# Stability of Axially Moving Piezolaminated Viscoelastic Plate Subjected to Follower Force

Authors: Yan Wang; Tao Jing; Jimei Wu; Min Xie
Journal: Advances in Materials Science and Engineering
 (2015)

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## Abstract

The stability of the moving viscoelastic plate with the piezoelectric layer subjected to uniformly distributed tangential follower force is investigated. The force excited by the piezoelectric layer due to external voltage is modeled as the follower tensile force. The differential equation of the axially moving viscoelastic rectangular plate with piezoelectric layer subjected to uniformly distributed tangential follower force is formulated on the basis of the Kirchhoff thin plate theory and the two-dimensional viscoelastic differential constitutive relation. The complex eigenvalue equations are established by the differential quadrature method. Via numerical calculation, the curves of real parts and imaginary parts of dimensionless complex frequencies versus uniformly distributed tangential follower force and dimensionless moving speed are obtained. The effects of nonconservative force, dimensionless axially moving speed, and dimensionless applied voltages on the stability of axially moving nonconservative viscoelastic plate with piezoelectric layer are analyzed.


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## Body

## 1. Introduction

Piezoelectric materials are used in the smart structure as sensors and/or actuators in the form of patches or layers. A plate with distributed, surface-bonded piezoelectric sheets has been widely used in active vibration, in structural health monitoring, in acoustic control, in buckling control, and among other applications. The behavior of laminated piezoelectric plates has gained much attention by researchers [1–3]. For example, numerous scholars have studied the piezoelectric effects on the dynamic behavior of composite structures. The influence of using smart materials on the free vibration and natural frequencies of laminated piezoelectric plates has also been addressed [4–8]. Structural systems lose their stability due to divergence, and such systems under partial follower compressive loading were investigated both qualitatively and quantitatively [9]. Chandrashekhara and Bhatia [10] investigated the active buckling control of smart composite plates based on finite element analysis. In [11], both cross force and voltage, together with the bending of the plate, were analyzed, and the bending of the plate was controlled through the application of different voltages to the actuators. In accordance with the inverse piezoelectric effect, the equivalent action of the actuators was obtained. Using Hamilton’s principle, the finite element formula of the bending deformation of the piezoelectric structure was derived. The bending of the annular plate made of piezoelectric material with four boundary conditions was also calculated. Chase and Bhashayam [12] used the piezoelectric sheet to increase the critical load of laminated beams and performed optimal stabilization on plate buckling. Wang [13] and Wang and Quek [14] studied the buckling of column structures with a pair of piezoelectric layers and improve the flutter and buckling capacity of the column using the piezoelectric layers. Ha et al. [15] examined a composite structure containing distributed piezoceramic sensors and actuators using finite element analysis.Stability and transverse vibrations of axially moving systems are present in various industrial applications, such as the paper webs and plastic sheets, steel strip in a thin steel sheet production line, band saw blade, conveyor belts and chain in power transmission lines, and aerial cableways. The axially moving considerations have been studied widely, and dynamics and stability have been reviewed by Wickert and Mote [16]. The research on two-dimensional axially moving plate can be dated back to Ulsoy and Mote Jr. [17], who analyzed the coupled transverse and torsional vibration of band saw blade. The stability and vibration characteristics of an axially moving plate with two simple supported edges and two free edges under homogeneous tension have been investigated by Lin [18, 19]. A few researches on transverse vibrations and stability of axially moving viscoelastic plates have also been done.  Zhou et al. [20, 21] studied transverse vibration characteristics of axially moving viscoelastic rectangular plates and parabolically varying thickness plate. Marynowski [22]  compared a three-parameter Zener model and a two-parameter Kelvin-Voigt model for the viscoelasticity. Saksa et al. [23] investigated stability and dynamic behaviour of axially moving viscoelastic panels with the help of the classical modal analysis. Tang and Chen [24] studied stability in parametric resonance of moving viscoelastic plates with time-dependent travelling speed.It is well known that instability may be induced in structures systems because of nonconservative forces. Indeed, much of the research in this field has focused on the stability of the structures subjected to different types of nonconservative forces. However, few papers have focused on the stability of axially moving viscoelastic plates with piezoelectric layers under nonconservative forces.Here, a relatively novel solution technique, the differential quadrature method (DQM), is used to analyze dynamic stability behavior of axially moving viscoelastic plates with piezoelectric layers subjected to nonconservative force. The current work focuses on the stability improvement of the axially moving nonconservative viscoelastic plate that is surface-bonded by piezoelectric layers. An analytical model that fully embodies the piezoelectric effects is first obtained based on the behavior of the follower force imposed by the external voltage on the piezoelectric layer. This paper aims to present the differential equation of the axially moving viscoelastic plate with piezoelectric layers subjected to uniformly distributed tangential follower force. Based on the 2D viscoelastic differential constitutive relation and Kirchhoff thin plate theory, the differential equation of the axially moving viscoelastic rectangular plate with piezoelectric layer subjected to uniformly distributed tangential follower force in the Laplace domain is formulated. And carrying out the Laplace inverse transformation, the differential equation of motion of the axially moving viscoelastic rectangular plate with piezoelectric layer subjected to uniformly distributed tangential follower force in time domain is yielded. The complex eigenvalue equations are established by the differential quadrature method. Via numerical calculation, the complex frequency and the instability type of the axially moving nonconservative viscoelastic plate constituted by elastic behavior in dilatation and the Kelvin-Voigt laws in distortion are obtained. The effects of nonconservative force, dimensionless axially moving speed, and dimensionless applied electric potential difference on the stability of axially moving nonconservative viscoelastic plate with piezoelectric layer are analyzed.


