Dynamic stability of an axially moving paper board with added subsystems

Abstract

Paper board with bending stiffness is usually used as the substrates on cigarette package printing industry. The vibration of these paper boards in high speed affects the printing precision. The dynamic characteristics and stability of moving paper board with finite interior elastic point supports and elastic edges restrained are investigated. First, the energy function of the system is established by using the extended Hamilton’s principle; second, the dimensionless equations of motion for the moving paper board are obtained using the element-free Galerkin method. The equations of motion and the eigenvalue equations of the system are established. The relationship between the first three complex frequencies of the system and the moving speed is then obtained by the numerical calculation. The effects of the elastic point supports, the elastically restrained edges, and the dimensionless speed of the motion on the dynamic stability of the paper board are analyzed. The critical speed when the paper board is in a stable state under different conditions is obtained. The results improve the dynamic stability of the paper board in printing process and provide the theoretical basis for the optimization of printing equipment.

Introduction

Moving paper board with certain thickness and bending stiffness is widely used in printing and packaging industry. The vibration characteristics of paper board have a great impact on printing quality. At present, the research of cigarette package with printing background is less. In this paper, based on industry background, the printing materials of Shaanxi Bei Ren PRC 250 cigarette package gravure press (see Figure 1) was used as the sample. When take it as the thin plate, the adjacent two rollers can be regarded as the boundary supports or intermediate supports. The transverse vibration characteristics and stability of moving paper board with added subsystems are investigated. In this paper added subsystems are the systems under elastic point supports and elastic edges restrained. The critical transferring speed of the paper board under different conditions is obtained.

Figure 1.

Shaanxi Bei Ren PRC 250 cigarette package gravure press.

Dynamics and mechanical stability of axially moving systems have been widely studied for a long time. The most common models for these systems include moving strings, beams, membranes, and plates. Ulsoy and Mote ¹ early researched the large band saw blade by two-dimensional plate model and investigated the coupled and torsional vibration of the plate. Lin ² investigated the stability and vibration characteristics of axially moving plates with two simply supported and two free edges. Marynowski ³ used a two-dimensional rheological element to model the moving viscoelastic web. Saksa et al. ⁴ studied the stability and dynamic behavior of axially moving viscoelastic panels with the help of a classical modal analysis. The nonlinear vibration in parametric and internal resonances of accelerating viscoelastic plates was studied by Tang and Chen. ⁵ The nonlinear dynamics for forced motions of an axially moving plate was investigated using Von Karman plate theory by Ghayesh et al. ⁶ Wang et al. ⁷ studied the dynamic stability of the moving viscoelastic plate with the piezoelectric.

Boundary control of axially moving system was investigated by Hong et al.^8–10 They proposed a set of effective boundary control laws for axially moving systems. An optimal delayed feedback control methodology was developed to mitigate the primary and super harmonic resonances of a flexible beam with piezoelectric sensor and actuator by Liu et al. ¹¹ They obtained the stable vibratory regions of the feedback gains by using the stability conditions of eigenvalue equation. Khanna and Singhal ¹² analyzed the vibration of a tapered isotropic rectangular plate under thermal condition, and they calculated the first two modes of frequency of rectangular plate for five boundary conditions. Nguyen and Hong^13,14 proposed a novel control algorithm for controlling the transverse vibration of an axially moving web system.

Ashour ¹⁵ presented an excellent review on the topic of plates with elastic edges restrained. The effects of elastic edge support on the natural frequencies of thin symmetric cross-ply laminates were studied by Hung et al. ¹⁶ Chung et al. ¹⁷ investigated the free vibration of orthotropic Mindlin plates with elastic edges restrained against rotation using the Rayleigh–Ritz method. Bert and Malik ¹⁸ applied a semianalytical differential quadrature method (DQM) to investigate the effects of rotationally restrained boundaries on the natural frequencies of laminated plates with two simply supported opposite edges. Liew et al. ¹⁹ studied the free vibration of symmetric cross-ply laminated plates that were elastically restrained against rotation and translation. Karamia et al. ²⁰ proposed a free vibration analysis of moderately thick symmetric laminated plates with elastic edges restrained using a DQM. The free and forced vibration of variable stiffness composite annular thin plates with elastically restrained edges was investigated by Tan and Nie. ²¹

