kitipornchai M A Bradford andY S Wang ldquoAxisymmetric nonlinear free vibration of size-dependent functionally graded annular microplatesrdquo Com-posites Part B Engineering vol 53 no 7 pp 207ndash217 2013

[17] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[18] M H Ghayesh H Farokhi and M Amabili ldquoNonlinearbehaviour of electrically actuated MEMS resonatorsrdquo

International Journal of Engineering Science vol 71 no 10pp 137ndash155 2013

[19] A G Arani and G S Jafari ldquoNonlinear vibration analysis oflaminated composite Mindlin micronano-plates resting onorthotropic Pasternak medium using DQMrdquo AppliedMathematics and Mechanics vol 36 no 8 pp 1033ndash10442015

[20] M Sobhy and A M Zenkour ldquoA comprehensive study onthe size-dependent hygrothermal analysis of exponentiallygraded microplates on elastic foundationsrdquo Mechanics ofAdvanced Materials and Structures vol 27 no 10pp 816ndash830 2019

[21] M Arefi and A M Zenkour ldquoSize-dependent free vibrationand dynamic analyses of piezo-electro-magnetic sandwichnanoplates resting on viscoelastic foundationrdquo Physica BCondensed Matter vol 521 pp 188ndash197 2017

[22] M Arefi M Kiani and A M Zenkour ldquoSize-dependent freevibration analysis of a three-layered exponentially gradednano-micro-plate with piezomagnetic face sheets resting onPasternakrsquos foundation via MCSTrdquo Journal of SandwichStructures and Materials vol 22 no 1 pp 55ndash86 2020

[23] H Farokhi and M H Ghayesh ldquoSize-dependent behavior ofelectrically actuated microcantilever-based MEMSrdquo In-ternational Journal of Mechanics amp Materials in Designvol 12 no 3 pp 1ndash15 2016

[24] M Tahani A R Askari Y Mohandes and B Hassani ldquoSize-dependent free vibration analysis of electrostatically pre-de-formed rectangular micro-plates based on the modifiedcouple stress theoryfied couple stress theoryrdquo InternationalJournal of Mechanical Sciences vol 94-95 no 22 pp 185ndash1982015

[25] A Veysi R Shabani and G Rezazadeh ldquoNonlinear vibrationsof micro-doubly curved shallow shells based on the modifiedcouple stress theoryfied couple stress theoryrdquo NonlinearDynamics vol 87 no 3 pp 2051ndash2065 2017

[26] B Jalalahmadi F Sadeghi and D Peroulis ldquoA numericalfatigue damage model for life scatter of MEMS devicesrdquoJournal of Microelectromechanical Systems vol 18 no 5pp 1016ndash1031 2009

[27] T S Slack F Sadeghi and D Peroulis ldquoA phenomenologicaldiscrete brittle damage-mechanics model for fatigue of MEMSdevices with application to LIGA Nirdquo Journal of Micro-electromechanical Systems vol 18 no 1 pp 119ndash128 2009

[28] A Basu R P Hennessy G G Adams and N E McGruerldquoHot switching damage mechanisms in MEMS contacts-ev-idence and understandingrdquo Journal of Micromechanics andMicroengineering vol 24 no 10 p 16 2014

[29] C Chen J Yuan and Y Mao ldquoPost-buckling of size-de-pendent micro-plate considering damage effectsrdquo NonlinearDynamics vol 90 no 2 pp 1301ndash1314 2017

[30] J Lemaitre A Course on Damage Mechanics Springer BerlinGermany 1992

[31] C Chen H Hu and L Dai ldquoNonlinear behavior andcharacterization of a piezoelectric laminated microbeamsystemrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1304ndash1315 2013

[32] A R Askari and M Tahani ldquoSize-dependent dynamic pull-inanalysis of geometric non-linear micro-plates based on themodified couple stress theoryrdquo Physica E Low-DimensionalSystems and Nanostructures vol 86 pp 262ndash274 2017

[33] A H Nayfeh and D T Mook Nonlinear OscillationsSpringer International Publishing New York NY USA 1979

12 Shock and Vibration

[34] R Ansari M Faghih Shojaei V Mohammadi R Gholamiand M A Darabi ldquoNonlinear vibrations of functionallygraded Mindlin microplates based on the modified couplestress theoryfied couple stress theoryrdquo Composite Structuresvol 114 no 18 pp 124ndash134 2014