## 2. Formulation

### 2.1. Differential Equation of Motion

The axially moving viscoelastic thin plate with surface-bonded piezoelectric layers subjected to uniformly distributed tangential follower forceq0 is shown in Figure 1. The piezoelectric layers are symmetrical to the midplane of the structure. The thin plate has a thickness hb in the z-direction, a material density ρb, and Poisson’s ratio μb. The piezoelectric layers have a thickness hp, a layer density ρp, and Poisson’s ratio μp.Figure 1
Configuration of axially moving viscoelastic plate with piezoelectric layers subjected to uniformly distributed tangential follower force.Using Kirchhoff’s hypothesis of classical thin plates, the strain and displacement relations for the viscoelastic plate can be written as(1)εxb=-z∂2w∗∂x2,εyb=-z∂2w∗∂y2,γxyb=-2z∂2w∗∂x∂y,where w∗ is the transverse displacement of the plate or deflection and z is the distance of the arbitrary point of the plate from the neutral plane.The strain of the piezoelectric bonding layers pertains to the strains of both surfaces of the viscoelastic plate, expressed as(2)εxp=∓hb2∂2w∗∂x2,εyp=∓hb2∂2w∗∂y2,γxyp=∓hb∂2w∗∂x∂y.Considering mechanical and electrical behavior of piezoelectric materials. Proper care is taken when constitutive equations for piezoelectric materials are used to model induced strain actuation, as they are poling direction dependent. The planar isotropy of poled ceramics is expressed by their piezoelectric strain constants, such thatd31=d32 [26, 27]. The constitutive equations of the piezoelectric layers for plane stress are(3)εxpεypγxyp=1Ep1-μp0-μp100021+μpσxpσypτxyp+d31hpVd31hpV0,where V denotes applied electric potential difference.Substituting (2) into (3) yields(4)σxpσypτxyp=Ep1-μp2×∓hb2∂2w∗∂x2-d31hpV+μp∓hb2∂2w∗∂y2-d31hpV∓hb2∂2w∗∂y2-d31hpV+μp∓hb2∂2w∗∂x2-d31hpV∓hb21-μp∂2w∗∂x∂y.The constitutive relations of linear viscoelastic materials in the Laplace domain are(5)P-0σ-xb=Q-0ε-xb+Q-1ε-yb,P-0σ-xb=Q-1ε-xb+Q-0ε-ybQ-′ε-xyb=P-′τ-xyb.In the Laplace domain, the constitutive relations of linear viscoelastic materials using Laplace transformationw- of deflection w∗ are expressed as (6)P-0σ-xb=Q-0-z∂2w-∂x2+Q-1-z∂2w-∂y2,P-0σ-yb=Q-1-z∂2w-∂x2+Q-0-z∂2w-∂y2,P-′τ-xyb=Q-′-z∂2w-∂x∂y,where the differential operators P′=∑k=0lpk′dk/dtk, Q′=∑k=0rqk′dk/dtk, P′′=∑k=0l1pk′′dk/dtk, and Q′′=∑k=0r1qk′′dk/dtk; pk′, qk′, pk′′, and qk′′ depend on the properties of the material; P-′, Q-′, P-′′, and Q-′′ are the Laplace transform of the differential operators P′, Q′, P′′, and Q′′. For convenience, let P-0=P-′P-′Q-′′+2Q-′P-′′, Q-0=Q-′(2P-′Q-′′+Q-′P-′′), Q-1=Q-′(P-′Q-′′-Q-′P-′′), and the polynomials P-0, Q-0, and Q-1 about Laplace variable s are independent of spatial coordinates.Assume that the piezoelectric sheets are of infinitesimal thickness and disregard its influence on the bending stiffness of the structure. The following termsσxp and σyp result in uniform stress in the piezoelectric layer. The force excited by the piezoelectric layer due to external electric potential difference is modeled as a follower tensile force, and the follower tensile force is a kind of nonconservative force. The tension force caused by the piezoelectric layer can be written as (7)Tx=∫hb/2hb/2+hpσxpdz=-Ep1-μpd31V,Ty=∫hb/2hb/2+hpσypdz=-Ep1-μpd31V.In the Laplace domain, the relations between internal torque and Laplace transformationw~ of deflection w∗ are given by(8)P-0M-xb=-hb312Q-0∂2w-∂x2+Q-1∂2w-∂y2,P-0M-yb=-hb312Q-1∂2w-∂x2+Q-0∂2w-∂y2,P-′M-xyb=P-′M-yxb=-hb312Q-′∂2w-∂x∂y.The transverse speed of moving plate is(9)v-t=dw∗dt=∂w∗∂t+v∂w∗∂x.The differential operator is given byd/dt=∂/∂t+v∂/∂x.The equilibrium equation of axially moving viscoelastic plate with piezoelectric layer subjected to uniformly distributed tangential follower force is given by(10)∂2Mx∂x2+2∂2Mxy∂x∂y+∂2My∂y2-Nx∂2w∗∂x2-Ny∂2w∗∂y2-2Nxy∂2w∗∂x∂y-ρbhb+2ρphp×∂2w∗∂t2+2v∂2w∗∂x∂t+v2∂2w∗∂x2=0,where(11)Nx=2Tx+q0a-x,Ny=2Ty,Nxy=0.Carrying out the Laplace transformation of (10) and (11) and multiplying it by P-0P-′. When the partial derivative is continuous, in the Laplace domain the differential equation of motion of the axially moving viscoelastic rectangular plate with piezoelectric layers subjected to uniformly distributed tangential follower force is obtained:(12)hb312Q-′2P-′Q-′′+Q-′P-′′∇4w-∗-P-′P-′Q-′′+2Q-′P-′′×2Epd31V1-μp∂2w-∗∂x2+∂2w-∗∂y2+P-′P-′Q-′′+2Q-′P-′′q0a-x∂2w-∗∂x2+P-′P-′Q-′′+2Q-′P-′′ρbhb+2ρphp×s12w-∗+2vs1∂w-∂x+v2∂2w-∗∂x2=0,where ∇4w-∗=∂4w-∗/∂x4+2∂4w-∗/∂x2∂y2+∂4w-∗/∂y4.Equation (11) is suitable for various viscoelastic differential models.Here, we assume that the material of the plate obeys elastic behavior in dilatation and the Kelvin-Voigt laws in distortion. Substituting the polynomialsP-′=1, Q-′=2G+2ηs, P-′′=1, and Q-′′=3K into (11) and carrying out the Laplace inverse transformation yield the differential equation of motion of axially moving nonconservative viscoelastic plate with elastic dilatation and Kelvin-Voigt distortion with piezoelectric layers in the time domain(13)hb312A3+A4∂∂t+A5∂2∂t2∇4w∗-A1+A2∂∂t2Epd31V1-μp∂2w∗∂x2+∂2w∗∂y2+A1+A2∂∂tq0a-x∂2w∗∂x2+A1+A2∂∂tρbhb+2ρphp×∂2w∗∂t2+2v∂2w∗∂t∂x+v2∂2w∗∂x2=0,where A1=3K+4G, A2=4η, A3=2G(6K+2G), A4=8Gη+12Kη, A5=4η2(14)K=Eb31-2μb,G=Eb21+μb,∇4w∗=∂4w∗∂x4+2∂4w∗∂x2∂y2+∂4w∗∂y4.