In addition, the paper board that can be supported at interior points is designed. Point-supported paper boards are plates that have prescribed displacements at a number of discrete locations within their domains. Rigid point supports have a prescribed displacement of zero while elastic point supports have displacements that are dependent on the stiffness at the support. Lee and Lee ²² studied a one-dimensional deformation function of a beam with point load, the admissible shape function of the plate was formed, and the free and flexural vibration for rectangular plates with arbitrary interior elastic point supports has been undertaken. Zhou et al. ²³ analyzed the free vibrations of thick, isotropic, and laminated composite rectangular plates with point supports by using the finite layer method. Gorman ²⁴ provided analytical solutions for the free vibration frequencies and shapes of thin corner-supported rectangular plates with symmetrically distributed reinforcing beams running along the plate edges. Taking into account the compatibility of displacements and the rotations between the plate and the columns, Zhou and Ji ²⁵ derived the coupled vibration of a thin rectangular plate with two opposite edges simply supported and additional column-supported interior points. Huang et al. ²⁶ developed a discrete method to analyze the free vibration problem of rectangular plates with point supports. In this method, the point supports were simulated using Dirac delta functions and the differential equations were transformed into integral equations; a Green’s function approach was used to obtain the characteristic equations. Bahmyari and Khedmati ²⁷ analyzed the free vibration of nonhomogeneous moderately thick plates with point supports resting on Pasternak elastic foundation by using element-free Galerkin method. Lopatina and Morozovb ²⁸ obtained the fundamental frequency of an orthotropic rectangular plate with a centrally located point support and free edges.

Therefore, we pay more attention to research the analytical problem of the transverse vibration stability of an axially moving rectangular paper board with added subsystems. In the present study, the variational equation for the transverse vibration of a moving paper board with finite elastic point supports and elastic restraints is formulated. This paper aims to study the transverse vibration stability of the paper board with added subsystems in axially moving condition. The research provides the theoretical basis for the optimization of printing equipment and improves the dynamic stability of the paper board in printing process.

Vibration model and solution of characteristic equation by element free Galerkin (EFG) method

The basic parameters of the Shaanxi Bei Ren PRC 250 cigarette package gravure press are shown in Table 1. The cigarette packaging paper board is fed through a pair of guide roller and transferred to printing color unit. The paper board between two adjacent rollers is studied. The two adjacent rollers can be regarded as boundary condition. This can be viewed as the axial system and its transverse vibration can be studied. A moving paper board with an internal elastic point support at $M\left(x_i,y_i\right)$ $\left(i=1,2,\mathrm{\dots }N\right)$ and an edge attached to a linear elastic restraint is considered, as is shown in Figure 2. The paper board has a length $a$ , width $b$ , and thickness $h$ in the $x$ , $y$ , and $z$ directions, respectively. The density of paper board is $\rho$ , the Young’s modulus is $E$ , the Poisson’s ratio is $\mu$ , the linear elastic stiffness along the edge is $K$ , and the spring constant of elastic support at the point $i$ is $K_{T_i}$ . The paper board is moving with a constant speed $v$ in the $x$ direction.

Figure 2.

Moving rectangular paper board with elastic point supports and elastic edges restrained.

Table 1.

The basic parameters of the Shaanxi Bei Ren PRC 250 gravure press.

Printed material	Length a (m)	Printing width b (m)	Printing speed v (m/min)	Thickness h (mm)	Poisson ratio $\mu$	Elasticity modulus E (MPa)	Density ρ (g/m2)
Paper board	0.75	0.82	250	0.95	0.15	$2.1\times {10}^3$	300

The transverse speed of the moving paper board can be expressed as

$$\overline{v}\left(t\right)=\frac{\text{d}{w}^{*}}{\text{d}t}=\frac{\partial w^{*}}{\partial t}+v\frac{\partial w^{*}}{\partial x}$$

(1)

where $w^{*}\left(x,y,t\right)$ is the deflection function and $\overline{v}\left(t\right)$ denotes the velocity in the transverse displacement direction.

