# Dynamic Stability of Axially Accelerating Viscoelastic Plate

Authors: Yin-Feng Zhou; Zhong-Min Wang
Journal: Scholarly Research Exchange
 (2009)

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## Abstract

The transverse vibration of an axially accelerating viscoelastic plate is investigated. The governing equation is derived from the
two-dimensional viscoelastic differential constitutive relation while the resulting equation is discretized by the differential quadrature method (DQM). By introducing state vector, the first-order state equation with periodic coefficients is established and then it is solved by Runge-Kutta method. Based on the Floquet theory, the dynamic instability regions and dynamic stability regions for the accelerating plate are determined and the effects of the system parameters on dynamic stability of the plate are discussed.


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## Body

## 1. Introduction

Transmission belts, band saw blades, elevator cables, magnetic types, paper webs are some technological examples of axially moving continua. There are comprehensive studies on such systems [1–5]. In recent years,
much attention has been paid to parametric
vibration of traveling
systems and transverse parametric vibration of axially accelerating systems has been extensively analyzed [6, 7]. Pakdemirli et al. [8]
conducted a stability analysis using the Floquet theory for sinusoidal
transporting velocity function. Oz [9] used the method of multiple scales to
calculate analytically the stability boundaries of an axially moving beam with
the velocity as a simple  harmonic function.
Based on the second-term Galerkin
discretization, Chen et al. [10] analyzed the stability of axially accelerating
linear beams. Chen and Yang [11] studied
transverse stability of axially moving viscoelastic beams with the speed that
is harmonically fluctuating about a constant mean value.All above-mentioned researchers considered elastic beams
or viscoelastic strings
and viscoelastic beams; however, the literature that is specially related to
axially accelerating viscoelastic plates is relatively limited. To address the
lack of research in this aspect, the present investigation is devoted to analytical study of axially
accelerating viscoelastic plate.In this paper, we take the axially
accelerating viscoelastic plate as research object; the axially moving speed of the plate is a constant mean
speed with small harmonic variations. The partial differential equation of
motion was discretized by DQM, then the stability analysis was made using the Floquet
theory and effects of the system parameters on dynamic stability of the plate are discussed.


## 2. The Governing Equation

Consider a uniform axially moving viscoelastic plate,
with densityρ, the
length a,
width b,
and thickness h in the x, y,
and z
directions, respectively. The plate
move with the time-dependent axial speed v(t),
as shown in Figure 1.Figure 1
The axially accelerating viscoelastic plate.Assuming that the material of the
plate obeys elastic behavior in dilatation and the Kelvin-Voigt law in
distortion, the constitutive equations are as follows [12]:(1)sij=2Geij+2ηe˙ij,σii=3Kεii, where G is shear elastic modulus, K is bulk elastic modulus, and η is viscosity coefficient. sij and eij are deviatoric tensor of stress and deviatoric
tensor of strain, σii and εii are spherical tensor of stress and spherical tensor of strain.The speed of
the plate in thex direction is v(t),
in the direction of transverse displacement w(x,y,t) is v¯(t),
that is to say (2)v¯(t)=dwdt=∂w∂t+v(t)∂w∂x, where the differential operator is(3)ddt=∂∂t+v(t)∂∂x,the transverse acceleration of the axially moving plate a¯ can be obtained as (4)a¯=dv¯dt=∂((∂w/∂t)+v(t)(∂w/∂x))∂t+v(t)∂((∂w/∂t)+v(t)(∂w/∂x))∂x=∂2w∂t2+2v∂2w∂x∂t+v2∂2w∂x2+dvdt∂w∂x. The geometry equations of viscoelastic plate are the same as the elastic plates.
Based on the thin plate theory and the
constitutive equations of the viscoelastic material in Laplace domain [13], the differential
equation of motion of axially moving viscoelastic plate constituted by the
Kelvin-Voigt model in time domain is (5)h312(A3+A4∂∂t+A5∂2∂t2)∇4w+ρhA1(∂2w∂t2+2v∂2w∂t∂x+v2∂2w∂x2+dvdt∂w∂x)+ρhA2∂∂t(∂2w∂t2+2v∂2w∂t∂x+v2∂2w∂x2+dvdt∂w∂x)=0,where A1=3K+4G,  A2=4η,  A3=2G(6K+2G),  A4=8Gη+12Kη, A5=4η2,
shear elastic modulus G=E/(2(1+μ)),
bulk elastic modulus K=E/(3(1−2μ)), μ is the Poisson ratio.The speed in thex direction is assumed to be
a small simple harmonic variation about the constant mean speed, namely, v(t)=v0+v1sinωt.Introduce the dimensionless variables and parameters:(6)ξ=xa,ψ=yb,W=wh,ϕ=abτ=tha2E12ρ(1−μ2),c=ah12ρ(1−μ2)Ev0,c1=ah12ρ(1−μ2)Ev1,H=ha2E12ρ(1−μ2)ηE,Ω=a2h12ρ(1−μ2)Eω, where H is dimensionless delay time, c and c1 are dimensionless axially moving speed, ϕ is aspect ratio of the plate, and τ is dimensionless time.Substituting (6) into (5), one obtains (7)[1+4(2−μ)(1+μ)3H∂∂τ+4(1−2μ)(1+μ)23H2∂2∂τ2]∇4W+[1+4(1−2μ)(1+μ)3(1−μ)H∂∂τ]×[∂2W∂τ2+2(c+c1sinΩτ)∂2W∂ξ∂τ+c1ΩcosΩτ∂W∂ξ+(c2+2cc1sinΩτ+12c12−12c12cos2Ωτ)∂2W∂ξ2]=0. The boundary conditions of the plate with four
edges simply supported are as follows: (8)ξ=0,1:W(ξ,ψ,τ)=∂2W∂ξ2=0,ψ=0,1:W(ξ,ψ,τ)=∂2W∂ψ2=0. The boundary conditions of the plate with two
opposite edges simply supported and other two edges clamped are as follows: (9)ξ=0,1:W(ξ,ψ,τ)=∂W∂ξ=0,ψ=0,1:W(ξ,ψ,τ)=∂2W∂ψ2=0.


