# Nonlinear Vibration Analysis of Damaged Microplate considering Size Effect

Authors: Jihai Yuan; Xiangmin Zhang; Changping Chen
Journal: Shock and Vibration
 (2020)

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## Abstract

Since microplates are extensively used in MEMS devices such as microbumps, micromirrors, and microphones, this work aims to study nonlinear vibration of an electrically actuated microplate whose four edges are clamped. Based on the modified couple stress theory (MCST) and strain equivalent assumption, size effect and damage are taken into consideration in the present model. The dynamic governing partial differential equations of the microplate system were obtained using Hamilton’s principle and solved using the harmonic balance method after they are transformed into ordinary differential equation with regard to time. Size effect and damage effect on nonlinear free vibration of the microplate under DC voltage are discussed using frequency-response curve. In the forced vibration analysis, the frequency-response curves were also employed for the purpose of highlighting the influence of different physical parameters such as external excitation, damping coefficient, material length scale parameter, and damage variable when the system is under AC voltage. The results presented in this study may be helpful and useful for the dynamic stability of a electrically actuated microplate system.


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## Body

## 1. Introduction

Since a microelectromechanical system (MEMS) received a considerable amount of attention in recent years, numerous works related to nonlinear responses and characteristics because of intrinsic existence of nonlinearity of these microdevices have been carried out in recent years. Using the multiple scale method, Younis and Nayfeh [1] studied nonlinear vibration of a resonant microbeam under dynamic electrostatic force. Based on the same method, Abdel-Rahman and Nayfeh [2] studied response of a microresonant sensor actuated by superharmonic and subharmonic electric forces and their results provide an analytical solution to predict the resonant response. Zhang and Meng [3] proposed a simpliﬁed model in order to study the resonant responses and nonlinear dynamics of microcantilever under electronic excitation. An electromechanical coupled nonlinear dynamic response was presented by Xu and Jia [4], who employed the perturbation method to discuss influence of mechanical and electric parameters on nonlinear natural frequencies and vibrating amplitudes of a microbeam. Vogl and Nayfeh [5] reported the response of the clamped plate with circular shape under primary resonance excitation using a reduced-order model, which is general enough for effectively designing capacitive micromachined ultrasonic transducers. Nayfeh et al. [6] developed a novel model for a resonant gas sensor and studied its nonlinear dynamic characteristics. Jia et al. [7] conducted a parametric study on forced vibration of microswitches to show effects of Casimir force, residual stress, and geometrical nonlinearity on the frequency response characteristics. Kim et al. [8] examined resonant behaviors of a microbeam subjected to axial force and electrostatic force. Saghir and Younis [9] conducted a study on nonlinear vibration behavior of rectangular microplate under static and dynamic load. They found an interesting phenomenon where the microplate shows a hardening behavior which switches to softening as the DC load increases. Sheikhlou et al. [10] investigated nonlinear resonant behavior of diaphragm-type micropumps.However, size-dependent behavior of microstructures has been experimentally validated [11]. With this consideration, some interesting studies related to vibration behavior are presented. Using strain gradient theory and shear deformation theory, Arefi and Zenkour [12] presented free vibration of a piezoelectric laminated microbeam. Based on nonlocal piezoelasticity theory, Arefi and Zenkour presented size-dependent bending and vibration response of a sandwich piezomagnetic nanobeam with curvature [13] and without curvature [14]. They also investigated thermo-electro-magneto-mechanical bending behavior of a Sandwich nanoplate with simplysupported boundary conditions [15]. A nonclassical microplate model was proposed by Ke et al. [16], who employed MCST [17] to study nonlinear free vibration of annular microplate. Using the same theory, Ghayesh et al. [18] analyzed the nonlinear behavior of a size-dependent resonator under primary and superharmonic excitations. Using Eringen nonlocal theory, Arani and Jafari [19] analyzed nonlinear vibration of sandwich microplates resting on an elastic matrix. With application of different kinds of materials such as exponentially graded, piezoelectric, and magnetic, Sobhy and Zenkour [20], Arefi and Zenkour [21], and Arefi et al. [22] investigated size-dependent free vibration of sandwich microplate resting on different foundations. The nonlinear size-dependent static and dynamic behaviors of a MEMS device were presented by Farokhi and Ghayesh [23] on the basis of MCST. The free vibration characteristics of microplate with size dependency were displayed by Tahani et al. [24] based on the MCST recently with consideration of various effects such as couple stress components, electrostatic attraction, and different boundary conditions on both mode shapes and natural frequencies. Unlike the previous research, Veysi et al. [25] presented vibrational behavior of microdoubly curved shallow shells incorporated with von-Kármán geometric nonlinearity, size dependence, and shear deformation by adopting multiple scale method.In the meantime, during manufacturing and operating process damage, that is, fatigue damage [26], discrete brittle damage [27] and contact damage [28] may occur due to the very existence of microcracks or abrasions which may evolve and develop [29]. However, most of these works ignored the significant role of size effect in mechanical behavior of microstructures. Actually, nonlinear size-dependent behavior of microstructure considering damage effect is limited in current literatures. Hence, a better understanding of damage and microscale effect on mechanical behavior of microstructure is necessary. With this motivation, this work aims to study size-dependent nonlinear free vibration and forced vibration behavior for microplates with damage under electrostatic actuation. In this research, the damage constitutive relations are established based on the strain equivalent assumption [30] and MCST. The governing differential equations were derived via Hamilton’s principle and solved numerically by Galerkin’s method and the harmonic balance method (HBM). The influence of various system parameters on nonlinear vibration of the microplate is studied comprehensively.