### 2.2. Differential Equation of Vibration Mode

Introducing dimensionless quantities and parameters,(15)ξ=xa,ψ=yb,w=w∗a,r=ab,τ=thba2Eb12ρb1-μb2,C=ahb12ρb1-μb2Ebv,q=12q0a31-μb2Ebhb3,λ=hphb,p=ρpρbλ,H=hba2Eb12ρb1-μb2ηEb,U=24Epd31a2V1-μb2Ebhb3.Substituting (15) into (13) yields the following dimensionless differential equation:(16)1+431+μb2-μbH∂∂τ+431-2μb1+μb2H2∂2∂τ2∇4w-11-μp+431-2μb1+μb1-μp1-μbH∂∂τ×U∂2w∂ξ2+U·r2∂2w∂ψ2+1+431-2μb1+μb1-μbH∂∂τq1-ξ∂2w∂ξ2+1+431-2μb1+μb1-μbH∂∂τ×1+2p∂2w∂τ2+2C∂2w∂ξ∂τ+C2∂2w∂ξ2∂2w∂τ2=0,where τ is the dimensionless time, H denotes the dimensionless time delay, C is the dimensionless moving speed, and U represents the dimensionless electric potential difference parameter. One has ∇4w=∂4w/∂ξ4+2r2∂4w/∂ξ2∂ψ2+r4∂2w/∂ψ4.Suppose that the solution to (16) takes the form(17)wξ,ψ,τ=Wξ,ψeiωτ,where i=-1 and ω is a dimensionless complex frequency. Substituting (17) into (16) gives (18)D1∇4W-D3U∂2W∂ξ2+Ur2∂2W∂ψ2+D2q1-ξ∂2W∂ξ2+D21+2p×-ω2W+2Ciω∂W∂ξ+C2∂2W∂ξ2=0,where(19)D1=1+i431+μb2-μbHω-431-2μb1+μb2H2ω2,D2=1+i431-2μb1+μb1-μbHω,D3=11-μp+i431-2μb1+μb1-μp1-μbHω.Boundary conditions for four-edge simply supported (SSSS) rectangular plates are(20)ξ=0,1:Wξ,ψ=∂2W∂ξ2=0,ψ=0,1:Wξ,ψ=∂2W∂ψ2=0.Boundary conditions for two opposite edges simply supported and other edges clamped (CSCS) are as follows:(21)ξ=0,1:Wξ,ψ=∂W∂ξ=0,ψ=0,1:Wξ,ψ=∂2W∂ψ2=0.