The bending strain energy of paper board is

$$V_1=\frac{1}{2}{\displaystyle \int \int \int \left({\sigma }_x{\epsilon }_x+{\sigma }_y{\epsilon }_y+{\tau }_{xy}{\gamma }_{xy}\right)\text{d}x\text{d}y\text{d}z}$$

(2)

where

$${\epsilon }_x=-z\frac{{\partial }^2{w}^{*}}{\partial x^2},\hspace{1em}{\epsilon }_y=-z\frac{{\partial }^2{w}^{*}}{\partial y^2},\hspace{1em}{\gamma }_{xy}=-2z\frac{{\partial }^2{w}^{*}}{\partial x\partial y}$$

The relations between stresses ( ${\sigma }_x$ , ${\sigma }_y$ , and ${\tau }_{xy}$ ) and the deflection ( $w^{*}$ ) are given by

$$\{\begin{array}{l}{\sigma }_x=-\frac{E}{1-{\mu }^2}\left(\frac{{\partial }^2{w}^{*}}{\partial x^2}+\mu \frac{{\partial }^2{w}^{*}}{\partial y^2}\right)\\ {\sigma }_y=-\frac{E}{1-{\mu }^2}\left(\frac{{\partial }^2{w}^{*}}{\partial y^2}+\mu \frac{{\partial }^2{w}^{*}}{\partial x^2}\right)\\ {\tau }_{xy}=-\frac{E}{1+\mu }\frac{{\partial }^2{w}^{*}}{\partial x\partial y}\end{array}$$

(3)

Substituting strains and stresses into equation (2) yields the following equation

$$V_1=\frac{D}{2}{\iint }_{\Omega }\left\{{\left(\frac{{\partial }^2{w}^{*}}{\partial x^2}+\frac{{\partial }^2{w}^{*}}{\partial y^2}\right)}^2-2\left(1-\mu \right)\left[\frac{{\partial }^2{w}^{*}}{\partial x^2}\frac{{\partial }^2{w}^{*}}{\partial y^2}-{\left(\frac{{\partial }^2{w}^{*}}{\partial x\partial y}\right)}^2\right]\right\}\text{d}x\text{d}y$$

(4)

where $D=E{h}^3/12\left(1-{\mu }^2\right)$ is the flexural rigidity of the paper board.

The strain energy induced by the internal elastic point support and the strain energy induced by the elastic restrain can be, respectively, expressed as

$$V_2={{\displaystyle \sum _{i=1}^N\frac{1}{2}{K}_{T_i}\left(w^{*}\left(x_i,y_i,t\right)\right)}}^2$$

(5)

$$V_3={\displaystyle {\int }_0^b\frac{1}{2}K{\left(w^{*}\left(a,y,t\right)\right)}^2}\text{d}y$$

(6)

The kinetic energy of the system can be expressed as

$$T=\frac{1}{2}\rho h{\displaystyle {\iint }_{\Omega }{\left(\frac{\partial w^{*}}{\partial t}+v\frac{\partial w^{*}}{\partial x}\right)}^2}+\frac{1}{2}\rho h{\displaystyle {\iint }_{\Omega }{v}^2\text{d}x\text{d}y}$$

(7)

The energy function can be expressed as

$$\begin{array}{c}{\Pi }^{*}={\displaystyle {\iint }_{\Omega }[-\frac{D}{2}{(\frac{{\partial }^2{w}^{*}}{\partial x^2}+\frac{{\partial }^2{w}^{*}}{\partial y^2})}^2+D\left(1-\mu \right)[\frac{{\partial }^2{w}^{*}}{\partial x^2}\frac{{\partial }^2{w}^{*}}{\partial y^2}-{\left(\frac{{\partial }^2{w}^{*}}{\partial x\partial y}\right)}^2]}\\ +\frac{\rho h}{2}{\left(\frac{\partial w^{*}}{\partial t}+v\frac{\partial w^{*}}{\partial x}\right)}^2+\frac{\rho h}{2}{v}^2]\text{d}x\text{d}y\\ -\frac{1}{2}{\displaystyle \sum _{i=1}^N{K}_{T_i}{\left(w^{*}\left(x_i,y_i,t\right)\right)}^2}-\frac{1}{2}{\displaystyle {\int }_0^{b}K{\left(w^{*}\left(a,y,t\right)\right)}^2}\text{d}y\end{array}$$

(8)

Based on Hamilton’s principle, the penalty function method is employed to impose the essential boundary conditions, where $\alpha$ is the penalty parameter. The system can be expressed as

$$\delta \Pi =\delta {\displaystyle {\int }_{t_1}^{t_2}\left({\Pi }^{*}+{\Pi }_{\Gamma }\right)}\text{d}t=0$$

(9)

$${\Pi }_{\Gamma }=\frac{1}{2}\alpha {\displaystyle {\int }_{\Gamma }\left(w^{*}-\overline{w}\right)\left(w^{*}-\overline{w}\right)\text{d}\Gamma }+\frac{1}{2}\alpha {\displaystyle {\int }_{\Gamma }\left(\beta -\overline{\beta }\right)\left(\beta -\overline{\beta }\right)\text{d}\Gamma }$$

(10)

where $\beta =\partial w^{*}/\partial n$ . $\overline{w}$ and $\overline{\beta }$ are the known displacements on the boundary $\Gamma$ .