## 2. Mathematical Modeling

Consider a movable microplate with mass densityρ over a stationary electrode under electronic force in Figure 1, where its width, length, and thickness are b,a, and h separately. The initial gap and external voltage between the electrode and the microplate are d and VC accordingly. The midsurface of the microplate is regarded as the reference plane z=0 as shown in this figure.Figure 1
Electrically actuated microplate.With the introduction of only one extra material parameter, material length scale parameter, Yang et al. [17] proposed MCST which claims the strain energy of an elastomer with volume V is expressed as(1)U=12∫Vσijεij+mijχijdV,i,j=x,y,z,where stress tensor σij, couple stress tensor mij, strain tensor εij, and symmetric curvature tensor χij, are given as follows:(2)σij=λtrεijδij+2μεij,(3)mij=2l2μχij,(4)εij=12ui,j+uj,i,(5)χij=12θi,j+θj,i,where ui represents displacement vector. There are two constitutive relations: one is between Cauchy’s stress and infinitesimal engineering strain represented by (2) and the other is between couple stress and symmetric curvature represented (3). The former one is traditional elastic constitutive relation with lame constant λ,μ. The latter one is extra constitutive relation accounting for size dependency using material length scale parameter l. The gyration vector θi is written as(6)θi=12eijkuk,j,where eijk is the permutation tensor.Kirchhoff’s displacement componentsu1,u2,u3 at a point of plate can be defined as(7)u1x,y,z,t=ux,y,t−zwx,u2x,y,z,t=vx,y,t−zwy,u3x,y,z,t=wx,y,t,where u,v,andw donate the displacements of the reference plane separately. For microplate encounters large deflection, the von Kármán’s plate theory is adopted to describe the geometric nonlinearity of the microplate. Thus, the strain-displacement relations can be given as(8)εx=εx0+zκx,εy=εy0+zκy,εxy=εxy0+zκxy,where stains εx0,εy0,andεxy0 and curvatures κx,κy,andκxy at a point of the middle surface are expressed as(9)εx0εy0εxy0=ux+12wx2vy+12wy2uy+vx+wxwy,κxκyκxy=−wxx−wyy−2wxy.Substituting (7) into (6), one has gyration vector:(10)θxx,y=wy,θyx,y=−wx,θzx,y=12vx−uy.Substituting (10) into (2) yields curvature tensor:(11)χxxχyyχxyχxzχyz=wxy−wxy12wyy−wxx14vxx−uxy14vxy−uyy.The stress-strain relations can be given with respect to Poisson’s ratioυ and Young’s modulus E as(12)σxσyσxy=Qd•εxεyεxy,(13)mxx,myy,mxy,mxz,myz=Fχxx,χyy,χxy,χxz,χyz,where(14)Qd=Q11dQ12d0Q22d0Q66d=E1−υ2υE1−υ20E1−υ20G1−D,F=2l2G1−D,where D denotes damage variable varying from 0 to 1 [30]; G is shear modulus.With Cauchy’s stress and couple stress expressed above, bending momentsMx,My,Mxy, membrane stress resultants Nx,Ny,Nxy, and couple moments Yxx,Yyy,Yxy,Yxz,Yyz can be, respectively, defined