### 2.3. Complex Eigenvalue Equation

The complex eigenvalue equation of the axially moving viscoelastic rectangular plate with piezoelectric layers subjected to uniformly distributed tangential follower force is derived by the DQM [28–31]. The DQM is used to approximate the partial derivatives of a function with respect to a spatial variable at any discrete point as the weighted linear sum of the function values at all the discrete points chosen in the solution domain of spatial variable.Postulating smooth functionf(x,y) in the region 0≤x≤a, 0≤y≤b, the rth order partial derivative of f(x,y) with respect to x, the sth order partial derivative of f(x,y) with respect to y, and the mixed partial derivative of the sth order with respect to y and the rth order with respect to x are, respectively, approximated as(22)∂rf(xi,yj)∂xr=∑k=1NAikrfxk,yj,i=1,2,…,N;r=1,2,…,N-1,∂sf(xi,yj)∂xs=∑m=1MAjmsfxi,ym,j=1,2,…,M;s=1,2,…,N-1,∂r+sf(xi,yj)∂xr∂ys=∑k=1NAikr∑m=1MAjmsfxk,ym,where N  and M are the number of grid points in the x and y direction, respectively, Aik(r) and Ajm(s) are the weighted coefficients, and they are defined by(23)Aik(1)=∏u=1,u≠i,kNxi-xu∏u=1,u≠kNxk-xui,k=1,2,…,N;k≠i,∑u=1u≠kN1xi-xui,k=1,2,…,N;k=i,Ajm(1)=∏u=1,u≠j,mMyj-yu∏u=1,u≠mMym-yuj,m=1,2,…,M;m≠j,∑u=1u≠mM1yj-yuj,m=1,2,…,M;m=j.In the case ofr=2,3,…,N-1, s=2,3,…,M-1, (24)Aik(r)=rAiir-1Aik1-Aikr-1xi-xki,k=1,2,…,N;k≠i,-∑u=1u≠iNAiu(r)i=1,2,…,N;1≤r≤N-1;k=i,Ajm(s)=sAjjs-1Ajm1-Ajms-1yj-ymj,m=1,2,…,M;m≠j,-∑u=1u≠jMAju(s)j=1,2,…,M;1≤r≤M-1;m=j.In this paper,N=M. The distribution forms of the grid points are nonuniform; the distribution forms of the grid points are(25)ξ1=0,ξN=1,ξi=121-cos⁡2i-32N-4π,i=2,3,…,N-1η1=0,ηN=1,ηi=121-cos⁡2i-32N-4π,i=2,3,…,N-1.According to the DQM procedures, (18) can be discretized into the following forms:(26)∑k=1NAik4Wkj+2r2∑m=1MBjm2∑k=1NAik2Wkm+r4∑m=1MBjm4Wim-11-μpU∑k=1NAik2Wkj+U·r2∑m=1MBjm2Wim+q1-ξi∑k=1NAik2Wkj+1+2pC2∑k=1NAik2Wkj+∑k=1NAik2Wkji431+μb2-μbH×∑k=1NAik4Wkj+2r2∑m=1NBjm2∑k=1MAik2Wkm+r4∑k=1MBjm4Wik-i431-2μb1+μb1-μp1-μbH×U∑k=1NAik2Wkj+Ur2∑m=1MBjm2Wim+i431-2μb1+μb1-μbHq1-ξi∑k=1NAik2Wkj+i1+2p2C∑k=1NAik1Wkj+i1+2p×431-2μb1+μb1-μbHC2∑k=1NAik2Wkjω-∑k=1NAik4Wkj+2r2∑m=1NBjm2∑k=1NAik2Wkm431-2μb1+μb2H2×∑k=1NAik4Wkj+2r2∑m=1NBjm2∑k=1NAik2Wkm+r4∑k=1NBjm4Wik+431+2p1-2μb1+μb1-μb2CH∑k=1NAikWkj+1+2pWij∑k=1NAik4Wkj+2r2∑m=1NBjm2∑k=1NAik2Wkmω2-i1+2p431-2μb1+μb1-μbHWijω3=0.The differential quadrature forms of the boundary conditions (20) are(27)W1j=WNj=Wi1=WiN=0,i,j=1,2,…,N,∑k=1NAik(2)Wkj=0,i=1,N,j=1,2,…,N,∑k=1NBjk(2)Wik=0,j=1,N,i=1,2,…,N.The differential quadrature forms of the boundary conditions (21) are(28)W1j=WNj=Wi1=WiN=0,i,j=1,2,…,N,∑k=1NAik(1)Wkj=0,i=2,N-1,j=2,3,…,N-2,∑k=1NBjk(2)Wik=0,j=1,N,i=1,2,…,N.After eliminating the boundary degrees of freedom from (22) using the boundary conditions, the equation can be written in the matrix form as(29)ω3Q+ω2R+ωG+KWkj=0.The complex eigenvalue equation of the axially moving viscoelastic rectangular plate with piezoelectric layers subjected to uniformly distributed tangential follower force is that coefficient determinant equal to zero; that is,(30)ω3Q+ω2R+ωG+K=0.In (30), ω is a generalized complex eigenvalue, where the matrices Q, R, G, and K involve such parameters as dimensionless moving speed C, the dimensionless follower force q, the dimensionless time delay H, the dimensionless time τ, the dimensionless electric potential difference parameter U, the dimensionless electric potential difference geometrical parameter λ, and aspect ratio r of the plate.