Substituting equations (8) and (10) into equation (9), according to the variational principle, the variational forms of the dynamic equations are obtained as follows

$$\begin{array}{l}\delta \Pi ={\displaystyle {\int }_{t_1}^{t_2}\{\rho h{\displaystyle \underset{\Omega }{\iint }\frac{{\partial }^2{w}^{*}}{\partial t^2}\delta w^{*}\text{d}x\text{d}y}}+\rho hv{\displaystyle \underset{\Omega }{\iint }(\frac{{\partial }^2{w}^{*}}{\partial t\partial x}\delta w^{*}}-\frac{\partial w^{*}}{\partial t}\frac{\partial \delta w^{*}}{\partial x})\text{d}x\text{d}y-\rho h{v}^2{\displaystyle \underset{\Omega }{\iint }\frac{\partial w^{*}}{\partial x}\delta \left(\frac{\partial w^{*}}{\partial x}\right)\text{d}x\text{d}y}\\ +D{\displaystyle \underset{\Omega }{\iint }\{{\nabla }^4{w}^{*}\delta w^{*}+\frac{\partial }{\partial x}\left[{\nabla }^2{w}^{*}\delta \left(\frac{\partial w^{*}}{\partial x}\right)\right]}+\frac{\partial }{\partial y}\left[{\nabla }^2{w}^{*}\delta \left(\frac{\partial w^{*}}{\partial y}\right)\right]-\frac{\partial }{\partial x}\left[\frac{\partial }{\partial x}\left({\nabla }^2{w}^{*}\right)\delta w^{*}\right]-\frac{\partial }{\partial y}\left[\frac{\partial }{\partial y}\left({\nabla }^2{w}^{*}\right)\delta w^{*}\right]\}\text{d}x\text{d}y\\ -D(1-\mu ){\displaystyle \underset{\Omega }{\iint }\{\frac{\partial }{\partial x}\left[\frac{{\partial }^2{w}^{*}}{\partial y^2}\delta \left(\frac{\partial w^{*}}{\partial x}\right)-\frac{{\partial }^2{w}^{*}}{\partial x\partial y}\delta \left(\frac{\partial w^{*}}{\partial y}\right)\right]}+\frac{\partial }{\partial y}\left[\frac{{\partial }^2{w}^{*}}{\partial x^2}\delta \left(\frac{\partial w^{*}}{\partial y}\right)-\frac{{\partial }^2{w}^{*}}{\partial x\partial y}\delta \left(\frac{\partial w^{*}}{\partial x}\right)\right]\}\text{d}x\text{d}y\\ +{\displaystyle \sum _{i=1}^N{K}_{T_i}}{w}^{*}\left(x_i,y_i,t\right)\delta w^{*}+{\displaystyle {\int }_0^{b}K}{w}^{*}\left(a,y,t\right)\delta w^{*}\text{d}y+\alpha {\displaystyle {\int }_{\Gamma }{w}^{*}}\delta w^{*}\text{d}\Gamma +\alpha {\displaystyle {\int }_{\Gamma }\frac{\partial w^{*}}{\partial n}}\delta \left(\frac{\partial w^{*}}{\partial n}\right)\text{d}\Gamma \}\text{d}t=0\end{array}$$

(11)

Introduce the dimensionless quantities

$$\xi =\frac{x}{a},\eta =\frac{y}{b},c=\frac{a}{b},w=\frac{w^{*}}{a},\tau =\frac{th}{a^2}\sqrt{\frac{E}{12\rho \left(1-{\mu }^2\right)}}$$

$$\lambda =\frac{a}{h}v\sqrt{\frac{12\rho \left(1-{\mu }^2\right)}{E}},k_{T_i}=\frac{12K_{T_i}{a}^3\left(1-{\mu }^2\right)}{E{h}^{3}b},k=\frac{12K{a}^3\left(1-{\mu }^2\right)}{E{h}^3}$$