as(15)MxMyMxy=∫−h/2h/2zσxσyσxydz,NxNyNxy=∫−h/2h/2σxσyσxydz,YxxYyyYxyYxzYyz=∫−h/2h/2mxxmyymxymxzmyzdz.Substituting (12) and (13) into (15) yields the following constitutive equations:(16)NxNyNxy=A11A120A12A22000A66εx0εy0εxy0,MxMyMxy=D11D120D12D22000D66κxκyκxy,YxxYyyYxyYxzYyz=Gχxxχyyχxyχxzχyz,where(17)Aij=hQijd,Dij=h312Qijd,G′=hF,i,j=1,2,6.With Hamilton’s principle, we have(18)δ∫t1t2T−U+Wdt=0.Substituting stresses and strains into (1), the virtual strain energy is obtained as(19)δU=∫VσijδεijdV+∫VmijδχijdV.The virtual kinetic energy is given as(20)δT=12ρ∫Vδwt2dV.The virtual work done by electric force and damping force can be expressed as(21)δW=∬sFz−cwtδwdS,where c denotes damping coefficient and Fz represents the electric force deﬁned as(22)Fz=εvVC22d−w2,where εv is the dielectric constant of air.By substituting (19)–(21) into (18), one has the nonlinear dynamic equations of the microplate system as(23)Nx,x+Nxy,y+12Yxz,xy+12Yyz,yy=0,Ny,y+Nxy,x−12Yxz,xx−12Yyz,xy=0,−Nxwx+Nxywyx−Nywy+Nxywxy+−Mx−Yxyxx+−My+Yxyyy+−2Mxy+Yxx−Yyyxy+Fe=ρhwtt+cwt.Boundary conditions are as follows:(24)Nx+12Yxz,y=0,orδu=0,Nxy−12Yxz,x−12Yyz,y=0,orδv=0,12Yxz=0,orδvx=0,Mx,x+2Mxy,y+Nxwx+Nxywy+Yxy,x+−Yxx+Yyyy=0,orδw=0,Mx+Yxy=0,orδwx=0,y=0,b,Ny−12Yyz,x=0,orδv=0,Nxy+12Yyz,y+12Yxz,x=0,orδu=0,12Yyz=0,orδuy=0,My,y+Nywy+Nxywx+2Mxy,x−Yxy,y−Yxx−Yyyx=0,orδw=0,My−Yxy=0,orδwy=0.The following dimensionless parameters are introduced:(25)ξ=xa,η=yb,U=ua,V=vb,W=wh,λ1=ha,λ2=hb,A¯ij=AijQ11h,l¯=lh,D¯ij=12DijQ11h3,G¯1=G′8Ga3,G¯2=G′8Gb3,G¯3=G′2Gh3,β=GQ11,Q11=E1−υ2,τ=tT,C¯=ca2b2Q11h3T,V¯C=VCV0,α1=εva2b2V022d2h4Q11,α2=εva2b2V02d3h3Q11,T=abhρQ11,ω=ΩT,where V0 is the unit voltage.With application of Taylor series expansion into (22) and ignoring nonlinear terms with respect to w [31], the nondimensional governing equations of the microplate are written as(26)A¯11Uξξ+λ12WξWξξ+A¯12Vξη+λ22WηWξη+βA¯66λ22λ12Uηη+Vξη+λ22WξWηη+λ22WηWξη+βλ1G¯1Vξξξη−λ22λ12Uξξηη+λ22λ12Vξηηη−λ24λ14Uηηηη=0A¯12Uξη+λ12WξWξη+A¯22Vηη+λ22WηWηη+βA¯66Uξη+λ12λ22Vξξ+λ12WηWξξ+λ12WξWξη−βλ2G¯2λ14λ24Vξξξξ−λ12λ22Uξξξη+λ12λ22Vξξηη−Uξηηη=0112D¯11+βG¯3λ12λ22Wξξξξ+16D¯12+13D¯66+2βG¯3Wξξηη+112D¯22+βG¯3λ22λ12Wηηηη+Wττ+C¯Wτ=A¯11Uξ+12λ12Wξ2+A¯12Vη+12λ22Wη21λ22Wξξ+A¯11Uξξ+λ12WξWξξ+A¯12Vξη+λ22WηWξη1λ22Wξ+2βA¯661λ12Uη+1λ22Vξ+WηWξWξη+βA¯661λ22Vξξ+1λ12Uξη+WηWξξ+WξWξηWη+A¯12Uξ+12λ12Wξ2+A¯22Vη+12λ22Wη21λ12Wηη+A¯12Uξη+λ12WξWξη+A¯22Vηη+λ22WηWηη1λ12Wη+βA¯661λ12Uηη+1λ22Vξη+WξηWη+WξWηηWξ+α1+α2WV¯C2.Due to the fact that a fully clamped rectangular microplate is extensively used in MEMS applications [32], this kind of boundary condition is also studied in this work and written in dimensionless formulation as follows:(27)A¯11U,ξ+12W,ξ2+βG¯38h2abV,ξξη−h2b2U,ξηη=0,ξ=0,1,V=0,W=0,V,ξ=0,W,ξ=0,A¯22V,η+12λ1W,η2−βG¯38h2a2V,ξξη−h2abU,ξηη=0,η=0,1,U=0,W=0,U,η=0,W,η=0.