## 3. Discretization of the Governing Equation by Differential Quadrature Method

The basic idea
of the DQ method is 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 [14]. Consider smooth function f(x,y,t) in region 0≤x≤a,0≤y≤b,
the partial derivative of the
rth order with respect to x of it at the point (xi,yi,t),
the partial derivative of the
sth order with respect to y,
the mixed partial derivative of the sth order with respect to y and the rth order with respect to x are defined as follows, respectively [15]: (10)∂rf(xi,yj,t)∂xr=∑k=1NAik(r)f(xk,yj,t),(i=1,2,…,N;r=1,2,…,N−1),(11)∂sf(xi,yj,t)∂ys=∑m=1MAjm(s)f(xi,ym,t),(j=1,2,…,M;s=1,2,…,N−1),(12)∂r+sf(xi,yj,t)∂xr∂ys=∑k=1NAik(r)∑m=1MAjm(s)f(xi,ym,t), where Aik(r) and Ajm(s) are weight coefficients, and they are defined by  (13)Aik(1)={∏μ=1,μ≠i,kN(xi−xμ)∏μ=1,μ≠kN(xk−xμ)(i,k=1,2,…,N;k≠i),∑μ=1μ≠iN1xi−xμ(i,k=1,2,…,N;k=i),(14)Ajm(1)={∏μ=1,μ≠j,mM(yj−yμ)∏μ=1,μ≠mM(ym−yμ)(j,m=1,2,…,M;m≠j),∑μ=1μ≠jM1yj−yμ(j,m=1,2,…,M;m=j), in the case of r=2,3,…,N−1;s=2,3,…,M−1, then  (15)Aik(r)={r(Aii(r−1)Aik(1)−Aik(r−1)xi−xk)(i,k=1,2,…,N;k≠i),−∑μ=1μ≠iNAiμ(r)(i=1,2,…,N;1≤r≤(N−1)),(16)Ajm(s)={s(Ajj(s−1)Ajm(1)−Ajm(s−1)yj−ym)(j,m=1,2,…,M;m≠j),−∑μ=1μ≠jMAjμ(s)(j=1,2,…,M;1≤s≤(M−1)), where N, M is the number of nodes in x and y
directions, respectively, in this paper M=N.There are two key points in the successful
application of the differential quadrature method, one is how to determine the
weighting coefficients and the other is how to select the grid points. The
natural and simplest choice of the grid points is equally spaced points in the
direction of the coordinate axes of the computational domain. It was
demonstrated that nonuniform grid points give a better results with the same number of equally
spaced grid points [16]. In this
paper, we choose these set of grid points in terms of natural coordinate directions ξ and ψ for the plate with
four edges simply supported and the plate with two opposite edges simply supported
and other edges clamped, respectively, as  (17)ξ1=0,ξN=1,ξi=12[1−cos(2i−32N−4π)](i=2,3,…,N−1),ψ1=0,ψN=1,ψi=12[1−cos(2i−32N−4π)](i=2,3,…,N−1),(18)ξ1=0,ξ2=δ,ξN−1=1−δ,ξN=1,ξi=12[1−cos(i−2N−3π)](i=3,4,…,N−2),ψ1=0,ψN=1,ψi=12[1−cos(2i−32N−4π)](i=2,3,…,N−1). According to the DQ method procedures, the governing
differential equation (7) with their associated boundary conditions can be
discretized into the following forms:  (19)4(1−2μ)(1+μ)3(1−μ)HW⃛+[43(1−2μ)(1+μ)2H2×(∑k=1NAik(4)+2ϕ2∑m=1NBjm(2)∑k=1NAik(2)+ϕ4∑k=1NBjk(4))+1+4(1−2μ)(1+μ)3(1−μ)H2(c+c1sinΩτ)∑k=1NAik(1)]W¨+{43(2−μ)(1+μ)H×(∑k=1NAik(4)+2ϕ2∑m=1NBjm(2)∑k=1NAik(2)+ϕ4∑k=1NAik(4))+2(c+c1sinΩτ)∑k=1NAik(1)+4(1−2μ)(1+μ)3(1−μ)H×[(c2+2cc1sinΩτ+12c12−12c12cos2Ωτ)×∑k=1NAik(2)+c1ΩcosΩτ∑k=1NAik(1)]}W˙+{∑k=1NAik(4)+2ϕ2∑m=1NBjm(2)∑k=1NAik(2)+ϕ4∑k=1NBjk(4)+(c2+2cc1sinΩτ+12c12−12c12cos2Ωτ)∑k=1NAik(2)+c1ΩcosΩτ∑k=1NAik(1)}W=0,(20)W1j=WNj=Wi1=WiN=0(i,j=1,2,…,N),∑k=1NAik(2)Wkj=0(i=1,N,j=1,2,…,N),∑k=1NAjk(2)Wik=0(j=1,N,i=1,2,…,N),(21)W1j=WNj=Wi1=WiN=0(i,j=1,2,…,N),∑k=1NAik(1)Wkj=0(i=2,N−1j=2,3,…,N−2),∑k=1NAjk(2)Wik=0(j=1,Ni=1,2,…,N). Equation (19) and boundary conditions (20) or (21)
can be written into the
matrix form  (22)[Q]W⃛+[R]W¨+[C]W˙+[K]W={0}, where matrices [Q],[R],[C],[K] involve such parameters as dimensionless delay
time H,
dimensionless axially moving speed c,c1,
aspect ratio ϕ of the plate, and dimensionless time τ,
furthermore, these matrices have the same period. So the
system of (22) is a third-order partial differential equation containing
periodic time-varying coefficient.