Equation (11) takes the form

$$\begin{array}{c}{\displaystyle {\int }_{{\tau }_1}^{{\tau }_2}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^1\frac{{\partial }^{2}w}{\partial {\tau }^2}}}}\delta w\text{d}\xi \text{d}\eta \text{d}\tau +\lambda {\displaystyle {\int }_{{\tau }_1}^{{\tau }_2}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^1[\frac{\partial }{\partial \tau }\left(\frac{\partial w}{\partial \xi }\right)\delta w-\frac{\partial w}{\partial \tau }\delta \left(\frac{\partial w}{\partial \xi }\right)]\text{d}\xi \text{d}\eta \text{d}}}}\tau \\ -{\lambda }^2{\displaystyle {\int }_{{\tau }_1}^{{\tau }_2}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^1\left[\left(\frac{\partial w}{\partial \xi }\right)\delta \left(\frac{\partial w}{\partial \xi }\right)\right]\text{d}\xi \text{d}\eta \text{d}\tau }}}+{\displaystyle {\int }_{{\tau }_1}^{{\tau }_2}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^1\{\left(\frac{{\partial }^{4}w}{\partial {\xi }^4}+2c^2\frac{{\partial }^{4}w}{\partial {\xi }^2\partial {\eta }^2}+c^4\frac{{\partial }^{4}w}{\partial {\eta }^4}\right)\delta w}}}+\frac{\partial }{\partial \xi }[(\frac{{\partial }^{2}w}{\partial {\xi }^2}\\ +c^2\frac{{\partial }^{2}w}{\partial {\eta }^2})\delta \left(\frac{\partial w}{\partial \xi }\right)]+c^2\frac{\partial }{\partial \eta }\left[\left(\frac{{\partial }^{2}w}{\partial {\xi }^2}+c^2\frac{{\partial }^{2}w}{\partial {\eta }^2}\right)\delta \left(\frac{\partial w}{\partial \eta }\right)\right]\\ -\frac{\partial }{\partial \xi }\left[\frac{\partial }{\partial \xi }\left(\frac{{\partial }^{2}w}{\partial {\xi }^2}+c^2\frac{{\partial }^{2}w}{\partial {\eta }^2}\right)\delta w\right]\\ -c^2\frac{\partial }{\partial \eta }\left[\frac{\partial }{\partial \eta }\left(\frac{{\partial }^{2}w}{\partial {\xi }^2}+c^2\frac{{\partial }^{2}w}{\partial {\eta }^2}\right)\delta w\right]\}\text{d}\xi \text{d}\eta \text{d}\tau -\left(1-\mu \right){\displaystyle {\int }_{{\tau }_1}^{{\tau }_2}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^1\{c^2\frac{\partial }{\partial \xi }[\frac{{\partial }^{2}w}{\partial {\eta }^2}\delta \left(\frac{\partial w}{\partial \xi }\right)}}}\\ -\frac{{\partial }^{2}w}{\partial \xi \partial \eta }\delta \left(\frac{\partial w}{\partial \eta }\right)]+c^2\frac{\partial }{\partial \eta }\left[\frac{{\partial }^{2}w}{\partial {\xi }^2}\delta \left(\frac{\partial w}{\partial \eta }\right)-\frac{{\partial }^{2}w}{\partial \xi \partial \eta }\delta \left(\frac{\partial w}{\partial \xi }\right)\right]\}\\ \text{d}\xi \text{d}\eta \text{d}\tau +{\displaystyle {\int }_{{\tau }_1}^{{\tau }_2}{\displaystyle \sum _{i=1}^N{k}_{T_i}}}w\left({\xi }_i,{\eta }_i,\tau \right)\delta w\text{d}\tau \\ +{\displaystyle {\int }_{{\tau }_1}^{{\tau }_2}{\displaystyle {\int }_0^{1}kw\left(1,\eta ,\tau \right)}}\delta w\text{d}\eta \text{d}\tau \\ +{\alpha }_1{\displaystyle {\int }_{{\tau }_1}^{{\tau }_2}\left({\displaystyle {\int }_{{\Gamma }_{\xi }}cw}\delta wd\xi +{\displaystyle {\int }_{{\Gamma }_{\eta }}w}\delta wd\eta \right)}\text{d}\tau \\ +{\alpha }_2{\displaystyle {\int }_{{\tau }_1}^{{\tau }_2}\left[{\displaystyle {\int }_{{\Gamma }_{\xi }}{c}^3\frac{\partial w}{\partial \eta }\delta \left(\frac{\partial w}{\partial \eta }\right)}\text{d}\xi +{\displaystyle {\int }_{{\Gamma }_{\eta }}\frac{\partial w}{\partial \xi }\delta \left(\frac{\partial w}{\partial \xi }\right)}\text{d}\eta \right]}\text{d}\tau =0\end{array}$$