## 3. Solution Methodology

First, note that, in the boundary conditions in (27), the items related to size effect are neglected due to the fact that they are negligible compared with other items [29]. With satisfaction of boundary conditions, a solution for (26) takes the following form:(28)Uξ,η,τ=U¯τcosπξsin2πη,Vξ,η,τ=V¯τsin2πξcosπη,Wξ,η,τ=W¯τsin2πξsin2πη.Submitting (28) into (26)fd25 and multiplying them by cosπξsin2πη,sin2πξcosπη, and sin2πξsin2πη, respectively, and integrating them from 0 to 1 with respect to ξ and η, the nonlinear governing differential equations with respect to time can be obtained as(29)-3π216A-11-βA-66π2λ224λ12-G-1βλ1π4λ224λ12+π4λ24λ14U-+-169A-12-169βA-66+G-1βλ164π29+16π2λ229λ12V-+π26λ12A-11+2π215λ22A-12-2π215βA-66λ22W-2=0,-169A-12-169βA-66+G-2βλ216π2λ129λ22+64π29U-+-3π216A-22-π2λ124λ22βA-66-G-2βλ2π4λ14λ24+π4λ124λ22V-+2π215λ12A-12+π26λ22A-22-2π215βA-66λ12W-2=0,3112D-11+βG-3λ12λ22+16D-12+13βD-66+2βG-3+3112D-22+βG-3λ12λ22π44W-+14W-ττ+C-4W-τ=π2A-113λ22-4π2βA-6615λ12+4π2A-1215λ12U-W-+4π2A-1215λ22-4π2βA-6615λ22+π2A-223λ12V-W--52.5π4λ121024λ22A-11+25π41024A-12+52.5π4λ221024λ12A-22+50π41024βA-66W-3+14α1+916α2W-V-C2.For convenience, (29) can be expressed as follows:(30)M11U-+M12V-+M13W-2=0,M21U-+M22V-+M23W-2=0,M31W-+14W-ττ+C-4W-τ=M32U-W-+M33V-W--M34W-3+14α1+916α2W-V-C2,where(31)M11=−3π216A¯11−βA¯66π2λ224λ12−G¯1βλ1π4λ224λ12+π4λ24λ14,M12=−169A¯12−169βA¯66+G¯1βλ164π29+16π2λ229λ12,M13=π26λ12A¯11+2π215λ22A¯12−2π215βA¯66λ22,M21=−169A¯12−169βA¯66+G¯2βλ216π2λ129λ22+64π29,M22=−3π216A¯22−π2λ124λ22βA¯66−G¯2βλ2π4λ14λ24+π4λ124λ22,M23=2π215λ12A¯12+π26λ22A¯22−2π215βA¯66,M31=3112D¯11+βG¯3λ12λ22+16D¯12+13D¯66+2βG¯3+3112D¯22+βG¯3λ12λ22π44,M32=π2A¯113λ22−4π2βA¯6615λ12+4π2A¯1215λ12,M33=4π2A¯1215λ22−4π2βA¯6615λ22+π2A¯223λ12,M34=−52.5π4λ121024λ22A¯11−25π41024A¯12−52.5π4λ221024λ12A¯22−50π41024βA¯66.After eliminatingU¯τ,V¯τ, the nonlinear differential equation only related to W¯τ can be obtained:(32)14W¯ττ+C¯4W¯τ+M31W¯−φW¯3=14α1+916α2W¯V¯C2,where(33)φ=M34+M32M22M13−M32M12M23M12M21−M11M22+M33M11M23−M33M13M21M12M21−M11M22.