## 4. Dynamic Stability Analysis

Introducing the state vectoru=[WW˙W¨]T,
then substituting it into (22) gives  (23)u˙=G(τ)u, where (24)G(τ)=[0I000I−Q−1K−Q−1C−Q−1R],IN×N is Nth-order identity matrix, and G3N×3N(τ+T)=G3N×3N(τ), that is to say, G3N×3N(τ) is periodic expression, T is the periodic of the function in the axially
accelerating speed.Based on the Floquet
theory, the solution matrix to express (23) is supposed as matrix U,
namely,  (25)U˙=G(τ)U. Because G(τ+T)=G(τ),
so U(τ+T) is also the solution matrix to (20), moreover,
the relation between U(τ+T) and U(τ) is  (26)U(τ+T)=AU(τ), where A(τ) is nonsingular matrix.Introducing transformation(27)U(τ)=PV(τ), then (26) can be expressed as  (28)V(τ+T)=P−1APV(τ)=BV(τ). The nonsingular matrix P can make matrix B has the Jordan  canonical form, the matrix B and matrix A are similar matrices, then they have the same
eigenvalues. The eigenvalues
of matrix A  can be obtained by  (29)|A−λI|=0. When (29) has different roots, matrix B will be diagonal matrix as  (30)B=[λ100⋯00λ20⋯000λ3⋯0⋯⋯⋯⋯00000λm]. So (28) can be
rewritten as the following component form:  (31)vi(t+T)=λivi(t). From format (31), we can obtain  (32)vi(t+nT)=λinvi(t), so when t→∞,
that is, n→∞(33)vi(t)→{0if|λi|<1,∞if|λi|>1. It can be seen if |λi|<1,
the system is stable, in case of, |λi|>1 the system is unstable, when |λi|=1 the system is critical condition of stability
and instability.It is known
from the analysis mentioned above that the eigenvalues of the dynamic stability
equation must be calculated before determining the dynamic instability region.We can use initial conditionU(0)=I to calculate the basic solution of  (25) in a vibrational periodic. From (26), one can get A=U(τ),
then solving (29), the eigenvalues are obtained. We can adopt the Runge-Kutta
method to get the matrix A.


## 5. Results and Discussion

Numerical
examples are presented to examine the effects of the dimensionless delay time,
the aspect ratio, and the mean axial speed on the instability region.