## 2.1. Differential Equation of Motion

The axially moving viscoelastic thin plate with surface-bonded piezoelectric layers subjected to uniformly distributed tangential follower forceq0 is shown in Figure 1. The piezoelectric layers are symmetrical to the midplane of the structure. The thin plate has a thickness hb in the z-direction, a material density ρb, and Poisson’s ratio μb. The piezoelectric layers have a thickness hp, a layer density ρp, and Poisson’s ratio μp.Figure 1
Configuration of axially moving viscoelastic plate with piezoelectric layers subjected to uniformly distributed tangential follower force.Using Kirchhoff’s hypothesis of classical thin plates, the strain and displacement relations for the viscoelastic plate can be written as(1)εxb=-z∂2w∗∂x2,εyb=-z∂2w∗∂y2,γxyb=-2z∂2w∗∂x∂y,where w∗ is the transverse displacement of the plate or deflection and z is the distance of the arbitrary point of the plate from the neutral plane.The strain of the piezoelectric bonding layers pertains to the strains of both surfaces of the viscoelastic plate, expressed as(2)εxp=∓hb2∂2w∗∂x2,εyp=∓hb2∂2w∗∂y2,γxyp=∓hb∂2w∗∂x∂y.Considering mechanical and electrical behavior of piezoelectric materials. Proper care is taken when constitutive equations for piezoelectric materials are used to model induced strain actuation, as they are poling direction dependent. The planar isotropy of poled ceramics is expressed by their piezoelectric strain constants, such thatd31=d32 [26, 27]. The constitutive equations of the piezoelectric layers for plane stress are(3)εxpεypγxyp=1Ep1-μp0-μp100021+μpσxpσypτxyp+d31hpVd31hpV0,where V denotes applied electric potential difference.Substituting (2) into (3) yields(4)σxpσypτxyp=Ep1-μp2×∓hb2∂2w∗∂x2-d31hpV+μp∓hb2∂2w∗∂y2-d31hpV∓hb2∂2w∗∂y2-d31hpV+μp∓hb2∂2w∗∂x2-d31hpV∓hb21-μp∂2w∗∂x∂y.The constitutive relations of linear viscoelastic materials in the Laplace domain are(5)P-0σ-xb=Q-0ε-xb+Q-1ε-yb,P-0σ-xb=Q-1ε-xb+Q-0ε-ybQ-′ε-xyb=P-′τ-xyb.In the Laplace domain, the constitutive relations of linear viscoelastic materials using Laplace transformationw- of deflection w∗ are expressed as (6)P-0σ-xb=Q-0-z∂2w-∂x2+Q-1-z∂2w-∂y2,P-0σ-yb=Q-1-z∂2w-∂x2+Q-0-z∂2w-∂y2,P-′τ-xyb=Q-′-z∂2w-∂x∂y,where the differential operators P′=∑k=0lpk′dk/dtk, Q′=∑k=0rqk′dk/dtk, P′′=∑k=0l1pk′′dk/dtk, and Q′′=∑k=0r1qk′′dk/dtk; pk′, qk′, pk′′, and qk′′ depend on the properties of the material; P-′, Q-′, P-′′, and Q-′′ are the Laplace transform of the differential operators P′, Q′, P′′, and Q′′. For convenience, let P-0=P-′P-′Q-′′+2Q-′P-′′, Q-0=Q-′(2P-′Q-′′+Q-′P-′′), Q-1=Q-′(P-′Q-′′-Q-′P-′′), and the polynomials P-0, Q-0, and Q-1 about Laplace variable s are independent of spatial coordinates.Assume that the piezoelectric sheets are of infinitesimal thickness and disregard its influence on the bending stiffness of the structure. The following termsσxp and σyp result in uniform stress in the piezoelectric layer. The force excited by the piezoelectric layer due to external electric potential difference is modeled as a follower tensile force, and the follower tensile force is a kind of nonconservative force. The tension force caused by the piezoelectric layer can be written as (7)Tx=∫hb/2hb/2+hpσxpdz=-Ep1-μpd31V,Ty=∫hb/2hb/2+hpσypdz=-Ep1-μpd31V.In the Laplace domain, the relations between internal torque and Laplace transformationw~ of deflection w∗ are given by(8)P-0M-xb=-hb312Q-0∂2w-∂x2+Q-1∂2w-∂y2,P-0M-yb=-hb312Q-1∂2w-∂x2+Q-0∂2w-∂y2,P-′M-xyb=P-′M-yxb=-hb312Q-′∂2w-∂x∂y.The transverse speed of moving plate is(9)v-t=dw∗dt=∂w∗∂t+v∂w∗∂x.The differential operator is given byd/dt=∂/∂t+v∂/∂x.The equilibrium equation of axially moving viscoelastic plate with piezoelectric layer subjected to uniformly distributed tangential follower force is given by(10)∂2Mx∂x2+2∂2Mxy∂x∂y+∂2My∂y2-Nx∂2w∗∂x2-Ny∂2w∗∂y2-2Nxy∂2w∗∂x∂y-ρbhb+2ρphp×∂2w∗∂t2+2v∂2w∗∂x∂t+v2∂2w∗∂x2=0,where(11)Nx=2Tx+q0a-x,Ny=2Ty,Nxy=0.Carrying out the Laplace transformation of (10) and (11) and multiplying it by P-0P-′. When the partial derivative is continuous, in the Laplace domain the differential equation of motion of the axially moving viscoelastic rectangular plate with piezoelectric layers subjected to uniformly distributed tangential follower force is obtained:(12)hb312Q-′2P-′Q-′′+Q-′P-′′∇4w-∗-P-′P-′Q-′′+2Q-′P-′′×2Epd31V1-μp∂2w-∗∂x2+∂2w-∗∂y2+P-′P-′Q-′′+2Q-′P-′′q0a-x∂2w-∗∂x2+P-′P-′Q-′′+2Q-′P-′′ρbhb+2ρphp×s12w-∗+2vs1∂w-∂x+v2∂2w-∗∂x2=0,where ∇4w-∗=∂4w-∗/∂x4+2∂4w-∗/∂x2∂y2+∂4w-∗/∂y4.Equation (11) is suitable for various viscoelastic differential models.Here, we assume that the material of the plate obeys elastic behavior in dilatation and the Kelvin-Voigt laws in distortion. Substituting the polynomialsP-′=1, Q-′=2G+2ηs, P-′′=1, and Q-′′=3K into (11) and carrying out the Laplace inverse transformation yield the differential equation of motion of axially moving nonconservative viscoelastic plate with elastic dilatation and Kelvin-Voigt distortion with piezoelectric layers in the time domain(13)hb312A3+A4∂∂t+A5∂2∂t2∇4w∗-A1+A2∂∂t2Epd31V1-μp∂2w∗∂x2+∂2w∗∂y2+A1+A2∂∂tq0a-x∂2w∗∂x2+A1+A2∂∂tρbhb+2ρphp×∂2w∗∂t2+2v∂2w∗∂t∂x+v2∂2w∗∂x2=0,where A1=3K+4G, A2=4η, A3=2G(6K+2G), A4=8Gη+12Kη, A5=4η2(14)K=Eb31-2μb,G=Eb21+μb,∇4w∗=∂4w∗∂x4+2∂4w∗∂x2∂y2+∂4w∗∂y4.


## 2.2. Differential Equation of Vibration Mode

Introducing dimensionless quantities and parameters,(15)ξ=xa,ψ=yb,w=w∗a,r=ab,τ=thba2Eb12ρb1-μb2,C=ahb12ρb1-μb2Ebv,q=12q0a31-μb2Ebhb3,λ=hphb,p=ρpρbλ,H=hba2Eb12ρb1-μb2ηEb,U=24Epd31a2V1-μb2Ebhb3.Substituting (15) into (13) yields the following dimensionless differential equation:(16)1+431+μb2-μbH∂∂τ+431-2μb1+μb2H2∂2∂τ2∇4w-11-μp+431-2μb1+μb1-μp1-μbH∂∂τ×U∂2w∂ξ2+U·r2∂2w∂ψ2+1+431-2μb1+μb1-μbH∂∂τq1-ξ∂2w∂ξ2+1+431-2μb1+μb1-μbH∂∂τ×1+2p∂2w∂τ2+2C∂2w∂ξ∂τ+C2∂2w∂ξ2∂2w∂τ2=0,where τ is the dimensionless time, H denotes the dimensionless time delay, C is the dimensionless moving speed, and U represents the dimensionless electric potential difference parameter. One has ∇4w=∂4w/∂ξ4+2r2∂4w/∂ξ2∂ψ2+r4∂2w/∂ψ4.Suppose that the solution to (16) takes the form(17)wξ,ψ,τ=Wξ,ψeiωτ,where i=-1 and ω is a dimensionless complex frequency. Substituting (17) into (16) gives (18)D1∇4W-D3U∂2W∂ξ2+Ur2∂2W∂ψ2+D2q1-ξ∂2W∂ξ2+D21+2p×-ω2W+2Ciω∂W∂ξ+C2∂2W∂ξ2=0,where(19)D1=1+i431+μb2-μbHω-431-2μb1+μb2H2ω2,D2=1+i431-2μb1+μb1-μbHω,D3=11-μp+i431-2μb1+μb1-μp1-μbHω.Boundary conditions for four-edge simply supported (SSSS) rectangular plates are(20)ξ=0,1:Wξ,ψ=∂2W∂ξ2=0,ψ=0,1:Wξ,ψ=∂2W∂ψ2=0.Boundary conditions for two opposite edges simply supported and other edges clamped (CSCS) are as follows:(21)ξ=0,1:Wξ,ψ=∂W∂ξ=0,ψ=0,1:Wξ,ψ=∂2W∂ψ2=0.