## 4. Results and Discussion

In this section, the nonlinear size-dependent vibration analysis of the microplate is studied under the influence of size effect and damage effect. Free vibration and forced vibration analysis of the microplate under DC voltage and AC voltage are discussed in detail accordingly. The frequency-response curves are presented for various system parameters such as material length scale parameter, damage variable, damping ratio, and external AC voltage. Note that geometric and physical parameters of the microplate are listed in Table1.Table 1
Geometric and material properties of the microplate.
PropertyValueLength,a300μmWidth,b150μmThickness,h2μmInitial gap,d2μmPoisson’s ratio,υ0.06Young’s modulus,E166×109Paβ0.2Dielectric constant,εv8.854×10−12F/m


### 4.1. Free Vibration Analysis under DC Voltage

For static analysis, (32) can be written as(34)W¯ττ+C¯W¯τ4+M31W¯−φW¯3=α1+9/16α2W¯V¯d24,where VD donates the DC voltage.The harmonic balance method is feasible for finding solutions of both weakly and strongly nonlinear problems. A general solution of (34) can be given as(35)Wτ=A0+A1cosωτ+θ.Submitting (35) into (34) and using harmonic balance method (HB) [33], we obtain(36)ω02A0−4φA03+32A0A12=α1V¯d2−A1ω2+ω02A1−4φ3A02A1+34A13=0,where A0 is so smaller than A1 that it can be neglected in the further step of solution, and then the following is obtained:(37)ω02=4M31−916α2V¯d2,ω2=ω02−3φA12.The variation of amplitude-frequency response of the microplate with various values of damage variable is presented in Figure2, where DC voltage V¯d and nondimensional material length scale parameter l/h are taken as 20 and 0.2, respectively. It should be pointed out that the current value of material length scale parameter is only a mean to illustrate microscale effect. Sophisticated experiments need to be conducted to measure material length scale parameter as it plays a significant role in microscale structures. As shown in Figure 2, the vibration amplitude becomes larger with increase of nonlinear frequency which indicates that the microplate exhibits a hardening-type behavior. It is also observed that the larger the damage variable is, the lower the nonlinear free vibration frequency becomes. Moreover, free vibration amplitude of the microplate midpoint increases as damage variable increases due to reduced stiffness of the microplate.Figure 2
Damage effect on frequency-response curves of the microplate.Influence of size effect on frequency-response curves is given in Figure3 in which damage variable D and DC voltage V¯d are taken as 0.2 and 20, separately. It is observed that large material length scale parameter results in small amplitude and high frequency of the microplate. When especially size effect is omitted, the classical microplate model is recovered. As can be observed from Figure 3, material length scale parameter increases the difference between the results of present microplate model and the classical microplate model increases, indicating that the microscale effect induce additional rigidity for microstructure. The extra stiffness contributed by couple stress together with the classical stiffness increased the total stiffness of the microplate. Furthermore, our results are consistent with the predictions obtained by Ansari et al. [34] who studied the nonlinear vibrations properties of functionally graded Mindlin’s microplates using MCST.Figure 3
Size effect on frequency-response curves of the microplate.