[35] R Ansari R Gholami and A Shahabodini ldquoSize-dependentgeometrically nonlinear forced vibration analysis of func-tionally graded first-order shear deformable microplatesrdquoJournal of Mechanics vol 32 no 5 pp 539ndash554 2016

Shock and Vibration 13

[2] E M Abdel-Rahman and A H Nayfeh ldquoSecondary reso-nances of electrically actuated resonant microsensorsrdquoJournal of Micromechanics and Microengineering vol 13no 3 pp 491ndash501 2003

[3] W Zhang and G Meng ldquoNonlinear dynamical system ofmicro-cantilever under combined parametric and forcingexcitations in MEMSrdquo Sensors and Actuators A Physicalvol 119 no 2 pp 291ndash299 2005

[4] L Xu and X Jia ldquoElectromechanical coupled nonlinear dy-namics for microbeamsrdquo Archive of Applied Mechanicsvol 77 no 7 pp 485ndash502 2007

[5] W G Vogl and A H Nayfeh ldquoPrimary resonance excitationof electrically actuated clamped circular platesrdquo NonlinearDynamics vol 47 no 1ndash3 pp 181ndash192 2006

[6] A H Nayfeh H M Ouakad F Choura and E M Abdel-Rahman ldquoNonlinear dynamics of a resonant gas sensorrdquoNonlinear Dynamics vol 59 no 4 pp 607ndash618 2010

[7] X L Jia J Yang S Kitipornchai and C W Lim ldquoResonancefrequency response of geometrically nonlinear micro-switches under electrical actuationrdquo Journal of Sound andVibration vol 331 no 14 pp 3397ndash3411 2012

[8] P Kim S Bae and J Seok ldquoResonant behaviors of a nonlinearcantilever beam with tip mass subject to an axial force andelectrostatic excitationrdquo International Journal of MechanicalSciences vol 64 no 1 pp 232ndash257 2012

[9] S Saghir and M I Younis ldquoAn investigation of the static anddynamic behavior of electrically actuated rectangularmicroplatesrdquo International Journal of Non-linear Mechanicsvol 85 pp 81ndash93 2016

[10] M Sheikhlou R Shabani and G Rezazadeh ldquoNonlinearanalysis of electrostatically actuated diaphragm-type micro-pumpsrdquo Nonlinear Dynamics vol 83 no 1-2 pp 1ndash11 2016

[11] P M Osterberg Electrostatically actuated micromechanicaltest structure for material property measurement PhD dis-sertation Massachusetts Institute of Technology CambridgeMA USA 1995

[12] M Arefi and A M Zenkour ldquoSize dependent electro-elasticanalysis of a sandwich microbeam based on higher ordersinusoidal shear deformation theory and strain gradienttheoryrdquo Journal of Intelligent Material Systems and Structuresvol 29 no 7 pp 27ndash40 2018

[13] M Arefi and A M Zenkour ldquoSize-dependent vibration andbending analyses of the piezomagnetic three-layer nano-beamsrdquo Applied Physics A vol 123 no 3 p 202 2017

[14] M Arefi and A M Zenkour ldquoSize-dependent vibration andelectro-magneto-elastic bending responses of sandwich pie-zomagnetic curved nanobeamsrdquo Steel and Composite Struc-tures vol 29 no 5 pp 579ndash590 2018

[15] M Arefi and A M Zenkour ldquoermo-electro-magneto-mechanical bending behavior of size-dependent sandwichpiezomagnetic nanoplatesrdquo Mechanics Research Communi-cations vol 84 pp 27ndash42 2017

[16] L L Ke J Yang S Kitipornchai M A Bradford andY S Wang ldquoAxisymmetric nonlinear free vibration of size-dependent functionally graded annular microplatesrdquo Com-posites Part B Engineering vol 53 no 7 pp 207ndash217 2013

[17] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[18] M H Ghayesh H Farokhi and M Amabili ldquoNonlinearbehaviour of electrically actuated MEMS resonatorsrdquo

International Journal of Engineering Science vol 71 no 10pp 137ndash155 2013

[19] A G Arani and G S Jafari ldquoNonlinear vibration analysis oflaminated composite Mindlin micronano-plates resting onorthotropic Pasternak medium using DQMrdquo AppliedMathematics and Mechanics vol 36 no 8 pp 1033ndash10442015

[20] M Sobhy and A M Zenkour ldquoA comprehensive study onthe size-dependent hygrothermal analysis of exponentiallygraded microplates on elastic foundationsrdquo Mechanics ofAdvanced Materials and Structures vol 27 no 10pp 816ndash830 2019