## 2.3. Complex Eigenvalue Equation

The complex eigenvalue equation of the axially moving viscoelastic rectangular plate with piezoelectric layers subjected to uniformly distributed tangential follower force is derived by the DQM [28–31]. The DQM is used to approximate the partial derivatives of a function with respect to a spatial variable at any discrete point as the weighted linear sum of the function values at all the discrete points chosen in the solution domain of spatial variable.Postulating smooth functionf(x,y) in the region 0≤x≤a, 0≤y≤b, the rth order partial derivative of f(x,y) with respect to x, the sth order partial derivative of f(x,y) with respect to y, and the mixed partial derivative of the sth order with respect to y and the rth order with respect to x are, respectively, approximated as(22)∂rf(xi,yj)∂xr=∑k=1NAikrfxk,yj,i=1,2,…,N;r=1,2,…,N-1,∂sf(xi,yj)∂xs=∑m=1MAjmsfxi,ym,j=1,2,…,M;s=1,2,…,N-1,∂r+sf(xi,yj)∂xr∂ys=∑k=1NAikr∑m=1MAjmsfxk,ym,where N  and M are the number of grid points in the x and y direction, respectively, Aik(r) and Ajm(s) are the weighted coefficients, and they are defined by(23)Aik(1)=∏u=1,u≠i,kNxi-xu∏u=1,u≠kNxk-xui,k=1,2,…,N;k≠i,∑u=1u≠kN1xi-xui,k=1,2,…,N;k=i,Ajm(1)=∏u=1,u≠j,mMyj-yu∏u=1,u≠mMym-yuj,m=1,2,…,M;m≠j,∑u=1u≠mM1yj-yuj,m=1,2,…,M;m=j.In the case ofr=2,3,…,N-1, s=2,3,…,M-1, (24)Aik(r)=rAiir-1Aik1-Aikr-1xi-xki,k=1,2,…,N;k≠i,-∑u=1u≠iNAiu(r)i=1,2,…,N;1≤r≤N-1;k=i,Ajm(s)=sAjjs-1Ajm1-Ajms-1yj-ymj,m=1,2,…,M;m≠j,-∑u=1u≠jMAju(s)j=1,2,…,M;1≤r≤M-1;m=j.In this paper,N=M. The distribution forms of the grid points are nonuniform; the distribution forms of the grid points are(25)ξ1=0,ξN=1,ξi=121-cos⁡2i-32N-4π,i=2,3,…,N-1η1=0,ηN=1,ηi=121-cos⁡2i-32N-4π,i=2,3,…,N-1.According to the DQM procedures, (18) can be discretized into the following forms:(26)∑k=1NAik4Wkj+2r2∑m=1MBjm2∑k=1NAik2Wkm+r4∑m=1MBjm4Wim-11-μpU∑k=1NAik2Wkj+U·r2∑m=1MBjm2Wim+q1-ξi∑k=1NAik2Wkj+1+2pC2∑k=1NAik2Wkj+∑k=1NAik2Wkji431+μb2-μbH×∑k=1NAik4Wkj+2r2∑m=1NBjm2∑k=1MAik2Wkm+r4∑k=1MBjm4Wik-i431-2μb1+μb1-μp1-μbH×U∑k=1NAik2Wkj+Ur2∑m=1MBjm2Wim+i431-2μb1+μb1-μbHq1-ξi∑k=1NAik2Wkj+i1+2p2C∑k=1NAik1Wkj+i1+2p×431-2μb1+μb1-μbHC2∑k=1NAik2Wkjω-∑k=1NAik4Wkj+2r2∑m=1NBjm2∑k=1NAik2Wkm431-2μb1+μb2H2×∑k=1NAik4Wkj+2r2∑m=1NBjm2∑k=1NAik2Wkm+r4∑k=1NBjm4Wik+431+2p1-2μb1+μb1-μb2CH∑k=1NAikWkj+1+2pWij∑k=1NAik4Wkj+2r2∑m=1NBjm2∑k=1NAik2Wkmω2-i1+2p431-2μb1+μb1-μbHWijω3=0.The differential quadrature forms of the boundary conditions (20) are(27)W1j=WNj=Wi1=WiN=0,i,j=1,2,…,N,∑k=1NAik(2)Wkj=0,i=1,N,j=1,2,…,N,∑k=1NBjk(2)Wik=0,j=1,N,i=1,2,…,N.The differential quadrature forms of the boundary conditions (21) are(28)W1j=WNj=Wi1=WiN=0,i,j=1,2,…,N,∑k=1NAik(1)Wkj=0,i=2,N-1,j=2,3,…,N-2,∑k=1NBjk(2)Wik=0,j=1,N,i=1,2,…,N.After eliminating the boundary degrees of freedom from (22) using the boundary conditions, the equation can be written in the matrix form as(29)ω3Q+ω2R+ωG+KWkj=0.The complex eigenvalue equation of the axially moving viscoelastic rectangular plate with piezoelectric layers subjected to uniformly distributed tangential follower force is that coefficient determinant equal to zero; that is,(30)ω3Q+ω2R+ωG+K=0.In (30), ω is a generalized complex eigenvalue, where the matrices Q, R, G, and K involve such parameters as dimensionless moving speed C, the dimensionless follower force q, the dimensionless time delay H, the dimensionless time τ, the dimensionless electric potential difference parameter U, the dimensionless electric potential difference geometrical parameter λ, and aspect ratio r of the plate.