### 4.2. Forced Vibration Analysis under AC Voltage

As for dynamic analysis, (32) can be written as(38)14W¯ττ+14C¯W¯τ+M31W¯−φW¯3=14α1+916α2W¯V¯a2cos2ωτ.Similarly, the harmonic balance method is employed for solving (38) and the general solution for this equation can be given as(39)Wτ=A1cos2ωτ+θ0+A2sin2ωτ+θ0=A1cos2Φ+A2sin2Φ,where(40)cos2ωτ=A3cos2Φ+A4sin2Φ,A3=cosθ0,A4=sinθ0.Substituting (39) into (38), we obtain(41)4M31−4ω2−932α2VA2A1+2ωC¯A2−3φA1A12+A22=α12V¯a2A34M31−4ω2−932α2VA2A2−2ωC¯A1−3φA2A12+A22=α12V¯a2A4.UsingA12+A22=A2 and A32+A42=1, the relationship between the amplitude of the microplate midpoint and the excitation frequency can be achieved as(42)9φ2A6−6φ4M31−4ω2−932α2V¯a2A4+4M31−4ω2−932α2V¯a22+4ω2C¯2A2=α124V¯a4.The frequency-response curves of the system with damage variableD, nondimensional material length scale parameter l/h, and damping coefficient C¯ are, respectively, taken as 0.2, 0.2, and 0.2 and are depicted in Figure 4 which reveals that the system still exhibits a hardening-type behavior. It is observed that the amplitude of microplate midpoint increases as excitation frequency increases from 1.1, actually from 0, until reaching a limit point (upper jump point), corresponding to the nonlinear resonance, and then the amplitude of microplate midpoint falls suddenly; for another, the amplitude of the microplate midpoint increases as the excitation frequency is decreased from 3.5, until reaching another limit point (lower jump point), and then the amplitude of the microplate midpoint rises abruptly. The region between the lower jump point and the upper jump point is regarded as the unstable region. This unique behavior is called as jump phenomenon which results from the nonlinearity of the dynamic system. Additionally, with the increase of the amplitude of AC voltage, the upper jump point tends to move towards the right side and the amplitude of the microplate midpoint increases. In other words, the nonlinear response region is enlarged by the greater external load. The jump phenomenon disappears when the amplitude of AC voltage is small, in this case, such as 10 (black line). Without enough energy input by external excitation, the dynamic system will remain stable.Figure 4
The frequency-response curves of the system for different amplitude of the AC voltageVAC.Figure5 indicates the effect of damping coefficient on the frequency-response curve of the microplate. In the numerical simulation, the nondimensional material length scale parameter l/h, damage variable D, and amplitude of AC voltage V¯a are taken to be 0.2, 0.2, and 20 accordingly. From this figure, we can see that, by increasing the damping coefficient, the unstable region decreases, and the peak of the curves goes down. It is attributed to the fact that the energy input by external load is consumed by the dissipation system. The larger the damping coefficient is, the more energy the dynamic system expends.Figure 5
The frequency-response curves of the system for different damping coefficientsC¯.Figure6 displays damage effect on frequency-response curves of the microplate with damping coefficient C¯, nondimensional material length scale parameter l/h, and amplitude of AC voltage V¯a are set to 0.2, 0.2, and 20, respectively. It is observed that the larger the damage variable, the bigger the peak of the curve and the wider the unstable region and nonlinear response region. What is more, unlike the first two cases, the unstable region moves towards the left side when the microplate suffers more serious damage. The system does not possess enough stiffness to resist external load. The reason for this is that the stiffness of the structure is reduced by damage effect.Figure 6
The frequency-response curves of the system for different damage variableD.Figure7 highlights the size effect on the frequency-response curves of dynamic system. The numerical calculations are performed by assuming D=0.2,C¯=0.2,V¯a=20. It is revealed that the peak of the curve goes down slightly and the hardening influence reduces mildly when nondimensional material length scale parameter increases. In addition, the unstable region moves to the right side and diminishes as size effect becomes more obvious. Furthermore, the distinction between the results predicted by classical theory l/h=0 and nonclassical theory gets more obvious. The reason for this change was already discussed in Section 4.1. Besides, the present result is parallel to that of Ansari et al. [35], who investigated the forced vibration of functionally graded microplate based on MCST. It should be noted that horizontal axis in this work represents nondimensional external frequency, while in Ansari et al.’s work it represents the frequency ratio (the ratio of external frequency to linear first frequency).Figure 7
The frequency-response curves of the system for different nondimensional material length scale parametersl/h.