(12)

The equations are derived using the element-free Galerkin method. Suppose that $n$ nodes are included within the supported region of the paper board. Then the dimensionless transverse displacement approximation function $w^h\left(x,y,\tau \right)$ can be expressed as

$$w\left(\xi ,\eta ,\tau \right)={\Phi }^T\left(\xi ,\eta \right)w\left(\tau \right)$$

(13)

The dimensionless shape function is given by

$$\Phi \left(\xi ,\eta \right)={\left[{\varphi }_1\left(\xi ,\eta \right),{\varphi }_2\left(\xi ,\eta \right),\mathrm{\dots },{\varphi }_n\left(\xi ,\eta \right)\right]}^{\text{T}}$$

(14)

The dimensionless generalized displacement is given by

$$w\left(\tau \right)={\left[w_1\left(\tau \right),w_2\left(\tau \right),\mathrm{\dots },w_n\left(\tau \right)\right]}^{\text{T}}$$

(15)

Substituting equation (13) into equation (12), according to the variational principle, the dimensionless equations for the transverse vibration of the moving rectangular paper board with finite elastic point supports and elastic restraints can be expressed as

$$M\ddot{w}\left(\tau \right)+C\dot{w}\left(\tau \right)+Kw\left(\tau \right)=0$$

(16)

where

$$\begin{array}{c}{M}^{\text{T}}={\displaystyle {\int }_0^1{\displaystyle {\int }_0^1\Phi {\Phi }^T\text{d}\xi \text{d}\eta }}\text{,}{C}^{\text{T}}=\lambda {\displaystyle {\int }_0^1{\displaystyle {\int }_0^1\left(\Phi {{\Phi }_{,\xi }}^T-{\Phi }_{,\xi }{\Phi }^T\right)\text{d}\xi \text{d}\eta }},\text{ and }\\ K^{\text{T}}=-{\lambda }^2{\displaystyle {\int }_0^1{\displaystyle {\int }_0^1{\Phi }_{,\xi }{{\Phi }_{,\xi }}^T\text{d}\xi \text{d}\eta +}}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^1[\left({\Phi }_{,\xi \xi }{\Phi }_{,\xi \xi }^T+c^2{\Phi }_{,\eta \eta }{\Phi }_{,\xi \xi }^T+c^2{\Phi }_{,\xi \xi }{\Phi }_{,\eta \eta }^T+c^4{\Phi }_{,\eta \eta }{\Phi }_{,\eta \eta }^T\right)}}\\ -c^2\left(1-\mu \right)({\Phi }_{,\eta \eta }{\Phi }_{,\xi \xi }^T-2{\Phi }_{,\xi \eta }{\Phi }_{,\xi \eta }^T+{\Phi }_{,\xi \xi }{\Phi }_{,\eta \eta }^T)]\text{d}\xi \text{d}\eta +{\displaystyle \sum _{i=1}^N{k}_{T_i}}\Phi \left({\xi }_i,{\eta }_i\right){\Phi }^T\left({\xi }_i,{\eta }_i\right)\\ +{\displaystyle {\int }_0^{1}k\Phi \left(1,\eta \right)}{\Phi }^T\left(1,\eta \right)\text{d}\eta +{\alpha }_{1}c{\displaystyle {\int }_{{\Gamma }_{\xi }}\Phi {\Phi }^T}\text{d}\xi +{\alpha }_1{\displaystyle {\int }_{{\Gamma }_{\eta }}\Phi {\Phi }^T}\text{d}\eta +{\alpha }_2{c}^3{\displaystyle {\int }_{{\Gamma }_{\xi }}{\Phi }_{,\eta }{\Phi }_{,\eta }^T\text{d}\xi }+{\alpha }_2{\displaystyle {\int }_{{\Gamma }_{\eta }}{\Phi }_{,\xi }}{\Phi }_{,\xi }^T\text{d}\eta \end{array}$$

Assuming the displacement vector is

$$w\left(\tau \right)=W{e}^{\text{j}\omega \tau }$$

(17)

where $\omega$ is the dimensionless complex frequency and $\text{j}=\sqrt{-1}$ . $W={\left[W_1,W_2,\mathrm{\dots },W_n\right]}^T$ denotes an $n$ -dimensional column vector.