[21] M Arefi and A M Zenkour ldquoSize-dependent free vibrationand dynamic analyses of piezo-electro-magnetic sandwichnanoplates resting on viscoelastic foundationrdquo Physica BCondensed Matter vol 521 pp 188ndash197 2017

[22] M Arefi M Kiani and A M Zenkour ldquoSize-dependent freevibration analysis of a three-layered exponentially gradednano-micro-plate with piezomagnetic face sheets resting onPasternakrsquos foundation via MCSTrdquo Journal of SandwichStructures and Materials vol 22 no 1 pp 55ndash86 2020

[23] H Farokhi and M H Ghayesh ldquoSize-dependent behavior ofelectrically actuated microcantilever-based MEMSrdquo In-ternational Journal of Mechanics amp Materials in Designvol 12 no 3 pp 1ndash15 2016

[24] M Tahani A R Askari Y Mohandes and B Hassani ldquoSize-dependent free vibration analysis of electrostatically pre-de-formed rectangular micro-plates based on the modifiedcouple stress theoryfied couple stress theoryrdquo InternationalJournal of Mechanical Sciences vol 94-95 no 22 pp 185ndash1982015

[25] A Veysi R Shabani and G Rezazadeh ldquoNonlinear vibrationsof micro-doubly curved shallow shells based on the modifiedcouple stress theoryfied couple stress theoryrdquo NonlinearDynamics vol 87 no 3 pp 2051ndash2065 2017

[26] B Jalalahmadi F Sadeghi and D Peroulis ldquoA numericalfatigue damage model for life scatter of MEMS devicesrdquoJournal of Microelectromechanical Systems vol 18 no 5pp 1016ndash1031 2009

[27] T S Slack F Sadeghi and D Peroulis ldquoA phenomenologicaldiscrete brittle damage-mechanics model for fatigue of MEMSdevices with application to LIGA Nirdquo Journal of Micro-electromechanical Systems vol 18 no 1 pp 119ndash128 2009

[28] A Basu R P Hennessy G G Adams and N E McGruerldquoHot switching damage mechanisms in MEMS contacts-ev-idence and understandingrdquo Journal of Micromechanics andMicroengineering vol 24 no 10 p 16 2014

[29] C Chen J Yuan and Y Mao ldquoPost-buckling of size-de-pendent micro-plate considering damage effectsrdquo NonlinearDynamics vol 90 no 2 pp 1301ndash1314 2017

[30] J Lemaitre A Course on Damage Mechanics Springer BerlinGermany 1992

[31] C Chen H Hu and L Dai ldquoNonlinear behavior andcharacterization of a piezoelectric laminated microbeamsystemrdquoCommunications in Nonlinear Science and NumericalSimulation vol 18 no 5 pp 1304ndash1315 2013

[32] A R Askari and M Tahani ldquoSize-dependent dynamic pull-inanalysis of geometric non-linear micro-plates based on themodified couple stress theoryrdquo Physica E Low-DimensionalSystems and Nanostructures vol 86 pp 262ndash274 2017

[33] A H Nayfeh and D T Mook Nonlinear OscillationsSpringer International Publishing New York NY USA 1979

12 Shock and Vibration

[34] R Ansari M Faghih Shojaei V Mohammadi R Gholamiand M A Darabi ldquoNonlinear vibrations of functionallygraded Mindlin microplates based on the modified couplestress theoryfied couple stress theoryrdquo Composite Structuresvol 114 no 18 pp 124ndash134 2014

[35] R Ansari R Gholami and A Shahabodini ldquoSize-dependentgeometrically nonlinear forced vibration analysis of func-tionally graded first-order shear deformable microplatesrdquoJournal of Mechanics vol 32 no 5 pp 539ndash554 2016

Shock and Vibration 13

[34] R Ansari M Faghih Shojaei V Mohammadi R Gholamiand M A Darabi ldquoNonlinear vibrations of functionallygraded Mindlin microplates based on the modified couplestress theoryfied couple stress theoryrdquo Composite Structuresvol 114 no 18 pp 124ndash134 2014

[35] R Ansari R Gholami and A Shahabodini ldquoSize-dependentgeometrically nonlinear forced vibration analysis of func-tionally graded first-order shear deformable microplatesrdquoJournal of Mechanics vol 32 no 5 pp 539ndash554 2016

Shock and Vibration 13