## 4.1. Free Vibration Analysis under DC Voltage

For static analysis, (32) can be written as(34)W¯ττ+C¯W¯τ4+M31W¯−φW¯3=α1+9/16α2W¯V¯d24,where VD donates the DC voltage.The harmonic balance method is feasible for finding solutions of both weakly and strongly nonlinear problems. A general solution of (34) can be given as(35)Wτ=A0+A1cosωτ+θ.Submitting (35) into (34) and using harmonic balance method (HB) [33], we obtain(36)ω02A0−4φA03+32A0A12=α1V¯d2−A1ω2+ω02A1−4φ3A02A1+34A13=0,where A0 is so smaller than A1 that it can be neglected in the further step of solution, and then the following is obtained:(37)ω02=4M31−916α2V¯d2,ω2=ω02−3φA12.The variation of amplitude-frequency response of the microplate with various values of damage variable is presented in Figure2, where DC voltage V¯d and nondimensional material length scale parameter l/h are taken as 20 and 0.2, respectively. It should be pointed out that the current value of material length scale parameter is only a mean to illustrate microscale effect. Sophisticated experiments need to be conducted to measure material length scale parameter as it plays a significant role in microscale structures. As shown in Figure 2, the vibration amplitude becomes larger with increase of nonlinear frequency which indicates that the microplate exhibits a hardening-type behavior. It is also observed that the larger the damage variable is, the lower the nonlinear free vibration frequency becomes. Moreover, free vibration amplitude of the microplate midpoint increases as damage variable increases due to reduced stiffness of the microplate.Figure 2
Damage effect on frequency-response curves of the microplate.Influence of size effect on frequency-response curves is given in Figure3 in which damage variable D and DC voltage V¯d are taken as 0.2 and 20, separately. It is observed that large material length scale parameter results in small amplitude and high frequency of the microplate. When especially size effect is omitted, the classical microplate model is recovered. As can be observed from Figure 3, material length scale parameter increases the difference between the results of present microplate model and the classical microplate model increases, indicating that the microscale effect induce additional rigidity for microstructure. The extra stiffness contributed by couple stress together with the classical stiffness increased the total stiffness of the microplate. Furthermore, our results are consistent with the predictions obtained by Ansari et al. [34] who studied the nonlinear vibrations properties of functionally graded Mindlin’s microplates using MCST.Figure 3
Size effect on frequency-response curves of the microplate.