## 3. Results and Discussion

In the case ofH=0, C=0, U=0, and p=0, (18) is reduced to the differential equation of motion of the elastic plate. Equation (30) is reduced to the complex eigenvalue equation of the elastic plate. In order to verify the DQM program, the first three-order natural frequencies of the transverse free vibration of elastic plate with four-edge simply supported boundary conditions are calculated firstly, and the results are in good agreement with those exhibited in [31], which can be seen from Table 1.Table 1
The first three-order natural frequencies of the uniform thickness elastic plate with different boundary conditions.
Aspect ratior
0.5
1
Present solutions
SSSS
12.3370
19.7401
31.8919
19.7392
49.3519
78.9647
SCSC
23.8184
28.9567
38.9379
28.9559
54.7609
70.1555
Results in [25]
SSSS
12.33
19.73
32.07
19.73
49.35
78.96
SCSC
23.82
28.95
39.09
28.95
54.74
69.33In the numerical examples, the number of grid points isN=M=9. Numerical studies have been conducted to investigate the effects of several key parameters on the dynamics stabilities of axially moving viscoelastic plates with the piezoelectric layer subjected to uniformly distributed tangential follower force. The critical moving speed of axially moving plates for H=10-5, q=0, r=1, U=0, and p=0 with different boundary conditions is seen from Table 2.Table 2
Critical moving speed of axially moving plates with different boundary conditions.
Boundary conditions
Divergence speed in first order
Regaining stability in the first order
Coupled-mode flutter speed
SSSS
6.31
7.81
8.22 (1st and 2nd couple)
CSCS
8.21
10.31
11.79 (1st and 3rd couple)


### 3.1. The Influence of the Dimensionless Electric Potential Difference

Figure2 displays the variation of the curve for the cases  H=10-5, q=50, r=1, U=0, and λ=0.05. It can be seen that, with increase of axially moving speed C, the real part Re(ω) of complex frequencies in the first mode becomes zero and the imaginary part Im(ω) of complex frequencies has two branches. This shows that the first mode behaves unstably by the divergence instability when the axially moving speed becomes equal to or larger than the critical speed C=3.13. When the moving speed increases further, the piezolaminated plate regains stability in the first-order mode. By maintaining an increase in the moving velocity, the plate undergoes divergence instability in the second mode. When the speed of motion is C=8.78, the first mode behaves unstably by the divergence instability again. When C=10.85, the second mode behaves divergence instability again.Figure 2
Dimensionless complex frequencies versus dimensionless axially moving speedC (H=10-5, q=50, r=1, U=0, and λ=0.05).In Figure3, the case of H=10-5, q=50, r=1, U=10, and λ=0.05 is shown. The first-order mode exhibits divergence instability when the moving speed C=5.77; at C=7.41 the first-order divergence instability is over. When the moving speed increases, the plate undergoes divergence instability in the second mode. By maintaining an increase in the moving velocity, the first mode behaves unstably by the divergence instability again. Then, the second mode behaves divergence instability again. In comparison with Figure 2, with the dimensionless electric potential difference increase, the critical speed increases, while the types of instability of the moving piezolaminated plate are in agreement.Figure 3
Dimensionless complex frequencies versus dimensionless axially moving speedC (H=10-5, q=50, r=1, U=10, and λ=0.05).Figure4 shows the variation of the first three-order dimensionless complex frequencies of the plate with dimensionless axially moving speed C for H=10-5, q=50, r=1, U=20, and λ=0.05. It can be seen that the influence of the dimensionless electric potential difference on the critical speed is obvious. With the dimensionless electric potential difference increase, the critical speed increases, while the types of instability of the moving piezolaminated plate subjected to uniformly distributed tangential follower force have not been changed.Figure 4
Dimensionless complex frequencies versus dimensionless axially moving speedC (H=10-5, q=50, r=1, U=20, and λ=0.05).


### 3.2. The Influence of the Dimensionless Moving Speed on Stability

Figures5 and 6 display the variation of the curve of the first-order dimensionless complex frequencies versus dimensionless follower force q for the cases U=10 and U=20. It can be seen that the viscoelastic plate exhibits divergence instability in the first-order mode under different moving speed C. This result indicates that increases in the dimensionless speed of moving piezolaminated viscoelastic plate subjected to uniformly distributed tangential follower force reduce the critical load but will not change the type of instability experienced by the moving plate.Figure 5
The first-order dimensionless complex frequencies versus dimensionless follower forceq (H=10-3, r=1, U=10, and λ=0.05).Figure 6
The first-order dimensionless complex frequencies versus dimensionless follower forceq (H=10-3, r=1, U=20, and λ=0.05).


### 3.3. The Influence of the Dimensionless Uniformly Distributed Tangential Follower Force on Stability

Figures7 and 8 indicate the variation of the first-order dimensionless complex frequencies with the dimensionless moving speed for the different values of U, respectively. In Figure 7, for dimensionless follower force q=0, it shows that the first-order mode behaves divergence instability; then the plate undergoes single-mode flutter. In the case of q=50 and q=100, it shows that the first-order mode behaves divergence instability first, when the dimensionless moving speed increases, the plate regains stability in the first-order mode. By maintaining an increase in the moving speed, the first mode behaves divergence instability again. It can be seen that the real part of the dimensionless complex frequencies decreases with the increase of the dimensionless follower force q. The critical speed of the first mode decreases when the dimensionless follower force q increases.Figure 7
Dimensionless complex frequencies versus dimensionless axially moving speedC(H=10-3, r=1, U=10, and λ=0.05).Figure 8
Dimensionless complex frequencies versus dimensionless axially moving speedC (H=10-3, r=1, U=20, and λ=0.05).In comparison with Figures7 and 8, respectively, it is observed that the same dimensionless follower force  q, the real parts of the dimensionless complex frequencies under U=20, is greater than that in the case of U=10. The critical speed of the first mode increases with the increase of electric potential difference.