Substituting equation (17) into equation (16) yields the following homogeneous equations

$$KW+\text{j}\omega CW-{\omega }^{2}MW=0$$

(18)

where the matrices $K$ , $C$ , and $M$ involve some parameters such as the dimensionless linear elastic stiffness $k$ , the dimensionless spring constant of elastic support $k_{T_i}$ , the dimensionless time $\tau$ , dimensionless moving speed $\lambda$ , and aspect ratio $c$ of the paper board. $K$ is stiffness matrix, $M$ denotes mass matrix, and $C$ is Coriolis matrix. The necessary and sufficient condition when $W$ has nonzero solution is that coefficient determinant is equal to zero, Then, the dimensionless characteristic equation of the axially moving with elastic point supports and elastically restrained edges can be expressed as

$$\left|K+j\omega C-{\omega }^{2}M\right|=0$$

(19)

In equation (19), $\omega$ is a generalized complex eigenvalue.

Using the EFG method the vibration model functions can be expressed as

$$U_i\left(\xi ,\eta \right)\approx U_i^h\left(\xi ,\eta \right)={\displaystyle \sum _{j=1}^n{\varphi }_j\left(\xi ,\eta \right)}{W}_{ij}={\Phi }^T\left(\xi ,\eta \right)W_i$$

(20)

Numerical calculation and discussion

In order to verify the present formulation, in the case of $\lambda =0$ and $k_{T_i}=0$ , the first three order natural frequencies of the transverse free vibration of paper board with four edges simply supported boundary conditions are calculated first, and the results are in good agreement with those exhibited in Gorman, ²⁹ which can be seen from Table 2.

Table 2.

The first three natural frequencies of the paper board with different aspect ratio.

Aspect ratio $c$		Present solutions			Results in Gorman29
$c=1$	SSSS	19.7392	49.3519	78.9647	19.73	49.35	78.96
$c=1.5$	SSSS	32.0762	61.690	98.7049	32.07	61.68	98.69
$c=2.0$	SSSS	49.3480	78.9608	127.5678	49.34	78.96	128.30

Paper board with four simply supported edges and one elastic point support

In Figure 3 when the dimensionless velocity $\lambda =0$ , imaginary part is zero, and the first three order frequency is real number. With the increase of velocity, real part of complex frequency is reduced, the imaginary part kept zero. When the dimensionless velocity of motion is $\lambda =6.31$ , the real part of the first order of the dimensionless complex frequency is reduced to zero and imaginary part appears as two branches. The paper board produces divergence instability in the first-order mode. When the speed increases to $\lambda =8.31$ , the paper board regains stability in the first-order mode. By maintaining an increase in the moving velocity, the paper board experiences coupled-mode flutter in the first and second modes. The paper board stable dimensionless working speed range is 0–6.31.

Figure 3.

The relation between the first three orders of the frequency and the moving speed λ. (a) Variation of the real part of the complex frequency and (b) variation of the imaginary part of the complex frequency.

In Figure 4 the case of elastic support stiffness $k_{T_i}=100$ is shown, the point support position is at $\xi =0.5$ _, $\eta =0.5$ . In comparison with Figure 3, the divergence speed in the first-order mode exhibits an obvious increase. This conclusion demonstrates that an elastic point support can increase the dynamic stability of the system.

Figure 4.

The relation between the first three orders of the frequency and the moving speed λ. (a) Variation of the real part of the complex frequency and (b) variation of the imaginary part of the complex frequency.

Paper board with two elastic point supports

Figure 5 displays the variation of the curve for the case where the elastic supports are located at ${\xi }_1=0.5$ , ${\eta }_1=0.5$ and ${\xi }_2=0.3$ , ${\eta }_2=0.3$ for $k_{T_1}=100$ and $k_{T_2}=100$ . The paper board exhibits a divergence instability in the first-order mode at first. By maintaining an increase in the moving speed, the paper board experiences coupled-mode flutter in the first and second modes. It indicates that by increasing the number of the elastic point supports, the critical speed can be increased. The stability region regains increase in the first-order mode, but will also change the type of instability experienced by the moving paper board. It can be seen that changes in the location of the elastic point supports on the nodal lines do not affect the natural frequencies of the paper board, and thus do not change the dynamic stability of the moving paper board.

Figure 5.

The relation between frequency and the moving speed λ. (a) Variation of the real part of the complex frequency and (b) variation of the imaginary part of the complex frequency.

