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The vibration model of moving membrane with variable density distribution is established, and the density distribution of the moving membrane varies along the lateral direction. The transverse vibration differential equations of moving membrane are established based on D’Alembert’s principle and discretized by using the differential quadrature method (DQM). The relationships of the first three dimensionless complex frequencies between dimensionless speed, density coefficient, and tension ratio of the membrane are analyzed by numerical calculation. The effects of the density coefficient and the tension ratio on transverse vibration characteristics of the membrane are investigated. The relationship between density coefficient and critical speed is obtained. The numerical results show that the density coefficient and the tension ratio have important influence on the stability of moving membrane. So the study provides a theoretical basis for improving the working stability of the membrane in the high-speed printing process.

The membrane including plastic film, paper web, cloth, metal foil, and other types of film-like material is widely used to make printing and packaging products. The processing schematic of axially moving membrane is shown in Figure

The axially moving web.

Kulachenko and his coworkers [

In these literatures above, the membrane chosen as sample was all supposed with constant density. But the membrane density is changing in many situations. In printing process, the plate is wetted and inked based on the image distribution, and then the ink is transferred to the substrate through rolling of the plate cylinder and impression cylinder. The ink and fountain solution which are absorbed into the paper web or membrane cause the change of density. Besides, the nonuniform thickness of membrane also can change the surface density [

Transverse natural vibration of the annular membrane with variable surface density and without axial velocity was studied by using modified perturbation method in [

In this study, the stability of the moving membrane with variable density in the lateral direction is studied. The transverse vibration differential equation of the moving membrane with variable density is established and discretized by using DQM. The relationships between the dimensionless vibration frequency and the dimensionless velocity, the density coefficient, and the tension ratio of the printing membrane are studied. The influence of the density coefficient and the tension ratio on vibration characteristics of the membrane is discussed.

The membrane between two guide rollers can be simplified to a moving membrane model shown in Figure

The model of printing moving membrane under the four edges fixed.

The density function of membrane is

Density function of printing moving membrane along the lateral direction.

The speed

Then the transverse acceleration

As shown in Figure

The force on an elemental area of the membrane.

It is assumed that the membrane is subjected to a transverse load

Irrespective of the transverse load, that is,

The piecewise functions

Substituting (

Introduce the dimensionless quantities as follows:

Substituting (

Supposes the solution to (

Substituting (

Figure

The boundary conditions of the printing moving membrane.

Four sides fixed

Three sides fixed and one free

Two sides fixed and two free

The boundary conditions of moving membrane with four sides fixed are expressed as follows:

The principle of DQM [

The value of each of the order partial derivatives of the unknown function can be described as

The weight coefficient of each order can be acquired by

The complex characteristic equation is established by using DQM. Then (

The boundary conditions are

Converting (

Matrix

When the density coefficient

Comparisons of the transverse vibration frequency with those in [

Aspect ratio |
1.0 | 2.0 | ||
---|---|---|---|---|

Frequency order | This study | Ref. [ |
This study | Ref. [ |

1 | 4.4429 | 4.4429 | 7.0248 | 7.0248 |

2 | 7.0251 | 7.0248 | 8.8896 | 8.8858 |

3 | 8.8862 | 8.8858 | 12.9636 | 12.9271 |

Comparisons of the transverse vibration frequency with those in [

Tension ratio |
0.5 | 0.8 | ||
---|---|---|---|---|

Frequency order | This study | Ref. [ |
This study | Ref. [ |

1 | 5.4414 | 5.4414 | 6.4383 | 6.4383 |

2 | 7.6997 | 7.6953 | 8.4338 | 8.4298 |

3 | 9.4319 | 9.4247 | 10.9043 | 10.9731 |

Figures

Dimensionless complex frequency varied with dimensionless velocity (

Dimensionless complex frequency varied with dimensionless velocity (

Figure

Relationship between density coefficient and critical speed.

Figure

Relationship between density coefficient and dimensionless complex frequency (

Figures

Relationship between tension ratio and dimensionless complex frequency (

The DQM is used to analyze the transverse vibration characteristics of moving membrane with variable density distribution along the lateral direction. The results show that the tension ratio and the density coefficient have important impacts on the stability of moving membrane. The conclusions are as follows:

The study provides a theoretical basis for improving the working stability of the membrane in the high-speed printing process.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This project is supported by National Natural Science Foundation of China (Grants nos. 11272253, 11202159, and 51305341) and Natural Science Foundation of Shaanxi (Grant no. 2016JM5023).