## 4.2. Forced Vibration Analysis under AC Voltage

As for dynamic analysis, (32) can be written as(38)14W¯ττ+14C¯W¯τ+M31W¯−φW¯3=14α1+916α2W¯V¯a2cos2ωτ.Similarly, the harmonic balance method is employed for solving (38) and the general solution for this equation can be given as(39)Wτ=A1cos2ωτ+θ0+A2sin2ωτ+θ0=A1cos2Φ+A2sin2Φ,where(40)cos2ωτ=A3cos2Φ+A4sin2Φ,A3=cosθ0,A4=sinθ0.Substituting (39) into (38), we obtain(41)4M31−4ω2−932α2VA2A1+2ωC¯A2−3φA1A12+A22=α12V¯a2A34M31−4ω2−932α2VA2A2−2ωC¯A1−3φA2A12+A22=α12V¯a2A4.UsingA12+A22=A2 and A32+A42=1, the relationship between the amplitude of the microplate midpoint and the excitation frequency can be achieved as(42)9φ2A6−6φ4M31−4ω2−932α2V¯a2A4+4M31−4ω2−932α2V¯a22+4ω2C¯2A2=α124V¯a4.The frequency-response curves of the system with damage variableD, nondimensional material length scale parameter l/h, and damping coefficient C¯ are, respectively, taken as 0.2, 0.2, and 0.2 and are depicted in Figure 4 which reveals that the system still exhibits a hardening-type behavior. It is observed that the amplitude of microplate midpoint increases as excitation frequency increases from 1.1, actually from 0, until reaching a limit point (upper jump point), corresponding to the nonlinear resonance, and then the amplitude of microplate midpoint falls suddenly; for another, the amplitude of the microplate midpoint increases as the excitation frequency is decreased from 3.5, until reaching another limit point (lower jump point), and then the amplitude of the microplate midpoint rises abruptly. The region between the lower jump point and the upper jump point is regarded as the unstable region. This unique behavior is called as jump phenomenon which results from the nonlinearity of the dynamic system. Additionally, with the increase of the amplitude of AC voltage, the upper jump point tends to move towards the right side and the amplitude of the microplate midpoint increases. In other words, the nonlinear response region is enlarged by the greater external load. The jump phenomenon disappears when the amplitude of AC voltage is small, in this case, such as 10 (black line). Without enough energy input by external excitation, the dynamic system will remain stable.Figure 4
The frequency-response curves of the system for different amplitude of the AC voltageVAC.Figure5 indicates the effect of damping coefficient on the frequency-response curve of the microplate. In the numerical simulation, the nondimensional material length scale parameter l/h, damage variable D, and amplitude of AC voltage V¯a are taken to be 0.2, 0.2, and 20 accordingly. From this figure, we can see that, by increasing the damping coefficient, the unstable region decreases, and the peak of the curves goes down. It is attributed to the fact that the energy input by external load is consumed by the dissipation system. The larger the damping coefficient is, the more energy the dynamic system expends.Figure 5
The frequency-response curves of the system for different damping coefficientsC¯.Figure6 displays damage effect on frequency-response curves of the microplate with damping coefficient C¯, nondimensional material length scale parameter l/h, and amplitude of AC voltage V¯a are set to 0.2, 0.2, and 20, respectively. It is observed that the larger the damage variable, the bigger the peak of the curve and the wider the unstable region and nonlinear response region. What is more, unlike the first two cases, the unstable region moves towards the left side when the microplate suffers more serious damage. The system does not possess enough stiffness to resist external load. The reason for this is that the stiffness of the structure is reduced by damage effect.Figure 6
The frequency-response curves of the system for different damage variableD.Figure7 highlights the size effect on the frequency-response curves of dynamic system. The numerical calculations are performed by assuming D=0.2,C¯=0.2,V¯a=20. It is revealed that the peak of the curve goes down slightly and the hardening influence reduces mildly when nondimensional material length scale parameter increases. In addition, the unstable region moves to the right side and diminishes as size effect becomes more obvious. Furthermore, the distinction between the results predicted by classical theory l/h=0 and nonclassical theory gets more obvious. The reason for this change was already discussed in Section 4.1. Besides, the present result is parallel to that of Ansari et al. [35], who investigated the forced vibration of functionally graded microplate based on MCST. It should be noted that horizontal axis in this work represents nondimensional external frequency, while in Ansari et al.’s work it represents the frequency ratio (the ratio of external frequency to linear first frequency).Figure 7
The frequency-response curves of the system for different nondimensional material length scale parametersl/h.


## 5. Conclusions

The nonlinear size-dependent vibration of a microplate with damage was explored by employing MCST and the strain equivalent assumption in this research. The nonlinear governing partial differential equations were transformed into nonlinear ordinary differential equations via Galerkin’s scheme and further solved numerically by the harmonic balance method. Numerical results indicate that damage effect and size dependency both have obvious influences on the static and dynamic behaviors of the microplate system. It was concluded that, on one hand, the hardening-type nonlinear behavior of the microplate system enhances when it encounters damage; on the other hand, the system exhibits a weaker nonlinear behavior, greater nondimensional frequency, and lower amplitude of the microplate midpoint as size effect gets obvious.



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*Source: 8897987-2020-08-26.xml*