## 3.1. The Influence of the Dimensionless Electric Potential Difference

Figure2 displays the variation of the curve for the cases  H=10-5, q=50, r=1, U=0, and λ=0.05. It can be seen that, with increase of axially moving speed C, the real part Re(ω) of complex frequencies in the first mode becomes zero and the imaginary part Im(ω) of complex frequencies has two branches. This shows that the first mode behaves unstably by the divergence instability when the axially moving speed becomes equal to or larger than the critical speed C=3.13. When the moving speed increases further, the piezolaminated plate regains stability in the first-order mode. By maintaining an increase in the moving velocity, the plate undergoes divergence instability in the second mode. When the speed of motion is C=8.78, the first mode behaves unstably by the divergence instability again. When C=10.85, the second mode behaves divergence instability again.Figure 2
Dimensionless complex frequencies versus dimensionless axially moving speedC (H=10-5, q=50, r=1, U=0, and λ=0.05).In Figure3, the case of H=10-5, q=50, r=1, U=10, and λ=0.05 is shown. The first-order mode exhibits divergence instability when the moving speed C=5.77; at C=7.41 the first-order divergence instability is over. When the moving speed increases, the plate undergoes divergence instability in the second mode. By maintaining an increase in the moving velocity, the first mode behaves unstably by the divergence instability again. Then, the second mode behaves divergence instability again. In comparison with Figure 2, with the dimensionless electric potential difference increase, the critical speed increases, while the types of instability of the moving piezolaminated plate are in agreement.Figure 3
Dimensionless complex frequencies versus dimensionless axially moving speedC (H=10-5, q=50, r=1, U=10, and λ=0.05).Figure4 shows the variation of the first three-order dimensionless complex frequencies of the plate with dimensionless axially moving speed C for H=10-5, q=50, r=1, U=20, and λ=0.05. It can be seen that the influence of the dimensionless electric potential difference on the critical speed is obvious. With the dimensionless electric potential difference increase, the critical speed increases, while the types of instability of the moving piezolaminated plate subjected to uniformly distributed tangential follower force have not been changed.Figure 4
Dimensionless complex frequencies versus dimensionless axially moving speedC (H=10-5, q=50, r=1, U=20, and λ=0.05).


## 3.2. The Influence of the Dimensionless Moving Speed on Stability

Figures5 and 6 display the variation of the curve of the first-order dimensionless complex frequencies versus dimensionless follower force q for the cases U=10 and U=20. It can be seen that the viscoelastic plate exhibits divergence instability in the first-order mode under different moving speed C. This result indicates that increases in the dimensionless speed of moving piezolaminated viscoelastic plate subjected to uniformly distributed tangential follower force reduce the critical load but will not change the type of instability experienced by the moving plate.Figure 5
The first-order dimensionless complex frequencies versus dimensionless follower forceq (H=10-3, r=1, U=10, and λ=0.05).Figure 6
The first-order dimensionless complex frequencies versus dimensionless follower forceq (H=10-3, r=1, U=20, and λ=0.05).


## 3.3. The Influence of the Dimensionless Uniformly Distributed Tangential Follower Force on Stability

Figures7 and 8 indicate the variation of the first-order dimensionless complex frequencies with the dimensionless moving speed for the different values of U, respectively. In Figure 7, for dimensionless follower force q=0, it shows that the first-order mode behaves divergence instability; then the plate undergoes single-mode flutter. In the case of q=50 and q=100, it shows that the first-order mode behaves divergence instability first, when the dimensionless moving speed increases, the plate regains stability in the first-order mode. By maintaining an increase in the moving speed, the first mode behaves divergence instability again. It can be seen that the real part of the dimensionless complex frequencies decreases with the increase of the dimensionless follower force q. The critical speed of the first mode decreases when the dimensionless follower force q increases.Figure 7
Dimensionless complex frequencies versus dimensionless axially moving speedC(H=10-3, r=1, U=10, and λ=0.05).Figure 8
Dimensionless complex frequencies versus dimensionless axially moving speedC (H=10-3, r=1, U=20, and λ=0.05).In comparison with Figures7 and 8, respectively, it is observed that the same dimensionless follower force  q, the real parts of the dimensionless complex frequencies under U=20, is greater than that in the case of U=10. The critical speed of the first mode increases with the increase of electric potential difference.


## 4. Conclusions

The formulation and results for the stability of the moving viscoelastic plate with the piezoelectric layer subjected to uniformly distributed tangential follower force are presented. The general complex frequencies are calculated. This paper analyzes stabilities of the moving viscoelastic plate with the piezoelectric layer subjected to follower force. The results of the analysis of the present study are summarized below.(1)
With the dimensionless electric potential difference increase, the critical speed increases, while the types of instability of the moving piezolaminated plate subjected to uniformly distributed tangential follower force have not been changed. The moving piezolaminated plate subjected to follower force behaves unstably by the divergence instability.(2)
By increasing in the dimensionless speed of moving piezolaminated viscoelastic plate subjected to follower force, the critical load (i.e., critical follower force) decreases but will not change the type of divergence instability experienced by the moving plate.(3)
The influence of the dimensionless uniformly distributed tangential follower force on stability of moving piezolaminated viscoelastic plate is obvious. By maintaining an increase in the dimensionless follower force, the critical speed of the first mode decreases.This research demonstrates the potential of using piezoelectric materials in enhancing the stability of axially moving viscoelastic plates subjected to uniformly distributed tangential follower force. The conclusions provide a theoretical basis and effective approach for improving structure designing and working stability.



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