### 5.1. The Plate with Four Edges Simply Supported (ssss)

Figure2 shows
the effect of the
dimensionless delay time for ϕ=1,c=2,
and H=10−4 (dotted line), 10−3 (solid line). Obviously, the larger the dimensionless
delay time, the smaller the instability region.Figure 2
The effect of the dimensionless delay time on the instability
boundary.Figure3 depicts
the effect of the aspect
ratio for H=10−3,c=2,
and ϕ=0.5 (dashed line),  1 (solid line). It indicates that with the increase of aspect ratio, the
instability region decreases.Figure 3
The effect of the aspect ratio on the instability boundary.Figure4
illustrates the effect of the mean axial speed for H=10−3,ϕ=1,
and c=2 (solid line),  5 (dash-dotted line).It can be seen
that the larger the mean axial speed, the smaller the
stable region.Figure 4
The effect of the mean axial speed on the instability boundary.


### 5.2. The Plate with Two Opposite Edges Simply Supported and Other Two Edges Clamped (cscs)

Figures5, 6,
and 7 demonstrate, respectively, the effects of  the  dimensionless delay time,
the aspect ratio, and the mean axial speed on the instability region for the
plate with two opposite edges simply supported and other two edges clamped. In
Figure 5, ϕ=1,c=2,
and H=10−4 (dotted line), 10−3 (solid line). In Figure 6, H=10−3,c=2,
and ϕ=0.5 (dashed line), 1 (solid line). In Figure 7, H=10−3,ϕ=1,
and c=2 (solid line), 5 (dash-dotted line). Similar to
the trends for the plate with four edges simply supported, the larger the dimensionless delay
time, the larger the
aspect ratio and the smaller the
mean axial speed, which
lead to the smaller the
instability region. That is, the increasing dimensionless delay time, the
increasing aspect ratio, and the decreasing mean axial speed make the instability
boundaries move toward the increasing direction of c1 in plane (Ω,c1) and the instability regions become small.Figure 5
The effect of the dimensionless delay time on the instability boundary.Figure 6
The effect of the aspect ratio on the instability boundary.Figure 7
The effect of the mean axial speed on the instability boundary.


## 5.1. The Plate with Four Edges Simply Supported (ssss)

Figure2 shows
the effect of the
dimensionless delay time for ϕ=1,c=2,
and H=10−4 (dotted line), 10−3 (solid line). Obviously, the larger the dimensionless
delay time, the smaller the instability region.Figure 2
The effect of the dimensionless delay time on the instability
boundary.Figure3 depicts
the effect of the aspect
ratio for H=10−3,c=2,
and ϕ=0.5 (dashed line),  1 (solid line). It indicates that with the increase of aspect ratio, the
instability region decreases.Figure 3
The effect of the aspect ratio on the instability boundary.Figure4
illustrates the effect of the mean axial speed for H=10−3,ϕ=1,
and c=2 (solid line),  5 (dash-dotted line).It can be seen
that the larger the mean axial speed, the smaller the
stable region.Figure 4
The effect of the mean axial speed on the instability boundary.


## 5.2. The Plate with Two Opposite Edges Simply Supported and Other Two Edges Clamped (cscs)

Figures5, 6,
and 7 demonstrate, respectively, the effects of  the  dimensionless delay time,
the aspect ratio, and the mean axial speed on the instability region for the
plate with two opposite edges simply supported and other two edges clamped. In
Figure 5, ϕ=1,c=2,
and H=10−4 (dotted line), 10−3 (solid line). In Figure 6, H=10−3,c=2,
and ϕ=0.5 (dashed line), 1 (solid line). In Figure 7, H=10−3,ϕ=1,
and c=2 (solid line), 5 (dash-dotted line). Similar to
the trends for the plate with four edges simply supported, the larger the dimensionless delay
time, the larger the
aspect ratio and the smaller the
mean axial speed, which
lead to the smaller the
instability region. That is, the increasing dimensionless delay time, the
increasing aspect ratio, and the decreasing mean axial speed make the instability
boundaries move toward the increasing direction of c1 in plane (Ω,c1) and the instability regions become small.Figure 5
The effect of the dimensionless delay time on the instability boundary.Figure 6
The effect of the aspect ratio on the instability boundary.Figure 7
The effect of the mean axial speed on the instability boundary.


## 6. Conclusions

Dynamic stability of axially moving viscoelastic plate is
studied in this paper. The axially moving speed of the viscoelastic plate is
assumed as harmonically fluctuating about a constant mean value. The differential quadrature method is applied to the partial-differential equation
governing the transverse parametric vibration.The results of the present investigation indicate that, for the SSSS plate and
the CSCS plate considered in the paper, (1) with the
increase of the delay
timeH, the
instability region decreases, (2) with the increase of the aspect ratio ϕ, the stability
region decreases, (3) with the increase of the mean axial speed c, the
instability region increases. In addition, for the same parameters, the instability region of CSCS plate is smaller
than that of SSSS plate.



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*Source: 856320-2009-03-17.xml*