Paper board with one elastic edge restrained

In Figure 6 the elastic restraint stiffness is $k=0$ . It can be seen that the paper board exhibits a divergence instability in the first-order mode, the first-order and the second-order divergence speed, respectively, are $\lambda =5.08$ and $\lambda =7.83$ .

Figure 6.

Dimensionless complex frequency versus the dimensionless moving speed λ. (a) Variation of the real part of the complex frequency and (b) variation of the imaginary part of the complex frequency.

In Figure 7 the elastic restraint stiffness is $k=10$ . The first-order mode exhibits a divergence instability at $\lambda =5.38$ , and the second-order mode exhibits this instability at $\lambda =8.06$ . When the speed of motion increases to $\lambda >8.71$ , the paper board regains stability in the second-order mode.

Figure 7.

Dimensionless complex frequency versus the dimensionless moving speed λ. (a) Variation of the real part of the complex frequency and (b) variation of the imaginary part of the complex frequency.

In Figure 8 the elastic restraint stiffness is $k=100$ . The first-order mode exhibits a divergence instability at $\lambda =6.07$ ; when the dimensionless speed of motion reaches $\lambda =8.3$ , the paper board regains stability in the first-order mode. The paper board experiences coupled-mode flutter between the first-order mode and the second-order mode when $\lambda =8.359$ . This result indicates that increases in the elastic stiffness of the restraints can increase the critical speed, but also change the type of instability experienced by the moving paper board.

Figure 8.

Dimensionless complex frequency versus the dimensionless moving speed λ ( $k=100$ , $k_{T_i}=0$ ). (a) Variation of the real part of the complex frequency and (b) variation of the imaginary part of the complex frequency.

Figures 9 and 10 show the variation of the critical speed of the paper board with the linear elastic stiffness of the elastically restrained edge. With increases in the elastic stiffness of the restraint, the critical speed increases. When the elastic stiffness of the restraint is in the range of $0<k<50$ , the paper board exhibits a divergence instability in both the first-order mode and second-order mode. For $50\le k\le 100$ , the paper board first exhibits a divergence instability in the first-order mode, and then a coupled-mode flutter of the first mode and the second mode occurs.

Figure 9.

Dimensionless critical moving speed λ versus the dimensionless linear elastic restraint $k$ ( $k_{T_i}=0$ ). (a) Variation of the first-order divergence speed λ and (b) variation of the second-order divergence speed λ.

Figure 10.

Dimensionless critical moving speed λ versus the dimensionless linear elastic restraint $k$ ( $k_{T_i}=0$ ). (a) Variation of the first order divergence speed λ and (b) variation of the couple-mode flutter speed λ of the first and second modes.

Figure 11 shows the first three vibration mode of the paper board with the dimensionless speed of motion without elastic point supports for an elastic restraint with $k_{T_i}=0$ .

Figure 11.

The first three vibration mode figures of the paper board. (a) The first mode, (b) the second mode, and (c) the third mode.

Conclusions

In this paper, the printing material of Shaanxi Bei Ren PRC 250 cigarette package gravure press is taken as the sample. The transverse vibration behavior and stability of moving paper board with finite interior elastic point supports and elastic edges restrained are investigated, considering the effects of the translating velocity. The equation of motion of the vibration system is derived according to the nonmoment theory and variational principle. The results of our analysis are summarized below.

For the paper board with four simply supported edges, the relationships between the dimensionless frequency and the critical dimensionless velocity are analyzed. The critical dimensionless velocity is obtained, and the critical dimensionless is $\lambda =6.31$ . By converting it to actual printing speed, the paper board stable working speed range is 0–361.08 m/min.

We find that the natural frequencies increase with increase of the elastic restraint on the boundary. However, increases in the elastic point supports do not always increase the natural frequencies of paper board. Furthermore, the natural frequencies are greatly influenced by the location of the point supports. Changes in the location of the elastic point supports along nodal lines do not affect the dynamic stability of the moving paper board.

In the case of $k<50$ , the paper board exhibits a divergence instability in both the first- and second-order modes. For $50\le k\le 100$ , the paper board exhibits a divergence instability in the first-order mode, and then the paper board experiences a coupled-mode flutter instability between the first- and second-order mode. With increasing in the stiffness of the elastic restraint, the critical speed increases, and the type of instability of the moving paper board is changed.

The study provides the theoretical basis for the optimization of printing equipment and improves the dynamic stability of the paper board in printing process. The study has important significance in practical engineering application.